UNIVERSITY OF CALGARY
FIGURE AND GROUND: CONSIDERATIONS ON THE METAPHYSICS AND
LOGICS OF TIME
by
JAMES ROY SCOTT
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF ARTS
DEPARTMENT OF PHILOSOPHY
CALGARY, ALBERTA
AUGUST, 2010
© JAMES ROY SCOTT 2010
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UNIVERSITY OF CALGARY
FACULTY OF GRADUATE STUDIES
The undersigned certify that they have read, and recommend to the Faculty of Graduate
Studies for acceptance, a thesis entitled "FIGURE AND GROUND:
CONSIDERATIONS ON THE METAPHYSICS AND LOGICS OF TIME" submitted by JAMES ROY SCOTT in partial fulfilment of the requirements of the degree of
MASTER OF ARTS.
Supervisor, Dr. Nicole Wyatt, Department of Philosophy
Dr. J. J. MacIntosh, Department of Philosophy
External Examiner, Dr. Bernard Linsky, Department of Philosophy, University of Alberta
Date
ii Abstract
The thesis provides a survey and analysis of three-dimensional and four- dimensional theories of time. These theories are considered both from the perspective of experienced time and of physical time. It is argued that experienced time is tensed and finite. It is further argued that physical time can be modeled by various theories, and that the choice of temporal theory is largely determined by the basic tenets of a particular theory of the external world.
Arthur Prior‟s tense logic and U-calculus are presented, and it is shown that tense logic models tensed time, while the U-calculus models tenseless time. Further, it is seen that the U-calculus can be reduced to tense logic. Following the development of a general hybrid modal logic, a temporal hybrid modal logic is presented which is shown to modal both tensed and tenseless time.
iii
Acknowledgements
I would like to begin by acknowledging the contribution to my work of the
University of Calgary Department of Philosophy faculty and staff. My work in philosophy occurred in two separate periods; one in the early 1990s and the second from
2007 until the present. During both these periods I benefited greatly from the willingness of members of the faculty to share so generously their expertise, passion for philosophy, and time. In particular, I would like to acknowledge the contributions of the following to my work: Brian Chellas, Brian Grant, Ish Haji, Ali Kazmi, Jack MacIntosh, Nicole
Wyatt, and Richard Zach. I am especially grateful to Brian Chellas for his willingness nearly twenty years ago to serve as my advisor and to Nicole Wyatt who so cheerfully and ably provided me with the structure and guidance that I needed to complete this project. I always left her office with fresh ideas and reduced confusion. I would, additionally, like to express my appreciation to Bernard Linsky from the University of
Alberta, Jack MacIntosh, and Nicole Wyatt for providing me with such helpful feedback in their role as examiners.
I have enjoyed the practical and personal support of my family through both stages of my study of philosophy. My parents, Laura and Jim Scott, have supported me unconditionally through this, as in all my adventures. My wife, Sandra Scott, has been enormously supportive while I have worked on this project. She has endured being a
“philosophy widow” with grace and good humour. I am very fortunate to have the support of my family in all I do.
iv Finally, I would like to acknowledge the support of the Calgary Board of
Education through the Professional Improvement Fellowship they provided during my
Education Leave during the 2009 – 2010 school year. Without this generous support I would have been unable to complete this work.
v Dedication
This thesis is dedicated to Sandra and the Hobo for their unconditional support and companionship through this entire project.
vi Table of Contents
Approval Page ...... ii Abstract ...... iii Acknowledgements ...... iv Dedication ...... vi Table of Contents ...... vii List of Figures and Illustrations ...... x
CHAPTER ONE: INTRODUCTION ...... 1
CHAPTER TWO: THREE-DIMENSIONAL THEORIES OF TIME ...... 6 2.1 McTaggart‟s Basic Metaphysics of Time ...... 7 2.1.1 Ontology and the C-Series ...... 7 2.1.2 The Temporal Relations ...... 8 2.1.3 The Series ...... 10 2.2 The Unreality of Time ...... 15 2.2.1 McTaggart‟s Argument ...... 15 2.2.1.1 Premiss One: Change If and Only If the A-Series ...... 16 2.2.1.2 Premiss Two: The A-Series Leads to a Contradiction ...... 18 2.2.2 Response to McTaggart‟s Argument ...... 20 2.2.2.1 The A-Theorists: Addressing Premiss Two ...... 20 2.2.2.2 The B-Theorists: Addressing Premiss One ...... 29 2.2.3 The Unreality of Time: Summary of Responses to McTaggart‟s Argument ..48 2.3 Concluding Comments on Three-Dimensionalism ...... 50
CHAPTER THREE: FOUR DIMENSIONALISM ...... 52 3.1 The Four-Dimensional View Overview ...... 52 3.2 Mereology ...... 54 3.2.1 Temporal Parts ...... 54 3.2.2 Instantaneous Temporal Parts ...... 56 3.2.3 Ersatz Temporal Parts ...... 57 3.3 Special Relativity and Geometric Representations of Four-Dimensional Space- Time ...... 58 3.3.1 Galilean Relativity ...... 58 3.3.2 Special Relativity ...... 59 3.3.3 Temporal Parts and Special Relativity ...... 67 3.4 Temporal Parts in Action ...... 70 3.4.1 The Ship of Theseus ...... 70 3.4.2 Temporary Coincidence ...... 71 3.4.3 McTaggart‟s Paradox ...... 72 3.5 Arguments Against Four-Dimensionalism ...... 73 3.6 The Four-Dimensional View Summary ...... 76
vii CHAPTER FOUR: CHARACTERISTICS OF TIME ...... 79 4.1 Now ...... 79 4.1.1 Now in Three-Dimensions ...... 79 4.1.1.1 The Specious Present ...... 79 4.1.1.2 McTaggart‟s Entity-X ...... 88 4.1.2 Now in Four Dimensions ...... 89 4.2 Time Series ...... 92 4.2.1 Overview ...... 92 4.2.2 Time Series ...... 93 4.2.2.1 Order and Direction in the Time Series ...... 93 4.2.2.2 Cardinality of the Set of Instants ...... 96 4.3 Ontologies of Time ...... 107 4.3.1 Eternalism ...... 108 4.3.1.1 Three-Dimensional and Four-Dimensional Perspectives on Eternalism ...... 112 4.3.2 Possibilism or Growing Universe ...... 116 4.3.2.1 Three-Dimensional and Four-Dimensional Perspectives on Possibilism ...... 120 4.3.3 Presentism ...... 122 4.3.3.1 Three-Dimensional and Four-Dimensional Perspectives on Presentism ...... 126 4.3.4 Reflections on the Ontologies of Eternalism, Possibilism and Presentism ...127
CHAPTER FIVE: PRIOR‟S TENSE LOGIC AND METAPHYSICS OF TIME ...... 129 5.1 Early Tense Logics ...... 129 5.2 Metric Tense Logic and the U-Calculus ...... 132 5.2.1 The basic U-Calculus ...... 132 5.2.2 The Metric U-Calculus ...... 135 5.3 Four Grades of Tense-Logical Involvement ...... 136 5.4 The Problem of Egocentric Logic ...... 148 5.5 Alternate Conclusion ...... 150
CHAPTER SIX: HYBRID MODAL LOGICS ...... 152 6.1 Introduction: Hybrid Logic ...... 152 6.2 Informal Semantics of Basic Hybrid Logic, H ...... 154 6.3 Formal Semantics of Basic Hybrid Logic, H...... 158 6.3.1 Objects of the Language: ...... 158 6.3.2 Forcing: ...... 158 6.3.3 Syntax ...... 160 6.3.4 Completeness ...... 161 6.3.5 Soundness ...... 169 6.4 Quantification ...... 175
viii CHAPTER SEVEN: TEMPORAL HYBRID LOGIC ...... 178 7.1 Introduction ...... 178 7.2 The Basic Temporal Hybrid Logic ...... 179 7.3 Fine-Tuning Temporal Hybrid Logic ...... 183 7.3.1 Relativity ...... 183 7.3.2 Now ...... 186 7.3.3 Time Series ...... 188 7.3.3.1 A-series and B-series ...... 188 7.3.3.2 The A-series ...... 190 7.3.4 Ontologies of Time ...... 190 7.3.4.1 Eternalism ...... 190 7.3.4.2 Possibilism and Presentism ...... 191 7.3.5 Logics of Experienced Time and Logics of Physical Time ...... 192 7.3.5.1 Experienced Time ...... 192 7.3.5.2 Physical Time ...... 198 7.4 Summary ...... 199
CHAPTER EIGHT: CONCLUSION ...... 201 8.1 Experienced Time ...... 201 8.2 Physical Time ...... 202 8.3 Logics of Time ...... 203
REFERENCES ...... 205
ix List of Figures and Illustrations
Figure 1 ...... 10
Figure 2 ...... 11
Figure 3 ...... 13
Figure 4 ...... 19
Figure 5 ...... 19
Figure 6 ...... 61
Figure 7 ...... 61
Figure 8 ...... 62
Figure 9 ...... 69
Figure 10 ...... 69
Figure 11 ...... 115
Figure 12 ...... 145
Figure 13 ...... 146
x 1
Chapter One: Introduction
This project begins with an examination of various views concerning the nature of time. Seeking to answer questions such as „Is there an absolute time?‟ and „Is time tensed?‟ is an exercise in speculative metaphysics. It involves a dance of a priori sentiments and empirical confirmation, in which arguments aspire to provoke rational persuasion and are not judged to fail if they fall short of certainty. The epistemological skeptic will always find a foothold in her attack, but this should not discourage us. As human animals we are by our nature speculative metaphysicians, seeking to glimpse the fundamental truths of the world and continually creating and refining and rejecting models of reality.
Our examination of the nature of time will begin with the presentation and evaluation of various arguments and positions found in the literature.
Views of time are divided into three-dimensionalist and four-dimensionalist theories. Three-dimensionalist theories are those that include the notion of an absolute time series. In three dimensionalist theories, objects and events exist in space and are related to instants in time. Change occurs in time on a three-dimensionalist view.
Objects persist through time either by enduring or perduring. An object is said to endure if it is wholly present through a span of time. We can think of objects as being at rest while time passes. Endurance theory easily supports identity of an object across time and describes an intuitive metaphysics, but has difficulty reconciling changes in the properties of an object with the principle of indiscerniblity of identicals. We can consider the example of a poker that is hot on one day and cool on all others. If the poker is seen
2 to persist through time by enduring, the poker is wholly present through all the time during which it persists. The problem with this model is that the poker has different properties at different times; it has the property of being hot one day and lacks that property on other days. The question for endurists is whether the identically same poker can have both the properties of being hot and of not being hot.
An object is said to perdure if it persists by having different parts, each of which exists for an instant, or a short interval of time. A play, organized into acts and scenes, has different parts, no two of which exist simultaneously. A play exists through the sequential existence of its parts. Perdurance theory more easily reconciles change in the properties of an object with the principle of the indiscerniblity of identicals, though it describes a less intuitive metaphysics. Our poker exists as the sum of its temporal parts, some of which have the property of being hot and others of which lack this property. The poker, itself, is the sum of these parts and, so, its properties do not change. The parts, which do not all have the same properties, and not considered to be numerically identical, so perdurance theory does not run afoul of the principle of the indiscerniblity of identicals.
Three-dimensional theories may hold that objects persist either by enduring or perduring. In either case, though, the notion of an absolute time series is central to every three-dimensional theory of time.
Four-dimensionalist theories hold that objects are made up of parts, and that each of these parts has a temporal component. In four-dimensionalist theories there is no absolute time that sits apart from objects or events. Each temporal part has its own temporal coordinate. Objects persist through time by perduring. The temporal parts that
3 comprise an object come into and go out of existence sequentially, like the frames of a motion picture film. Under a relativized four-dimensionalist view, each object occupies a unique frame of reference that defines its own space-time coordinate system. The space- time coordinate system of each frame of reference includes a stipulated origin and an axes scale set by the velocity of the object relative to the velocity of other objects. In a relativized four-dimensionalist theory, any frame of reference can be selected as the
„privileged‟ perspective, with each object and event then described according to the space and time scales of this frame of reference. The description of each object and event will vary when viewed from the perspective one frame as opposed to another.
With the division of theories into three-dimensionalist and four-dimensionalist, time itself is divided into experienced time and physical time. When we consider how we experience time, we can list several of its characteristics. Experienced time has a particular direction, and is divided into past, present, and future. As we shall argue in
Chapter Two, we experience a three-dimensionalist time, with an absolute time series.
We experience change as occurring against the backdrop of this time series. Physical time, on the other hand, is the time of the objective universe – time as it would be even in the absence of sentient creatures. Our theory of physical time needs to be consistent with our theories of other elements of the external world. Accordingly, we shall see in
Chapter Three that physical time is often described by a relativized four-dimensionalist theory that is consistent with current theories of physics. In Chapter Five, however,
Arthur Prior reminds us that we should not be too quick simply to follow science in developing our metaphysics.
4
A further distinction with respect to theories of time is whether or not time is tensed. Theories of tensed time, called A-theories, describe time as we experience it and account for the various logical differences that we hold between objects and occurrences in the past, as compared with those in the present and with those in the future. Generally, the past is seen to be fixed in the sense that events that occurred in the past can not be
„undone,‟ and objects that existed cannot retrospectively be erased from the ontology of the past. Objects and events that are present are real and open to change. Future objects and events exist as possibilities. Under A-theories of time this division into past, present and future exists and is metaphysically prior to the tenseless temporal relations of „earlier than‟ and „later than.‟ Tenseless theories, called B-theories, hold that tense is reducible to the metaphysically prior relations of „earlier than‟ and „later than.‟ As will be seen, the choice between tensed and tenseless theories is made within the broader metaphysical context of the existence and nature of the external world.
A logic is a structure that helps, in the words of Prior “…to expose and eliminate philosophical „pseudo-problems,‟ and in order to distinguish real objects from mere
„logical constructions.‟” [Prior 1996, p. 45] Tense logics formalize tensed language and provide us with the means to analyze our tensed expressions. Prior holds that logic is a tool that extends beyond analysis of language to the real world. “Philosophy, including logic, is not primarily about language, but about the real world.” [Ibid] Prior represents the view that a sound logic will reveal aspects of the nature of objects in the world. He develops two temporal logics: tense logic and the U-calculus. Tense logic describes a tensed view of time while the U-calculus describes a tenseless or, in Prior‟s words, a
5 tapestry view of time. Prior shows that the U-calculus can be reduced to tense logic, and draws from this the conclusion that real time is tensed time.
In his U-calculus, Prior introduces into a normal modal logic a type of „special- proposition,‟ that we call an „instant-proposition.‟ An instant-proposition is true at one and only one instant. It acts syntactically like an ordinary proposition, but allows for a kind-of direct reference to instants from within a sentence in the language. Prior also quantifies over instant-propositions. This device of Prior‟s has recently been brought into the syntax and Kripke-frame semantics of modal logic to form a class of logics known as
„hybrid logics.‟ Hybrid logics contain special formulae named „nominals‟ that function in the manner of Prior‟s special-propositions. In Chapter Seven of this thesis, a basic tensed hybrid logic is developed. Possible modifications to the semantics and syntax are discussed that allow the logic to model both tensed and tenseless theories of time.
No claim is made that these temporal hybrid logics reveal truths of physical time.
They do, however, reveal truths of our own experience of time and of our understanding of physical time.
6
Chapter Two: Three-Dimensional Theories of Time
In his 1907 paper The Unreality of Time and then again in his 1927 book The
Nature of Existence, McTaggart offers an influential argument with the conclusion that time is unreal. Additionally, on McTaggart‟s account, change is real only if time is real.
Unless we can separate change from time, if we accept McTaggart‟s argument, we must accept the unreality of change.
As was common until the second third of the Twentieth Century, McTaggart held a three-dimensional theory of time in which objects and events occupy space and hold relations to an absolute time series that exists separate from, but in some manner parallel to, physical space.1 This three-dimensional notion of absolute time is largely consistent with our common intuitions regarding time, although, when considered carefully, we find that our experienced time does not always appear to „pass‟ at a constant rate.
When we are engaged in a challenging and enjoyable enterprise, time seems to pass quickly. Time seems almost to cease during even the best educational leadership lecture. In the midst of a crisis, each instant seems to stop before yielding to the next.
And, of course, as we age the rate at which time seems to pass accelerates. Yet, while the rate at which time passes seems to vary, the rough intuition that drives such metaphors as
„the passage of time,‟ remains one in which events occur against a background of absolute time. As I will discuss later in this chapter, the qualities of our experience of
1 Throughout the thesis I will use the short-hand of referring to a theory built on a three-dimensional [or four-dimensional] view of the external world as a „three-dimensional [or four-dimensional] theory of time,‟ rather than „a theory of time based on a three-dimensional [four-dimensional] view of the external world.‟
7 time can be considered separately from the actual qualities of physical time. Our experience of time can certainly be captured by a three-dimensional theory. It is not surprising, then, that our intuitions of time are informed by a three-dimensional view, and that many of the theories of physical time are three-dimensional theories.
We find in McTaggart‟s philosophy of time a clear and comprehensive examination of the most important elements of time as viewed through three-dimensional theories. While, on the surface, McTaggart‟s conclusion makes his work an unlikely place to begin our structured examination of time, his terminology and models of time have framed nearly all subsequent studies of time. Our look at three-dimensional theories of time will be centred on McTaggart‟s argument.
2.1 McTaggart’s Basic Metaphysics of Time
2.1.1 Ontology and the C-Series
McTaggart was an ontological idealist. His approach to metaphysics was to argue deductively from an a priori foundation. The foundation of his metaphysics can be reduced to three ontological principles: (1) the indiscernibility of identicals; (2) the infinite divisibility of substances; and (3) the principle that a substance is fully determined by its properties. While it is well beyond the scope of the present project to explicate McTaggart‟s entire metaphysics, a brief overview will place his temporal argument in its proper context.
For McTaggart, reality is entirely comprised of a community of selves, connected by the relation of love. Nothing exists beyond this community. Each of these selves experiences perceptions. For any given self and event, the set of perceptions of the event that is experienced by the self form a C-series of that event in that self. The series exists
8 as part of that self. The C-series are series of perceptions individuated both by the self that perceives and by that which is perceived.
For example, consider Maci‟s perception of the event that is the paddling from
Tobacco Caye to South Water Caye. The perception of the total event is divisible into parts. For instance, the perceptions of launching the kayak form a proper part of the entire perception of the total event. The smallest parts of the perception correspond to individual moments in time. These smallest parts represent the terms of the C-series.
A non-temporal relation of inclusion obtains between the terms of a C-series.
Consider perceptions c1 and c2 , where is the perception of the kayak being launched from Tobacco Caye and is the perception of the kayak landing on South Water Caye.
Because is the smaller part of the total perception, bears the inclusion relation to perception . This transitive non-symmetric relation of inclusion is mistakenly understood by Maci to be temporal. As will be seen, this mistake is the reason that we often make the common error of interpreting events as occurring in time. While
McTaggart‟s C-series does not play a featured role in his arguments concerning time unreality of time, it does help McTaggart to reconcile the metaphysical unreality of physical time with the undeniable experience of time.
2.1.2 The Temporal Relations
McTaggart begins his analysis of time by stating that there are positions in time, which he calls „moments.‟ Other writers use „instants‟ to name positions in time. I will use „instants‟ through most of this thesis, except when referring directly to McTaggart‟s position. McTaggart‟s moments are occupied by events, and are distinguished in two
9 ways. First, each moment is distinguished by its A-relation; that is, by being past, present, or future. Consider the moment occupied by the event of the death of Neil
Young. That moment, which sixty-four years ago was in the distant future, is, in August of 2010, in the relatively-near future. This moment will then be present and afterwards be in the recent past. Long after Neil Young has died, the moment of his death will continue to recede into the increasingly distant past. A metric relative to some measurement scale can be attached to an A-relation. For example, in November of 2009 the A-relation of the moment of Neil Young‟s birth was sixty-four years in the past.
The A-relation, or tense, of each moment is ever-changing, while the moment and the events that occupy it remain unchanged. Consider, for example, that Neil Young was born on November 12, 1945. The moment on the A-series that contains that event, say noon on November 12, 1945, will always contain that event. The event itself occurs and then remains unchanging. What does change is the A-relation of that moment. On
January 25, 1627 that moment held the A-relation of „future‟ with the then-Now. At noon on November 12, 1945, that moment held the A-relation of „present‟ with the then-Now.
Finally, at 1400 on August 4, 2010 that moment holds the A-relation of „past‟ with the then-Now. The value of a moment‟s A-relation changes along a continuum as the tense advances from distant future through near future, present, recent past, and towards increasingly distant past.
The second distinction between moments exists as a static relation between
moments. McTaggart names this distinction the B-relation. For any two moments, M1
and M 2 , either (i) is earlier than , or (ii) is later than , or (iii) and
10
M 2 are simultaneous. Consider moments dRV, dBH, dJS, and dNY at which, respectively, the events of the deaths of Ritchie Valens [1959], Buddy Holly [1959], Joe
Strummer [2002], and Neil Young [future] occur. The moment dRV, which holds the death of Richie Valens, bears the B-relation of being „earlier than‟ the moment dNY, which holds the death of Neil Young. Similarly, the moment dRV is simultaneous with the moment dBH, the death of Buddy Holly [Figure 1]. As is the case with A-relations, a metric relative to some measurement scale can be attached to a B-relation. For example, the B-relation between the moment dBH and the moment dJS, the death of Joe Strummer, is „16 040 days earlier than.‟
dBH dRV dJS dNY
1959 2002 ?
Figure 1
2.1.3 The Series
McTaggart defined the A-series as consisting of all moments and the temporal A- relation. The A-relation holds between each moment and some entity-X, outside of the series. A particular moment, M, will be future, present or past in relation to Entity-X.
The value of the relation is the tense of the moment. Entity-X is dynamic while the moments are static. It is the dynamism of Entity-X that allows for change.
An intuitive way to see the A-series is to imagine that Entity-X represents Now.
As Now „moves,‟ the relation between a „stationary‟ moment, M, and Now changes and,
11 as the relation changes, the tense of M changes [Figure 2]. This picture accords fairly closely to our intuitive notion of tense.
M is Future M is Present M is Past
Now Now Now
M A-series
Figure 2
McTaggart‟s first distinction among moments, the A-relation, is this changing relation between each stationary moment and the moving Entity-X. His second distinction, the B-relation, arises from the combination of the A-Series and the C-Series.
A C-series is a non-temporal series. Each C-series forms part of a self and consists of perceptions of an event. For each event, there is a one-to-one correspondence between terms of the C-series and terms of the A-series and of the B-series. While the terms of the C-series are related by the non-temporal inclusion relation, the series provides no relational temporal information concerning the moments. The temporal information that can be inferred from the C-series is limited. Importantly, the cardinality of the C-series will correspond to the number of moments in the related event. If the C- series is finite, there are finitely many moments of time. If the C-series is a countable set, then there is a countable infinity of moments. Finally, if the C-series is an uncountable set, then there is an uncountable infinity of moments. Because one of the primary principles of McTaggart‟s ontology is that all substances are infinitely divisible, and since he considers the C-series to be part of its self, it is likely McTaggart would consider
12 the inclusion relation of the C-series to be dense; that is, given any two terms in the C- series, c and c , where is included in I c, c there is a third term, c , such i j ij 0
that I ci , c0 and I c0 , c j . Accordingly, since there is a one-to-one correspondence between the terms of the C-series and the relevant terms in each of the A-series and B- series, McTaggart would hold that each the A-relation and B-relation is dense.
McTaggart argues that C-series are real, as they are simply comprised of perceptions. Because the inclusion relation is not temporal, McTaggart‟s argument for the unreality of time does not have direct implications for the C-series.
The moments of the A-series correspond to the perceptions of a C-series. When the C-series, with its inclusion relation, is combined with the A-series, which contains the temporal A-relation, the B-series is produced. The B-series represents the second type of temporal relations, those of being earlier than, simultaneous with, and later than. To see
how these B-relations arise from the A-relations, consider moments M1 and M 2 . is earlier than just in case is future when is present [Figure 3]. Similarly, is simultaneous with just in case is present when is present, and is later than just in case is past when is present. At the moment of Buddy Holly‟s death [dBH], the value of the A-relation between the moment of Joe Strummer‟s death
[dJS] and X is „forty-three years in the future.‟ Accordingly, the moment of dJS holds the
B-relation of „forty-three years later‟ to the moment of dBH. The A-relation provides the direction of time, which is necessary in the assignment of values of the B-relation. It is
13 the moving Entity-X that is responsible for the direction of the time series, so the A- relation is necessary for the creation of the B-series.
M2 is Future
M1 M2
Figure 3
The B-series, then, consists of the same moments of time found in the A-series.
All pairs of moments on the B-series are related by the B-relation, which is derivative of the A-relation. McTaggart holds that because the A-relation is metaphysically prior to the B-relation, the B-Series exists only if the A-Series exists. He also emphasizes that, while the value of the A-relation for any moment is continually changing, the value of the
B-relation between any two moments does not change. If it is ever the case that M1 is n
time units earlier than M 2 , it is always the case that is n time units earlier than .
It is important to notice that it is possible to define the A-relation in terms of the B
-relation. For example, suppose at , we seek to determine the A-relation of . If the B-relation of and is known, we can determine the A-relation of . If, for
example, holds the B-relation of „twenty minutes prior‟ to M 2 , holds the A-
relation of being „twenty minutes in the future‟ at M1 . McTaggart would not deny this translatability between values of the A- and B-relations. What he would deny is that the
B-relation could be metaphysically prior to the A-relation.
There is no change in the B-relation or in the B-series. Because the value of the
B-relation between any two moments is unchanging, the B-relation on its own is
14 insufficient to facilitate change. The B-relation lacks the „animation‟ inherent in the ever-changing A-relation. The value of the B-relation between any two moments does not change because the temporal order between the moments does not change. The death of Joe Strummer was, is, and will always be 16 040 days after the day the music died.
There is no change in the B-series.
The A-relation and the A-series continually change. The A-relation is determined between moments and the entity-X. Entity-X is a theoretical construct that can be seen as representing the ever-changing present moment, or Now. Because Now is continually changing, the A-relation is continually changing. It is this change in designation of the present that facilitates the change in the A-series.
While the static B-relation between any two moments can yield a „snapshot‟ of a moment‟s A-relation, it cannot generate the ever-changing nature of the A-relation. In contrast, the ever-changing A-relation can produce the non-changing values of the B- series. It is for this reason that the A-relation is seen to be metaphysically prior to the B- relation.
The A-series consists of the set of all moments together with the A-relation, which holds between each moment and an ever-changing Entity-X. The B-series consists of the set of all moments, together with the B-relation. The B-relation, which requires information determined by the A-relation, provides an ordering of the moments in the B- series. Because the B-relation between any two moments, once set, is unchanging, the position of each moment in the B-series, once set, is permanent.
15
2.2 The Unreality of Time
2.2.1 McTaggart’s Argument
McTaggart will argue that time is unreal; that is, that nothing existent can possess the characteristic of being in time. “I believe that nothing that exists can be temporal, and that therefore time is unreal.” [McTaggart 1927, §304] This metaphysical claim does not deny the place of time and tense in our mental processes. He acknowledges that “we have no experience which does not appear to be temporal,” [McTaggart 1927, §303] and that even the very experience of considering whether time is real appears to be in time.
This error arises from our mistakenly taking the non-temporal C-relation of inclusion to be the temporal B-relation. That we perceive tensed time is made clear by considering the different ways in which we apprehend events. Our direct perceptions are only of present events. We recall from our memory those events that are past, and anticipate with our reason and imagination those that are future. McTaggart suggests that we separate experienced time from physical time. It is at physical time, and not experienced time, that McTaggart‟s argument is directed.
McTaggart‟s argument begins with his view that time involves change. He holds both that time is a necessary condition for the occurrence of change and that change is a necessary condition for the existence of time. He proceeds to argue that the A-relation provides the changing facts that determine the truth-values of statements that describe change. Without the A-relation, then, there are no facts to make statements of change either true of false, so there is no change. Finally, he finds that the A-relation leads to a contradiction, and concludes that time cannot be real.
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2.2.1.1 Premiss One: Change If and Only If the A-Series
On McTaggart‟s view, events are the objects of change, and it is the changing values of the A-relation that constitutes the change. He contrasts his position with that of
Bertrand Russell in Principles of Mathematics. McTaggart quotes Russell from §442.
“Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and the time T, and the proposition concerning the same entity and the time T , provided that these propositions differ only by the fact that T occurs in one where occurs in the other.” [McTaggart 1927, §313]
McTaggart considers the example of a poker that is hot on a particular Monday, but cool at all other times. „The poker is hot,‟ is true on that particular Monday, while the same sentence is false at all other times. The truth-making facts responsible for these truth-values arise from the B-series and, for Russell, represent the change that has occurred. These facts are sometimes referred to as „B-facts.‟
Recall that McTaggart holds the indiscerniblity of identicals as an a priori truth, and that he, further, holds that a substance is fully determined by its properties. With these principles at the base of his metaphysics, he rejects the view that poker can have the contradictory properties of being „hot‟ and „not hot,‟ or that it can gain or lose any property and remain the same poker. Rather, he holds that the poker has the non-
contradictory property of being „hot at tM and not hot at tT .‟ The problem for McTaggart is that, without the changing values of the A-relation, nothing changes. The poker always has the property of being „hot at and not hot at .‟ The B-series with its B-relation cannot account for the change in the poker.
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McTaggart‟s view is that, on Russell‟s analysis, there is no change. The poker always has the property of being hot on that particular Monday and of being cool at all other times. The B-facts responsible for the truth values of the relevant propositions are unchanging. The poker being hot on Monday occupies some set of moments on the B-
Series. The poker not being hot on other days occupies some other set of moments on the
B-Series. Suppose tM is a moment in the B-series during which the poker is hot, and tT is a moment on the B-series during which the poker is not hot. There is no change in the properties of the poker. The poker always has the property of being hot on and of being cool on . There is no change in the facts to support the position that there has been a change in the poker because there is no change in the B-series. McTaggart holds that changing facts are required to support change.
What does change is the value of the A-relation of each of and . Consider the value of the A-relation of each of these points as Now changes. When Now is simultaneous with a moment earlier than , the A-relation of is future and the A- relation of is more distant future. When Now is simultaneous with a moment between
and , the A-relation of is past and the A-relation of is future. Finally, as the
Now is simultaneous with a moment later than both of and , these moments have the
A-relations of more distant past and past, respectively. The ever-changing Now causes the A-relation to change, and this change in the value of the A-relation provides the changing facts that are required to support change.
We have seen, though, that because of the ever-changing Now, the A-relation provides the changing facts required for change. Change consists in the event being in
18 the future then being in the present then being in the past. There is no change within the event of the poker becoming hot, or in the event that is the history of existence of the poker. These events themselves, together with their constituent objects, remain totally unchanged. It is only their tense that changes.
Change can occur only if the A-series, with its A-relation, is real.
Besides establishing that change requires the sort of time available through the A- series, McTaggart holds that time would not exist in the absence of change. “A universe in which nothing ever changed would be a timeless universe.” [McTaggart 1908, 459].
The possibility of some thing undergoing no change over an interval of time does not represent a counterexample to this claim. When we consider moments of time during which some object stays the same, we recognize that this object remains the same while other things are changing. If all change ceased, this would include the change that is the passage of time from one moment to the next.
The first premiss of McTaggart‟s argument is that change occurs if and only if the
A-series exists.
2.2.1.2 Premiss Two: The A-Series Leads to a Contradiction
To illustrate McTaggart‟s argument that the A-series cannot exist, we will consider the events represented by the deaths of three individuals: Joe Strummer [dJS] in
2002, Warren Zevon [dWZ] in 2003, and Michael Jackson [dMJ] in 2009 [Figure 4].
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dWZ dJS dMJ
2002 2003 2009
Figure 4
Consider the death of WZ. As JS dies, the death of WZ is future; as WZ dies, the death of WZ is present; and as MJ dies, the death of WZ is past [Figure 5]. Past, present, and future are incompatible relations in the sense that, for example, nothing that is past can also be future. But we see in our example that the death of WZ is future, present, and past.
dWZ is Present dWZ is Future dWZ is Past
2002 2003 2009
Figure 5
McTaggart anticipates and addresses a response to this concern. The response acknowledges that these relations are incompatible with one another, but maintains that nothing holds one of these relations at the same time that it holds either of the others. In our example, the death of WZ is future at one time [2002], present at a second time
[2003], and past at a third time [2009]. The death of WZ never holds incompatible relations simultaneously. At the moment of the death of dWZ, the death of Warren Zevon is present, was future, and will be past. These tensed verbs, though, get their meaning from the changing A-relation. For example, what was is distinguished from what is only
20 by its being in the past. We have assumed the existence of a time series to explain the way in which moments can be past, present, and future.
We are left with an unattractive choice. Either events hold incompatible relations simultaneously or we enter an infinite regress represented by escalating cycles of assuming time to explain time. Because neither alternative is tenable, McTaggart concludes that tense, and hence the A-series, cannot exist.
McTaggart‟s full argument is as follows: There is no A-series. Because the temporal B-series arises from the temporal A-series and the non-temporal C-series, without the A-series, there is no B-series. In the absence of the A-series there are no A- relations of future, present, or past; in the absence of the B-series, there are no B-relations of earlier than, simultaneous with, or later than. Without either series there are no temporal relations and, so, there is no time. As we have seen, as well, without time, there is no change.
2.2.2 Response to McTaggart’s Argument
2.2.2.1 The A-Theorists: Addressing Premiss Two
2.2.2.1.1 Admitting Now
There is an alternative available to McTaggart, but this alternative comes at a cost in terms of our ontology, and would not be compatible with his ontological idealism.
McTaggart does not offer any attempt at explicating his Entity-X, which is one of the two relatants of the A-relation. The alternative that McTaggart does not consider is to allow this Entity-X, or Now, an ontological standing. If we do so, the response McTaggart offers to the problem of an event holding incompatible tenses would not be susceptible to his counter. Now would simply never be in a position at which any moment would hold
21 more than one tense. At the death of JS, Now would be in such a position that the death of WZ is future relative to that-Now. The A-relation would have just this one value at the moment of JS‟s death. Similarly, at the death of WZ, Now would be in such a position that the death of WZ is present relative to that-Now. Again, the tense relation would have just this one value while Now is in that particular position. There is not even the illusion that an event holds more than one tense simultaneously. This manoeuvre preserves the
A-series and, in so doing, preserves time and change by providing the changing facts.
While some will surely object that Now is too vague a notion, any metaphysics will either be founded in some primitive, irreducible, axioms or be widely circular. This proposal sees Now as being one of the primitives of the metaphysics. Now is a theoretical posit, not unlike forces in physics or numbers in mathematics. At some point in our metaphysics we need to stand behind entities that we cannot fully explain.
McTaggart‟s argument for the unreality of time suggests that the analysis of time cannot be reduced beyond tense. From this point, one can follow McTaggart and conclude that tensed time is not real. Alternatively, one can part company with
McTaggart and allow that Now be admitted into our metaphysics as a theoretical posit and, with it, the A-relations of tense between Now and each moment. This is the position of the A-theorists.
2.2.2.1.2 Reasons for Accepting Now
While it is evident that accepting Now into our ontology helps address
McTaggart‟s paradox, this is not reason enough for taking this metaphysical step.
Perhaps the most compelling reason for accepting Now is that it is so basic to our lived experience. That there is something unique or privileged in our experience of time about
22 the moment we call Now is undeniable. Of course, this may be entirely caused by a peculiarity of the human brain and not reflective of any property of the external world, but it seems to underlie the foundation of our experience as much as, say, that space is extended. The centrality of Now is seen in the irreducibility of tensed indexicals in our language.
In The Unreality of Time, McTaggart writes that “It is possible, however, that [the
A-series] is merely subjective. It may be the case that … the distinction of past, present, and future is simply a constant illusion of our minds, and that the real nature of time only contains the distinction of the B-series – the distinction of earlier and later.”
McTaggart goes on to state that, while it is the view of some that tense is part of experienced time but not part of physical time, this is a position he finds untenable. For
McTaggart, the “A-series is essential to time,” and that any argument that finds the A- series not to be real is an argument that finds time is unreal. Notwithstanding
McTaggart‟s view, I will consider the existence of tense in experienced time and in physical time. Our language offers insight into the way in which we experience the world, and provides a solid starting point from which to consider the role of tense in experienced time.
Before examining the content of tensed indexicals, it is important that some terms and concepts be clarified. Following John Perry, I will use „utterance‟ to refer to any spoken or written or other token of a sentence. The „content‟ of an utterance is what is said by the utterance. The content of a name is the object that the name designates. The content of a predicate or a definite description is the condition they express. The content of a statement is the proposition that carries the truth conditions of that statement.
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Consider, for example, the utterance “Jaxon plays soccer.” The content of the name „Jaxon‟ is the person named – the content of „Jaxon‟ is Jaxon. The content of
„plays soccer‟ is the condition of playing soccer. This might involve being a member of a team, attending games that the team plays, and running around and kicking the ball in appropriate ways. The content of the full statement is a proposition comprised of two parts:
„true‟ just in case the object named „Jaxon‟ meets the conditions expressed by „playing soccer‟ and holds the truth value of „false‟ otherwise.
An „indexical‟ is a term whose content depends on the context of the utterance.
Consider, for example, Kris saying to Jaxon following a soccer game, “You played well.”
„You‟ is an indexical; in order to set its content, the context of the utterance must be known. In this case, because Kris is addressing Jaxon, the content of „You‟ is Jaxon.
The proposition expressed by this utterance in the described context is
If this same utterance, “You played well,” is spoken to the red-haired boy, the content of the statement is
We see in these examples that determining the content of utterances that contain indexicals depends on the context of the utterance. Once the content of the indexical is set, the proposition expressed by an utterance containing that indexical is identical with the proposition expressed by a suitable alternate utterance. Returning to the example of
24 the utterance “You played well,” when spoke by Kris to Jaxon, we have seen that the content of the statement is the proposition
Consider, now, the utterance “Jaxon played well.” The content of this utterance is also the proposition
An important general observation here is that, on a conventional analysis of an utterance containing an indexical, we use the context of the utterance to replace each occurrence of an indexical with an appropriate non-indexical content. The resulting proposition is identical to the proposition of some other utterance that does not contain any indexicals; in our example, in a particular context the proposition expressed by “You played well” is identical to the proposition expressed by “Jaxon played well.”
Consider the utterance “Jaxon plays tomorrow,” made on January 1st. The context of the utterance is needed to determine the content of „tomorrow.‟ Clearly „tomorrow‟ is an indexical and, because a temporal quality of the context is required, I will refer to
„tomorrow‟ as a tensed indexical.
On our first analysis, we might set the content of „tomorrow‟ in the context of this example as January 2nd. The content of the statement, then, would be
On further analysis, however, there is an element of “Jaxon plays tomorrow” when uttered in this context, that is not captured by “Jaxon plays on January 2nd.” The element that is not captured is the „tomorrowness‟ of the original utterance. Following
Perry, we can see that this difference is significant in that it can be responsible for actions or belief-states that do not follow from the non-indexical utterance. For example, in
25 virtue of the tomorrowness, I may charge my camera battery. I would not do so if
January 2nd were, for example, ten days hence, rather than the following day.
To capture the tomorrowness, the content of the utterance in this context could be seen to a proposition as follows:
We might replace its explicit presence with some typographical convention such as
nd
Any attempt to explicate this indexical will result either in an alternate unanalyzed indexed element or in the loss of „tomorrowness‟ from the content.
This point is seen, as well, in Arthur Prior‟s „Thank Goodness That‟s Over!‟ In this essay, Prior argues that part of the content of “Thank goodness that‟s over!” when it is uttered following the conclusion of an unpleasant circumstance, is lost when we restrict ourselves to B-facts. That the speaker is grateful that the agony that has ended on, say,
Friday, June 15, 1954 is only part of the content of the utterance; that it is now Friday,
June 15, 1954 is very much a part of the content. The nowness of the utterance can not be reduced. Thank goodness that‟s over now.
In both McTaggart‟s metaphysical analysis and in the linguistic analysis patterned after Perry‟s, we are unable to explicate all temporal relations in terms on non-temporal concepts or objects. That tense cannot be eliminated from the content of our utterances
26 supports the view that experienced time is tensed, since our language reflects the way in which we experience.
Recalling McTaggart‟s argument, we see that he defines the tense of moment M as the relation between M and Entity-X, or the moving Now. To speak of the tense of M without reference to the present moment is nonsensical on this analysis, because the tense relation requires two arguments: M and Now. M is past in relation to Now. M cannot be past simpliciter. But this seems to be what McTaggart is suggesting. “These characteristics [past, present, and future], therefore, are incompatible. But every event has them all.” [McTaggart 1927, p. 20] McTaggart is conflating „characteristics‟ with the values of relations. On his own analysis, tense is not a characteristic of anything; rather, it is the value of a two-place relation. There is only one value of the tense relation between a moment and any particular instance of Now.
Of course, McTaggart sees events as occupying moments. A single event occupies each moment in some sense. An event could be said to have the property of being, for example, past, but only relative to a particular instance of Now. The property of an event being past is inherited from the tense relation between the moment it is occupying and a particular instance of Now. Just as it makes no sense to consider a moment being past simpliciter, it makes no sense to consider an event to be past simpliciter. When Joe Strummer died, the death of Warren Zevon had the characteristic of being future relative to the-Now when Joe Strummer died.
So, on McTaggart‟s account, the tense-relation for a moment requires Now as its second argument, and this implies that the tense-characteristic of an event must be indexed to a Now. His putative infinite regress shows only that Now is required for a
27 cogent explanation of temporal relations. This suggests that Now cannot be eliminated as an element of physical time.
Similarly, when determining the content of an occurrence of a tensed indexical in an utterance, the indexical Now is uneliminable. The context of an utterance is used to determine the content of indexical terms that occur in that utterance. The content of an utterance, or of an element of an utterance, is what is said by that utterance or element.
In the case of some indexicals, the context allows for the replacement of the indexical term by a non-indexical term such that nothing in what is said by the indexical in that context is lost in the replacement by a non-indexical term. In contrast, if a tensed indexical is replaced by a non-indexical, some of what is said by the tensed indexical is lost2. That tensed indexicals cannot be eliminated from our language suggests that Now cannot be eliminated as an element of the experienced time.
In the utterance, “Sam had a flu vaccine two weeks ago.” The phrase “two weeks ago” is a tensed indexical. If the utterance is made on November 30th, some of the content of “two weeks ago” is captured in “November 16th.” But, while some of what is said is captured, some of the content of the tensed indexical is lost. That Sam received the vaccine two weeks ago assures him that, at the time of the utterance, he has sufficient antibodies to consider himself protected from the flu virus; that is, when the utterance occurs, Sam is now protected from the flu. As we have seen, there is no way to eliminate an indexical reference to the present moment from the content of a tensed indexical.
2 This phenomenon is not limited to tensed indexicals. For example, Perry argues persuasively that the same is true of the indexical „I.‟
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The ineliminability of Now from our metaphysics and the ineliminability of tensed indexicals from the content of utterances of our language can be seen to connect in at least two possible ways. First, to the extent that our natural language models the world as it is, the ineliminability of tensed indexicals can be seen simply to reflect the metaphysical ineliminability of Now. In this way, the parallel in our language provides support for a metaphysical position, contra McTaggart, that tense and time are both real.
The second possible connection between these findings is less bold, perhaps, but more certain. Even if it cannot be said of our language that it models the world as it is, it certainly models the world as we experience it. McTaggart writes “we have no experience which does not appear to be temporal.” [McTaggart 1927 §303] Being subjective in this way does not, for McTaggart, make time real. But, even in this second connection, the parallel in our language provides support for the position that tense is real at least in the world as we experience it. Even if we could conceive of a tenseless world, our language would be incapable of modeling its tenselessness.
These two possibilities can be seen to converge in one conclusion. Whatever assumptions we make with respect to the connection between the natural language we use in our metaphysical analyses and understanding, and the „metaphysical facts‟ themselves, these mutually supportive findings that tense is not eliminable, provide compelling reason to reject McTaggart‟s view that time is unreal. They overwhelmingly support the metaphysical positions that time is real and that time is tensed. That is to say, they support the position that Now is real in experienced time and, under a metaphysics similar to McTaggart‟s, in physical time.
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2.2.2.2 The B-Theorists: Addressing Premiss One
One response to McTaggart is to reject his first premiss; that is, to deny that change occurs if and only if the A-series is real. This position is held by B-theorists, including Russell, who maintain that the B-series alone provides the necessary and sufficient conditions for change to occur.
2.2.2.2.1 Russell
Russell‟s theory of time arises from his views on change. For Russell, “[c]hange is the difference, in respect of truth or falsehood, between a proposition concerning an entity at a time T and a proposition concerning the same entity and another time T , provided that the two propositions differ only by the fact that T occurs in one where occurs in the other.” [Russell 1937, §442] Two important principles underpin this notion of change. Change always involves, first, a fixed entity and, second, a relation between this entity, other entities and some moments of time.
Russell‟s notion of a fixed entity is that of an entity that perdures. An entity perdures if it persists by having different temporal parts, each of which exists at a single moment and none of which exists for more than a single moment. A perduring entity is seen as being the mereological sum of all its temporal parts. In contrast, an entity is said to endure if it persists by being wholly present at more than one time.
For Russell, an entity is a class whose particulars are parts, each of which holds an existence relation with exactly one moment in time. For Russell, then, entities persist
30 by perduring. In more contemporary terms, an entity [object, event, etc.] is the mereological sum of all its temporal parts3.
Consider the Calgary Zoo. Incorporated in 1929, the 1929-temporal-part housed
36 mammals. The 1937-temporal-part sees the construction of the 120 ton „Dinny the
Dinosaur‟ statue. A prehistoric park was introduced in the 1984-temporal-part. The
2010-temporal-part does not include „Dinny the Dinosaur,‟ but is still a part of the
Calgary Zoo. Each temporal part is distinct from each other part, and none is identical with the Calgary Zoo. The Calgary Zoo is the mereological sum of each of its temporal parts. In this way, the entity that is the Calgary Zoo is seen to be of a different logical type than the entities that make up its individual temporal parts.
Russell writes “Thus we may say that a term changes, when it has a fixed relation to a collection of other terms, each of which exists at some part of time, while all do not exist at exactly the same series of moments.” [Russell 1937 §442] This fixed relation must be that of “a simple class-concept to simple particulars contained under it.” [Ibid]
So, change is due “to the fact that many terms have relations to some parts of time which they do not have to others.” [Russell 1937 §443]
On Russell‟s account, then, there are two relations that are involved in change.4
At the lower level, there is the relation between temporal parts of an entity and individual
3 There is a lack of uniformity in the literature concerning the application of the phrases „Three Dimensionalism‟ and „Four Dimensionalism.‟ Writers whose primary interest is the existence of things in time tend to draw the divide between theories that hold that objects endure [Three Dimensionalism] and those that hold that objects perdure [Four Dimensionalism]. These writers would include Russell amoung the Four Dimensionalists. I will follow the convention of many writers whose primary concern is over theories of time and who draw the divide between theories in which spatially extended objects hold a relation to time [Three Dimensionalism] and those in which objects are spatio-temporally extended and
31 moments in time. Change, itself, is a relation between the class-concept, which is the mereological sum of all the temporal parts, and individual parts. The change in the
Calgary Zoo between 1929 and 1937 consists in the relation between a temporal part related to some moment in 1929 and a temporal part related to some moment in 1937.
Reverting to the more intuitive language of the truth of propositions, the proposition of
“Dinny the Dinosaur is a feature of the Calgary Zoo” was false in 1929 and true in 1937.
What accounts for this difference in truth values are the facts that (i) a temporal part of the Calgary Zoo that does not contain Dinny is related to some moment in 1929 and (ii) a different temporal part of the Calgary Zoo that does contain Dinny is related to some moment in 1937.
When we compare Russell‟s view of time to McTaggart‟s, we first note the similarity between Russell‟s relation that holds between an entity and its temporal parts and McTaggart‟s C-relation. While the relata of Russell‟s relation are parts of entities in the world and those of the C-relations are perceptions in a self, the relations themselves exhibit a resounding similarity. Consider the entity, E, at time, t. For Russell, E is comprised of the mereological sum of all the temporal parts of E that hold the B-relation of earlier-than to t. For McTaggart, a self‟s perception of E is the mereological sum of all the perceptions of E that hold the C-relation of inclusion to E. The basic difference relevant to McTaggart‟s argument is that Russell sees the B-relation as being temporal and McTaggart sees the C-relation as being non-temporal.
whose extension in time is inseparable from their extension in space. Under this distinction, Russell is included among the Three Dimensionalists. 4 Russell refers to these as being a “three-cornered relation” [Russell 1937 §442].
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We recall that McTaggart‟s response is that there is no change on Russell‟s account, since there are no changing facts. With respect to our example, McTaggart‟s view would be that the change from the Calgary Zoo without Dinny to the Calgary Zoo with Dinny consists in the changing A-relation of an event, say the completion of the
Dinny statue. In 1929, this event was in the future; at some moment in 1937, this event was present; and in 2010, this event is in the past. The change is animated by the continual change in the A-relation which, in turn, is animated by the continually changing
Now. In contrast, McTaggart would argue, the B-facts in virtue of which Russell claims this change occurs are unchanging non-temporal facts, and change cannot arise from unchanging facts.
Russell denies that his B-relation is non-temporal. He holds that it is the time- relation of „physical time,‟ while the A-relation is the time-relation of „mental time.‟ “In a world in which there was no experience, there would be no past, present, or future but there might well be earlier and later.” [Russell 1915, p.212] Thus, Russell and
McTaggart begin with incompatible positions concerning which of the A-relation or B- relation is primary. For Russell, the B-relation is metaphysically prior to the A-relation; for McTaggart, the A-relation is metaphysically prior to the B-relation.
Besides his complaint that Russell‟s B-relation is non-temporal, McTaggart denies that a proposition can be true of an entity at one time and false of the same entity at a different time. The Calgary Zoo cannot have the property at some time of featuring
Dinny and the contradictory property at another time of not featuring Dinny and remain the same entity. McTaggart sees this as a contravention of the first principle of his own
33 metaphysics: the indiscerniblity of identicals. But, in adopting this view, it appears
McTaggart is making a category mistake with respect to his own C-series.
Recall McTaggart‟s example of poker, and consider the related C-series in self A.
In particular, suppose a particular poker is hot at 1200 and cool at 1300. A‟s perception
p1 at 1200 is included in A‟s perception p2 at 1300. Perception p1 has the property of
„being of a hot object,‟ while does not. By the principle of indiscerniblity of
identicals, then, and p2 are distinct. But on McTaggart‟s account it makes no sense to think of either of these individual perceptions as being the poker. Moreover, neither needs to correlate with A‟s idea of the poker, so the fact that and are not identical is not problematic. It is the C-series itself that most naturally represents the poker.
Because McTaggart sees objects as holding their properties timelessly, he would see a C- series timelessly in its entirety. Viewing the C-series of the poker in this way, with all its elements timelessly in place, it is seen not to lose or gain properties. From this point, were he not an Idealist, McTaggart could build a model of change that would look just like Russell‟s.
At the bottom, it is their respective positions on idealism and realism that separates McTaggart‟s model of the C-series from Russell‟s B-series model of change.
McTaggart admits as much when he claims that Russell‟s B-series is really his own C- series. His claim that only an ever-changing relation, like the A-relation, can be a temporal relation is simply stated without argument. For McTaggart, it seems, this is self-evident. Its apparent self-evidence likely comes from the centrality of the A-relation in what Russell calls „mental time.‟ For an Idealist like McTaggart, it could be argued
34 that mental time is the „real time.‟ Of course, for a Realist like Russell, it is physical time that matters. McTaggart‟s argument that the A-series is necessary for change, then, is simply a corollary to his Idealism. As such, the question of whether to support
McTaggart‟s first premiss reduces to the question of whether to accept Idealism.
The disagreement between Russell and McTaggart with respect to the nature of time is more properly seen as being a particular consequence of their respective metaphysical positions than as being centrally about time. They are not beginning from some common point and then arguing to different conclusions. If this had been the case, one argument might be seen to be the superior of the other, with one philosopher finally seeing his mistake and moving to the other‟s position. But, in this case, neither argument triumphs over the other. The reader will simply follow the philosopher whose basic metaphysics he shares. A reader who favours Realism will follow Russell while the
Idealist will follow McTaggart. Their respective positions follow soundly from their respective foundational metaphysical perspectives.
So, being a Realist, Russell rejects McTaggart‟s first premiss. By arguing that the
B-relation is all that is required for change, he denies the first premiss of McTaggart‟s argument: that change occurs if and only if the A-series exists.
2.2.2.2.2 Old, New, and Even Newer B-Theories
B-theorists all agree that, while tense plays a role in experienced time, it does not exist in physical time. While this position is framed in various ways among three- dimensionalists, it amounts to the claim that the B-relations of „earlier than‟ and „later than‟ are sufficient for change, as well as for any other effects of time. Often, these relations, together with measuring and naming conventions, are used to create B-times.
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For example, „2315 GMT on August 5th, 1945‟ names a B-time. A B-sentence expresses a B-proposition, such as „The most recent test was on February 15th.‟ The truth value of this proposition is determined by a B-fact. A B-fact is a temporal fact, or state of affairs, that arises from a B-relation. The state of affairs that is a certain test being administered on February 15th is a B-fact, in virtue of which the proposition expressed by the earlier B- sentence is true. Notice that the B-fact can be reduced to a relation between a state of affairs and a B-time. This is a fundamentally three-dimensional conception, with „time‟ being apart from the „physical‟ state of affairs. The three physical, or spatial, dimensions stand in some relation to a separate temporal dimension.
In maintaining their support of the various elements of B-time, B-theorists reject the existence of the corresponding A-elements. The B-theorist holds that the A-relation of tense does not exist in physical time. With the elimination of the A-relation comes the rejection of A-facts, which would be required to determine the truth values of the A- propositions expressed by A-sentences. But, because no B-theorist even suggests the abandonment of A-sentences, any credible B-theory must include some framework for determining the meaning of tensed A-sentences and assessing their truth values.
The most obvious approach for a B-theorist to take when faced with a tensed A- sentence is simply to translate it into a tenseless B-sentence. Borrowing Mellor‟s example from Real Time II, suppose that on June 1, the A-sentence „Sydney will race tomorrow‟ is uttered. This A-sentence, uttered on June 1, can be translated into the B- sentence „Sydney races on June 2.‟ The claim of the early B-theorists is that this tenseless B-sentence translation has the same meaning as the tensed A-sentence.
36
While this was the approach taken by early B-theorists, such as Russell, it is clear that this translation does not capture all of the content of „Sydney will race tomorrow.‟
Recalling the point made by Prior and Perry, nothing in this B-theory analysis would account for Sydney „carb-loading‟ by eating a plate full of pasta on the evening of the utterance. There is a clear sense of today being June 1 and the race being tomorrow that is lost in the translation. This lost sense seems an undeniable part of the meaning of
„Sydney races tomorrow.‟
Nathan Oaklander, who is, himself, a B-theorist, notes that adherents of the „old‟
B-theory were unsuccessful in their attempts to translate A-sentences into B-sentences without losing any of the meaning of the A-sentences. [Oaklander 2004, p. 268ff] He argues that the reason for their mistake is that they conflate various notions of what constitutes the meaning of a sentence. Oaklander writes that there are four relevant senses of meaning.5
The first, M1 , is the intentional meaning; that is, the „thought‟ that the speaker intends to communicate. The meaning of „Sydney races tomorrow‟ would include, but not be limited to, the fact that his race is on June 2nd. It might also include the suggestion of nervousness or excited anticipation together with a notice of the necessity to prepare in the appropriate way for an event that will soon occur. This is the meaning, or content, of sentences such as „Thank goodness that is over!‟ that A-theorists argue is missing from any tenseless view of the world.
5 The first three of these would be considered as the „content‟ of the sentence according to our conventions. Since Oaklander follows a different convention and does not distinguish between „meaning‟ and „content‟,
37
Oaklander‟s second sense of meaning, M 2 , asserts not what is thought or intended; rather, it is what is the case. This sense of meaning is the state of affairs
described by the sentence. The M 2 meaning of „Sydney races tomorrow‟ is the state of affairs that sees the sentence uttered [today] on June 1st and Sydney racing [tomorrow,] on June 2nd. The A-theorist will claim that the bracketed words are necessary parts of the
meaning; that the todayness and tomorrowness are parts of the state of affairs that the sentence describes. The B-theorist will respond that, because tense is not part of our ontology, there is no todayness or tomorrowness to be part of the state of affairs. The only temporal elements of the described state of affairs are that the race is June 2nd and that the sentence is uttered on June 1st.
The third sense of meaning described by Oaklander, M 3 , is simply the set of truth conditions of the sentence. „Sydney races tomorrow‟ contains a tensed indexical. A B- theorist might use a date analysis form of truth conditions, such as: when uttered on June
1st, „Sydney races tomorrow‟ is true just in case „Sydney races on June 2nd‟ is true. Then, the truth conditions of the tenseless „Sydney races on June 2nd‟ will obtain.
Oaklander‟s final sense of meaning, M 4 , is the set of linguistic rules that regulate
the correct usage of the sentence. According to its M 4 sense, the sentence „Sydney races tomorrow‟ means something like: (1) find the referent of „Sydney;‟ (2) note the date of utterance; (3) look into the state of affairs on the day following the date of utterance and,
I will temporarily adopt his convention in this brief section of exegesis in an attempt to avoid or, at least, minimize confusion.
38 in particular, see that the referent of „Sydney‟ meets the criteria for „is racing‟ on that day.
Returning now to the putative B-sentence translation „Sydney races on June 2nd‟ of the A-sentence „Sydney races tomorrow,‟ we can see that it serves, at best, to preserve
the truth-conditions meaning, M 3 . The truth conditions of the translation accord with those of the A-sentence, when uttered on June 1st. It certainly fails to capture the full
intention or thought that is meant, as is required by the intentional meaning, M1 .
Further, the state of affairs evoked by the A-sentence is fundamentally tensed. The A- sentence describes a tomorrowed state of affairs. Clearly, then, the translation fails to
respect Oaklander‟s M 2 sense of the meaning, the described states of affairs. Finally,
with respect to the linguistic rules, M 4 , the A-sentence contains an indexical, which is absent from the translation. The rules that govern indexical sentences are clearly different from those that govern sentences without indexicals.
Recent efforts directed at a more complete tenseless translation of A-sentences have been made by a number of philosophers, including Mellor. Mellor considers an A-
sentence, S, of the form „It is now t1 .‟ Sentence S cannot be translated as S , „“It is now
” is true at and only at ,‟ because, as an analytic sentence, S is always true, whereas
„It is now ‟ is true at only at . But, while is not a translation of S, it does state the truth conditions of S, and, as such, does express the sense of the content of S. In the
sense, the content of S is a function from moments, t, to truth-values, where the value
of the function is „true‟ if and only if tt 1 . This shows, Mellor argues, that A-sentences
39 have B-time truth-makers. That is, there are B-facts in virtue of which A-sentences express true [or false] propositions. In this case, it is the B-fact that S is uttered at the B-
time t1 that makes S true.
Mellor acknowledges that this „old‟ B-theory leaves unexplained the content of S
in the senses of M1 [speakers‟ intention] and M 2 [tensed states of affairs]. Shifting to a more robust example, let us reconsider „The meeting starts now‟ which is true on a certain day when uttered at noon. The sentence „The meeting starts at noon‟ will have the same truth-value at noon as our sentence. Even when uttered at noon, „The meeting starts at noon,‟ however, will fail to provoke the action that „The meeting starts now‟ will provoke. What provokes the action is the tensed component of the content of „The meeting starts now;‟ it is the nowness of the sentence that leads the speaker to rush to the conference room. As we have seen, this component is lost in the attempt to create a tenseless translation of the sentence. Mellor recognizes the importance of this intentional content of the sentence, and makes no attempt to deny its existence. He does not, however, posit that this intention has any existence beyond the thoughts of an agent.
There is no tensed fact in the world that it is now now that prompts the action, only the agent‟s tensed belief that it is now now. Mellor introduces the label „A-belief‟ in reference to such a mental state [Mellor 2002, p. 40ff]. It is the speaker‟s A-belief that it is now noon that leads her to rush away, and not some tensed states of affairs in the world outside of her mental states.
The tensed part of the content in the [speakers‟ intention or thought] sense is provided by the A-belief. The tomorrowness in „Sydney will race tomorrow‟ is found in
40 an A-belief. It is this A-belief that prompts Sydney to get a good sleep that night in anticipation of his race the following day. Similarly, the tensed part of the content in the
M 2 [tensed states of affairs] sense is also provided by the A-belief. It is not a tensed state of affair of the world that is expressed by the sentence, rather a state of affair of an
A-belief that is expressed.
For a B-theorist, the object of the A-belief cannot be an A-fact; it must be an A- proposition. Further, this A-proposition must have B-fact truth-makers. The proposition
that is the object of my A-belief at t1 that „t2 is past‟ is the B-fact that t2 holds the B-
relation of „earlier than‟ to t1 .
This „new‟ B-theory purports to provide tenseless content, in all three senses, to
A-sentences. The tensed intentional content, M1 , consists in A-beliefs, the objects of which are A-propositions. The truth-makers, , of these A-propositions are tenseless
B-facts. Finally, the truth conditions, M 3 , are simply carried forward from the putative translations of the „old‟ B-theory. In „Sydney will race tomorrow‟ the intentional content is the A-belief that Sydney will race the following day; the truth-makers of this A-belief are the B-facts that the day of the B-time at which the sentence is uttered holds the B- relation of „one day prior‟ to the day of the B-time at which „Sydney runs the race‟ is true; and the truth conditions of the sentence are „“Sydney will race tomorrow” is true
when uttered at d1 if and only if Sydney races at d2 and holds the B-relation of „one
day earlier‟ to d2 .‟
41
These three senses exhaust the content of any A-sentence. Oaklander‟s fourth
sense, M 4 , is just the meaning of the A-sentence and, as such, is really a part of the philosophy of language and beyond the scope of the current project. It is important to note, though, that it is seen in the discussion of presented earlier in this section that the meaning of an A-sentence does not, at least in any obvious way, require A-facts.
Mellor‟s view, then, is that we have tensed A-beliefs, many of which explain our actions. These A-beliefs express A-propositions, whose truth values are determined by
B-facts. The M1 [intentional] content of the A-proposition is provided by the A-belief;
the M 2 [reference] content is provided by the B-facts; and the M 3 [truth conditions] content is provided by Mellor‟s tc-function, which involves only B-facts and the B-series.
For example, Sydney‟s belief on Sunday that he races today leads to his leaving the hotel in the morning and travelling to the starting area. Sydney‟s belief that he races today expresses a tensed A-proposition that needs a truth-maker which, for Mellor, is provided by the B-fact that Sydney races Sunday. He introduces a truth condition function [tc-function] to explain the truth conditions of a tensed proposition. A tc- function takes places and B-times as arguments and assigns to them the truth conditions of A-sentences at those places and times. If „Sydney races today‟ is uttered on Sunday, the tc-function takes Sunday as an argument and assigns a value of „true‟ if there are B- facts that make the B-sentence „Sydney races on Sunday‟ true and assigns a value of
„false‟ otherwise. But, a closer look at our example shows that Mellor‟s analysis is insufficient to explain the impetus for those of Sydney‟s actions that are based on his A- belief that „Sydney races today.‟
42
The B-fact that „Sydney races Sunday‟ contributes to the truth value of the A- proposition „Sydney races today,‟ but does not determine it. This B-fact needs to be accompanied by the additional B-fact that Sunday is the day of utterance. The A- proposition expressed by the sentence „Sydney races today,‟ when expressed on Sunday, is made true by the B-facts „Sydney races Sunday,‟ and „the sentence „Sydney races today‟ is uttered on Sunday.‟ Together, these B-facts determine the truth value of the A- proposition.
Consider, though, Sydney‟s actions that arise from this A-belief. He leaves the hotel now, not because the race is Sunday or because the utterance was made on Sunday.
He leaves the hotel now because the race is today. His A-belief is not that the race is
Sunday, but that the race is today. Mellor‟s analysis can take us part of the way to an explanation of Sydney‟s actions, but not the full way. In addition to Mellor‟s B-facts that
„Sydney races Sunday,‟ and „the utterance is made on Sunday,‟ the A-fact „Today is
Sunday‟ is required to explain Sydney‟s successful action of leaving the hotel when he does and proceeding to the starting line. As we saw earlier in the discussion of tensed indexicals, there is an un-eliminable role played by tense in explaining Sydney‟s actions.
Sydney‟s action of leaving the hotel when he does is prompted by his belief that the A- proposition expressed by A-sentence „Sydney races today‟ has a truth value of „true.‟ For
Sydney, that truth value requires at least three facts; two B-facts and one A-fact.
Returning to Oaklander‟s discussion of the different types of meaning, when viewed from a perspective in which agency is considered, because Mellor‟s analysis does not produce a truth-maker for an A-proposition, it does not provide an explanation even
for the M 3 type of meaning. In contexts apart from those involving agency, however, he
43 does succeed in eliminating the need for A-fact truth-makers and, in doing so, provides
an explanation for the M 3 type of meaning. For example, the A-sentence „It is raining today,‟ when uttered on Sunday, expresses an A-proposition whose truth value is determined by the B-facts „It is raining Sunday,‟ and „The sentence is uttered on Sunday.‟
Because of the failure of Mellor‟s B-theory to explain fully the meaning of an A- sentence from a perspective that includes agency, and its success in explaining the meaning of an A-sentence from a perspective that does not include agency, we can see it as being a failed theory of experienced time and a potentially successful theory of physical time.
Oaklander points out, in addition, that this model fails to distinguish between the truth conditions of tensed A-beliefs and related un-tensed B-beliefs. The B-belief that
„Sydney races on Sunday‟ has the same truth conditions when expressed on Sunday as does „Sydney races today.‟ Yet, we have seen that the propositions these express have different intentional content. Mellor‟s attempt to reduce A-beliefs to B-beliefs flows from his attention to the type of content. His focus, though, fails to account for their different intentional content. To the extent that he has succeeded, Mellor has only managed to provide an explanation for the truth-conditional content of a tensed sentence when considered in the context of physical time. A robust B-theory of time needs to be much broader in its application.
In presenting his “even newer” B-theory, Oaklander suggests that B-theorists‟ attempts to provide truth-makers for A-propositions from within a B-theoretic ontology will all prove futile. Instead, he proposes a B-theory under which all A-propositions are
44 false. Oaklander writes that “A-sentences express subjective contents that are their
intentional meaning [ M1 content] … .” [Oaklander 2004, p. 284] These subjective contents are responsible for the actions that arise from our tensed beliefs. But, while they
have intentional content, A-sentences do not have reference content [ M 2 content]. There are no tensed states of affairs for A-sentences to reference. Accordingly, there are no A- facts to serve as truth-makers for A-propositions. Since B-facts cannot act as truth- makers, assuming bivalence, all A-propositions are false.
Oaklander claims this newer B-theory avoids Mellor‟s mistake of failing to
distinguish the reference and truth-condition content [ and M 3 content] of A- sentences from that of B-sentences. This theory has the virtue of recognizing that tenseless sentences cannot provide the full meaning of tensed sentences. Unfortunately, it shares with the new B-theory a failure to identify the object of A-beliefs. Both theories acknowledge the importance of A-beliefs in explaining action, but neither provides a satisfactory tenseless account of them. This failure arises from an inability to reconcile the uneliminability of tensed beliefs with a B-theoretic ontology. These theories are in need of an ontologically-acceptable Now.
Using our analysis from earlier in this chapter, the sentence „Sydney races today‟ is seen to express the proposition
45 proposition has no referent. With no referent, there is no truth-maker. Because of this, under Oaklander‟s view, it is simply false. While Oaklander, unlike Mellor, provides a means for determining the truth-value of the A-belief that „today is d,‟ he does so at the expense of reducing every tensed proposition to a truth-value of „false.‟ „Sydney races tomorrow‟
„today is d-1‟ and „today is d‟ both fail to refer to any fact and so, according to
Oaklander, are simply false.
Oaklander has addressed the issue he found in Mellor‟s approach. „Sydney races today,‟ when uttered on Sunday does have different truth conditions than the corresponding B-sentence „Sydney races Sunday.‟ This newer B-theory, however, does not provide an explanation for Sydney‟s action of leaving the hotel when he does.
Further, it offers an analysis that seems to deny that there is anything in the world in virtue of which Sydney‟s action proves to be successful or unsuccessful.
Both Mellor and Oaklander seem to view beliefs as being relations between individuals and propositions. For Sydney to leave the hotel when he does, he must believe that „Sydney races today‟ is true. On Oaklander‟s account, though, the tensed proposition expressed by „Sydney races today‟ is false, no matter when it is evaluated.
On Sunday morning, then, there is no difference in truth value between the proposition expressed by „Sydney races today‟ and that expressed by „Sydney races tomorrow.‟
Because „today‟ and „tomorrow‟ fail to refer, on Oaklander‟s view, these propositions are both false. But if the truth conditions are identical, there are no facts in any particular
46 state of affairs that would suggest that Sydney should leave the hotel one day as opposed to the next.
Not only, though, does Sydney leave the hotel at a particular time on a particular day based on his own A-beliefs, his decision to leave when he does will prove either to be successful or unsuccessful. If, believing today is Sunday, Sydney leaves Saturday for the race he will arrive to an empty starting line and his action will prove unsuccessful. If, believing today is Sunday, Sydney leaves Sunday for the race, his action will prove successful. There are facts in virtue of which these actions prove, respectively, to be unsuccessful or successful. These actions, on both Mellor and Oaklander‟s account, are connected to A-beliefs.
Oaklander‟s position certainly fails as a theory of experienced time. It also fails as a theory of physical time, at least insofar as it offers no effective analysis of A- sentences. „It is raining today,‟ and „It is not the case that it is raining today.‟ are contradictory. Yet, for Oaklander, they have the same truth value. Just as an agent will be moved to different actions based on the respective contents of different A-sentences, a third-person description of a state of affairs will vary according to the respective contents of different A-sentences. We intend for the sentence „It is raining today,‟ when uttered on Sunday, to have the same truth-conditions as „It is raining on Sunday.‟
This leaves us in a worse position than we were under Mellor. While Oaklander, rightly, points out that Mellor cannot make do with just tenseless truth-makers, his own attempt to solve this problem by making do with no truth-makers at all is even more unsatisfying.
47
Perhaps ironically, one means of providing a tensed truth-maker within a B- theoretical ontology is suggested by Russell‟s old B-theory. In On the Experience of
Time, Russell writes that “past, present, and future arise from time-relations from subject to object,” [Russell 1915, p. 212] and that the time which arises from subject to object is mental time. Mental time is much less simple than physical time. The essence of B-time is simply succession, which is similar to McTaggart‟s B-relation. The experience of succession may include remembered objects or objects given to sense and, so, analyses of memory and sensation are necessary to an understanding of mental time.
Objects from memory, together with objects simultaneous to them which are absent from memory, form the past. The present is inhabited by the objects of sensation.
Russell defines Now as “„simultaneous with this,‟ where „this‟ is a sense-datum.” [Russell
1915, p. 221]. Now is the moment of sensation. Without begging many questions in the
Philosophy of Mind, it is clear that we collect sense-data from a single moment at a time.
As we collect sense-data from a moment, that moment is Now. Now moves to another moment when that moment becomes the object of our attention.
The A-theorist holds that Now is something in the world independent of our minds, and that we are immediately aware only of the moment that holds the A-relation of „present‟ with Now. For the B-theorist, there is no Now in the world independent of our minds. So, for the B-theorist, Now can just be the moment of which we are immediately aware.
Understanding Now in this way, we return to our analysis of the proposition expressed by „Sydney races today.‟ The first three components of
48 component will be determined relative to the „witness(es)‟ of the utterance. Consider the speaker of the sentence. When the speaker‟s immediate awareness is focussed on a particular moment during the utterance of the sentence, say the moment at which the final sound is made, that moment is Now for the speaker. That Now is the referent of the
„today is d‟ part of the proposition. A similar analysis is available for any other witness to the utterance.
Both Mellor and Oaklander acknowledge that tense is uneliminable from our experience of time. It is an inescapable part of the intentional and truth-conditional content of A-sentences. They face the difficulty, however, of trying to satisfy this requirement from a B-theoretical ontology. By separating experienced time from physical time, Russell offers a way through this difficulty. While tense is not a part of physical time, it is a part of experienced time. If we define Now relative to any agent as being the moment of immediate awareness – the moment at which sensations are experienced – we have a notion of Now that provides all of the intentional, referential, and truth-conditional content of A-sentences without following the A-theorist and positing a Now in physical time. A-facts belong only to experienced time, because Now arises from experience – it is the moment of sensation. The B-theoretical ontology of physical time remains uncontaminated, as tense is seen to supervene on consciousness.
2.2.3 The Unreality of Time: Summary of Responses to McTaggart’s Argument
McTaggart‟s argument that time is unreal – that nothing has the characteristic of being in time – rests on two main premises. First, McTaggart argues that change requires the A-series with its tensed A-relation. His basic claim is that, without a changing element in time itself, there would be no medium in which change could occur. The B-
49 theorists hold that the B-series with its B-relation is sufficient to explain change. They maintain that the apparent uneliminability of tense is, in fact, confined only to our minds and our beliefs. They argue that, while tense is a part of experienced time, it is not a part of physical time. In this way, they are not committed to an A-theoretic ontology. The B- theorist attacks McTaggart‟s first premiss by arguing that tensed physical time is not a necessary condition of change.
McTaggart‟s second premiss is that the A-series and A-relation cannot exist, that their existence would lead either to the objects of change possessing contradictory properties or to a viscous regress. The A-theorist attacks this premiss by admitting the A- relation into her ontology. By positing the existence of an actual „moving‟ Now, the A- theorist provides the dynamic element of time needed by McTaggart to explain change, while avoiding the dilemma between offending the indiscerniblity of identicals and causing a vicious regress. The A-theorist does not need to meet any greater burden in positing this theoretical entity than does the physicist who believes in forces. Now is known by its effects; it can be described, measured, and forms the basis of accurate predictions and explanations.
In the end, McTaggart‟s position with respect to the unreality of time is a clear and unavoidable consequence of his Idealism. Evaluated from within a context of his other metaphysics, McTaggart‟s argument is sound. The counter-arguments brought against his argument do not attack it from within this system. Rather, they serve more to elucidate the variously-held Realist positions which find a place in their ontology for temporal entities. One clear value of the arguments to a contemporary reader is that they
50 provide opportunities for thorough examinations of these various positions and, in doing so, offer comment on many important issues concerning Realist metaphysics of time.
2.3 Concluding Comments on Three-Dimensionalism
Three-dimensionalist theories of time vary widely in their ontologies and in the characteristics they attribute to many of the elements of time. While these factors distinguish individual three-dimensionalist theories, they all hold that temporal relations exist between objects, or events, and moments in time. Space and time are separable, though related, on three-dimensional accounts of time. A-theorists who hold three- dimensionalist views argue that there is a temporal relation between the object of change and some changing thing that corresponds to the present – McTaggart‟s Entity-X or our
Now. B-theorists who hold three-dimensionalist views argue that objects, or temporal parts of objects, are related to moments on the static B-series.
There is a clear difference between experienced time and physical time.
Experienced time is unquestionably tensed, with the moment of awareness being identified with Now. Experienced time is A-time, with a Now that resists McTaggart‟s argument. Objects do not have contradictory temporal properties and there is no viscous regress when the moment of awareness is taken as Now.
The most simple ontology for physical time is tenseless B-time. As B-theorists argue, change can be accounted for by B-time. Those instances of agency which cannot be accounted for by B-time are explained by our theory of experienced time, so they do not contribute counter-examples to a B-theory of physical time.
At this point, we are left with at least the possibility that there are two „times‟ we reference. Experienced time, which is tensed, and physical time, which is not. We will
51 now turn our attention to theories of time in which the external world is taken to be four- dimensional.
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Chapter Three: Four Dimensionalism
3.1 The Four-Dimensional View Overview
Four-dimensionalists often appeal to our familiarity with the whole-part relation in non-temporal contexts when trying to develop an intuitive foundation of their theory of time. The Lego castle is made of Lego block parts. A tree has leaves, branches, a trunk and roots. That some of a tree‟s parts are below the ground and others above the ground does not lead to the conclusion either that some of the parts are not parts of the tree or that the tree is guilty of the paradox of being both above and below the ground. We gracefully accommodate different, even contradictory, properties of various spatial parts of an object. Temporal four-dimensionalists hold that, in addition to these physical parts, objects are made of temporal parts. McTaggart‟s poker has some temporal parts that are hot and others that are cool. They are numerically distinct temporal parts of the same object. The object, the poker in this case, is nothing other than the mereological sum of all its temporal parts.
Four-Dimensionalists hold the view that objects are made of „parts.‟ Often, these are loosely referred to as „temporal parts,‟ but this blurs an important element of the four- dimensionalists‟ position. On the four-dimensionalists‟ view, each part of an object is represented by a set of spatiotemporal points, each of which can be described by an ordered four-tuple whose first three elements are spatial coordinates and whose fourth element is a temporal coordinate. When we refer to a „temporal part‟ we identify a set of spatiotemporal points that share a temporal coordinate. This is only one way to organize
53 the parts of an object, though it happens to correspond with the way we experience change. We could equally identify sets of spatiotemporal points that share the first coordinate, these parts would be collinear; or that share the whole ordered triple of spatial coordinates, these parts would be coincident. While we favour the temporal organization in a study of time, we do not need to invest in it any metaphysical priority over any other organization. Because this thesis is concerned with temporal properties of objects, however, our focus here will primarily be on temporal parts.
Many four-dimensionalists arrive at their position through an interest in identity.
Four-dimensionalism offers solutions to a number of problems in identity theory, such as the Ship of Theseus. The Ship of Theseus is really a family of hypothetical problems that question identity through time. The planks of Ship A are removed and replaced, with the removed planks being reassembled to form of Ship B. Even as the first plank is removed and replaced, there is a question of whether the ship after the replacement is numerically identical with Ship A. A four-dimensionalist could say that Ship A is a collection of temporal parts, some of which include the original plank and other of which include the replacement plank. No two temporal parts are numerically identical6, but the collection of which these parts are elements would, even when considered at different times, be identical with itself. The object named „Ship A‟ contains parts, some of which have the original plank and some of which do not, just as my right hand and my left hand, spatially separated though they are, are parts of the same body. The temporal parts of Ship B are all distinct from those of Ship A, even though they contain some of the same matter that
54 was found in Ship A. When we hold the perspectives of the respective ships, we encounter no identity crisis under a temporal parts analysis.
Now, consider a plank, Plank p, which is the first plank to be removed from Ship
A, and then is the first plank set in place for Ship B. Some temporal parts of Plank p have the property of „being an element of Ship A,‟ and that some other temporal parts of Plank p have the property of „being an element of Ship B.‟ Of course, other temporal parts of
Plank p have the property of being part of the trunk of a certain tree, while others have the property, perhaps, of being barnacle-encrusted while Ship B sits quietly in its slip after its productive years are past.
A body is extended in space and, so, its parts are spatially separate. Four- dimensionalism simply extends this notion to temporally separated parts of an object that extend in time. Parts of objects are seen, not only to be spatially separated, but temporally separated as well. Strictly, the four-dimensionalist simply sees objects to be made up of numerically distinct spatiotemporal parts that extend in space-time.
Four-dimensionalists view objects to be the mereological sums of their temporal parts and hold that they persist by perduring through time. In the following sections, we will examine these principles in greater detail.
3.2 Mereology
3.2.1 Temporal Parts
In 1940, Henry Leonard and Nelson Goodman introduced a calculus of individuals [Leonard and Goodman 1940], which served to explicate formally the part-
6 Notice that, even with no change in matter [if this is really possible] or position, by definition of temporal part, no two parts would have the same temporal coordinates.
55 whole relation. Leonard and Goodman define a fusion relation that holds between an individual and a non-empty class. They take a two-place relation „D‟ as a primitive7.
The natural interpretation of „D‟ is „is distinct from.‟ Then, given a set, , there is a fusion, x, such that x is comprised of all and only the elements of . That is, if „F‟
represents the fusion relation, FDDxDf z zx y y zy . On the surface, there appears to be little difference between the set and the fusion x. The important difference lies in their being of different types. is a set while x is not a set, but an individual.
In addition to the fusion relation, we will require a two-place overlap relation, „O‟
between objects, defined as OPPyxDf z zx zy. That is, objects x and y overlap just in case they have a common part.
This definition assures us that nothing distinct from the containing object will fail to be distinct from any contained part. In a positive frame, it asserts that all of each part of an object is fully present in that object. In a four-dimensionalist context, we see that all of a temporal part is contained in any fusion of which it is a part.
Before defining temporal part on this account, we need one final two-place relation. The part relation, „P,‟ holds between two objects subject to:
PDDyxDf z zx zy . This definition assures us that nothing distinct from the containing object will fail to be distinct from any contained part. It asserts that all of each part of an object is fully present in that object. Because relations O and P are such that
7 Leonard and Goodman built their calculus on the language of Principia Mathematica. Here, I will follow
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y xOD yx yx , objects overlap just in case they are not distinct, we may more
intuitively define the part relation as POOyxDf z zy zx. Object y is a part of object x if everything that overlaps y also overlaps x.
We can now define „temporal part8‟ as follows: y is a temporal part of x at t if and only if (1) y is a part of x at t; and (2) y overlaps everything at t that is part of x at t. [Sider
2001, p. 59] Indexing our two-place relations to times, and using the symbol „T‟ to
represent the relation „is a temporal part,‟ we have TPPOyxtDf yxt z zxt yzt .
Object y is a temporal part of fusion x at instant t if and only if y is a part of x at t and, for all objects z, if z is a part of x at t, then y overlaps z at t. So, besides being a part of an object at a particular time, a temporal part is a complete part of the object at that time.
3.2.2 Instantaneous Temporal Parts
Often, we find it useful to think of thick temporal parts. One may reflect on what kind of person Madonna was in her thirties, or be interested in the plank while it is part of
Ship A. The definition of a temporal part in the previous section is sufficient for these common uses, but these do not characterize the more robust notion of temporal parts that we often employ in our thoughts concerning time. In an adequate four-dimensionalist theory of time, temporal parts must shoulder the work that instants perform in three- dimensionalist theories. Accordingly, we need to define an instantaneous temporal part.
the equivalent symbolization found in Kazmi, 1990. 8 While four-dimensionalists hold a theory of spatiotemporal part-hood simpliciter, in the context of our discussion, it is often more clear to speak of temporal part-hood. Before doing so, however, it should be emphasized again that four-dimensionalism need not assume any metaphysical priority to temporal parts over spatial parts; while it is the case, though, that our experience seems to favour temporal parts.
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We write y is an instantaneous temporal part of x at instant t if and only if (1) y exists at t and only at t; (2) y is a part of x at t; and (3) y overlaps everything at t that is
part of x at t. Given a two-place existence predicate, „E,‟ we define the relation „ TI ‟ as:
TEEPPOI yxtDf yt u yu u t yxt z zxt yzt . The temporal part, y, still has the properties of a temporal part described in the previous section but, under this definition, has a duration of only an instant and, so, constitutes only an instantaneous temporal part of the fusion x.
3.2.3 Ersatz Temporal Parts
Sider names ordered pairs of objects and times „ersatz temporal parts.‟ [Sider
2001, p. 61] While these, as his name suggests, are not real temporal parts, they can be helpful in some contexts. For example, Russell‟s temporal parts are really ersatz temporary parts. They were relations between objects and times and, for this reason, form part of a three-dimensional theory. But, while ersatz temporal parts have some usefulness, they do not reflect a four-dimensional view.
We recall that, for the four-dimensionalist, objects are made of parts and that these parts are collections of spatiotemporal points. Time, as represented by the fourth coordinate of each point, is as inextricably tied to the part as are its spatial dimensions.
There is no need to look for time outside of the object or its temporal parts. The time is in each of the spatiotemporal points that comprise the object and its parts.
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3.3 Special Relativity and Geometric Representations of Four-Dimensional Space- Time9
3.3.1 Galilean Relativity
Galilean Relativity illustrates the principle of symmetry in physics. Consider two identical sets of apparatus that allow the various properties of a falling object to be observed and measured. Suppose that each apparatus is „relevantly complete‟ in the sense that everything contributing to the properties of the falling object is contained in the apparatus. Finally, suppose that the two apparatus are in independent uniform motion.
Galilean Relativity assures the observer that all properties of the falling object will be indistinguishable between the apparatus. One consequence of Galilean Relativity is that the physics on a rotating Earth in a heliocentric solar system will be indistinguishable from the physics on a stationary Earth. The question, then, of where the centre of the
Universe is to be found ceases to be a meaningful question with respect to physics. The centre could be anywhere and nothing would be affected.
Galilean Relativity ushered the end to the notion in physics of absolute space.
The idea that there is an absolute space within which physical systems are situated, whether at rest or in motion, was replaced by the view simply that there are physical systems whose positions and motion are understood only relative to the positions or motions of other systems. With no absolute space, there can be no privileged and common „coordinate system‟ relative to which the positions and motions of systems can be described. Each system defines its own centre and corresponding coordinate system relative to which the position and motion of other physical systems can be described.
9 The material in this section derives primarily from DiSalle 2010, Hawking 1988, Norton 2010,Putnam
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But, while Galilean Relativity does away with the notion of absolute space, it leaves untouched the notion of absolute time. Prior to Einstein‟s Special Theory of Relativity, the emerging understanding was one of relative space against a backdrop of absolute time. As we have seen, three-dimensionalist theories of time all presuppose some foundation of absolute time – of time being apart from objects and events. For three- dimensionalists, change occurs within time.
3.3.2 Special Relativity
Special Relativity is built on both Galilean Relativity and the principles of symmetry in physics. The principle of Special Relativity that is of primary concern to us is that the speed of light relative to any object remains constant across physical systems.
Recall our two identical falling-object apparatus. Again, all the properties of the falling object will be the same between the two apparatus but, under Special Relativity, the speed of light relative to the motion of one apparatus will be the same as the speed of light relative to the other.
The significance, and seeming paradox, of this principle can be seen through dis- analogous examples of relative velocities not involving light. Suppose Roy is standing in place and throws a baseball with sufficient force that it leaves his hand at a velocity of 50 km/h. The velocity of the ball as it leaves his hand is 50 km/h when measured relative both to Roy‟s hand and to the ground. Suppose, now, that Roy is standing in his car with his torso extended through the moon roof, while his wife drives the car along a straight stretch of the highway at a velocity of 100 km/h relative to the ground, and that he throws
1967, and Savitt 2010.
60 the ball with sufficient force that it leaves his hand with a velocity of 50 km/h relative to his hand. As it is released, the ball has a velocity of 50 km/h relative to Roy‟s hand, the car, and everything in the moving car. But we notice that it has a velocity of 150 km/h relative to the point on the ground above which the ball was released.
Now, if the example is changed so that, rather than throwing a ball, Roy turns on a flashlight, we see the difference that Special Relativity introduces. If, instead of throwing a ball, Roy switches on a flashlight, the light leaves the flashlight at a velocity of c 299792458 m/s relative to Roy and the moving car. It also leaves at a velocity of c relative to the point on the ground above which the flashlight was turned on. The velocity of the light relative to Roy and his moving car is equal to the velocity of the light relative to ground.
To make the example a bit more instructive, we will imagine Roy travelling inside a wide spacecraft that speeds past a space-buoy on which his wife is standing. Suppose the following: Prior to his boarding the spacecraft, Roy builds two identical clocks which, when placed side by side, were exactly synchronous. He also constructs two measuring rods whose lengths when held together are identical. Roy gives one clock and one rod to his wife and keeps the others. His wife then takes her clock and rod to the space-buoy past which Roy‟s spacecraft will fly. When Roy boards the spacecraft he stands against the side wall, facing away from the wall at a right-angle to the direction of motion and holds a flashlight. Directly opposite him, at a distance of d units as measure by his rod, there is a mirror. With the spacecraft travelling at a uniform velocity of v m/s relative to the space-buoy on which Roy‟s wife is standing, Roy flies past his wife
[Figure 6].
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Space Buoy
Roy
d
Mirror
Direction of Spacecraft‟s Motion [v m/s]
Figure 6
Path of Light Relative to Spacecraft
Figure 7
Suppose that, while the spacecraft is in motion, Roy shines his flashlight and uses his very accurate and responsive clock to measure the time it takes for the light to travel to the mirror and then return. From the perspective of Roy and everything moving with the spacecraft, the beam of light will travel straight ahead d units to the mirror and then
straight back d units to Roy [Figure 7]. Suppose the time this takes measures ts seconds on Roy‟s clock.
From the perspective of Roy‟s wife on the space-buoy, the path of the light ray will appear different. Relative to this position, both the light ray and the spacecraft are in
62 motion. While the light travels from Roy‟s flashlight to the mirror, the spacecraft remains in motion, so the spacecraft – and the mirror with it – will have moved in the time it takes for the light to reach the mirror. Similarly, Roy will have moved in the time it takes for the light to return [Figure 8].
Path of Light Relative to Space-Buoy
d
Direction of Spacecraft‟s Motion [v m/s]
Figure 8
Relative to the space-buoy, the light travels a longer path to the mirror and back than it does relative to the spacecraft. Since the speed of light is always c m/s relative to the motion of any body, the time it takes for the light‟s round trip will, also, be longer when measured from the space-buoy.
d Consider, first, the time it takes relative to the spacecraft. In general, t , v where t, d, and v are time, distance, and velocity, respectively. When measured on the spacecraft, the light travels a total distance of 2d on its path to the mirror and back.
2d Relative to the spacecraft, the time for the round trip will be t . s c
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2 The distance travelled by the light relative to the space-buoy is 4d vtB ,
where v, is the velocity of the spacecraft and tB is the time it takes for the light to complete the round-trip, both measured from the space-buoy. Notice that the distance the light travels increases as the relative velocity of the spacecraft increases, and is equal to
2d when the spacecraft is stationary relative to the space-buoy. Relative to the space-
4d2 vt buoy, then, the time it takes for the light to complete its return trip is t B . B c
We see that, if v 0 , ttBS . This means that, if Roy and his wife both measure the event with their clocks, Roy‟s clock will run slower relative to his wife‟s.
The light emitted from the flashlight has a velocity of c 299792458 m/s relative to Roy‟s hand and relative to the space-buoy. Because the velocity of the light is constant relative to any perspective, the difference in the length of the path of the light between the perspectives of the space-buoy and the spacecraft forces a corresponding difference in the measured time of the event. To accommodate this effect of special relativity, we consider two frames, one for Roy and his spacecraft and the other for the his wife and her space-buoy. Each frame can be thought of as consisting of an independent space-time coordinate system for the respective systems. In our example, let
FS be the frame of the system that consists of everything moving with the spacecraft and
FB be the frame of the system that includes the space-buoy. The symmetry of physics requires that the physical properties in one system will be the same as those in the other; that is, the physical laws will yield the same results in one system that they will in the other. By adding the principle that the speed of light as measured in one system will be
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identical to the speed of light as measured in the other, Special Relativity requires that FS
and FB have different space-time coordinate systems.
Without loss of generality, we will consider the relative motion of the spacecraft to be occurring along the x-axis of each of and . Recall that the position of spatiotemporal points can be described by an ordered four-tuple, where the first three coordinates describe the spatial position of the point and the fourth coordinate describes the temporal position of the point. The Lorentz Equations provide a mathematical model that relates the coordinates of any spatiotemporal point in one frame to its coordinates in another. In our current example, consider spatiotemporal point, p. Let the coordinates of
p in be xSSSS,,, y z t , and in be xBBBB,,, y z t . If the velocity of relative to
is v, the Lorentz Equations tell us that:
xBB vt xS ; v2 1 c2
yySB ;
zzSB ; and
vx t B B c2 tS v2 1 c2
The calculations converting coordinates relative to one frame into coordinates relative to another will not concern us here. We can see easily from these equations, though, that if v 0 ; that is, the frames are not in motion relative to each other, then
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xSSSSBBBB,,,,,, y z t x y z t . More generally, they show that the faster one frame is moving relative to the other, the greater the difference will be between a point‟s temporal coordinates10 in the respective frames. An intuitive consequence of this is that clocks that would run synchronously in the same frame will seem to run at different rates in different frames.
Special Relativity implies that there is no absolute time; there is no „master clock‟ or „ideal timeline‟ that objects or events can be related to. Each independently-moving system has its own „clock‟ that runs according to that system‟s frame. The notorious
„Twins Paradox,‟ illustrates one consequence of this feature of Special Relativity.
Consider monozygotic twins named Pollux and Castor who were born at the same instant by Caesarean section, and have lived together until their twentieth birthday, when Pollux steps aboard Roy‟s spaceship. Leaving his twin Castor behind on Earth, Pollux cooks a three-minute egg on board the spaceship while it travels on a circular path through space at a very high rate of speed. The spaceship returns to the Earth just as Pollux‟s egg is cooked. He checks his watch to see that three minutes have elapsed since he left, and disembarks to find his twin Castor, surrounded by friends and family, celebrating his twenty-third birthday. Castor‟s watch and calendars attest to his having aged three years, while Pollux has aged just three minutes.
There is no single answer to the question “How long was the spacecraft away.” It was away three minutes relative to Pollux‟s frame and three years relative to Castor‟s frame. There is no absolute time to reference in response to the question. Any frame of
10 We notice the corresponding observation concerning the spatial x-coordinate, but will keep our primary
66 reference can be chosen and designated as being the privileged frame. Objects and events can then be measured with respect to the spatial and temporal coordinate system of that frame. We often make such a selection. For example, we choose the frame of reference of the Earth to measure those objects and events that we are most directly involved with. The objects and events in our experienced world all rest in a frame similar to that of the Earth, so it serves a useful frame to privilege. It is important, though, to see that this – or any other frame we may choose as privileged – is not ontologically privileged.
Castor and Pollux‟s world lines sat in very similar, though non-identical, frames prior to their twentieth birthday. Until that time, any difference between their watches would have been miniscule. Then, when he sped away on the spacecraft, Pollux engaged with a significantly different frame than the one in which Castor remained. Their different frames while Pollux was away had significantly different time scales from each other. As a result, while Pollux was away, their personal clocks ran at very different rates. When they were reunited, they, once again, found their world lines in virtually identical frames. As long as neither travelled again at a high velocity relative to the other, they would, again, age at virtually identical rates, although the three years that
Castor lost to Pollux would never be regained. From the reunion onward, Pollux will always have aged nearly three years less than his twin.
focus on the temporal coordinate.
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3.3.3 Temporal Parts and Special Relativity
The four-dimensionalist, by analyzing objects as being made up of four dimensional spatiotemporal points, can accommodate the temporal outcomes of special relativity fairly gracefully. We recall that the four-dimensionalist sees objects and events as being fusions of parts and, in temporal contexts, think in particular of these parts as being temporal parts.
For example, Castor and Pollux can be seen as fusions of their respective temporal parts. Interestingly, though mostly incidental to our point here, as monozygotic twins, they share a set of temporal parts between the instant of their conception and the duplication of their zygote. If we examine Castor and Pollux‟s respective parts, we notice that, prior to Pollux‟s space adventure, there would have been a rough correspondence between the temporal coordinates of their spatiotemporal points. This correspondence would have approximated a relation of simultaneity between points. At any instant until his twentieth birthday, Pollux could have looked to his brother and said truthfully, “We are the same age.” That is, at any moment in his life until then, Pollux‟s then-current temporal part will have approximately the same temporal coordinate relative to his frame as Castor‟s temporal part will have relative to Castor‟s frame. This changes once Pollux‟s spacecraft lifts off. When Pollux returns, his then-current temporal part has a temporal coordinate relative to his frame that is nearly three years earlier than
Castor‟s then-current temporal part relative to Castor‟s frame.
In the absence of absolute time and space, spatiotemporal points acquire their coordinates from their frame. Castor and Pollux begin life in the same frame, and then occupy nearly identical frames until their twentieth birthday, when their frames are
68 radically different from one another until Pollux returns. Each fusion of parts occupies a frame with its own four-dimensional spatiotemporal coordinate system. It would seem often to be the case that a fusion occupies a series of frames. Pollux occupied one frame that included his proximal patch of the Earth until he boarded the spacecraft and then began to occupy its frame. This allows the fusion that is Pollux to experience time differently from the fusion that is Castor.
We often speak of an object‟s world line. A world line is a two dimensional representation of that object‟s path through space-time. The vertical dimension of a world line curve represents the passage along the temporal dimension and the horizontal dimension represents passage along the spatial dimensions. The horizontal dimension compresses three spatial dimensions to one; it may be thought of as representing the scalar distance from some stipulated point of origin. Because they represent passage through space-time, each world line must be set relative to a particular frame of reference. The frame of reference will set the scales for the axes of the world line.
Consider the portions of Castor and Pollux‟s world lines that begin at some point prior to their twentieth birthdays and end when they are reunited, first, from the perspective of Castor‟s frame [Figure 9].
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World Lines With Respect to Castor‟s Frame
Castor Pollux
- 20th Birthday - Reuinion
Figure 9
With respect to Castor‟s frame, Pollux‟s world line between his twentieth birthday and the reunion will feature dramatic horizontal displacement, while Castor‟s world line will continue to meander much as it did prior to the birthday. When drawn with respect to Pollux‟s frame [Figure 10], the vertical distance between the twentieth birthday and the reunion is very short and Pollux exhibits moderate horizontal displacement, it would reflect only as far as he could travel in three minutes.
World Lines With Respect to Pollux‟s Frame
Castor Pollux
Figure 10
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An object‟s world line can be seen as a two-dimensional shadow of its four- dimensional space-time manifold. We can think of it as being a visual representation of a fusion. Consider a translation of a fusion in which, for each spatiotemporal point
x,,, y z t in the fusion, an ordered pair x2 y 2 z 2 , t is created whose first coordinate describes the square of the points‟s distance from the origin and whose second coordinate is equal to the point‟s fourth coordinate. Each point on a world line corresponds, then, to a point in this transformation.
3.4 Temporal Parts in Action
The utility of a theory is not justification for it. The following examples are included to serve two purposes. First, they help elucidate the four-dimensionalist view of time and, second, they provide support, if not a substantive argument, for the position of four-dimensionalism.
3.4.1 The Ship of Theseus
In the introduction to this chapter, the puzzle of the Ship of Theseus was untangled by the intuitive use temporal parts. When viewed as fusions of their respective temporal parts, Ships A and B are seen clearly both to be non-identical to each other and self-identical continuants in spite of the changes they undergo. Plank a, similarly, presents no real difficulties with respect to identity when it is viewed both as a fusion of its own temporal parts and as a part of a temporal part of each of Ships A and B at different times. Consider some time, t, at which Plank a is a physical part of Ship A. The temporal part of Ship A at t is simply the set of all spatiotemporal points defined as Ship
A whose fourth coordinate is t. Because the spatiotemporal points that comprise Plank a would all be included in this set, Plank a is a part of Ship A at t. The temporal part of
71 plank a at t is a part of the temporal part of Ship A at t. This is most simply seen by considering each of these parts to be a set of points; the set of points that is the temporal part of Plank a at t is a subset of the set of points that is the temporal part of Ship A at t.
Other temporal parts of Plank a will be parts of temporal parts of Ship B but, critically, there is no t at which the temporal part of Plank a is a part of both the temporal parts of Ships A and B. Again, the problem of the Ship of Theseus dissolves under the four-dimensionalist view.
3.4.2 Temporary Coincidence
The problem of temporary coincidence can, similarly, be seen to be managed effectively by a four-dimensionalist. Consider a lump of clay that is delivered to an artist‟s studio on Monday. The clay, which we name „Lump,‟ remains a lump until
Tuesday when the artist creates from it a statue, which we name „Statue.‟ After the creation it is clear that, while Statue has been created, Lump has not been destroyed.
Both Statue and Lump exist and seem to be the same object. But, if they are identical with each other, we seem to run afoul of the principle of the indiscerniblity of identicals, as Lump has the property of „existing Monday,‟ while Statue does not.
Statue and Lump are both objects and, as such, are fusions of their respective temporal parts. It is plain to see that there are not identical objects. Lump was delivered and lived in a box Monday, before Statue even existed. But, they seem to be identical for a certain duration once Statue is created. The identity here is of the temporal parts and not the fusions. For every t while the clay is in the form of a statue, Lump‟s temporal part at t is identical with Statue‟s temporal part at t. The identity relation obtains between
72 temporal parts, but not between fusions. Again, an apparent problem in identity dissolves under the gaze of a four-dimensionalist.
3.4.3 McTaggart’s Paradox
McTaggart‟s argument for the unreality of time holds that since tensed time is necessary for change and tensed time leads either to a contradiction or to a vicious regress, time is not real. Four-dimensionalism presents an alternative view of change and, thereby, avoids McTaggart‟s conclusion that time is unreal.
Consider McTaggart‟s example of a poker that is hot on Monday and cold at all other times. On McTaggart‟s account, change cannot be explained in terms of the poker having, at one time, the property of being hot and, at another time, the contrary property of not being hot. The poker cannot have and then not have a particular property and remain the same poker. Instead, McTaggart holds that the poker always has the B- property of being hot on Monday and the B-property on every other day of being not-hot.
Change for McTaggart requires the A-Series. Change is accounted for by the change in the tense of the instant at which the poker is hot from being „future‟ to being „present‟ to being „past.‟ As we saw in the previous chapter, this account leads three-dimensionalists to one of three places: (1) an A-theory of time in which tense is seen as real, (2) a B- theory of time in which tense is denied but the B-relation of „earlier than‟ is sufficient to explain change; or (3) the conclusion that time is unreal.
A four-dimensionalist position, however, can address the problem of the identity of the poker through the change. The poker is seen as a fusion, with a temporal part that
has the property of being hot at some time, t1 , on Monday and a temporal part that has
the property of not being hot at some time, t2 , on Tuesday. The parts are not identical,
73 but they are both parts of the same fusion. The four-dimensionalist response to
McTaggart‟s paradox is that change is explained without resort to tense.
An A-theorist will maintain, however, that every B-theory lacks animation. The
B-theorist, whether three-dimensionalist or four-dimensionalist, can account for the
difference between an object‟s properties at t1 as opposed to t2 , but has nothing to say about the mechanism that „takes‟ us from to . The A-theorist has an answer: the ever-moving Now. But because, as McTaggart notes, defining Now is not easy
[McTaggart 1927, p. 328], the A-theorist offers little more than the B-theorist in terms of explanation of the mechanism of animation. While the A-theorist gives it a name, neither offers a satisfactory explanation of how subsequent states of affairs arise, or of how one instant succeeds another.
3.5 Arguments Against Four-Dimensionalism
One criticism found against four-dimensionalism is that it does not provide a sufficient explanation of the existence of temporal parts. According to four- dimensionalism, a perduring object consists of a collection of parts, each of which will have at least some different properties than all others. The parts are separate and not
identical with one another. In this way, for example, the-poker-at- t1 can be hot while the-poker-at- is not hot. Because they are acknowledged being non-identical, their having different properties is unproblematic. The poker, itself, is a fusion consisting of these, and other, parts.
Consider, though, two contiguous or, at least, sequential parts: the poker-at-ti and
the poker-at- t j . If, at , we observe the poker, what we see is not the poker simpliciter,
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but the poker-at- ti . Then, at t j , we are looking at something new: the poker-at- . The appearance of this new thing seems to occur without explanation.
Judith Thomson makes this point clearly in Parthood and Identity Across Time.
She illustrates why she sees this as being a “crazy metaphysic” [Thomson, 1983. p. 213] with the example of her holding a piece of chalk. “As I hold the bit of chalk in my hand, new stuff, new chalk keeps constantly coming into existence ex nihilo.” [Ibid] Of course, while she limits her example, for the four-dimensionalist, not only new chalk is coming into existence, but so, too, is new Judith. The problem that Thomson‟s example points to is essentially the problem that the three-dimensionalist B-theorist faces in supplying that which animates change and creates the „flow‟ from one instant to the next.
The poker-at- does not come into existence ex nihilo, it comes into existence from the poker-at- . The physical laws that guide this may be unknown, but they represent no more of a „crazy metaphysic‟ than do the physical laws of motion or force.
This criticism raises again a basic difficulty in arbitration among competing theories of time. The version of four-dimensionalism that Thomson brands a „crazy metaphysic‟ is a B-theory of time. We recognize that B-theories do not have a mechanism for movement from instant to instant. This is one reason many philosophers remain A-theorists. Without this mechanism, it very much appears that each temporal part of the chalk comes into existence ex nihilo. A-theories have a mechanism that moves us from instant to instant. The chalk is not repeatedly created. The chalk endures and what changes is the identity of the privileged instant that is Now. There is no sense in which this „crazy metaphysic‟ of serial coming-into-beingness is required in an A-
75 theory analysis. What is required, however, is the equally „crazy metaphysic‟ of a moving Now that holds the „present‟ relation with exactly one instant at a time. It comes down to our choice of which crazy metaphysic we are to hold. We need to be mindful not to throw stones in this regard.
Peter van Inwagan‟s oppostion to four-dimensionalism is that not only is the notion of temporal parts unnecessary, it is barely intelligible.
Beginning with van Inwagan‟s argument that the notion of temporal parts is unnecessary, consider a picture depicting Plato at the Academy. “[The label „Plato‟] will be a description of the intentional content of the picture, like „the mechanism of the watch‟ or „the structure of RNA.‟” [van Inwagan 2000, 454] For van Inwagan, nothing more is required for the name „Plato‟ to refer, whether used as a label for a picture or as a name in a sentence. It simply refers to Plato, even though Plato is neither physically nor temporally available. It does not refer to some „at the Academy‟ temporal part of Plato.
It refers to Plato. For van Inwagon, objects wholly endure while they exist and are wholly available as referents of definite descriptions while they do not exist. Four- dimensionalism is often presented as being motivated by problems of reference and by problems of identity across change. With van Inwagan‟s realism, there is no need for perduring objects comprised of temporal parts.
To support his position that the notion of temporal parts is barely intelligible, van
Inwagon introduces the phrase modally inductile to refer to that property of temporal parts in virtue of which their duration cannot be extended. To say that a particular temporal part of Descartes could extend a year and a half would be inconsistent with
76 mainstream four-dimensionalism. Because this temporal part of Descartes cannot be so extended, it is modally inductile.
Van Inwagan then notes that Descartes is made of modally inductile temporal parts, and proceeds to claim that Descartes, as a sum of all these temporal parts, is a temporal part of himself. But to say that Descartes‟ life could not have been extended beyond fifty-four years is obviously false, so the temporal part that is all of Descartes is not modally inductile.
The simple, and correct, response to van Inwagan‟s putative contradiction is that
Descartes himself, holus bolus, is not a temporal part. He is a fusion of his temporal parts. That the parts are each modally inductile seems a reasonable position; Descartes could have grown to be older, but not through his parts expanding. Differences in
Descartes‟ lifespan would result in his having a greater, or lesser, number of temporal parts – not the same numbers of fatter or thinner parts.
The most compelling response to all the objections to four-dimensionalism is to point to the resounding acceptance of general relativity and, with it, Minkowski space- time. The consistent empirical support of this family of theories is resounding. Modern science is grounded in the notion of four-dimensional space time.
3.6 The Four-Dimensional View Summary
Four-dimensionalists view all objects as being mereological fusions of parts which, themselves, are comprised of points in space-time. Each point can be described by an ordered four-tuple, with the first three coordinates referring to the point‟s location in space and the fourth coordinate referring to the point‟s position in time. In the language of special relativity, a fusion is described as a world line, which is made of
77 points, each identified by an ordered pair. The first coordinate of each point of a world line is a spatial coordinate and the second is a temporal coordinate.
Because temporal coordinates are contained immediately within the spatiotemporal points, there is no absolute time series external to physical space to which points in space are related. Additionally, a result of special relativity is that there is no absolute space or absolute time; that both are relative to frames of reference. A frame of reference contains scales of space and time. Each world line is governed by a frame of reference. The particular frame is determined by the velocity of the system that comprises the world line relative to the velocity of others. A world line may be governed by a series of frames, as the velocity of the system changes.
So the picture we have is one of objects moving about that are comprised of spatiotemporal parts. The coordinates that describe the spatiotemporal location of each point are determined relative to the scale set by the particular frame of reference for that point. Change occurs in an object when two of its temporal parts have different properties.
Returning to our division between experienced time and physical time, a four- dimensional theory of time does not seem to fit our experience of time. We experience objects as enduring over time, not as being comprised of a sequence of temporal parts. It is the poker that goes from being hot to being cool, and not a part of the poker that is hot and another part that is cool. The separation of objects into spatial parts does not, in our experience, find a parallel in the separation of objects into temporal parts.
In contrast, a four-dimensional B-theory suggests itself as a likely model for physical time. The theories of contemporary science consistently and clearly support a
78 four-dimensional view of the external world. A B-theory built on the notion of perduring objects that are fusions of temporal parts dissolve many temporal puzzles and provides a cogent model that respects principles such as the indiscerniblity of identicals. While this does not constitute proof, it is persuasive.
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Chapter Four: Characteristics of Time
4.1 Now
Three-dimensionalist theories fall into either A-theories, in which tense is irreducible, or B-theories, in which tense is reduced to non-tensed B-relations. Four- dimensionalist theories are principally B-theories insofar as they are tenseless. An exception to this, introduced by Tooley, will be discussed after our consideration of Now and tense in three-dimensional theories. Any four-dimensionalist theory of tense will either have to reject General Relativity or be amenable to being relativized to a frame of reference, as there is no absolute time apart from individual fusions or world lines. A global Now would entail simultaneity of the individual frame-relative Nows.
We begin with a look at Now in three-dimensionalist theories.
4.1.1 Now in Three-Dimensions
4.1.1.1 The Specious Present
Apart from concerns arising from McTaggart‟s paradox, A-theorists must address questions concerning the nature of Now in particular, and of tensed time in general. One important question is whether Now has duration.
In a very broad sense, there are many „Nows‟. As I write, it is now Friday, it is now Spring, it is now 2010, it is now the Holocene epoch. Many of these Nows overlap and some are contained in others. It may appear that a McTaggart-like infinite regress lurks here, as well. In March of 2010, January of 2010 is past. But, since it is Now 2010,
January 2010 is also present. Hugh Mellor, in Real Time II, suggests that, when we are using Now in this broad sense in any particular context, we apply it only to an event that
80 includes the actual Now. In our current example set in March 2010, notwithstanding the fact that it is now 2010, January 2010 does not include the actual Now. Accordingly, it is not now January 2010. January 2010 is past. Mellor suggests a sort of theory of types here, in which pseudo-regress turns out to be a confusion of types of events. In this case, we confuse months with years.
In moving towards a more clear articulation of the characteristics of the Now that
A-theorists take to be real, we need to consider events with durations shorter than epochs.
When we speak metaphysically of Now, we are speaking more of instants, or of
McTaggart‟s indivisible moments. A further sense of Now, and one more obviously connected to instants, is captured in what is referred to as the „experienced present‟ or the
„specious present.‟
Setting aside the question of whether an event can occur in a single instant, there are certainly some events that have duration. Consider these events. Using McTaggart‟s framework, they occupy a set of [likely contiguous] instants. Suppose event occurs
over the set of instants E e1,,,, e 2 e 3 ej. When the instant eEn is present, the elements of set E are divided into three subsets:
Ep e i E: A relation e i , e n past (the instants that have the A-relation of „past‟
when Now is simultaneous with en ); Eenn (the instant that is simultaneous with
Now); and Ef e i E: A relation e i , e n future (the instants that have the A- relation of „future‟ when Now is simultaneous with ). No instant in E holds more than one A-relation with . Instants over which an event occurs hold all three A-relations,
81 because part of the event is past, part is present, and part is future. All three relations are found among the instants occupied by an event.
For example, suppose is the Rolling Stones‟ performance of „Honky Tonk
Woman‟ in BC Place Stadium on November 1, 1989, and E is the set of instants
occupied by . The event of is present at every instant eE . At any instant during the performance of this song, an audience member would be correct to report that “They
are playing „Honky Tonk Woman‟ now.” In this sense, is present throughout all of E .
Suppose, further, that is the event of the singing of the lyric “I met a gin-soaked bar
room queen in Memphis” and EE is the set of instants occupied by . When the opening chord of the song is played, is in the future and, as the final refrain of the
chorus is sung, is in the past. At all instants, EE , during the song when the lyric is not being sung, events and have different A-relations. For example, after the lyric is sung, the playing of the song is present, while the singing of the lyric is past. Because the
A-relation is a relation between instants and Now, and not between durations and Now,
this would really entail that, after the lyric is sung, the instants in E have the A-relations of „past‟ when considered with respect to being Now, and „present‟ with respect to being Now. During , the event that is the singing of „Honky Tonk Woman‟ is all of past, present, and future.
This sense of Now, however, is more connected to the experience of time than it is to physical time. As McTaggart writes, “[the] present, however, of which we speak is not an indivisible point. It has a certain duration, which comprehends more than one term. …
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There is no fixed magnitude for such presents. … the present of which we are speaking is a duration, and a duration which may vary in length.” [McTaggart 1927, §611].
The term „specious present‟ is used by many, including McTaggart, to refer to the experienced present. William James most famously picked the term „specious present‟ from E. R. Clay. As cited by James, Clay‟s notion is that the present we experience is really the most recent part of the past. He names this most recent past, that which we experience as the present, the „specious present.‟ Clay uses the example of watching a meteor streak across the sky. The entirety of the streak appears to the observer as being present while, in fact, the streak represents the most recent past of the meteor‟s path.
Given the nature of our apprehension, argues Clay, we should divide time into four parts: the obvious past, the specious present, the actual present, and the future.
James uses Clay‟s argument to support his own view that the „sensible present‟ has duration. He writes “In short, the practically cognized present is no knife-edge, but a saddleback, with a certain breadth of its own on which we sit perched, and from which we look in two directions in time.” [James 1890, p. 609] What is the „breadth‟ of the specious present? James view is that the duration of the specious present is variable, ranging from as long as twelve seconds to as brief as 0.00205 seconds. This range of durations was determined experimentally. The former is equal to the maximum duration of sound that a subject would apprehend as a single measure. The latter is equal to the minimum elapsed time between two snaps of electric sparks that could be distinguished as distinct by the subject of a certain experiment. When less time elapsed between the sparks, this subject heard the separate snaps as one. While it is occurring, the singing of lyric would all arguably be in the specious present. A member of the audience would
83 likely experience it as a unit that occurs in a single present. But even if is too extended to be experienced entirely in a specious present, the singing of the word „queen‟ certainly is not.
James maintains that we cannot attend to the present instant. Instead, we apprehend the specious present which, James holds, is an interval of time that has vague backward and forward boundaries. James‟ specious present includes an interval in the past of the present instant, as well as an interval in its future. The duration of the specious present will vary from instance to instance.
This characterisation differs somewhat from that described by Clay, who does not view the specious present as including any future instants. Clay‟s specious present consists of the instants immediately prior to the present instant and the present instant.
For Clay, the backward boundary, which the earliest part of the recent past, may well be vague, but the forward boundary, which is the last instant of the past, is sharp.
Prior to Clay and James, in an essay written in response to Locke‟s account of the idea of duration, Thomas Reid provides a clear explication of the difference between experienced time and physical time. “Philosophers give the name of present to that indivisible point of time, which divides the future from the past, but the vulgar find it more convenient in the affairs of life, to give the name present to a portion of time, which extends more or less, according to the circumstances, to the past or the future.” [Reid
1785, p. 326] According to Reid, we apprehend change through an interaction of sense and memory. Sense extends only to the [Philosophers‟] present while memory extends only to the past. We apprehend change when we sense the present place of an object and remember its successive advance. So, for Reid, while the experienced present has
84 duration, the actual present is an indivisible point. Because our awareness is active only on this point, we require our memory to place the present instant, and all that the present instant contains, into its proper context. Reid‟s point is, perhaps, more subtle than Clay‟s or James‟ in that it effectively reconciles the physical present with the experienced, or specious, present. An important difference is that James claims we cannot sense the indivisible present instant; while Reid claims that we sense only the indivisible present instant.
In New Essays on Human Understanding, Leibniz offers an interesting variation on the instantaneous nature of the present. “… [S]trictly speaking, points and instants are not parts of time and space, and do not have parts either. They are only termini.” [ p.
153] He does not seem to take a position on whether the present instant is the last instant of the past or the first instant of the future, but either position would suggest that the present does not exist independently of the past or future. This position is reminiscent of
Aristotle‟s definition of the present being the instant between the past and the future.
A significant metaphysical point emerges in Reid‟s discussion of Locke‟s account of the idea of duration. In §16 of Book II in An Essay Concerning Human
Understanding, Locke expresses the view that the idea of duration arises from the constant succession of ideas. A member of the Rolling Stones‟ audience perceives a succession of sounds as the lyric is sung. This succession is measurable as a succession by its duration. It is the succession that is primary. Time, as measured by duration, is secondary.
In contrast, for Reid and James, time is primary. Against Locke‟s view that succession is prior to duration, Reid‟s argument is that each impression that forms a
85 succession already has duration. The succession occurs in time. Duration is still a measure of amount of time the succession occupies.
Locke‟s position is suggestive of the view that there is no absolute time; that is, for Locke, without change there would be no time. Time simply measures change. Reid and James‟ position is suggestive of the view that there is absolute time that would exist even in the absence of change. On this account, time is the medium within which change occurs.
Reid‟s argument clarifies the important difference between a conception of time in which instants have duration and one in which instants are divisible. We can see that if a time series is comprised of a finite set of instants, each instant must have duration. If instants are infinitely divisible, any duration will be found to consist of an infinite set of instants.
Reid‟s overall position on the present is very similar to that expressed by
Augustine in Book XI of The Confessions. Augustine addresses the question of what it means when we speak of „a long time‟ as opposed to „a short time.‟ The present, he holds, is real but has no duration. The past, in contract, has duration but is not real.
Augustine sees the present as being an instantaneous point between the past and future, neither of which is real in the sense that the present is real. The past exists only in one‟s mind. Accordingly, for Augustine, duration is the measure of something mental.
Returning now to the example of the performance of „Honky Tonk Woman,‟ we can separate the experienced present from the physical present and, in so doing, hopefully more fully characterize the nature of the actual Now. As the performance progresses, a member of the audience will experience a progressing sequence of specious presents,
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each with a finite duration. Each instant, eE , will hold the A-relation of present with
Now exactly once during the performance of . Following Clay, as a particular instant,
ei , holds the A-relation of present with Now, some set of instants, ej,,,, e i3 e i 2 e i 1,
will constitute the specious present. The earliest instant e j in this set will be determined in some manner dependent upon the circumstances. The specious present is what we experience as the present instant. We can see how it is related to, but distinct from, the actual present. This is the Now of experienced time.
4.1.1.1.1 Prior‟s Fugitive Truth
In his 1969 paper Fugitive Truth, Prior attributes to Anthony Kenny a paradox concerning present-tensed utterances. It would seem, Kenny pointed out, that it is impossible to utter a true present-tensed sentence that reports an instantaneous event.
Consider „Eclipse is just now past the winning post.‟ By the time the utterance is made,
Eclipse will be some distance past the winning post. It takes some time to complete any utterance and states of affairs can change during this time. Prior attributes to John
Buridan the observation that „Socrates is sitting and Socrates is standing.‟ can be true if
Socrates is sitting during the first part of the utterance and standing during the second part.
Clearly, what is needed is some convention. The utterance must be understood to be evaluated at a particular stage of the utterance. It could be that it reports something that is supposed to be the case at the instant the utterance begins, or the instant at which it concludes, or the median instant of the utterance of the verb. In any case, we solve this putative difficulty in a practical way in common conversation by inferring the speaker‟s
87 intention in the context of the utterance. The solution will vary according to the context.
For example, if Millie is directing her brother Willie to start his stop-watch to time an event, and says, “Start the watch … Now.” The understood convention is that he should start the watch as he hears the word „now.‟ Of course, it still needs to be worked out which part of the utterance of „now‟ is meant but, because the short amount of time it takes to utter the single syllable, even if Willie and Millie assume different conventions, this is likely to be a difference that does not make a difference. While this reliance on in situ convention may result in relatively few significant misunderstandings in practice, we find the same problem in our formal inferences, where appeal to participants‟ judgement- based ad hoc conventions is not sufficient.
We can take Buridan‟s Socrates example and convert it into a formal syllogism:
1. If Socrates is sitting, he is not standing.
2. Socrates is sitting.
3. Therefore, Socrates is not standing.
The first is a non-tensed sentence that is a necessary truth. The second and third sentences are tensed and contingent. They may be true at some times and false at others.
A parallel concern to that which Kenny raises in the instance of the present-tensed sentence about Eclipse can be raised here if we suppose that the second and third sentences are evaluated at different times. In response to this problem, Buridan added to the now-standard definition of validity of an argument the requirement that the premises and conclusion be simul formatis. [Prior 1969, p. 7] Prior supports this view: “the state of affairs we are arguing about must not alter while we are arguing about it.” [Ibid] This is an important point in considering any metrified temporal logic.
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4.1.1.2 McTaggart‟s Entity-X
The A-theorist‟s position of admitting McTaggart‟s ever-changing Entity-X, our
Now, into our ontology defeats the second premiss of McTaggart‟s proof of the unreality of time by avoiding the infinite regress. This, as a position regarding experienced time, is supported by our inability to completely eliminate tense from our language. Now serves to „privilege‟ each instant in the temporal series exactly once, when that instant holds the
A-relation of present with Now. But what is this Now?
Now is usually presented metaphorically. It is a „moving spotlight‟ that pivots to shine sequentially on each instant in the time series. It is the crest of a wave that we ride into the future on the wake of the past. It is the point opposite on the river-bank as we are swept forward by the current of time. McTaggart‟s device of positing an ever-changing entity with which instants in time hold an A-relation creates a helpful imagery and provides a useful conceptual model but, still, leaves one holding a metaphysical black box. The heuristic utility of Now is important and worthwhile – even to the B-theorist when considering experienced time – but should not be confused as being a metaphysical theory.
While we are no closer than McTaggart to describing the raw nature of Now, for the A-theorist it is a real posited entity. When a two-year-old asks why the ball rolls down the hill, we answer, “Gravity.” What‟s gravity? It‟s a force. What‟s a force? It‟s what moves things. So, the ball is moved by that which moves things. An A-theorist reaches a similar point in her explanation of Now. It is an entity that changes constantly is such a way as to determine the A-relation for each instant in the way described above.
It is by its effect on the A-relation of the instants in the time series that we know Now.
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4.1.2 Now in Four Dimensions
In Time, Tense, and Causation, Tooley proposes a model of four-dimensionalism built on what might be seen as a tensed B-theory. Unlike many philosophers of time, but in a manner similar to Locke, Tooley does not set a time series as being a primitive, or foundational, concept in his theory.
As a four-dimensionalist, Tooley sees objects as being fusions of temporal parts.
For Tooley, the past and present are real, but the future is open. Tense is important in such an ontology, as it is used as a primary filter to distinguish what exists from what does not. But tense is hard to work out in a system with no absolute time, especially if its effect is to reach beyond any given world line. If tense is going to be used to distinguish objects that exist from those that do not, it would be advantageous for there to be a universal Now. Nows relative to frames of reference would lead to objects with existence relative to frames of reference. Without tense, though, we lose a natural mechanism to describe the notion of temporal becoming that is important in Tooley‟s metaphysics.
Tooley begins with temporal becoming as his foundational notion. Each temporal part can be seen as being a state of affairs. To form a part of a fusion, a given state of affairs must come into being. This provides Tooley with his metaphysical basis of tense.
The states of affairs that comprise the temporal parts of a perduring object come into being sequentially, one at a time. As a particular state of affairs comes into being, it is
Now.
It is the coming into being that privileges a particular temporal part, not its relation to an absolute Now. This new part, in virtue of its privilege, acts in many ways like a present part would. Those parts that are already contained in the fusion are fixed;
90 they are unchanging. Subsequent parts have not yet come into being and, so, are not fixed. The notion also accords well with the Now of experienced time. We are directly aware through our senses of the immediate state of affairs as it comes into being. The
Now of our experienced time, that instant of which we are conscious, coincides with a particular state of affairs that is coming into being.
So, we can see that there is a functional notion of Now relative to any given world line. In four-dimensionalism, there is no absolute time, so there is no immediately obvious absolute Now. But this does not seem consistent with our experience. Two people talking with each other are unlikely to disagree on the truth-value of the sentence
“It is now.” One possibility is that there is a simultaneous coming into being of a state of affairs in their respective world lines, and that both new states of affairs include the utterance of “It is now.” While this seems persuasive, we need to pause and consider the notion of simultaneity under general relativity.
Variations of a travelling train thought experiment are used to demonstrate the relativity of simultaneity. Consider a train travelling along a track at some velocity carrying Milla and a light source at the centre of a car. Suppose Carlos stands at the side of the track and that Milla‟s light source emits a 360° flash exactly as she and the centre of the car passes Carlos at the side of the track. From Milla‟s perspective inside the train, the light will reach the opposite ends of the car simultaneously. This is because Milla, the light source and the train are all contained in the same frame of reference. From the perspective of Carlos, who stands in a different frame of reference, the light will reach the back end of the car before it reaches the front end. Since in Carlos‟ frame the car is in motion, the rear end of the car is rushing towards the light source as the front end is
91 rushing away. Because the speed of light is constant relative to all frames, in Carlos‟ frame, the light will strike the ends of the car at different times. What is simultaneous in one frame is not simultaneous in another.
There is no absolute simultaneity between instants in separate frames. Our example, though, of two people talking is different from the train example in degree, at least as concerns physical time. While the two conversants are separate fusions and, so, are represented by separate world lines existing in separate frames of reference, these separate frames are, at least during the conversation, so similar that they can be considered to be virtually identical. For the conversation to be effective, both people must be in nearly-identical motion in a nearly-identical direction and at nearly-identical velocities. The time scales of their respective frames would, accordingly, be nearly- identical. In the case of Milla and Carlos, because the difference in the scales of their respective frames of reference is so small relative to their perceptual sensitivity, the lack of simultaneity between their perspectives would be indistinguishable, at least in terms of experienced time. The Now of one would be almost exactly simultaneous with the Now of the other.
The four-dimensionalist picture of Now, then, is based on the coming onto being of states of affairs; tense supervenes on the metaphysically-prior coming into being of states of affairs. Now is, strictly speaking, relative to a frame. In our experienced world, because all the objects and other people we encounter are in such similar frames, there is an apparent simultaneity of the Nows of everything in our immediate and direct awareness. Of course, one may look longingly towards the Andromeda Galaxy, which is the most distant object regularly accessible to unaided human sight. Not only has the
92 light from the Andromeda Galaxy seen by an Earth-bound observer travelled about 2.5 million years to get here, so that what we see is 2.5 million years old, the velocity of the galaxy is about 300 km/s greater than ours. At least, this difference in relative velocities is measured between current observations of our motion and 2.5 million year old observations of the Andromeda Galaxy‟s. That there is no common Now between an
Earth-bound observer and an Andromeda Galaxy-bound observer does not upset our intuition concerning the simultaneity of Now in our immediate experience. The importance of recognizing the relativity of simultaneity at a distance is not paralleled by the importance of recognizing the relativity of simultaneity in similar frames.
With respect to physical time, Tooley‟s theory provides an attractive model of
Now relative to a frame of reference. Under general relativity there can be no absolute
Now, and no absolute simultaneity between frames. We can, however, set Now relative to a chosen privileged frame and consider events in other frames to be simultaneous just in case they appear from the privileged frame to be included in a state of affairs that is coming into being Now.
4.2 Time Series
4.2.1 Overview
Three-dimensionalist views of time are defined by relations between objects and events and an absolute time series. Four-dimensionalist views of time are defined by mereological sums of parts. These parts are usually thought of as temporal parts, each of which has a temporal coordinate. The set of temporal coordinates of the parts of a fusion or world line can be seen to form a series. In what follows, we will consider that there is a time-series that corresponds to any given fusion or world line. Our discussion will
93 often focus on three-dimensional time series. What is said about these will, unless otherwise indicated, be supposed to apply to this series associated with four-dimensional fusions or world lines.
4.2.2 Time Series
Temporal theories nearly all include the B-series of instants. A-theories begin with an A-series, which can be reduced to form a B-series. Because Now is continually changing, the A-series is continually changing, while the B-series is static.
4.2.2.1 Order and Direction in the Time Series
The B-series is seen as an ordered set of instants. Objects, temporal parts of objects, and events are seen as „occupying‟ these instants. This occupation amounts to a relation between the occupying entity and instants of the time series. The instants themselves form a strict partial order; that is, they form an asymmetric transitive closure.
As such, the series of instants has the following characteristics:
1. For any instants, m1 and m2 , if is earlier than , then is not
earlier than [asymmetry]11;
2. For any instants, , , and m3 if is earlier than , and is
earlier than , then is earlier than [transitivity]; and
3. For every instant, m, it is not the case that m is earlier than m
[irreflexivity].
So, the B-series is the strict partial order of the set of instants under the relation „is earlier than.‟ That the ordering relation is the B-relation „is earlier than‟ rather than its
94 complement „is later than‟ is determined by the direction of time. Three-dimensionalists largely agree that the direction of time is from earlier to later. Some argue this position from the order of causality, while some argue from the other physical principles such as the Fourth Law of Thermodynamics [that entropy increases], the expansion of the universe, and the outward expansion of radiation from its source. These arguments all base their conclusion concerning the direction of time on the „direction‟ of observed change. Once broken, an egg does not reassemble, in spite of all efforts of the King‟s horses and men. All theories of time, and of change, correlate the direction of time with the direction of change.
These arguments for a certain direction, however, may not be as compelling as they first seem. We certainly seem to experience the world in a particular direction – from earlier to later – but this is no assurance that the world is ordered in this way.
Experienced time seems undeniably to progress from earlier than to later than, but it is not as clear that this is the case with respect to physical time. When we carefully consider even common examples of causality as our guide concerning the direction of time, we find that efficient causes do not precede their effects. What causes a leaf to fall to the ground? It is not the application of an initial force, followed by a spell of falling.
Rather, the leaf is moved by a continual force through its entire descent. Causes coincide with their effects and, so, it is not the case that the direction of time is set by the direction of causation.
11 Notice that this implies that the ordering relation is irreflexive. No instant is earlier than itself.
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The other arguments for the direction of time can be seen, at best, to be generalizations from the asymmetry with which we experience the world. Suppose Sam stands on a pivoting platform holding a slingshot and stone. Suppose, further, that the platform can pivot through twelve discrete steps and that there is a different pane of glass directly in front of the platform for each of these twelve steps. Now, suppose Sam steps onto the platform, which is then pivoted to rest at one of the twelve steps. There is now one pane of glass directly in front of Sam. Finally, suppose Sam uses the slingshot to fling a stone through the pane of glass that is in front of the position to which he has pivoted. Consider the following two counterfactual statements:
1. If the platform had been pivoted differently, the stone would have
followed a different path.
2. If a different pane of glass had been broken, the stone would have
followed a different path.
The first counterfactual accords with our intuition, whereas the second does not.
To evaluate the first conditional, we imagine that everything in the world prior to the pivoting of the platform is held constant. We then imagine the platform being pivoted differently, and consider how the path of the stone in the counterfactual future will be affected by this change. To evaluate the second counterfactual in an analogous manner would require that we imagine everything in the world after the breaking of the glass is held constant. We would then imagine that a different pane is broken, and consider how the path of the stone would have differed in this counterfactual past. The preference of the first approach over the second is seen by some as a matter of convention, determined by our perceptual proclivities and not by any mind-independent fact about the world.
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Huw Price [Price 1997] claims that physicists are increasingly adopting a symmetrical view of consequence which would see as a matter of convention the choice between interpreting a counterfactual statement by fixing the past and looking at a counterfactual future and fixing the future and looking at a counterfactual past. For our purpose, it will be enough to note three points: (1) our intuition is asymmetric, as is the direction of experienced time; (2) the world may be causally symmetrical, in which case there would be no determined direction of physical time; and (3) in the event the world is causally symmetrical, the B-series could be ordered by either the relation „is earlier than‟ or the relation „is later than.‟
Having addressed the ordering of the set, we now consider the cardinality of the
B-series by discussing each of three possibilities.
4.2.2.2 Cardinality of the Set of Instants
4.2.2.2.1 Finite or Non-Dense Denumerable Set of Instants
The first possibility we will examine is that the B-series could be comprised of a finite set of instants or a non-dense denumerably infinite set of instants. These would not affect either the ordering just discussed, or the nature of the relations between objects or events and instants. Either would affect, however, the duration of instants, as well as
„boundaries‟ between instants. If the set of instants is finite, it would require that time has a beginning and an end. If the set is a non-dense denumerable set, time could be infinite in either or both directions. There could be a first instant or a last instant, but not both.
If the B-series is comprised of a finite or non-dense denumerable set of instants, each instant will have to have some actual, non-infinitesimal, duration. Otherwise, no
97 sequence of finitely-many instants will form an actual, non-infinitesimal, duration. This can be seen with the help of a spatial analogy. Geometric points have no dimension. A finite collection of points, similarly, has no dimension. Yet, an infinite set of points does have a spatial dimension.
Any defined interval over either a finite or non-dense denumerable set of instants would have a first and last instant and a finite collection of instants between. Notice that in a partially ordered denumerable set, the subset of elements between any two given elements is infinite only if the set is dense. Otherwise, as is the case under the present assumption, any two elements define a finite subset. If there are finitely-many instants in an interval, then, there must be an indivisible atomic duration that measures each instant.
The duration of any span of time would simply be the product of the number of instants in that span and this atomic duration.
A momentary fear may arise that allowing individual instants to have duration opens the possibility of McTaggart-like contradictions, in which any single instant is all of past, present and future. It might be thought that there will be some part of the duration of an instant that is past while another part of the duration is present and still another part is future. But this is to overlook the indivisibility of the duration of an instant. Being indivisible, an atomic duration has no parts and, so, is protected from this kind of attack.
An ordered finite set will have a first element, a last element, and, for every element except the last, a next element12. A finite B-series with n instants, then, will
12 Similarly, with the exception of the first element, each will have a previous element.
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have: (1) an instant, m1 , which is the first instant in the time series; (2) an instant, mn ,
which is the final instant in the time series; and (3) for each instant, mi , where in , an
instant, mi1 , such that is the instant immediately following [ ‟s next instant].
An ordered non-dense denumerable set will exhibit property (3), and may exhibit either property (1) or property (2), but not both. Each of these properties corresponds with significant metaphysical claims.
With respect to there being a first or last instant, there is an Aristotelian argument that holds that every instant is, at some point, „now‟ and that „now‟ is an intermediate instant between other instants. No instant can be „now‟ if it is not flanked on both sides by other instants. A first instant or a last instant would not be an intermediate instant; therefore, there can be no first instant and there can be no last instant.13 Although one is always cautious when finding disagreement with Aristotle, this is not an especially strong argument, as it really just begs the question.
Many modern cosmologies hold that this universe began with a singularity. Time began as the singularity started its expansion. These theories are consistent with either the assumption of a finite B-series or a non-dense denumerably infinite B-series. That there is a first and a last instant is consistent with a finite B-series or a dense infinite
[either denumerable of innumerable] B-series.
One important feature of a finite or a non-dense denumerable B-series is that every instant [except the final instant] has a „next instant.‟ Similarly, every instant
[except the first instant] has a „previous instant.‟ This view of discrete time, whether or
99 not it has a beginning or an end, allows the metaphorical reduction of events in the world to frames of a film. There are discrete steps from one instant in a sequence to any other instant in that sequence. Such a conception is free from Zeno-like paradoxes which point to the apparent difficulty in transversing an infinite collection of instants one-by-one.
This is almost certainly the time series of experienced time, and is a viable time series for physical time. The application of Okham‟s Razor might suggest it is the preferred time series for a theory of physical time.
4.2.2.2.2 Dense Denumerable Set of Instants
A second possibility concerning the cardinality of the set of instants in the B- series is that it contains a dense denumerable infinity of elements. If the B-series is comprised of a dense denumerably infinite set of instants, the instants must each have an infinitesimal duration. If the instants were to have a duration of zero, then no sequence of instants, not even an innumerably infinite sequence of instants, would amount to a duration greater than zero. On the other hand, if the instants were to have a finite duration, no matter how short, the interval between any two instants would have an infinite duration. If the instants form a dense denumerable infinity, then for any instants
m1 and m2 , where is earlier than , there is an instant, mi , such that is earlier than and is earlier than . So, between any two instants there is another instant.
Density ensures not only that there is an instant, , between and , it ensures that
there are further instants, mi and mi that respectively mediate and , and and
13 Aristotle argues further that, since time is the metric of change, there can be no first or last change, either.
100
m2 . Of course, this continues until we find a countable infinity of instants between any two given instants.
So, if each instant has a finite duration, then for any instants m1 and , where
is earlier than , the duration of the interval from to would be equal to the sum of an infinite series of positive non-zero numbers. Such a series has an infinite sum, regardless of how small the numbers in the series are. So, the interval between and
would have an infinite duration.
With a dense denumerable B-series, then, the duration of each instant must be greater than zero and less than any finite number. An infinitesimal is often defined as a
1 1 1 number greater than zero but less than every member of the sequence 1, , , , . It is 2 4 8 a positive non-zero value that is smaller than any positive number. Finite sums of infinitesimals are infinitesimal. Innumerably infinite sums of infinitesimals, however, have finite values14. While this may or may not hold for denumerably infinite sums of infinitesimals, a recently developed arithmetic of infinite and infinitesimal numbers provides a clear and intuitive model for determining the duration of an interval between instants on a dense denumerable B-series.
Yaroslav Sergeyev15 developed an applied mathematical method for undertaking calculations with infinite and infinitesimal numbers by introducing the numeral „‟ to
14 This is a well-established principle of mathematics, which will be discussed more fully later in this section. 15 The duration of an interval over a dense denumerable time series can more easily be established by mapping the instants to elements of the Rational Numbers. This observation, which should have been
101 represent an infinite number „grossone,‟ and the Infinite Unit Axiom [Sergeyev, 2008, p.
572ff]. The Infinite Unit Axiom [IUA] is stated in three named parts:
1. Infinity: Any finite natural number, n, is less than one grossone; that is:
nn ;
2. Identity: The following identities hold: (a) 0 0 0 ; (b) 0;
(c) 1; (d) 0 1; and (e) 00 ; and
3. Divisibility: For any finite n , sets kn, ,1kn, have the
n cardinality of , where: kn, k, k n , k 2 n , Note: kn, . n k1
This third part of the axiom is made clearer by an example. 2,3 is the set whose first element is „2‟ and whose subsequent elements increase in increments of 3. So,
2, 5, 8, 11, . The cardinality of is . We can see, then, that the 2,3 3
cardinality of 2,3 2, 5, 8, 11, is greater than the cardinality of, say,
2, 7, 12, 17, ; that is . This matches out intuition more closely than 2,5 35
Cantor‟s arithmetic of the transfinite, in which the cardinalities of these two sets is equal.
We see by Infinity that is an infinite number that has the property of being larger than the largest Natural Number. Identity will provide the necessary foundation to ensure that the arithmetic of infinite numbers based on follows the same „rules‟ as
obvious to me, was gently suggested by Bernard Linsky during my oral examination. The present section remains in the text principally for its interest value.
102 does the arithmetic of finite numbers. Divisibility entrenches, perhaps, the most philosophically controversial principle of Sergeyev‟s system; that “the part is less than
the whole.” [Sergeyev 2008, p. 571]. We can see that, in the example of 2,3 , the
cardinality of 2,3 is one-third the cardinality of the set . This matches our intuition, since two out of every three numbers in are missing from . But, because of this principle, we need to be careful not to employ certain concepts in our method for computing the duration of an interval. These include many properties of denumerable, innumerable and bijection sets. These concepts can be used elsewhere in our analysis, just not in the context of calculating the duration of intervals16.
While there is much to this arithmetic of infinite numbers, what is of most concern for our purpose is the definition and arithmetic of infinitesimals. A „base‟ infinitesimal number can be immediately derived from grossone as 1 . We notice that, as the rules of arithmetic all apply, that and are multiplicative inverses; that is
10 1. Here, we see that the product of an infinite number and an infinitesimal number is a finite number.
To illustrate the usefulness of this system in describing the duration of an interval on a dense denumerable B-series, we need to establish a metric along the series. Without loss of generality, we can consider the interval of time measured by 1 second. Under our
16 Interestingly, with the help of this principle, Sergeyev offers a solution to Hilbert‟s Grand Hotel paradox. Hilbert asks us to imagine the Grand Hotel, which has a denumerably infinite collection of numbered rooms. Without the assumption that „the part is less than the whole,‟ a visitor who arrives at the hotel and finds a „no vacancy‟ sign can still be accommodated. The hotel manager can simply move the occupant of Room 1 into Room 2 and, in general, for every Room n, move the occupant from Room n into Room n+1. The newly-arrived visitor can then be accommodated into Room 1. For Sergeyev, „no vacancy‟ means no
103 current assumption concerning the B-series, there are instants in one second, each with a duration of 1 . The product of the number of instants on an interval and the duration of each instant will equal the duration of the interval. Here, the interval of one second is found to have a duration of one second. If we, now, measure the duration of an interval
55 of two and a half seconds, we find the product 1 seconds. 22
Sergeyev‟s arithmetic of infinite numbers provides a useful and intuitive model for seeing that, if the instants in the B-series form a dense denumerably infinite set, the duration of each instant is infinitesimal; that is: .
Under this second possibility, in which the instants that comprise the B-series form a dense denumerably infinite set, we find that, since any two given instants have a third instant between them, instants do not have a previous instant or a following instant.
There is no „next‟ instant if the B-series is dense. We find, additionally, that each instant must have a non-zero, non-finite duration. Each instant has an infinitesimal duration and, with Sergeyev‟s calculus, we find that the infinitely many instants that span any interval will constitute a finite duration. We turn now to the third possibility for the make-up of the B-series.
4.2.2.2.3 Continuous Series of Instants
The third possibility for the B-series is that its elements form a continuous series; that the instants form an ordered innumerably infinite set. Such a B-series would, like our previous possibility, be dense. Between every two instants there would be a third;
vacancy. If the manager were to attempt and shuffle his guests, he would find he is unable to relocate the guest in Room , since there is no Room +1.
104 moreover, iterating this principle, we find that between any two instants there are an innumerable infinity of instants. The difference between this property here and the corresponding property for the dense, denumerably infinite set, is that here the interval contains a gapless continuum of instants. Although an ordered denumerable set is dense, it contains gaps; whereas an innumerable set does not. So, with an innumerable set, we get all the instants that come from a denumerable set without the gaps.
As with any other infinite B-series, there may be a first instant or there may be a last instant and, because it is dense, there can be both. Of course, like any infinite set, an innumerably infinite B-series could in infinite in both directions.
The duration of instants needs to be infinitesimal if the B-series is an innumerably infinite set, as well. With continuity, there is a long-established principle that an infinite sum of infinitesimals is finite17. While the model differs, the outcome is the same as we found with a dense denumerably infinite B-series. Instants each have infinitesimal duration and the interval between any two given points is finite.
With a strict partially ordered innumerably infinite B-series, we have density of instants without gaps, and instants with infinitesimal duration.
4.2.2.2.4 Summary of Cardinality of the Set of Instants
To summarize this section, it might be helpful to summon analogies to number lines. Our first possibilities correspond to the Integers. A finite B-series would map to a subsection of the Integers number line. This would have a first and last point and each
17 Cauchy stated this as an unproven lemma in 1823 as part of his proof of the dependence of integration on the infinite subdivision of the entire interval [Lauwitz, 1989]. That an infinite series of infinitesimals had a finite sum is a principle that sits at the foundation of the differential and integral calculus.
105 point, with the familiar caveats about first and last points, each instant would have both an immediate predecessor and an immediate successor. Each instant would be indivisible and have a finite atomic duration. The duration of an interval would be the finite sum of all the atomic durations. A non-dense denumerably infinite B-series would correspond either to the entire Integer number line or to a ray of the line with a single endpoint. Such a series would be one of (1) infinite in both directions; (2) infinite in the future with a first instant; or (3) infinite in the past with a final instant. Otherwise, it would share all the characteristics of a finite B-series.
Our second possibility corresponds to the Rational number line. Between any two given instants of a dense denumerably infinite B-series, there rests a denumerably infinite set of intermediary instants. There is no immediate predecessor or immediate successor to any instant. While the series is dense, it has gaps. In fact, it has more gaps than it has instants. Each instant would have an infinitesimal duration, and each interval between two given instants would have a finite duration. A dense denumerably infinite B-series would correspond either to the entire Rational number line or to a ray of the line with a single endpoint. Such a series would be one of (1) infinite in both directions; (2) infinite in the future with a first instant; or (3) infinite in the past with a final instant.
Finally, our third possibility corresponds to the Real number line. Between any two given instants of a continuous B-series, there rests an innumerably infinite set of intermediary instants. There is no immediate predecessor or immediate successor to any instant. The series does not have gaps. Each instant would have an infinitesimal duration, and each interval between two given instants would have a finite duration. A continuous B-series would correspond either to the entire Real number line or to a ray of
106 the line with a single endpoint. Such a series would be one of (1) infinite in both directions; (2) infinite in the future with a first instant; or (3) infinite in the past with a final instant.
Now that we have three possibilities, our question is which accords with the experienced time and which with physical time. We should be guided, in part, by
Ockam‟s Razor and accept the possibility that provides all we need with the least ontological commitment.
We experience a finite world. When we try to think of an infinitely large number, we think of a number so large that no other number can be larger. But that is to apply a finite number model to describe an infinite number. We can imagine very small amount, but not an amount smaller than any other amount, yet more than nothing. Our experience of time rolls out like the frames of a film. Instants are very short in duration, but each instant is „wide‟ enough to contain a state of affairs, and one moment gives on to the next. The time-series of experienced time is non-dense. Whether experienced time has a first or last instant is less clear. We speak often of the „beginning of time‟ and wonder
„how did this all get started?‟ While we can talk about a universe without beginning, what that really amounts to is a universe that began a really, really, long time ago – so far in that past that there may as well not have been a first instant. But this is not to say that we have an easy intuition of what it would be for there not to be a universe. The same applies to the deep future; our experience does not guide us into an understanding of a future without end beyond a simple notion that it will go on for a very long time. What is clear, however, about experienced time, is that our own „personal‟ times have first and last moments. The past that is between my own first instant and the present has a
107 different quality than the past that precedes my own first instant. In this way, we can see that there is a particular privilege that this first personal instant holds in experienced time.
With respect to physical time, the time series would most reasonably be expected to mirror physical space. This clearly must the case in any four-dimensional theory.
That is, if space is made of discrete points, then the time series would be made of discrete instants. A continuous space would be matched by a continuous time series. It is unlikely that space, or the time series, is actually comprised of a denumerably infinite set of points. Such space would feature „gaps‟ that correspond to the absent irrational, or incommensurable, points. Continuous space and time would appear to us to be indistinguishable from finite space and time, as long as if the finite space and time is large enough. A finite model that employs a notion of „arbitrarily large‟ or „arbitrarily small‟ rather than „infinitely large‟ or „infinitesimally small‟ can have the same explanatory and predictive power as an continuous model. In the absence of a clear argument in favour of continuous space and time over finite space and time, Okham‟s
Razor urges us to adopt a theory of physical time in which the time series is comprised of a finite set of instants.
4.3 Ontologies of Time
McTaggart argued that physical time is unreal. While Realists reject McTaggart‟s conclusion, they disagree among themselves about what parts of time are real and even about what sense of „real‟ is in play. Some favour Eternalism, which holds that all of past, present and future are equally real; others favour Possibilism, also known as the
Growing Block or Growing Universe view, which holds that the past and present are real but that the future is not [yet] real; and still others favour Presentism, the view that only
108 the present is real. We will consider each of these positions from the perspectives of both three-dimensionalism and four-dimensionalism.
4.3.1 Eternalism
Eternalism is the view that all of what we consider to be the past, present and future is equally real. This is a popular view among B-theorists who also hold that persisting objects endure by being wholly present while they exist, as opposed to perdure by having temporal parts. For Eternalists, Now is no more privileged than its spatial analogue here. Because of this, Eternalists hold that objects existing in distant times are no less real for their temporal distance than objects existing in distance places are less real for their spatial distance. To extend the comparison, we can expect that temporally distant objects to be less accessible than temporally proximal objects, just as spatially distant objects are less accessible than are those that are spatially proximal.
B-theorists reject the reality of the A-relation of tense. Because of this, tense is not available to a B-theorist as the basis on which to distinguish temporal regions. It would be inconsistent for the B-theorist to maintain that a real distinction in physical time is made on the basis of tense. Certainly, there is what we perceive to be past, or what is past in our experienced time, but nowhere in a B-theory can there be a division of physical time that is irreducibly based on tense. This makes Eternalism, in which tense plays no role in the ontological status of objects, well-suited to B-theories of time.
Because there is no distinction based on tense in Eternalism, objects and their properties cannot be grouped according to their tense. In particular, the property of being
„real‟ cannot be determined by an object‟s A-relation. Under Eternalism, Socrates,
Kripke, and, Sam, the first person born one hundred years after Kripke‟s death, are all
109 equally real in the same sense that spatially-distant objects are real. Socrates is no longer accessible in the way that Kripke is, but he is no less real for his inaccessibility. This certainly comes as good news to philosophers of language who may be uneasy with reference to non-existing objects, but it appears to present a problem for advocates of agency and free-will. If Sam is real, then so are his properties and his relations to other objects. For example, Sam‟s parents must also be real and must conceive him, and his mother must birth him. It is not possible, for example, for Sam‟s parents not to meet each other or for Sam‟s father to decide not to spend that fateful night. It would seem that if everything exists, then everything happens.
Eternalism does not, however, imply that everything has happened or even that everything will happen. For this to be the case, physical time would need to be tensed which, under the assumption of Eternalism, it is not. At the time of Kripke‟s birth, there is no fact that makes the sentence „Sam is born to Ethyl and Gilbert‟ either true or false.
The Eternalist can still maintain, a la Aristotle, that the truth value of this sentence is undetermined at the time of Kripke‟s birth. That there is a fact at the instant of Sam‟s birth in virtue of which „Sam is born to Ethyl and Gilbert‟ is true does not speak to the truth-value of the sentence at the time of Kripke‟s birth. Moreover, this fact at the time of Sam‟s birth does not need to control Gilbert or Ethyl‟s behaviour on that night nine months earlier. Under Eternalism, we can retain the metaphysical position that, from any chosen temporal perspective, earlier events are fixed and later events, that are not already
110 present in their causes, are contingent18. From any chosen temporal perspective, sentences about earlier facts are, if true, necessarily true per accidens. But this necessity is relative to a particular temporal perspective and need not, in general, extend to earlier temporal perspectives. From a three-dimensional Eternalist perspective, an absolute, or universal, B-series exists, independent of perspective. The B-relation „sorts‟ all objects and events according to the B-relations of „earlier than‟ and „later than.‟ All objects and events are related to instants on the B-series and are real at those B-instants during which they occur. These events are possible from the perspective of instants holding the „earlier than‟ relation and necessary from the perspective of those holding the „later than‟ relation.
Eternalism‟s egalitarianism with respect to the existence of objects does not imply that, in 2010, Socrates is real in every sense in which Kripke is real. We should not expect to meet Socrates at the Stones concert any more than we should expect to run into our seventh-generation granddaughter at the market. Events and „real‟ existence of objects still are relative to B-relations. It is the case that Socrates‟ death precedes
Kripke‟s birth. Because of this, their lives do not overlap; their respective intervals of existence as real-world objects are separate, even under Eternalism. When Socrates exists as a real-world object, Kripke does not. Similarly, when Kripke exists in this way,
Socrates does not.
18 Though a robust three-dimensionalist Eternalist may well consider all objects and events real in a stronger sense, and maintain that Socrates simply is as real as Kripke regardless of when the matter is considered, this is really a position tantamount to fatalism. While Fatalism would be implied by a sufficiently strong version of Eternalism, it is outside the scope of this project to enter into this discussion. For the purpose of considering its implications for the ontology of time, the principle of Eternalism is adequately represented in our interpretation of „real at a time.‟
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In what sense under Eternalism, then, are the existence of Socrates and the existence of Kripke equally real? The answer to the question “What exists?” depends entirely on the context of the utterance; in particular, on the answer to the clarifying question “What exists, when?” Eternalists reject the idea that there is an absolute, or perspective-independent, privileged Now. As I write, I might think that Kripke is real in some sense in which Socrates is not. But, if I were writing in 400 BC, I would think that
Socrates is real in some sense in which Kripke is not. When an Eternalist maintains that
Socrates and Kripke are both real, she does not necessarily mean that they are both real at the same time. She means, instead, that it is only a matter of the arbitrary selection of a temporal perspective that distinguishes the fact of the existence of Socrates from that of
Kripke. Because there is no Now in physical time, no privileged present instant, for an
Eternalist selecting 400 BC as our temporal perspective is no more and no less acceptable than selecting 2010 AD. When considered before a perspective is chosen, Socrates is as real as Kripke.
The distinction between the various views concerning the ontology of time are often thought of in terms of the set of values over which variables in a first-order sentence can range. Consider the sentence x Txr ySyx , where „r‟ is a name for
Ringo, „T‟ is a two-place „is taller than‟ relation, and „S‟ is a two-place „is shorter than‟ relation. For the Eternalist, the variables x and y range over all persons, regardless of when they exist. Instantiations might include „If Aristotle is taller than Ringo then Ringo is shorter than Aristotle,‟ and „If George‟s first-born great-great-grandson is taller than
Ringo, then Napoleon is shorter than George‟s first-born great-great-grandson,‟ even if
George‟s children do not yet have any children of their own. This will not figure
112 prominently in the temporal logics we explore in later chapters, as our focus will be on propositional languages. This first-order implication of Eternalism, though, does fuel the accusation that B-theories imply fatalism. As we have argued, though, this is not a fair criticism of any but a radically strong version of Eternalism.
4.3.1.1 Three-Dimensional and Four-Dimensional Perspectives on Eternalism
A non-relativized four-dimensional Eternalist theory will differ little in effect from its three-dimensionalist counterpart. Because there is no absolute time series in a four-dimensionalist theory, it is not immediately clear to what we refer when we indicate a particular instant, a. When we speak of an object at a particular instant, we are really speaking of a particular temporal part of the object. We can think of a temporal part, say of the piece of chalk at instant a, as being described by a collection of points in space- time. These points would all have the temporal coordinate a. For the B-theorist, the chalk exists at instant a, so it is real. Another temporal part of some other object, a chalk- board for example, that exists simultaneously with the piece of chalk at a should, likewise, have a temporal coordinate of a.
Because there is no absolute time series, though, there is no „standard‟ by which to set the coordinates of space-time points. What we can do, though, is set the coordinates relative to some object that exists co-temporally with the other objects we are interested in. For many purposes we use parts of the Earth for this purpose. Degrees of longitude and latitude set a spatial coordinate system based on the physical features of the poles and equator. A particular temporal part of the Earth is stipulated to be „0.‟ The temporal part at which the Earth has concluded exactly one rotation from 0 is given a temporal coordinate of „1 day.‟ The „day‟ unit is divided into smaller pieces, each
113 corresponding to a temporal piece of the Earth that occurs between 0 and 1. The unit of year is set between temporal parts at the same position in the Earth‟s revolution of the sun. These crude coordinate systems suffice to illustrate how space-time points can be assigned coordinates. To speak of the chalk and chalk-board at a, then, really begins with the Earth at a. The chalk at a is the temporal part of the chalk that exists simultaneously with the temporal part of the Earth that has the temporal coordinate a. The chalk at a and chalkboard at a can both be viewed from the Earth‟s frame of reference. If we select the
Earth‟s frame of reference as being privileged, we can easily talk in terms of simultaneity of the temporal parts of the chalk and the chalkboard at a.
While we cannot name an instant „a‟ without making reference to a particular temporal part of some object or to the coordinate system of the frame of reference occupied by that object, we generally speak as though we can. But this talk can simply be thought to rest on the understanding that „the chalk at instant a‟ is a short-hand representing something like „the temporal part of the chalk that is simultaneous with the temporal part of the Earth with the temporal coordinate a when considered from the
Earth‟s frame of reference.‟
With these conventions in place in non-relativized four-dimensionalism, we can say that every instant, a, defines the class of temporal parts that have a as their temporal coordinate. Eternalism will now make the same claims as it did under three-dimensional models. Objects with temporal parts at a are real at a. Because there is no privileged instant, though, any instant can be chosen and all those parts existing at that instant will be real. It is, as it is in three-dimensionalism, in the sense that no instant is ontologically privileged as Now that all objects and events are equally real under Eternalism.
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A relativized four-dimensional Eternalist theory will not feature an absolute B- series. Each world line has its own temporal dimension, which will be unique among the temporal dimensions of all world lines. Considered from the perspective of a given world line, given Eternalism, there is no ontologically privileged Now instant . Any given point, p, on a world line divides the line into three parts: p, the points prior to p and the points subsequent to p. If we interpret p as Now, it divides the world line into past, present, and future. From the perspective of p, the reality of objects and events represented by points prior to p differs from the reality of objects and events represented by points subsequent to p. Under Eternalism, though, there is nothing that picks p. With no tense in physical time, there is no objective Now, whether absolute or referenced to a world line, to identify p. In the same sense as in three-dimensional Eternalism, then, any p will do as well as any other for dividing the world line and, because the split into preceding p and succeeding p is set by the selection of p, there is no non-arbitrary division of the points in the world line. Since there is no metaphysical condition that determines the division, it would be unreasonable to attribute metaphysical properties such as existence on the basis of this division.
The issue of modality is somewhat more complicated in connection with four- dimensionalism because, from any point, a boundary is formed such that events outside of this boundary are not possible relative to that point. This can be best seen with the help of a Minkowski cone diagram [Figure 11].
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t
y
x
p
Figure 11
An event, depicted as point p, is at the centre of the diagram. The horizontal axes of this three dimensional diagram represent two spatial dimensions and the vertical axis represents time. The spatial dimensions are often thought of as representing the object‟s shadow. The sides of the cones are set at 45° to the axis; with the time scale in seconds and the spatial scales in light-seconds, the sides of the cone create boundaries of what is possible. All points in the possible past of p will be found inside of the bottom cone.
Any point prior to p that is outside of the cone cannot possibly be in p‟s past. Similarly, only points inside the upper cone are possible in p‟s future. This is because, to get to any point outside of the cone from p would require increasing the spatial coordinate by a greater amount than the temporal coordinate is increased by. But, because of the scales of the axis and the 45° measure of the sides of the cone, this would require a velocity greater than the speed of light. A similar analysis explains why p is not in the possible future of any point or event outside of the bottom cone.
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The Minkowski cone diagram helps us to see a constraint on modality from the perspective of any given instant, p. This constraint on de re modality, while different from the constraint on per accidens modality that arises from the „earlier than‟ and „later than‟ relations, is not an absolute constraint but is relative to a position in time. The events possible de re in the future of p are all inside the upper cone.
Under a relativized four-dimensional theory of time, there is no simultaneity between world lines. Privileging a point, p, in a particular world line divides that world line into a past, present and future relative to p, but it does not divide the world beyond that world line. In the following section, we consider a notion of simultaneity between world lines that could be applied to an Eternalist view of a relativized four-dimensional theory of time.
Eternalism is the standard ontology of B-theories. Whether viewed from a three- dimensional perspective or a four-dimensional perspective, under Eternalism, no instant is privileged as Now. Objects and events are real relative to their spatial or temporal locations. All objects and events are equally real in the sense that there is nothing privileged about the particular B-times at which any given object exists or any given event occurs.
4.3.2 Possibilism or Growing Universe
Possibilists maintain that tense is a feature of physical time and that it represents the fundamental difference between space and time. They agree with the Eternalists that here is relative, so that a point in space can be both near to point of origin A and far from point of origin B. In contrast, Possibilists hold that Now is absolute. An instant cannot simultaneously be both past relative to one privileged Now and future relative to another.
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An instant can simultaneously hold contradictory B-relations; it can be earlier than some instant and later than another. An instant cannot, however, simultaneously hold contradictory A-relations, because tense is relative to Now and not relative to other instants.
In “Time, Tense, and Causation,” Tooley introduces his notion of Possibilism by writing “The account of the nature of time that I am defending here involves the claims, first, that the world is dynamic, rather than static; secondly, that the present is the point at which states of affairs come into existence; and, thirdly, that, once a state of affairs is actual, it remains part of reality from that point on.”19
In Possibilism, the past and present are real, while the future is open. Now serves to divide time into three parts: the past, the present, and the future. Possibilism matches our common intuitions, first, that Socrates and Kripke are both real, but in different senses and, second, that the first person born in 2020 is simply not real in any direct sense. The first person born in 2020 might be considered by real as a „concept‟ or description, but not as an actual ontological entity. The reason for this is that, for the
Possibilist it is not Now the case that there is a time at which the first person born in 2020 exists. In contrast, because the past has a positive ontological standing under Possibilism, there is Now the case that there is a time at which Socrates exists.
Possibilism, like Eternalism, allows for asymmetries between objects and events
„earlier than‟ a given instant and objects and events „later than‟ that same instant.
Possibilism, unlike Eternalism, allows for the given instant to be identified with Now.
19 Tooley [2000], P. 33
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For most Possibilists, objects and events in the past are real and fixed, while those in the future are neither. This allows for the easy analysis of “Kripke quoted Socrates” when uttered in 2010. Assuming Possibilism, „Kripke‟ refers to Kripke and „Socrates‟ refers to
Socrates. The truth-maker for this tensed sentence is some past event in which Kripke quotes Socrates. Relative to the context of the utterance, one referent and the truth- making event are found in the past, while one referent is found in the present. Because objects and events in the past are real, they can serve as referents and truth-makers.
This is connected to the domain of first-order language quantifiers. Possibilism can be thought of formally in terms of the set of values over which variables in a first- order sentence can range. Consider again the sentence x Txr ySyx , where „r‟ is a name for Ringo, „T‟ is a two-place „is taller than‟ relation, and „S‟ is a two-place „is shorter than‟ relation. For the Possiblist, the variables x and y range over all persons who exist at the time of evaluation of the sentence, and who existed in the past of the time of evaluation. Instantiations would include „If Aristotle is taller than Ringo then Ringo is shorter than Aristotle,‟ but not „If George‟s first-born great-great-grandson is taller than
Ringo, then Napolean is shorter than George‟s first-born great-great-grandson,‟ assuming
George‟s children do not yet have any children of their own.
A satisfactory analysis of some, but not all, future-tensed sentences is available to a Possiblist. First, consider “It will be the case tomorrow that it rained yesterday,” or
“The one hundredth anniversary of the death of Elvis will be August 16, 2077.” These are future-tensed sentences, but their truth-makers are found in the present. It might be argued that these are only pseudo-future-tensed sentences, since their truth could have been established, but this just begs the question against future-tensed sentences with
119 present truth-makers. Of greater interest, though, are future-tensed sentences without present truth-makers or referents.
The sentence “On August 16, 2077 at Graceland, Elvis‟ first-born granddaughter will attend the official ceremony commemorating the one hundredth anniversary of Elvis‟ death” is a future-tense sentence which, in 2010, has no truth-maker. The name „Elvis‟ refers to Elvis because the past is real, and the definite description „Elvis‟ first-born granddaughter‟ refers to Danielle Keough because the present is real. But, since the truth-maker is not in the past or the present, it is not real. The sentence expresses a proposition, but this proposition has no truth value prior to August 16, 2077.
The greater problem than a presently-undeterminable truth-value of this sentence might seem to be the absence of a referent for the definite description „the official ceremony commemorating the one hundredth anniversary of Elvis‟ death.‟ This is a problem of reference that extends beyond issues involving tense. Though we are not making direct reference in the way in which we are with „Elvis,‟ the sentence becomes no less meaningful than „The largest prime is greater than seven.‟ „The largest prime‟ does not refer to any particular number or object, and yet we seem to have used it in a true sentence. While it does not imply the existence of an object or event, reference of future objects or events presents no additional difficulty.
Most of what can be said of Eternalism can be said of Possibilism. The fundamental difference between these ontological positions is that Possibilism includes
Now in its ontology while Eternalism does not. The import of this is that the instant that divides real from not-real in Possibilism is determined by Now, while we are free to stipulate any instant in Eternalism.
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4.3.2.1 Three-Dimensional and Four-Dimensional Perspectives on Possibilism
A three-dimensional Possibilist ontology includes the A-relation and resulting A- series, together with the ever-changing Now. Because the future is not real, no existing object or event holds the future relation to Now. As a result, with each incremental movement in Now, a further element of the A-series is populated by existing objects and events, as well as by new objects and events that come into being as this next instant becomes the present instant. As each instant becomes the present instant, it sees the emergence of a new state of affairs, which consists of all the objects, events and relations that exist in that instant. Once a state of affairs is „born‟ its objects and events remain part of existence, even as they pass from being in the present instant to being in the past.
A relativized four-dimensional Possibilism requires a notion of tense that is consistent with general relativity. Because time is relative and not absolute, there cannot be the Now. Now must be relative to a frame of reference, since it is in frames of reference that time scales are set. If we assume a tensed view, the tense must be relative to a frame. Consider Now in Frame F. It would seem reasonable to suppose that the simultaneous instant in every other frame is also Now. But not only are scales of time relative to frames of reference, as was seen earlier, the relation of simultaneity is relative as well.
Because simultaneity is relative, we cannot identify Now in a preferred frame, F, and then use this to define a universal Now as being the instant in each frame that is simultaneous with this instant in F. The Possibilist can begin with Now in a particular frame of reference, and make a claim about the ontology of any system described under that frame, but this falls far short of being a fully satisfying metaphysical position. For a
121 more general and powerful theory of time that extends across multiple frames of reference, we cannot simply begin with Now relative to a particular frame.
Instead, we begin with the coming into existence of states of affairs. Consider an arbitrary world line, W, through the lens of Possibilism. W is comprised of a collection of points, each of which describes a point in space-time relative to a frame of reference.
Because Possibilism is an A-theory, one of these points, say n, is privileged as Now. The points in W up until n are real; there are no points in W beyond n. At n a new state of affairs comes into existence, while already existing states of affairs continue in existence, attached in some way to previous points in W, all of which remain real. A state of affairs relative to W and n, is made up of facts concerning W, in addition to facts concerning all world line segments accessible from n. A segment of a world line is accessible from n if and only if it is within either the past or future light cone of n. Under Possibilism, the state of affairs may contain facts only from accessible world line segments in the past cone of n.
Now is still relative to a particular point in space-time, but the state of affairs that comes into existence Now is comprised of all the facts in the accessible past of n. This means that, for example, in Tony Bennett‟s Now there are facts about the Andromeda
Galaxy, Greek philosophers, landings on the Moon, Las Vegas performances, the coalescence of gasses to form the Earth, and the birth of Wayne Newton‟s mother.
Although there may be a great many facts that lie outside of this collection, they are all so distant from Tony Bennett in space or time so as to exert no influence over his world line.
Any fact that does exert influence over his world line is found in this collection.
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Returning to an earlier example, Castor‟s world line is distinct from Pollux‟s, and
Castor‟s Now is not simultaneous with Pollux‟s. But, until their twentieth birthdays, the difference between their respective Nows and the resulting states of affairs was a difference so slight that it made no difference.
Ontologies in a relativized four-dimensional Possibilism are, also then, relative to particular frames of reference. What there is, exists relative to a spatiotemporal point and the frame of reference it occupies. Still, even with the relativity of ontologies and of Now and of states of affairs, the experience of observers in reasonable spatiotemporal proximity who are moving with similar velocities are going to effectively indistinguishable.
4.3.3 Presentism
Presentism is the view that only objects in the present are real. The intuition of
Presentism is compelling; the past is no more, the future is yet to come, so only the present is real. Like Possibilism, Presentism is an A-theory.
Prior is a proponent of Presentism. In the paper “Thank Goodness That is Over!” he examines the circumstances under which one would utter the expression “Thank goodness that is over!” This expression would coincide with a feeling of relief that some unpleasant thing, that was once experienced, is no longer being experienced.
Fundamental to the expression is that the present – the context in which the utterance is made – is distinct from the past. Moreover, and this is the important part for Presentism, the present is, and the past is no more. The relief must be actually felt to prompt the expression; it must be real. If the unpleasant thing continued to exist, it would not be over and the relief would not be felt.
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What Prior‟s example demonstrates most clearly is that there is a stark difference in our experience of the present as compared with our experience of the past. This difference is based on the present being real in a very strong sense that is absent in the past. The relief is real because the unpleasant thing is not real. We recall in this context
Perry‟s example of the 12 o‟clock meeting that is starting now. That we believe it is presently 12 o‟clock is what causes our anxiety at not yet being in the meeting. That the commencement of the meeting is in the present and not in the past or the future, is what produces the anxiety. Again, the experience marks the difference between the present and the future; that is, the present is real and the future is not.
With these complementary examples, the Presentist demonstrates the uniqueness of the present as compared with the past and the future, and attributes this distinction to the fact that the present, alone, has the property of being real. The Presentist takes the intuition of the Possibilist towards the future and extends it towards the past.
One of the first things we notice about Presentism is that is takes the problem of existence of future truth-makers we encountered with Possibilism and adds to it a problem of the existence of past truth-makers. A sentence like “Dinosaurs once roamed the plains.” is problematic for the Presentist because there does not seem to be any present truth-maker for the sentence.
Under Presentism, truth-makers must be present-tensed. One available approach is to suppose that all facts about the present yield present-tensed facts about the past as the instant that is present changes. For example, the fact „It is raining at 2:00‟ is a present-tensed fact at 2:00 that yields the new present-tensed fact „It is the case that it was raining at 2:00‟ at 2:01. So, the fact that serves as truth-maker to “Dinosaurs once
124 roamed the plains.” is the present fact „It is now the case that it was once the case that dinosaurs roamed the plains.‟ More particularly, we can see the accumulation of B-facts that will serve as present truth-makers. „A dinosaur is on the plain at 0936 on September
12, 144 000 017 BC.‟ Because B-facts are tenseless, they are, once established, always available. This B-fact is created at 0936 on September 12, 144 000 017 BC and then is added to the set of facts created in the next instant. In this way it is carried, instant by instant, from its creation to the present. Each present instant contains all the new facts created at that instant, together with the collection of all B-facts carried forward from the previous instant. But, besides the bloated population of facts that this approach would require, this is a tenseless solution to a tensed problem. There is no escaping that
Presentism is an A-theory; it is a theory that bases ontology on tense. These B-facts, then, must be reducible to A-facts. What we have in this approach is a short-hand of B- facts in place of A-facts. The danger in using this short hand is that the hidden A-facts may turn out not to be purely present-tensed.
A second and, perhaps, more direct approach to the problem of finding present truth-makers for sentences about the past, an approach that does not demand this replication of B-facts, is to reduce past- and future-tensed sentences to functions of present-tensed sentences. The present-tensed sentences will then have present-tensed A- facts without any reliance on a B-theory. Our sentence “Dinosaurs once roamed the plains.” would be translated as “WAS[Dinosaurs roam the plains].” The sentence inside the scope of the WAS operator is a present-tensed sentence and, so, has a present-tense truth-maker. The operator pulls the truth-value of the present-tensed sentence „Dinosaurs roam the plains‟ forward to the present.
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Both these solutions to Presentism‟s requirement of present-tensed truth-makers rely on the past being, in some sense, available. The required facts are somehow carried forward to the present. If we were to take Presentism absolutely seriously, there would be no past to carry facts forward from. In fact, there would be no difference between the unreal past and the unreal future; it would make no more sense to talk about an open future than it would an open past. Nothing in our metaphysics would justify the fixity of the past if the past is not real. Under strict Presentism, the past would be exactly as open and changeable as the future.
It is interesting and, perhaps, instructive to consider the spatial analogue of
Presentism. We can transform Prior‟s “Thank goodness that is over!” to “Thank goodness that is there!” This is something that one might express upon reading that a devastating earthquake has occurred on a distant continent. Certainly, our relief that the catastrophic event happened at a spatial distance is real, and is reflective of the different way in which spatially proximal objects and events seem real to us than those that are distant. When I look out my window at the elm trees on our boulevard and houses across the street, and consider these in relation to the palm trees and cabanas I have recently seen on a caye in Belize, those objects outside my window seem more real. Part of this is that I can confirm their presence in a more direct way than I can confirm the continued existence of the palm trees and cabanas, and that I am apprehending the former through my senses and the latter through my memory. There would seem to be more to this difference; something to explain my relief that the earthquake happened there as opposed to here. If this extra difference is simply that I feel less personally threatened by a distant earthquake than I do by a proximal earthquake, this spatial example really is analogous to
126 the temporal case. In both, the relief is the result of the unpleasant thing being at a safe remove and not present and, in this sense, not real. There is nothing about the temporal case that persuasively allows the conclusion that the past is not real that would not lead to the conclusion in the spatial case that the distant continent is not real.
4.3.3.1 Three-Dimensional and Four-Dimensional Perspectives on Presentism
A three-dimensional Presentism requires an A-theory of time with an absolute A- series on which just the instant holding the „present‟ relation with Now is occupied by objects and events. As in Possibilism, a new state of affairs is created with each movement of Now. These states of affairs contain some of facts that will serve in the new present instant as truth-makers for sentences about the past.
Each world line in a four-dimensional Presentism will have just one point that is real. As each new point is added, the previous point vanishes. The same problems of simultaneity are faced here as were faced in four-dimensional Possibilism, with the same solutions. The „present‟ is relativized to a frame of reference. All world lines in comparable frames will come close on agreeing on what is present, though no two will exactly agree. As was the case with Possibilism, for these world lines, this is a difference that does not make a practical difference.
Like Eternalism and Possibilism, whether viewed from a three-dimensional perspective or a four-dimensional perspective, Presentism can be thought of formally in terms of the set of values over which variables in a sentence can range. Consider for a final time the sentence x Txa ySyx , where „a‟ is a name for Arnold, „T‟ is a two- place „is taller than‟ relation, and „S‟ is a two-place „is shorter than‟ relation. For the
Presentist, the variables x and y range only over persons who exist at the time of
127 evaluation of the sentence. Instantiations would include „If Willis is taller than Arnold then Arnold is shorter than Willis.‟ The following, however, are not instantiations: „If
Aristotle is taller than Arnold then Arnold is shorter than Aristotle,‟ and „If George‟s first-born great-great-grandson is taller than Arnold, then Napolean is shorter than
George‟s first-born great-great-grandson,‟ assuming George‟s children do not yet have any children of their own.
4.3.4 Reflections on the Ontologies of Eternalism, Possibilism and Presentism
Considering, especially, the moderate versions of these three ontologies of time, we find, first, that they separate naturally with Eternalism on one side of the A-theory –
B-theory divide and both Possibilism and Presentism on the other. Eternalism is the natural ontology of any B-theory and is easily compatible with four-dimensionalist views.
Possibilism and Presentism are A-theory ontologies and strain somewhat under a relativistic interpretation.
The separation between Possibilism and Presentism is significant only in proportion to the degree of severity with which we interpret „real.‟ Under a liberal notion of „real,‟ the separation collapses. It is only when we adopt a strict interpretation that there Presentism is significantly different from Possibilism.
While Eternalism manages more gracefully with questions of reference and truth- making than its A-theory counterparts, it stumbles in any attempts to model temporal passage. As we saw with McTaggart, without tense the B-series is static. The model that
Eternalists propose is, likewise, static. That there are big questions about agency and modality sit beside the question of how we get from one instant to the next. In contrast, both Possibilism and Presentism serve both the present us with a split between the real
128 and unreal and, just as importantly, accommodate that wave of time on which the present seems to ride. Of course, the B-theorists will maintain that the wave is in the way we experience physical time and not in the physical time itself. The likely reason there remain adherents of B-theories and adherents of A-theories is that neither is able to deliver a decisive blow the other. Both theories carry some desperately-packed baggage;
B-theorists and A-theorists alike carry some lumpy items that they would rather no one notice.
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Chapter Five: Prior’s Tense Logic and Metaphysics of Time
5.1 Early Tense Logics
A sentence such as “Socrates is sitting” is interesting to philosophers, who note that the statement describes a state of affairs that may obtain at some times and not at others. Accordingly, the proposition expressed by the sentence has different truth values at different times. Even more problematic is the determination of the truth values of propositions that are expressed by sentences that make assertions concerning future contingent events. Aristotle considered propositions that express future contingent truths to be neither true nor false when they are uttered. Prior‟s work in temporal logic grew from his analysis of sentences whose truth is undetermined at some times and determined at others.
Prior holds that logic enjoys a tight connection to the states of affairs over which its sentences range. “Philosophy, including logic, is not primarily about language, but about the real world.” [Prior 1996, p. 45] This connection, for Prior, arises directly out of the metaphysics of these states of affairs. To use his example, the logical fact that „if
John is sick then John is sick,‟ is not a truth about the sentence „John is sick.‟ Rather, it is a truth about John. The logic is about objects in the world, not about sentences. Prior rejected the formalism so popular today in which logic is about the abstract structures.
For Prior, logic is about objects in the world.
Within the context of this view concerning logic, Prior‟s temporal realism led to his development of tense logics. The tense logics he developed describe a three-
130 dimensional A-theory of time. Prior referred to B-theories of time as a tapestry view of time, and held that this view failed to capture the physical nature of time. He considered tense to be real, in the sense that time is divided by a real Now into past and present. The past is fixed; what existed and what occurred in the past will now always have existed and have occurred in the past. The future is, at least somewhat, open and this openness affords us some measure of freedom. This division is fundamental to his metaphysics.
The B-relations of „earlier than‟ and „later than‟ are helpful, but metaphysically subsequent to the A-relations of „past,‟ „present,‟ and „future.‟
Prior argues that conclusions from general relativity need not preclude the possibility of an absolute, or common, time series. The edifice of modern physics is built on the tapestry view of time, in which the relations of „earlier than‟ and „later than‟ are taken as primary. He holds that this mistaken view leads to erroneous conclusions. In effect, the physicists have begged the question of the nature of time, at least with respect to tense. This is reminiscent of the differences we saw earlier between McTaggart with his idealism, and Russell with his realism. At the very least, Prior reminds us to be cautious of following any „crazy metaphysic‟ too blindly.
Prior wanted to formalize his metaphysics and, believing that tense is a type of modality, began by adapting a standard modal logic. He introduced an operator, F, which carried the interpretation „it will be the case that … .‟ He added a second operator, G, which is the dual of F, such that Gp F p 20. G is interpreted as „it will always be
20 Prior used Polish notation in all the papers and books cited. I have chosen to translate these formulae into Peano notation so as to maintain some uniformity in this work and, likely, to help Prior‟s ideas to be more transparent to contemporary readers.
131 the case that … .‟ His earliest logics were future-focussed and were not concerned with the past. The following axioms were standard in Prior‟s early tense logic:
A1 FFp Fp „it will be the case that it will be the case that p iff it will be
the case that p‟
A2 G p q Gp Gq „if it will always be the case that if it is the case
that p then it will be the case that q, then, if it is the case that it will always
be the case that p, then it will always be the case that q.‟
A3 Gp Fp „if it will always be the case that p then it will be the case that
p.‟
We can see that F is really a „diamond‟ operator while G is a „box‟ operator. Fp is true at w just in case p holds at some accessible point in time, w, and Gp is true at w just in case p holds at every accessible point in time. Of course, the set of times, or instants, is assumed to be ordered and the accessibility relation is „<.‟ Under this interpretation, A1 represents the dual of schema 4 in modern modal logic, A2 represents the schema K, and A3 represents schema D.
When Prior introduced past-tensed operators, P, „it was the case that … ,‟ and H,
„it has always been the case that … ,‟ he needed axioms to relate past and future, and chose:
A4 p GPp „if it is the case that p, then it will always be the case that it was
the case that p.‟
A5 p HFp „if it is the case that p then it has always been the case that it
will be the case that p.‟
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The P and H operators do act, respectively, like the diamond and box, but from a different perspective. If, for example, Pp is evaluated at w, we see that Pp is true just in case p is true at some w from which w is an accessible time. Similarly, Hp is true at w just in case p is true at all s from which w is an accessible time.
These basic additions to propositional modal logic provided some structure for analyzing the preservation of truth in tensed sentences. The logic is limited, though, in particular by its inability to capture truth at a particular instant or to quantify over times.
Prior expanded his basic propositional tense logic by developing levels of metric tense logics.
5.2 Metric Tense Logic and the U-Calculus
5.2.1 The basic U-Calculus
In Stratified Metric Tense Logic [Prior 1968a, pp. 88 – 97.] Prior introduces a hybrid modal logic with levels of metric tensed logic built upon a basic non-metric tensed logic. A hybrid modal logic is a standard modal logic augmented with a set of special formulae that, in effect, refer directly to individual „instants.‟ Prior‟s U-Calculus includes the propositional modal tense logic described in Section 5.1, with the addition of a set of „instant-propositions,‟ denoted by abc,,, . Each instant-proposition is true at exactly one instant. In addition to these instant-propositions, the U-calculus still includes the set pqr,,, of regular propositions.
Besides the set of instant-propositions, Prior adds three relations in the U- calculus. The relation Uab, „instant a is earlier than instant b,‟ takes two instant- propositions as its arguments. The second relation, Iab, „instant a is identical with instant
133 b,‟ also operates on two instant-propositions. The third relation, Tap, „it is the case at instant a that p,‟ takes an instant-proposition as its first argument and a regular proposition as its second.
He derives postulates U1 – U4 from the rules of substitution and detachment and the following rules and axioms:
R1 „if is a theorem, then it is not the case that it will be the case F
that not-.‟ [notice that this is equivalent to ] G
R2 „if is a theorem, then it is not the case that it was the case that P
not-.‟ [notice that this is equivalent to ] H
A1.1 F p q Fp Fq [notice that this is equivalent to
G p q Fp Fq] „if it is always going to be the case that if p
then q, then if it will be the case that p then it will be the case that q.‟
A1.2 P p q Pp Pq [notice that this is equivalent to
H p q Pp Pq] „if it has always been the case that if p then q,
then, if it was the case that p, then it was the case that q.‟
A2.1 F P p p [notice that this is equivalent to FHp p ] „if it will be
the case that it has always been the case that p, then it is the case that p.‟
A2.2 P F p p [notice that this is equivalent to PGp p ] „if it was the
case that it always will be the case that p, then it is the case that p.‟
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From these rules and axioms, together with the basic propositional tense logic, propositions U1 – U4 follow:
U1 Ta p Tap „it is the case at a that not-p iff it is not the case at a that
p.‟
U2 Ta p q Tap Taq „it is the case at a that if it is the case that p
then it is the case that q, iff if it is the case at a that p then it is the case at a
that q.‟
U3 TaFp x Uax Txp „it is the case at a that it will be the case that p,
iff there is an instant, x, such that a is earlier than x and it is the case at x
that p.‟
U4 TaPp x Uxa Txp „it is the case at a that is was the case that p, iff
there is an instant, x, such that x is earlier than a and it is the case at x that
p.‟
This much allows for the evaluation of a proposition at a time. It also allows us in the language to express conditions on the time series through the relation U. For example, if the time series is dense, we have the condition x y Uxy z Uxz Uzy on U and the corresponding postulate Fp FFp ; if the time series is transitive, we have the condition x y z Uxy Uyz Uxz on U and the corresponding postulate
FFp Fp .
Prior introduces two additional modal operators, M and L, interpreted roughly as
„it is true at some time that … ,‟ and „it is true at all times that … ,‟ respectively. L can
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be defined as LDef xTx , for all formulae that contain no free occurrences of x.
M is then just the dual of L; that is, ML .
5.2.2 The Metric U-Calculus
For his metric U-calculus, Prior introduces interval-variables for numbers
mn,, and “as much number theory as we will need.” [Prior 1968a, p. 89] For example, Fnp will be read as „it will be the case in n time units that p.‟ Syntactically, the interval-variables can take the place of instant-propositions in the U-calculus. A two- place S relation on the domain of intervals is added, so that Smn represents the sum of m and n. Finally, the U relation becomes a three-place relation. Uabn is read „a is earlier than b by an interval of n time units.‟ This results in the following adjustments to U3 and
U4:
MU3 TaFnp x Uaxn Txp „it is the case at a that it will be the case in n
units of time that p, iff there is an instant, x, such that a is earlier than x by
n units for time and it is the case at x that p.‟
MU4 TaPnp x Uxan Txp „it is the case at a that is was the case n units of
time earlier that p, iff there is an instant, x, such that x is earlier by n units
of time than a and it is the case at x that p.‟
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The following two postulates cover quantification of intervals:
MU5 Ta x x x Ta x , where is a formula containing an
unbound occurrence of interval-variable x. „it is the case at a that there is
an interval, x, such that in x units of time it will be the case that x , iff
there is an interval, x, such that it is the case at a that in x units of time it
will be the case that .
MU6 UabSmn x Uaxm Uxbn „if a is earlier than b by mn units of
time, the there is an instant, x, such that a is earlier than x by m time units
and x is earlier than b by n time units.‟ This makes it clear that the sum of
two intervals is meant to be the sum of two contiguous intervals.
5.3 Four Grades of Tense-Logical Involvement
In Tense Logic and the Logic of Earlier and Later, Prior acknowledges that, as in the case of modal logic21, there may be elements, or grades, of tense logic with which philosophers may be more or less comfortable. Prior did not see his tense logic as being simply an abstract formal system that one could operate within to establish abstract formal results which might, incidentally, have some practical utility. Instead, he saw tense logic as being an actual representation of the way in which we soundly reason in tensed language. Even more, he saw it as representing the way in which temporal truth functions. For Prior, tense logic had to have a solid metaphysical grounding. Each grade of tense-logical involvement requires greater metaphysical commitment.
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In the first grade of tense-logic involvement, Prior considers an interpretation of the basic U-calculus in which the propositional variables, p, q, etc stand as predicates of the instants at which they are true. So, Tap, which is normally interpreted as „p is true at a,‟ is interpreted as „a is p-ish,‟ or „a ps.‟ p simply acts as a predicate of a to indicate that it is a property of the instant a; that a is an instant when p obtains.
At this level, the elementary forms of Tap and Uab are „chronologically definite‟ propositions, whose truth value is independent of the time at which they are evaluated. If
Tap is true, then it is true at any time. If Tap is interpreted as it is true on Friday that Ken burns the dinner, then it is true on Saturday that it is true on Friday that Ken burns the dinner; that is, TbTap .
We notice, as well, that Tap expresses the tenseless B-truth that p is true at instant a, while Fp expresses the tensed A-truth that p will be true in the future. In general, we can view Prior‟s U-calculus to be descriptive of B-theories of time and his tense logic to be descriptive of A-theories. As we shall see, this distinction is very important to Prior.
Prior suggests that few philosophers will express much uneasiness about this basic U-calculus under this interpretation of predication in which Tap is interpreted as pa ; the instant a falls under the predicate p. If a is „Friday‟ and p is „The dinner is burned,‟ then under the interpretation of predication, Tap [alternatively, ] would be read as „Friday is a dinner-burning day.‟ This, argues Prior, should create little discomfort for any philosopher.
21 See Quine Three Grades of Modal Involvement in Quine, 1976.
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For the second grade of tense-logic involvement, Prior focuses on the relation between Tap and p. He proposes that formulae can be substituted for p to create sentences such as TaTbq, where Tbq is substituted into Tap for p. In Prior‟s U-calculus, formulae such as Tap and xTxp are at the same syntactic level as p and, so, may occur as arguments of a single truth function as follows: Ta xTxp and Tap. We can also have formulae such as Tap q and xTxp p . The addition of the following rule and three axiom schema to the basic U-calculus formalizes this equivalence, which we saw intuitively in the first grade of involvement:
R3 , where a is not already in ; „if is a theorem, then Ta is also a Ta
theorem;‟
U622 „if p is true at all instants, then p is true;‟
U7 x Txp yTyTxp; „for any instant, x, if p is true at x then, for any
instant y, it is true at y that p is true at x;‟
U8 Tap TbTap „if p is true at a, then it is true at b that p is true at a.‟
Here, Prior notes that, if we introduce Ldef xTx , where x does not occur unbound in , as the universal modality „it is always the case that ,‟ we can prove for L the following:
RL „if is a theorem, then L is a theorem;‟ L
22 The numbering of axioms and rules in this work seeks to parallel Prior‟s numbering. Unfortunately, since Prior develops his ideas over several papers, there are some inconsistencies in his numbering. To avoid unnecessary confusion, I am not using „U5‟ as an axiom name.
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L1 L p q Lp Lq „if it is always the case that if p is true then q is
true, then if it is always the case that p is true then it will always be the
case that q is true;‟
L2 Lp p „if it is always the case that p is true, then p is true;‟ and
L3 Lp L Lp „if it is not the case that it is always the case that p is
true, then it is always the case that it is not the case that it is always the
case that p is true.‟
Notice that these are Gödel‟s postulates for Lewis‟ S5 system.
At this second level of tense-logic involvement, Tap and p are at the same syntactic level. If a formula of the form Ta, where a does not occur unbound in α, can be proven in the U-calculus, then the formula can be proven. This is achieved by, first, applying universal generalization on Ta to derive xTx and then U6 to derive .
Because Ta can always, subject to the restriction on universal generalization, be reduced to , any sentence proved in the U-calculus can be reduced to a sentence in the tense logic. For example, TaFp can be reduced to Fp. TbTaGp can be reduced, first, to
TaGp and, then, to Gp. Recalling that the U-calculus describes B-truths while the tense logic describes A-truths, we can see this as a reduction from B-sentences to A-sentences.
Because of Prior‟s views concerning the connectedness of logic to metaphysics, he interprets this as a reduction from B-theory to A-theory.
Prior notes that philosophers still apprehensive about the metaphysics that may underlie the temporal logic or U-calculus can still take refuge in the first grade interpretation of predication.
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So, in the first grade Tap, a, and p are at different syntactic levels. In the second grade Tap and p are at the same syntactic level, but a is at a different level. In the third grade of tense-logical involvement, Tap, a, and p are all at the same syntactical level.
In his third grade of tense-logical involvement, Prior urges the move past the predicate interpretation of the first two grades. In the third level, the instant-variables represent propositions. We might think of the instant-variable „a‟ as representing the conjunction of all propositions true at a, or as a proposition that is true at, and only at, a.
With this more robust semantics, however, we need to revisit our interpretation of Tap. If a is considered to be a proposition, Tap would be read „proposition p is true at proposition a.‟ To speak of a proposition being true at a proposition makes no sense.
But Tap is equivalent to L a p ; „it is true at all times that if a then p.‟ Recall that, when viewed as a proposition, a is true at, and only at, instant a. Beginning with the left to right part of the equivalence, we note that if p is true at instant a, then it is always the case that, when proposition a obtains – that is, at and only at instant a – then p obtains as well. In the right to left direction, holds if it is always the case that ap holds; that is either a is false or a and p are true together. If proposition a is false, then
sets no conditions on instant a or, by extension Tap. If propositions a and p are true together, then proposition p is true at instant a and, so, Tap is true. This establishes the logical equivalence Tap, where „a‟ denotes an instant, and L a p , where „a‟ denotes a proposition.
Prior acknowledges that this appears to be a “highly artificial procedure,” [Prior
1968a, p. 123] but notes that instants are artificial entities anyway. We are reminded here
141 again of the tight connection between metaphysics and logic for Prior, who considers the
B-series to be reducible to the A-series. For Prior, the B-series is a useful fiction; it “is just disguised talk about what is and has been and will be the case.” [Ibid]
In the third grade of modal involvement, quantification over instants becomes quantification over a certain type of propositions. There is no quantification indicated over propositions of the usual „the cat is on the mat‟ sort, represented syntactically by variables pqr,,, . Prior quantifies just over the instant-propositions, represented syntactically by the instant-variables, abc,,, . This is something that Prior recognizes and acknowledges as being potentially controversial, but accepts without any argument.
A consequence of this dual-duty of the set abc,,, of state-variables is that we can now place them into either the „instant‟ place or the „proposition‟ place in T_ _. For example, Taa is now a theorem in the U-calculus. Proposition a is the case at instant a.
Besides adding theorem T6: Taa, Prior adds T7: Tap a p . If proposition p is the case at instant a, then if proposition a is the case then proposition p is true.
In addition to being a „truth‟ relation when one argument is an instant and the other is a proposition, T can now be seen also to be an „equivalence‟ relation when both arguments are instant-propositions. Consider Tab. Here, „a‟ represents and instant and
„b‟ represents a proposition. Recalling the first of the two intuitive ways Prior offers for making sense of this third grade of temporal-logic involvement, we would read Tab as
„the conjunction of all the propositions true at instant b is true at instant a.‟ Under the second interpretation, Tab would be read as „the proposition true at instant b and only at instant b is true at instant a.‟ Under this reading, we see Tab obtaining when and only
142 when a and b are associated with the same instant. We can easily show that Tab Tba and Tab Tbc Tac, and these, together with T6: Taa , establish that, under this interpretation, T is an equivalence relation. Really, then, when both arguments are filled by instant-propositions, Tab is just Iab.
This enhanced interpretation of Prior‟s state-variables extends to formulae containing his temporal modal operators. For example, Uab TaFb and TaFb Uab are now theorems of the U-calculus. Instant a is earlier than instant b if and only if it is true at instant a that it will be the case that proposition b is true. We begin to see that there is an attractive result to the „blurring‟ of the distinction between instant and proposition. The logic assumes a fluid quality and, if we speak quickly, we find ourselves reading as „if a is earlier than b, then it is true at a that b is in the future,‟ and, here, thinking of the second b as an instant, even though it is playing the syntactical part of a proposition. We are encouraged to think in this intuitive way, with the assurance that the logic will „work out.‟
With the third grade of tense-logical involvement, Tap, a, and p are all at the same syntactical level. The metaphysical commitment has been increased; we have provisionally accepted the existence of instants. For Prior, this is tantamount to accepting a B-theory of time. But, this affirmative ontological status of instants will be short-lived.
Prior will bolster his tense logic, which supports an A-theory of time, by defining it in terms of L, the universal modality. He will then move to the fourth grade by reducing the
B-theory U-calculus to the A-theory L-calculus.
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But, unlike with the first two grades, Prior does not demonstrate yet that this full
U-calculus reduces to basic tense logic. In preparation for this reduction, which will lead to the fourth grade of tense-logical involvement, Prior reformulates the U-calculus with the modal operator L its the basis rather than the relation T.
This reformulation begins with the addition of the definitions Tapdef L a p
and Uabdef L a Fb to tense logic. The first of these definitions is already familiar.
The intuition behind the second definition is that to say „a is earlier than b‟ is no different than to say „b is always in the future of a.‟
With these definitions added to tense logic, some interpretation needs to be given in this new L-calculus for the „special‟ propositions, abc,,, , that are not present in the tense logic. Prior introduces a new one-argument function, Q, such that Qp is true just in case there is a unique point at which p is true. If Qp is true, p will act like an instant- proposition. So, for example, we can rewrite L a p as Qq L q p . This manoeuvre eliminates the instant-propositions at the cost of introducing a new function.
The L-calculus, then, consists of tense logic, with the addition of two definitions that define T in terms of L, and the addition of a function, Q that facilitates the replacement of the instant-propositions found in the U-calculus by ordinary propositions.
Recalling RL, L1, L2, and L3, we see that the L-calculus is an S5 system. Because the L- calculus is really just a reformulation of the U-calculus, the full U-calculus is reducible to the L-calculus. In the fourth grade of tense-logical involvement, Prior will offer a modest augment to the L-calculus which will render it reducible to tense logic. As a result, the full U-calculus with its instant-propositions will be found, through the L-calculus, to be
144 reducible to tense logic. This, for Prior, demonstrates the metaphysical primacy of the A- theory of time over the B-theory.
In the fourth grade of tense-logical involvement, Prior offers two paths to the reducibility of the L-calculus to tense logic. The first is based on setting certain metaphysical properties of the time series. For example, if time has no branching in the
past, Lpdef Hp p Gp . It is always the case that p iff it always has been the case that p and it is the case that p and it always will be the case that p. With this definition in place, the universal modality, L, could be replaced throughout the L-calculus, thus reducing it to tense logic.
This definition of L is sufficient given our intuitive notion of time, it is sufficient for experienced time, but is not general enough for all models of time. For example, suppose that time has a branching past. Consider the three instants indicated in Figure
12. Instant t1 is earlier than instant t2 and instant t3 is earlier than instant . Because of the way in which time branches, however, there is no „earlier than‟ relation that obtains between instants and . Suppose that the sentence Hp p Gp is true at . In that case, p must be true at . But, because is not connected by the „earlier than‟ relation with , the truth value of p at is entirely independent of the truth of Hp or Gp at . In our counter-model we assume that p is true at ; p is not true at every instant, so it is not the case that Lp. While is true at , it is not the case that it is true at all times that p. So, we have and Lp .
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t1
p t2 Hp p Gp p
t 3 p
Figure 12
Of course, there are other examples of basic assumptions with regard to the structure of time that will scuttle this tense logic definition of the universal relation. If we begin with an alternative metaphysics, defining L in terms of tense logic may be problematic23.
The second path Prior offers to reduce the L-calculus to tense logic does not require particular assumptions concerning the nature of the time-series, but does involve bringing the Natural Numbers into tense logic and, as Prior writes, “enlarging our symbolic apparatus a little.” [Prior 1968a, p. 128] We begin by inductively defining the form Lpn as follows:
L0 p p ;
Ln1 p HL n p GL n p .
For example, L1 p Hp Gp ; it always has been the case that p and it always will be the case that p; and L2 p H Hp Gp G Hp Gp; it always has been the case that it always has been the case that p and that it always will be the case that p, and it
146 always will be the case that it always has been the case that p and that it always will be the case that p.
n Prior defines the universal modality, L, as Ldef nL p , where nN .
If we now reconsider our earlier example, as we see in Figure 13, that while
is true at t , Lp is not. 0 and L1 Hp Gp are true at , as was seen Hp p Gp 1 Lp earlier, but L2 H Hp Gp G Hp Gp is not true at .
is not true at because Hp is not true at t2 as there is
a time, t3 , in the past of at which p is true. Since Hp is not true at , it is not the case at that „it will always be the case that it always was the case that p.‟ So, HGp does not obtain at and, because of this, neither does the first conjunct of
. Some further consideration of similar counter-models will show that they can be resolved with Ln for appropriate values of n.
t1
p t2 Hp Gp p GHp Hp
t 3 p
Figure 13
23 See The Logic of Ending Time in Prior 1968a pp. 98 – 115.
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We see that L0 , while it yields Lp p , is insufficient to capture L because it does not give us, for example, Lp Hp and, while Lp1 yields Lp Hp Gp it fails to imply . Because the L-calculus is an S5 system, we know it must contain
. Prior satisfies all these needs in his definition of Lp as nLn p . To show
, we offer the following deduction:
1. Lp [premiss]
2. nLn p [definition]
3. Lp0 [UI]
4. p [definition]
5. Lp p [PL]
Similarly, we can deduce L p q Lp Lq , which is schema K, and
MLp p , which is the dual of scheme B. This gives us a Brouwersche system. Prior proves Lp LLp , which is schema 4. The addition to schema 4 to the Brouwersche system yields an S5 system.
The importance of all this is, first, that Prior has now established that the L- calculus can be reduced to tense logic. Since the U-calculus can be reduced to the L- calculus, this means that the U-calculus, with its instant-propositions, can be reduced to tense logic. Because tense logic is the logic of the A-theory of time, and the U-calculus, which is the logic of the B-theory of time, can be built from tense logic, tense logic is more basic than the U-calculus. Since Prior so closely connects metaphysics to logic, for
148 him, this provides support for the view that an A-theory of time is more basic than a B- theory. It supports his view that physical time is tensed.
Prior has also established that, without the assumption of infinite time, the L- calculus can still be reduced to tense logic, as long as we include in tense logic the inductive definition of Lp as nLn p , where nN . In this case, we eventually arrive at the conclusions that the U-calculus can be reduced to tense logic and that physical time is tensed.
The parallel nature of the U-calculus, with its instant-propositions that identify instants, with the metaphysically-safe tense logic provides a system in which the
„usefulness‟ of instants can be exploited without our having to bring them into our ontology. This was vitally important for Prior. We have access to all the analytic and heuristic devices of B-time without any of the metaphysical commitment. For an adherent of an A-theory of time, this is very good news.
5.4 The Problem of Egocentric Logic
After building his U-calculus and establishing its reducibility to tense logic, Prior considered other examples of modality; in particular, those he called egocentric.
As an example, Prior proposes a possible language that lacks proper names and pronouns. In such a language, an individual would say „Sitting,‟ to express what „I am sitting,‟ would mean in English. A person would use modalizations relative to themselves to speak about others and whether or not they are sitting. For example, the modality F being interpreted as „someone taller than I is … ,‟ and the modality G being interpreted as „everyone taller than I is … .‟ If p is „sitting,‟ Fp is read as „someone taller
149 than I is sitting.‟ Of course, the modalities of P and H and, if we like, L and M, would be interpreted appropriately.
This egocentric interpretation of tense logic will support a corresponding U- calculus, where the instant-propositions will not connect to instants but, rather, to persons. In our example, Tap would be read „person a is sitting,‟ and Uab would be read
„person a is taller than person b.‟ On the surface, this is a powerful result, since it provides a robust formalization for logics that cover a broad range of language. For
Prior, though, it presented a problem.
Prior maintained that instants, though useful fictions, were still fictions. There are no instants in the world. By committing to the fourth grade of tense-logical involvement, he was able to reduce the logic talk of instants to talk of tense, to reduce B-theories of time to A-theories. This was desirable because, while B-theory instants do not exist, A- theory tense does exist. It is important for Prior that the logic supports the metaphysics.
In the case of our egocentric example, however, Prior does not doubt the existence of persons. But, because the parallel logics behave in an analogous manner to the U-calculus and tense logic, the reasoning he employs with respect to the ontology of instants would apply equally to the ontology of individuals. It seems he faces an unfortunate dilemma: search for a flaw in his earlier findings or accept the non-existence of individuals.
Prior holds simply that instant-individuals are not real while person-individuals are. The principle of ontological commitment that he proposes is that the types of individuals that exist are those we are comfortable with as subjects of predicates. Those that do not exist are those we prefer to qualify with modal operators. “If we prefer to
150 handle instant-variables , for example, or person-variables, as subjects of predicates, then we may be taken to believe in the existence of instants, or persons. If, on the other hand, we prefer to treat either of these as propositional variables, i.e. as arguments of truth- functions and modal functions, then we may be taken as not believing in the existence of instants, etc.” [Prior 1968a, p. 142]
Certainly, this position seems to be a little ad hoc, but it addresses a difficult problem for Prior.
While this is not an entirely satisfactory position, it must be remembered that, in
1968, this was a work-in-progress for Prior. Unfortunately, we do not benefit from knowing where Prior would have taken this work.
5.5 Alternate Conclusion
Through the previous three chapters, I have consistently made a distinction between experienced time and physical time. While unable to offer a definitive characterization of physical time, I did argue the view that experienced time is tensed. In the current context, we can now see that, while Prior does not bring it to our attention, experienced time is egocentric.
When considered strictly with respect to experienced time, what Prior articulated in discussing the four grades of tense-logical involvement suggests that experienced time can be reduced to tensed time. While tense logic may not reflect the metaphysics of physical time, it does reflect a formalization of our thinking about time and, in this way, is seen to describe our experience of time. It is not surprising that our experience of time is irreducibly tensed; it is manifest in our language specifically, but also in our seeming inability to leave tensed-thought behind in any consideration of the way in which the
151 world works. It is only with abstract formal systems, like Prior‟s U-calculus, that we seem to be able to explore a world without tense – our intuition will not take us on that adventure. As will be seen in the next two chapters, we can develop a hybrid temporal logic which can model both our experienced time and a variety of physical time theories.
The importance of a logic of experienced time is that it affords a tool for confirming our intuitions and conclusions regarding experienced time, and for ensuring consistency in our reasoning. With respect to physical time, a logic will allow us to extend our reasoning beyond our intuitions and examine consequences of various physical time theories.
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Chapter Six: Hybrid Modal Logics
This chapter provides a general background of hybrid logics24. It is not a specific examination of tensed hybrid logic; that is the topic of Chapter Seven. A reader familiar with hybrid logic, or with little interest in a formal presentation of general hybrid logic, can proceed directly to the following chapter without missing any discussion of theories of time or of tense logics.
6.1 Introduction: Hybrid Logic
Prior‟s full fourth-grade U-calculus, with its special propositions, is a hybrid modal logic. A hybrid modal logic consists of a normal propositional modal logic with the addition of a class of objects called nominals. A nominal is a formula that has the property of being true in a model at exactly one point. Prior‟s special propositions are nominals. Generally, in a hybrid modal logic, we would say that a nominal is a formula that is true in exactly one world; in our hybrid temporal logic we will say that a nominal is true at exactly one instant. Following Prior, we will use abc,,, to denote nominals.
As we saw with the U-calculus, nominals expand the expressive capacity of a language. If p is „the meeting starts,‟ and a is „noon on March 1, 2010,‟ then ap will be true if the meeting starts at noon on March 1, 2010. The sentence ap asserts that it is noon on March 1, 2010 and the meeting is starting. This sentence will be true at
24 The Hybrid Logic semantics and syntax presented in this chapter represent a synthesis of the material in Areces et al 2001, Areces and Blackburn 2001, Blackburn and ten Cate 1998, Blackburn 2006, and Blackburn and ten Cate 2006. The completeness and soundness results are original, following a strategy presented by Richard Zach in a course on Modal Logic at the University of Calgary in 2007.
153 exactly one instant, the instant at which a is uniquely true. That instant is not formally given any name in the logic, but it is convenient to refer to it as instant „a‟ since it is uniquely identified as the instant at which formula a is true. Following Prior, we will avail ourselves of this expediency, except in cases in which confusion might result.
We will build a temporal hybrid modal logic, similar to the U-calculus but utilizing Kripke semantics. In addition to a propositional normal modal logic augmented with the a set of nominals, our temporal hybrid logic will contain a satisfaction operator which directly facilitates the evaluation of a sentence, or sub-sentence, at an instant, and a local quantifier which assigns the nominal true at the current instant to a variable.
The satisfaction operator, symbolized as „@‟ acts a little like Prior‟s T relation.
We would write @a p to read „proposition p is true at instant a.‟ As we shall see, this satisfaction operator is more versatile than the T relation. The local quantifier, symbolized as „ ‟ binds a variable that ranges over nominals to the instant of evaluation.
From our previous example, the sentence x x p will only be true when it is evaluated on noon on March 1, 2010, when it will read „it is noon on March 1, 2010 and the meeting starts.‟ If it is evaluated at noon on February 28, 2010, it will assert „it is noon on February 28, 2010 and the meeting starts,‟ and, so, be false.
In this chapter, the basic hybrid logic H will be developed which will, with a suitable temporal interpretation, serve as our hybrid temporal logic, T. In the following chapter, we will examine ways in which this basic temporal logic can be adapted to reflect the characteristics of the various theories of time that were examined in Chapters 2 through 4.
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6.2 Informal Semantics of Basic Hybrid Logic, H
Hybrid logics are modal logics with formulae that refer to worlds. These formulae, called nominals, combine with other objects of the logic (propositions, formulae, connectives, modal operators) to form sentences that make reference to particular worlds, or classes of worlds.
Nominals are denoted by the symbols abc,,, . The formula ap is true at world w just in case nominal a is true at w and the proposition p is true at w. So,
a p a q p q is a valid sentence, unlike
r p r q p q. If we evaluate from a world, w, we see that the antecedent is true if, at some accessible world, w, rp is true and at some accessible world, w , rq is true. A counter-model for this sentence will exist if there is no world accessible from w at which pq is true. A simple counter-model is found by limiting the accessibility of w to just and , and by supposing that p and r are true and q is false at and that q and r are true and p is false at . In this model, the antecedent is true because the first conjunct is true at accessible world and the second conjunct is true at . The consequence, however, is false at both accessible worlds. Because there is a counter-model,
is not a tautology.
If we now evaluate at world w, we see that the antecedent is true if, at some accessible world, , is true and at some accessible world, , aq is true. But, since a is a nominal, it is true at only one world.
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Accordingly, ww . We have from the antecedent‟s first conjunct that p is true at this world, and from the antecedent‟s second conjunct that q is true at this world, so it must be the case that pq is true at this world. Because a is a nominal and is true at exactly one world, there can be no counter-model for a p a q p q , so it is a tautology.
In addition to nominals, hybrid logics contain satisfaction operators, denoted by the symbol @. A satisfaction operator acts on a nominal to shift the evaluation to the
world at which that nominal is true. @a A reads: “Formula A is forced at the world at
25 which a is uniquely true. ” To evaluate @@ab ABA , the formula AB is evaluated at the world denoted by a and the formula A is evaluated at the world denoted
by b; so @@ab ABA is true just in case is true at world a and A is true at world b.
Accessibility characteristics of a particular world can be expressed by sentences
in the logic. @a b reads that “b is possible from a,” so there is a world accessible to the world denoted by a such that b obtains at that world. That accessible world, then, is
denoted by b. This construction is similar to the U relation in the U-calculus. @a a
states that a is accessible to itself; that is, the relation on a is reflexive. If @a a then the relation on a is irreflexive.
25 In the interests of expediency, we will begin to read „the world at which a is uniquely true‟ as „the world denoted by a‟ or, simply, „world a‟ without losing sight of the fact that „a‟ names a formula and not a world.
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Quantification across nominals can be introduced into hybrid logic. The familiar
global quantifiers and operate in the expected manner. x@x p q p
states that p q p is forced at all worlds in W. xx@a states that there is a world accessible from a.
Hybrid logics with global quantifiers allow the introduction of sentences that make direct reference to the frame. For example, the following formulae determine the corresponding classes of frames:
xx@x reflexive
xx@x irreflexive
x y@@xy y x symmetric
x y@@xy y x antisymmetric
x y z@@@x y y z x z transitive
x y@@@x y z x z z y dense
x y z@@@x z y z x y Euclidean
In addition to the global quantifiers, a local quantifier is available in hybrid logic. x assigns the current world to x. When evaluated at a, x x p is
instantiated as ap . The sentence xx@x is forced at a world just in case that world is accessible to itself. Quantifiers can be used to express formulae that contain information about worlds, their relations to each other, and the sentences that are forced.
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The formula x y A @x z A y is true at a world w just in case there are two distinct successors of w at which A is forced.
As is shown in more detail a following section, hybridization of temporal logics extends the range of sentences with temporal information that can be formalized. It allows reference to particular instants. For example, if a is „1830 on May 9, 2010‟ and A is „The plane leaves the gate,‟ then we can express „The plane leaves the gate at 1830 on
May 9, 2010‟ as @a A. While it might be argued that this is expressible in a basic temporal logic as M, a A , this construction is not „portable.‟ With the hybridized logic, „The plane leaves the gate at 1830 on May 9, 2010‟ can be expressed as a constituent of a more comprehensive formula, or be evaluated at a world other than a. If
B is „The passenger arrives to the gate,‟ and C is „The passenger arrives late to the gate,‟ we can express „If the passenger arrives to the gate after 1830 on May 9, 2010 then the
passenger arrives late.” as @@@aA b B a b C . x@a x C is true when evaluated after 1830 on May 9, 2010 and false otherwise, assuming that instants in time are ordered and that the accessibility relation in the logic is such that later instants are accessible from earlier instants.
We will now develop the formal semantics and syntax of the basic hybrid logic,
H. This will be followed by proofs of the completeness and soundness of H. A trusting reader can proceed directly to Chapter Seven.
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6.3 Formal Semantics of Basic Hybrid Logic, H.
6.3.1 Objects of the Language:
O1 P p,,, q r is a denumerable set of propositions;
O2 N a,,, b c is a denumerable set of nominals;
O3 G A,B,, C is a denumerable set of well-formed formulae, comprised of nominals, propositional variables, propositional connectives ,,,, , modal operators , , and a satisfaction operator@;
O4 SPW: P is a function that assigns one element of the power set of
W to each element of P;
O5 g: N W is a function that assigns one element of W to each element of
N;
O6 FWR , is a frame consisting of a denumerable set of „worlds,‟ W, and a binary accessibility relation, RW 2 ; and
O7 MWRS ,, is a model consisting of frame F and assignment function
S.
6.3.2 Forcing:
M,, g w A is read “A is forced under assignment g at world w in model M,” subject to the following definitions:
F1 If Ap , for some pP , then M,, g w A iff w S p ;
F2 If Aa , for some aN , then iff w g a;
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F3 If A , then M,, g w A;
F4 If AB , then M,, g w A iff M,, g w B ;
F5 If ABC, then iff M,, g w B and M,, g w C ;
F6 If ABC, then iff or ;
F7 If ABC, then iff or ;
F8 If AB , then iff w: wRw , M,, g w B ;
F9 If A , then iff w: wRw and ; and
F10 If AB @a , then iff M,, g g a B .
Lemma 2.1: wRv M,@ ga b , for any w, v W and some nominals a and b, where g a w and g b v .
Proof (Lemma 2.1):
LR Suppose that wRv , where and , for some and some nominals a and b. From , M,, g v b. Because , M,, g w b.
Since , M,@ ga b .
LR Suppose . Let . Then . So, there is a vW such that and . Then .
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6.3.3 Syntax
The following axioms are contained in H, where A, B are any well-formed formulae in H and a, b are any nominals in N:
A1 All PL and normal ML tautologies
A2 (K): ABAB
A3 @@@a ABAB a a
A4 @@aaAA
A5 @a a
A6 @@@a bAA b
A7 a A@a A
A8 @@aaAA
The following Rules of Inference are contained in H:
AAB R1 (MP): B
A R2 (Nec): A
A R3 (Sub): , where M,, g w B iff M,, g w B . A B B
A R4 @a A
@ A R5 a , if a does not occur in A A
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@@bA R6 ab, if ab and b does not occur in A @a A
@@b A B Lemma 2.2: ab , where and a does not occur in A or B. @a AB
Proof (Lemma 2.2):
Suppose (1) @@abb A B , and (2) @a AB, where and a does not occur in A or B.
@a b or @b A or B
a b: g a R gb or b: g b A or
M w:, M w B
(since a and b are unbound)
@a A and B
a w: g a Rw and wA and M w:, M w B
@@b A B But this is a contradiction, so ab , where and a does not @a AB occur in A or B.
6.3.4 Completeness
Proposition 3.1 (Completeness): Let be any set of sentences in H, and let H be the logic obtained by adding all theorems in to the axioms of H and closing the
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resulting set of sentences under the rules of H. Then H is complete with respect to the class of frames defined by .
Lemma 3.1 (Modified Lindenbaum’s Lemma): If is an -consistent set, then there is a such that is an -maximally consistent set and:
1. at least one of the elements of is a nominal; and
2. for all @a , there is a b such that @a b and @b A.
Proof (Lemma 3.1):
The proof will consist in the description of a method of producing , followed by an argument that possesses the characteristics set forth in Lemma 3.1.
Let AAA1,,, 2 3 be an enumeration of all formulae well-formed in H.
Let a1,,, a 2 a 3 be an enumeration of all nominals.
Define n as follows:
0 H aii, where a is a nominal not in H ; and, for n1
nn11 if AHn is inconsistent; nn11 AAHAB if is consistent and is not of the form @ ; or n n n n a nn11 A,@ a ,@ B if A is H consistent and A is of the form @ B , n a m am n n a n1 where am is the first nominal that does not occur in or An . Let m . m0
Lemma 3.1.1: m is -consistent, for any m 0.
Proof (Proposition 3.1.1): Proof by induction.
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[Base Step] For m 0: Let ai be a nominal not in H . Suppose Ha i is
-inconsistent. Then there is a set of formulae, AAAAH1,,,, 2 3 k such that
a A A A A . By R4, @ a A A A A H i 1 2 3 k H ai i 1 2 3 k and by A3, A1 (tautologies), and R1 (MP), @@a A A A A . H aii i a 1 2 3 k
By A5, @ a and by R1 (MP), @ AAAA . Finally, by R5, H ai i H ai 1 2 3 k
AAAA . But , and is consistent. By Hk 1 2 3 contradiction, the assumption fails, so is -consistent.
[Inductive Step] Suppose m1 is -consistent. Prove m is -consistent.
This will need to cover three cases.
m1 Case 1: AHm is inconsistent .
If , then mm 1 . Since, by hypothesis, is -consistent, is -consistent.
m1 Case 2: AHABm is consistent and m is not of the form @ a .
If , then
mm1 Am . So, is -consistent.
m1 Case 3: AHABm is consistent and m is of the form @ a .
If , then
mm 1 A,@,@ a B , where a is the first nominal not to occur in or m a k ak k
Am . @amBA and is substituted for clearer explication. Suppose
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mm 1 A,@,@ a B is H -inconsistent. Then there is a set of formulae m a k ak
m1 AAAA1,,,, 2 3 k such that
@@@B a B A A A A . By A1 (PL) and Lemma H a a k ak 1 2 3 k
2.2, @ BAAAA . But, since, by hypothesis, H a 1 2 3 k
m1 @ BH is consistent , @ BAAAA . a H a1 2 3 k
This concludes the proof of Lemma 3.1.1: that m is -consistent, for any m 0.
Lemma 3.1.2: is -consistent.
Proof (Lemma 3.1.2):
Since m , if were -inconsistent, there would be a set of formulae m0
AAAA,,,, such that AAAA . If 1 2 3 k Hk 1 2 3
n , then there is some n such that AAAA1,,,, 2 3 k . But then
n would be -inconsistent, which is contrary to Lemma 3.1.1. So, is - consistent.
Lemma 3.1.3: is maximally -consistent.
Proof (Lemma 3.1.3):
Assume is not maximally -consistent. Then there is some formulae Aj
such that Aj and Aj is -consistent. If , then it was not added at
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j1 stage j of the construction of . So, Aj is H -inconsistent. But, since
j1 AAjj , is -inconsistent. This contradicts Lemma 3.1.2, so is maximally -consistent.
Because part (a) of Lemma 3.1 is satisfied by the n 0 case of the definition of
n and part (b) is satisfied by the third part of the n 1 case of the definition of , this concludes the proof of Lemma 3.1.
HH Lemma 3.2 (Truth Lemma): M,,@ g a Aa A , for all formulae,
A, and nominals, a, where MWRSHHHH ,, is a canonical model of
MWRS ,, as described above in O1 – O7 and F1 – F10, such that, for all formulae,
A, and nominals a and b:
a b:@a b
WH a: a
H R a,:@ b a b
H S p a:@a p
gH a a
HH Proof (Lemma 3.2 – Truth Lemma): M,,@ g a Aa A , for all formulae, A and nominals, a.
The proof will be by induction on the complexity of A.
Base Cases:
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Ap
MHH,, g a p a SH p
@a p
Ab
MHH,, g a b gH b a
ba Since gH b b
ba Since bb
@a b
HH Inductive Hypothesis: Assume M,,@ g a Ba B for all formulae,
B, where B is a subsentence of A.
Inductive Cases:
AB
MHH,, g a B MHH,, g a B
@a B (IH)
@a B ( is a Maximal Complete Set [MCS])
@a B
B ABB4 :@aa @ ; closed under MP A
ABC
MHH,, g a B C MHH,, g a B and MHH,, g a C
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@a B and @a C (IH)
@a BC ( is MCS – closed under PL)
ABC
MHH,, g a B C MHH,, g a B or MHH,, g a C
or (IH)
@a BC ( is MCS – closed under PL)
ABC
MHH,, g a B C MHH,, g a B or
@a B or (IH and result for AB )
@a BC ( is MCS – closed under PL)
AB @a
HH HHH M,,@ g aa B M,, g g a B
gH a a
(IH)
AB
MHH,, g a B b such that a RH b and MHH,, g b B
H b such that @a b and (Def‟n R )
and @b B (IH)
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@ B ( is MCS and Con @b ,@ B ,@ B by Lemma a H a b a
3.1)
@,@,@ab b B a B , for some b. (Lemma 3.1)
a RH b (Lemma 2.1) and MHH,, g b B (IH)
MHH,, g a B
AB
HH LR Let M,, g a B and suppose @a B . Then b if
H then . By the definition of R , for all b, if @a b then
. By the inductive hypothesis, for all b, if then @b B .
Since, by hypothesis, , and is a MCS set closed under R1 through R6,
@a B . By A4 and BB , @a B . So, for all b such that
, @,@,@aB a b b B . But, by Lemma 2.1, for all b such that
, Con @B ,@ b ,@ B , so for all b such that , H a a b
Con @,@,@B b B . But since is a MCS set, it cannot have an H a a b
inconsistent subset. By this contradiction, @a B
LR Let and consider any b such that . Suppose
@b B . Then, since is a MCS set closed under R1 through R10, @b B .
Since , by A4 and , @a B . So, for all b such that
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@a b , @,@,@a B a b b B . By Lemma 2.1, for all b such that
, Con @B ,@ b ,@ B , so for all b such that , H a a b
Con @,@,@ B b B . But since is a MCS set, it cannot have an H a a b
inconsistent subset. By this contradiction, @b B , for all b such that .
So, for all b such that , @,@,@aB a b b B .
By the definition of RH , a RH b , for all b such that . By the inductive hypothesis, MHH,, g b B , for all b such that . So,
MHH,, g a B
This concludes the proof of Lemma 3.2.
Lemma 3.3: If is an H -consistent set, then there is an M such that
Mw, .
Proof (Lemma 3.3): Since is an -consistent set, by Lemma 3.1, , where is a MCS set. In Lemma 3.2, canonical model M H is defined such that, for all A, MHH,, g A.
This completes the proof of Proposition 3.1 (Completeness).
6.3.5 Soundness
Proposition 4.1 (Soundness): If H A, then A is valid on every H-frame.
Lemma 4.1: Each of A2 through A8 is valid on every H-frame.
Proof (Lemma 4.1):
A2: ABAB
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Suppose for some w, a, and g in a model, M: (i) M,, g w A B and (ii)
M,, g w A B . By (i), F8 and F7, v: wRv , M,, g v A or M,, g v B . By (ii),
F7 and F8, u: wRu , M,, g u A and M,, g u B . But this is a contradiction, so there is no w, a, and g in a model, M such that M,, g w A B A B , so
ABAB .
A3: @@@a ABAB a a
Suppose for some w, a, and g in a model, M: (i) M,,@ g wa A B and (ii)
M,,@@ g w aa A B. By (i), F10 and F7, M,, g g a A or M,, g g a B . By
(ii), F7 and F10, M,, g g a A and M,, g g a B . But this is a contradiction, so there is no w, a, and g in a model, M such that
M,, g w @@@a ABAB a a , so @@@a ABAB a a .
A4: @@aaAA
LR Suppose for some w, a, and g in a model, M: (i) M,,@ g wa A and (ii)
M,, g w @ A. By (i) and F10, . By (ii), F4, F10 and (again) F4,
. But this is a contradiction, so there is no w, a, and g in a model, M such
that M,, g w @@aaAA , so @@aaAA .
LR Suppose for some w, a, and g in a model, M: (i)
M,,@ g wa Aand (ii) M,, g w @ A. By (i) F4, F10 and F4, . By
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(ii), F4 and F10, M,, g g a A. But this is a contradiction, so there is no w, a, and g in
a model, M such that M,, g w @@aa AA , so @@aa AA .
Because @@aaAA and , @@aaAA .
A5: @a a
Suppose for some w, a, and g in a model, M: M,, g w @a a . Then, by F4, F10 and F4, M,, g g a a . But, by F2 M,, g g a a . But this is a contradiction, so there
is no w, a, and g in a model, M such that M,, g w @a a , so @a a .
A6: @@@a bAA b
Suppose for some w, a, and g in a model, M: (i) M,,@@ g wab A and
M,, g w @b A. By (i) and F10, M,,@ g g a b A and, by F10 again, M,, g g b A.
By (ii), F4 and F10, M,, g g b A . But this is a contradiction, so there is no w, a, and g
in a model, M such that M,, g w @@@a bAA b , so @@@a bAA b .
A7: a A@a A
Suppose for some w, a, and g in a model, M: (i) M,, g w a and either (iia)
M,, g w A and M,, g w @a A or (iib) M,, g w A and M,,@ g wa A . By (i) and w g a. Then, if (iia), by F4 and F10, M,, g g a A and M,, g g a A . This is a contradiction, so (iib) must obtain. If (iib), by F4 and F10, and
. But this, too, is a contradiction. So, so there is no w, a, and g in a
model, M such that M,,@ g w a Aa A, so a A@a A.
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A8: @@aaAA
Suppose for some w, a, and g in a model, M: (i) M,,@ g w a A and (ii)
M,,@ g w a A. By (i) and F9, v: wRv and M,,@ g va A, and then by F10,
M,, g g a A . By (ii), F4, F10 and F4, M,, g g a A . But this is a contradiction, so
there is no w, a, and g in a model, M such that M,,@@ g w aa A A, so
@@aaAA.
This concludes to proof of Lemma 4.1.
Lemma 4.2: Rules R1 through R6 preserve validity in H.
AAB R1: B
Suppose (i) A and AB and (ii) B . By (ii), there are some w, a, and g in a model, M such that M,, g w B . By (i), M,, g w A B and M,, g w A. Since
, by F7, M,, g w A or M,, g w B . But the first disjunct contradicts
(i) and the second disjunct contradicts (ii). So, if and , there are no w, a, and g in a model, M such that and, so, B .
A R2: A
Suppose (i) and (ii) A. By (ii), there are some w, a, and g in a model, M such that M,, g w A and by F8, such that M,, g v A . But, by (i)
M,, g v A . So, there are no w, a, and g in a model, M such that and, so,
.
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A R3: , where M,, g w B iff M,, g w B . A B B
We prove this rule by induction on the complexity of A. Suppose Ap . Then the proof is trivial, since AA B . Assume that, for all sentences shorter than A, the B rule of replacement holds.
Suppose ACD and A. By the inductive hypothesis, C iff C B B and D iff D B . By F5, since CD , B and . But, since iff B
and iff , and . By F5 again,
CDBB . So, the rule holds when A is a conjunction. BB
The rule also holds when A is a negation, disjunction, or conditional. The respective proofs are parallel [with the use of F4, F6, and F7] to that where A is a conjunction and, in the interest of brevity, will be omitted here.
Suppose AC and . By the inductive hypothesis, iff . By
R2, If then C and if then C B . But, since, iff B
, if then . So, the rule holds when A is a necessitation.
The rule also holds when A is a modal sentence with a „@‟ operator. The proof is parallel [with the use of R4] that where A is a necessitation and, in the interest of brevity, will be omitted here. Similarly, the rule holds when A is a modal sentence with a „ ‟
174 operator. The proof in this case uses F9 to show that C iff C . The proof then simply invokes the results form negation and necessitation.
A R4: @a A
Suppose (i) A and (ii) @a A . By (ii) F4, F10 and F4, there are some w, a, and g in a model, M such that g a w and M,, g g a A . But by (i),
M,, g g a A . So, there are no w, a, and g in a model, M such that and
and, so, @a A .
@ A R5: a , if a does not occur in A A
Suppose , if a does not occur in A. Then, since a is an unbound variable, for any w, a, and g in any model, M, . But this is equivalent to .
@@bA R6: ab, if ab and b does not occur in A @a A
Suppose (i) @@abbA, where and b does not occur in A, and (ii)
@a A . Because a and b are unbound, by (i), F7, F10 and Lemma 2.1, for any
a, b N , either g a, g b R or @b A . By (ii), F4, F10 and F8, v: g a Rv and
M,, g v A . Without loss of generality, let v g b . Then, g a, g b R which by (i) implies . But then, in particular, M,, g v A , which would represent a contradiction. So, if , where and b does not occur in A, then
@a A .
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This concludes the proof of Lemma 4.2.
Definition 4.1: Consider AAAA1,,,, 2 3 n to be a finite sequence of formulae in
H, such that for each term, Ai , either is an axiom of H or follows from some prior
subsequence of terms by one of the rules of inference of H. Then, HnA .
Proof (Proposition 4.1):
This proof will proceed by induction.
Base Case: A is a single-term sequence. By definition 4.1, A is an axiom and, by
Lemma 4.1, A is valid on every H-frame.
Inductive Hypothesis: Suppose LAAAA 1,,,, 2 3 n is a finite sequence as
described in Definition 4.1 and An is valid on every H-frame.
Inductive Case: Let AAAAA1,,,,, 2 3 n be the sequence formed by appending A to , where A follows from some prior subsequence of L by one of the rules of inference of H. By Lemma 4.2, A is valid on every H-frame.
This conludes the proof of Proposition 4.1 (Soundness).
6.4 Quantification
The expressive capacity of hybrid logic can be increased through the introduction of local and global quantifiers that range over nominals.
The local quantifier ranges just over the immediate formula in which it is imbedded. Particularly, it „fixes‟ the present world of evaluation. Formally, if
A x B , then M,, g w A iff M,, g w B , where gg and w g x . Informally, x
xB is forced at a world, w, just in case B is forced at w. This is, of course, really only
176 interesting when B contains unbound occurrences of x. Suppose, for example, that
Bb@x . Then B is forced at w just in case the world at which nominal b is true is accessible to w.
This local quantifier facilitates the creation of formulae that contain particular information concerning the immediate world of evaluation. For example, if
M,, g w x x , then wRw.
The quantifier effectively attaches the current point of evaluation to a state variable. The satisfaction operator @ can work with the quantifier by shifting the point of evaluation. For example, consider an „until‟ operator, UAB , , that is true at an instant, t, if there is a future instant, t , where A holds and such that B holds at all instants between t and . For instance, let A be “the needle is found” and B be “the needle is lost.” is true at an instant, , if, at , the needle is lost at and then it is found at
.
This can be expressed in H as follows: M,,@ g t x y B x y A .
This first assigns the value t to the state variable x and to y, where is the instant at which B obtains (the needle is found). It then returns to t and finds every successor of t that precedes and states that A (the needle is lost) obtains at those instants.
The global quantifiers and range over all wW , over all the worlds in the frame.
If A xB , then M,, g w A iff there is a g such that gg and x
M,, g w A.
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If A xB , then M,, g w A iff, for all g such that gg , M,, g w A . x
These global quantifiers facilitate the creation of formulae that contain
information concerning the accessibility relation. For example, xx @x states that
there is a world that is not accessible to itself. x y@x y states that the model is serial.
The quantifier can also work with the global quantifiers.
x y@@xy y A states that A is forced at all worlds accessible from the world at which the formulae is being evaluated.
This completes the development of the basic hybrid logic. In the following chapter, this logic will provide the basis for our basic temporal hybrid logic.
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Chapter Seven: Temporal Hybrid Logic
7.1 Introduction
We are now in a position to bring together Prior‟s tense logic and U-calculus with a modern hybrid logic. Before examining how this can be done, however, it is important to pause and ask why it is important.
In his rejection of Logical Formalism, the position that logic is about abstract constructions, Prior emphasizes the importance of logical formalizations. “It is necessary to pay attention to the structure of our language in order to expose and eliminate philosophical „pseudo-problems,‟ and in order to distinguish real objects from mere
„logical constructions.‟” [Prior 1996, p. 45] Our understanding of anything is expressed in language; by formalizing our language we are formalizing the expressions of our thought. Logics, in this sense, are concerned with objects and occurrences in the world, but the direct objects of logic are linguistic objects. An appropriate logic formalizes our language, which is about objects and occurrences in the world. Importantly, though, the logic formalizes the expressions of our thought and not the objects of our thought. It allows us to test the structure and consistency of our temporal theories.
In Chapters Two, Three, and Four, we looked at the characteristics of various theories of experienced time and physical time. Following Prior, we will now develop a logic with which to formalize the language we use in expressing these characteristics of time. I will part company, somewhat, with Prior in that I will propose a single logical structure for both tensed and tenseless theories. Though mediated through language, the
179 logic is still fundamentally about objects and occurrences in the world. To be „about the world,‟ a logic does not need to possess the characteristics of the world. A logic is a model, a metaphor. The relationship between the logic and the objects and occurrences of the world is not one of direct correspondence. Rather, the way in which the logic functions bears a metaphorical connection to the way in which the world functions. The results we obtain in the logic will correspond to particular occurrences in the world, without directly describing those occurrences. Because of this, a single logic can be seen to model, for example, both tensed and tenseless theories.
7.2 The Basic Temporal Hybrid Logic
As we saw with the U-calculus, hybrid logic is well-suited to the representation of sentences that contain non-tensed B-theory temporal information. A basic temporal modal logic might be „hybridized‟ by the addition of nominals, satisfaction operators, local quantifiers, and global quantifiers.
Consider a basic temporal modal logic, with the following features:
MWRS ,, , where:
W is a denumerable set of worlds, or „instants,‟ such that W and
W ,,,,,, wi2 w i 1 w i w i 1 w i 2 .
RW 2 is a binary accessibility relation, read “is prior to,” such that,
w w w W :
R is irreflexive, that is: w w R w (no instant is prior to itself);
R is asymmetric, that is: w w w Rw w R w (if a is prior to b, then b is not prior to a); and
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R is transitive, that is: ww w wRw wRw wRw (if a is prior to b and b is prior to c, then a is prior to c).
S p P W , where P W is the power set of W.
Modal tense operators, F and P, are introduced with the following forcing conditions:
MwPA, wwRw and Mw , A
MwFA, wwRw and Mw , A
Intuitively, FA is read, „it will be the case that A,‟ and PA is read, „it was the case that A.‟
The basic temporal logic is limited by the inability of its formulae to make reference to any particular instant. For example, it cannot formalize a statement such as
“The program failed at midnight on January 1, 2000.”
In hybrid logic, “The program fails at midnight on January 1, 2000.” Can be
formalized as @a A , where a midnight on January 1, 2000 and A is „the program fails.‟ The verb tense in the original statement: “The program failed at midnight on
January 1, 2000.” can be formalized generally as xA@a .
If the basic temporal logic is hybridized, “The program failed at midnight on
January 1, 2000” can be formalized as P a A .
The hybridized temporal logic accommodates the formalization of sentences with verbs in all tenses, as illustrated in Table 1:
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Tense Expression Formalization
Past in the Past at a it was the case that it had been A. P a PA
Simple Past at a it was the case that A. P a A
Future in the Past at a it was the case that it would be that A. P a FA
Perfect at a it would have been the case that A. a PA
Present at a it is the case that A. aA
Prospective at a it is the case that it will be A. a FA
Future Perfect at a it will have been the case that A. F a PA
Future at a it will be the case that A. F a A
Future in the Future at a it will be the case that it will be the case that A. F a FA
Table 1
The basic temporal logic can be extended with the addition of two further modal tense operators. H and G are introduced with the following forcing conditions:
MwHA, wwRw Mw , A
MwGA, wwRw Mw , A
Intuitively, HA is read, „it has always been the case that A’, and GA is read, „it will always be the case that A‟. These operators allow for the formalization of some natural language sentences that contain continuous tense verbs.
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Hybridization of this extension of the basic temporal logic allows the formalization of additional classes of sentences that relate to truth over a span of time.
Beginning at midnight of January 1, 2000, the program will always fail. Can be expressed
by the formula @a A GA.
Formulae in hybridized temporal logic can determine a class of frames that describe particular metaphysical notions of time. Some of these formulae, together with a description of the class of frames they determine and a sentence in the basic temporal logic valid in each such frame are presented in Table 2:
HL Formula [TL Formula] Notion of Time
x y@y x HA PA Unbound in past
x y@x y GA FA Unbound in future
x y@@xz y z x HA PA GA FA Unbound in past and future
xP@x There is a first instant
xF@x There is a last instant
x y@@@x y z x z z y FA FFA Dense
x y z@@@x z z y x y FFA FA Transitive
x y z@@@@@x z y z x y x y y x Linear in past FPA PA A FA
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x y z@@@@@z x z y x y x y y x Linear in future PFA PA A FA
Table 2
7.3 Fine-Tuning Temporal Hybrid Logic
7.3.1 Relativity
Prior observed that we can interpret a sentence „it is [really] the case that p‟ as „it appears from this point of view that p.‟ [Prior 1968a, p. 134] We can follow this intuition and interpret out temporal hybrid logic as being a logic relative to a particular frame of reference.
Recall our earlier example the twins Castor and Pollux. The twins were virtually inseparable until their twentieth birthdays, when Pollux flew away for a journey aboard a very fast spacecraft. When they were reunited, three minutes had elapsed in Pollux‟s frame of reference, while three years had passed in Castor‟s frame of reference. Consider the following interpretation: a the instant Pollux leaves, b the instant the twins are reunited, p the proposition „Pollux was born twenty-three years ago.‟ The sentence
@b p , „it is the case at the instant of their reunion that Pollux was born twenty-three years ago,‟ would be false from Pollux‟s frame of reference, while from Castor‟s, it would be true. In an application in which it may be unclear from whose frame of reference a sentence is stated, an extra-logical denotation can be included. For example,
Pollux Castor @b p if false, while @b p is true.
While our logic will assume a particular, or privileged, perspective or frame of reference, we want to be able to express sentences that contain elements from multiple
184 frames of reference. We can accomplish this be making the following additions and adjustments to the semantics and syntax.
To the semantics, we add a denumerable set of frames of reference and a function from instants in the non-privileged frame to instants in the privileged frame.
FoR i1,,, i 2 i 3 is a denumerable set of frames of reference;
j j j j W w1,,, w 2 w 3 is a denumerable set of denumerable sets of instants, one
for each ij FoR ; and
:WWj is a function that assigns one element of W to each element of each set in W j . Function serves to translate instants from non-privileged frames of reference into instants in the privileged frame of reference. The Lorentz equations provide us with the assurance that such a function exists.
The syntax is only subtly augmented by the addition of formulae of the form
a j , where a j is a nominal true at a unique instant in frame of reference .
Formulae of this form are nominals.
The function provides a translation of instants from a second frame of reference into instants of the „privileged‟ frame of reference. In our Castor and Pollux example, if we set Castor‟s as the privileged frame of reference, function translates instants from
Pollux‟s frame of reference into instants in Castor‟s. We will use the convention of placing a subscript to the left of a nominal to indicate that it is true at an instant in the non-privileged frame, and of placing a subscript to the right of a nominal to indicate that it is true at an instant in the privileged frame. For example, if Pollux was in frame of
185
reference i1 while he travelled, bb11 , where b1 is a nominal true at and only at the instant in Castor‟s frame of reference that corresponds to the instant in Pollux‟s frame of
reference at which the nominal 1b is uniquely true. Of course, Pollux is not a frame of reference, but he occupies a frame of reference and it is this frame of reference that is being referenced as the argument of the function. The identity can be thought of as stating that „what was three minutes in Pollux‟s frame of reference was three years in Castor‟s.‟ With Castor‟s the privileged frame, we can write @ p [equivalently, b1
@ p ] to read „it is the case at the instant b that not-p.‟ It is the case at the instant 1b 1 three minutes after the instant of his twentieth birthday [instant ] that Pollux was not born twenty-three years ago.
The function identifies the instant in the privileged frame that corresponds to the perspective of the other frame of reference. It allows us to shift frames of reference.
In our example, it allows us to shift to Pollux‟s frame of reference. The sentence
@ p can be thought of as expressing „from Pollux‟s point of view at the time of the b1 reunion, he was not born twenty-three years ago.‟
Notice that the instants, when seen as representing the fourth coordinate of space- time points, are relative to a particular frame of reference. Each frame of reference has its own scale by which the passage from instant to instant is measured, so the temporal coordinate of each space-time points is relative to a frame of reference.
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7.3.2 Now
Prior‟s tense logic is an A series logic. Sentences are read from the perspective of
Now; for example, FA PB is read casually as „it will be the case that A and it was the case that B,‟ but more accurately as „it is Now the case that, both, it will be the case that A and it was the case that B.‟ The „it is Now the case that … ,‟ which precedes every sentence is normally left unsaid, but is always meant to be implied. While Now is an implied part of every sentence in tense logic, it cannot be expressed in the U-calculus.
Our temporal hybrid logic has the capacity to express Now. Using the local
quantifier „ ‟, we write x@x A B to read „at the instant of evaluation, it is the case that, both, it is the case that A and it is the case that B.‟ The instant of evaluation serves as Now, in the way that the instant of expression serves as Now in tense logic. The local quantifier allows the instant of evaluation to be brought explicitly into sentences of the
language. Under the appropriate interpretation, the sentence x@x a A reads „it is
Now noon on March 1, 2010 and the meeting is starting.‟ This sentence will be true when, and only when, it is evaluated at noon on March 1, 2010. The sentence
x@x a FA reads „it is Now earlier than noon on March 1, 2010 and it will be the case that the meeting is starting;‟ that is, „the meeting has not started yet.‟ Similarly,
x@a x PA, „the meeting has already started.‟
We can express Now on its own as xx . This sentence can be read as „the value of the instant of evaluation is true.‟ That value, x, will be true only when it expresses the current instant; that is, when x is Now.
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The Now captured by this temporal hybrid logic is the instantaneous Now. To capture the sense of Now suggested by the specious present, a „specious simultaneity‟ function could be introduced, which would hold between each instant and a set of proximal instants.
:WWP is a function that assigns an element of the power set of W to each element of W. Intuitively, assigns to each instant the set of instants that represent its specious present. Notice that, for James, the value of w will include instants earlier than w and later than w, while for Reid, will include only values earlier than w.
We introduce the symbol „ ‟ as a specious present modal operator, and set the forcing condition as follows:
If A , then M,, g w A iff w: w w and M,, g w B ; that is, B is forced at an instant w just in case B is forced at some instant in the set defined by .
The set defined by is just the collection of instants that join w in the specious present.
The relation of „specious simultaneity‟ could then be expressed syntactically like
the accessibility relation expressed by „ ‟ in @a x . The sentence @a b would be read as „it is true at instant a that instant b belongs to the specious present.‟ The sentence
x@a x PA is true if it is the case that it is instant a and it was the case that A, with
the present understood as the specious present. Finally, x@x a PA expresses „the
188 meeting has already started,‟ with Now to be understood as representing the specious present.
For Reid, x y@@@x y x y y x, „for any two instants x and y, if y is part of the specious present with x, then either x is identical with y or y is earlier than x,‟ would be a theorem. Introducing this sentence as an axiom would fix this principle as part of the logic. This would not be a theorem for James; no similar condition is entailed by his notion of specious present.
7.3.3 Time Series
7.3.3.1 A-series and B-series
The instants that form either the A-series or the B-series form a non-strict partially ordered set. Accordingly, the accessibility relation, R, must be irreflexive, asymmetric, and transitive. These can be captured either by conditions on R or by the introduction of a set of appropriate axioms. The conditions on R will entail the axioms, and the axioms will entail the conditions.
Conditions on R:
1. x xRx, irreflexive;
2. x y xRy yRx, asymmetric; and
3. x y z xRy yRz xRz, transitive.
Axioms:
1. xx@x , irreflexive;
2. x y@@xy y x , asymmetric; and
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3. x y x@@@x y y z x z , transitive.
The series may have a first instant or a last instant. The conditions on R for these, respectively, are:
4. x y y R x , there is an instant, x, such that for any instant y, it is not the
case that y is earlier than x; and
5. x y y R x , for any instant, x, there is an instant, y, such it is not the
case that y is earlier than x.
We would introduce the symbols and , respectively, to name the first and last instants. The corresponding axioms, then, are:
6. xx@@x , is earlier than every other instant; and
7. x@@xx , Every other instant is earlier than .
In addition to being ordered, the instants in the series may form a denumerable dense set. Again, this can be set in the logic through a condition on R or the introduction of an axiom.
8. x y xRy z xRz zRy , for any two instants x and y such that x is
prior to y, there is an instant, z, such that x is prior to z and z is prior to y,
i.e. z is between x and y; or
9. x y@@@x y z x z z y .
If the instants in the series are not dense, then the following condition and axioms will apply:
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10. x y xRy y xRy x xRz zRy , for any instant, x, if x has a
successor, then there is an instant, y, such that x is related to y and there is
no instant z such that x is related to z and z is related to y.
11. x @@@@x y x y z x z z y , if x is not the last
instant, then there is an instant, y, such that and x is earlier than y, and
there are no other instants between x and y.
7.3.3.2 The A-series
In the A-series, it is always the case that exactly one instant has the A-relation
„present,‟ while all others have either the A-relation „past‟ or the A-relation „future.‟ Of course, these values change as Now changes. The sentence
x x y@@ x Py y Fy captures this quality of the A-series. It can yx be read, „the instant of evaluation is such that it is present, and any other instant, y, is such that, if y is earlier that x then y is past and if x is earlier than y then y is in the future.‟
With its capacity to describe Now, the temporal hybrid logic can model an A- theory of time. This distinguishes it from the U-calculus. The nominals and satisfaction operator allow it to model a B-theory of time. Because of this, it is a logic that can model both experienced time and physical time.
7.3.4 Ontologies of Time
7.3.4.1 Eternalism
Eternalism is the view that all of what we consider to be the past, present and future is equally real. Eternalism is the ontology of B-theories of time. There is nothing
191 inherent in Eternalism that suggests whether or not the elements in the time series are dense, or whether the series has a first or last term.
Because Eternalism is not an A-theory, though, the sentence xx will not be a theorem.
7.3.4.2 Possibilism and Presentism
Possibilism and Presentism represent the views that objects and events in the future are not real and that, to the extent that objects and events in the past can be considered real, they are, nevertheless, real in a slightly different sense than present objects and events. They are the ontologies of A-theories of time. Because of this, they contain the theorems xx and x x y@@ x Py y Fy , in yx additional to all that any B-theory would contain.
Beyond those that arise from Eternalism being the ontology of B-theories and
Possibilism and Presentism being the ontologies of A-theories, the differences between these ontologies are not apparent in a propositional logic. The other significant difference is found in the domains of objects over which the predicate variables range. In propositional logics, this difference will be accounted for by the function S that assigns sets of instants to propositions. The set of instants assigned to a proposition is interpreted as the set of instants at which that proposition is true. It is in the result of this assignment that we would also find differences under the respective ontologies.
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7.3.5 Logics of Experienced Time and Logics of Physical Time
7.3.5.1 Experienced Time
Our temporal hybrid logic of experienced time is built from the relativized temporal hybrid logic. Experienced time is tensed and includes a Reid-like specious present in which the experience of Now combines the present instant and the set of immediately preceding instants.
Recall that the relativized temporal hybrid logic, itself, contains the basic temporal hybrid logic with the addition of the sets frame of reference and the corresponding sets W i and the function :WWj . Four axioms are added to the relativized temporal hybrid logic to create the experienced time hybrid logic.
Because experienced time is tensed, the sentences xx and
x x y@@ x Py y Fy will be theorems in any logic of yx experienced time. To extend the logic to include the experience of the specious present, we need to add the function :WWP and the symbol „ .‟ With respect to density of experienced time, there is not the same certainty as there is with tense. We often speak in terms of „the next instant,‟ but this is really just metaphorical. On the other hand, „infinitely divisible‟ is arguably not a property that can be directly apprehended by a human brain; it is really a phrase used as much metaphorically as is
„the next instant.‟ While allowing that there may be some question, there is certainly reason to allow that experienced time is not dense and, so,
x @@@@x y x y z x z z y is also a theorem of experienced time.
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Experienced time has the same beginning and end as does the experiencer. This is true whether the frame of reference of our logic is occupied by a person or by an object or by just a point. The world-line will have a beginning and an end. This is clearly true for all of our logic in which we are examining statements from a human perspective within a frame of reference. Accordingly, in experienced time we find both schema
@ a and schema @a to be theorems. This raises a difficulty, though, for sentences that contain nominals that are true outside of the range of instants included in a particular
frame of reference. For example, suppose W,,,, w12 w is the set of instants frame of reference occupied by Pat. Suppose, further, that Pat was born at midnight on
May 1, 1980. Finally, suppose that Pat‟s mother, Quinella, coincidentally, was born at midnight on May 1, 1950 and that Pat‟s daughter, Ophelia, was born at midnight on May
1, 2010. Pat does not have a nominal that is true on May 1, 1950, since that instant is not contained in his W, though, since he is alive for the birth of his granddaughter, he does have a nominal true at the instant of her birth. Consider the following syllogism:
1. Ophelia was born on May 1, 2010;
2. Pat was born on May 1, 1980;
3. Therefore, Ophelia was born after Pat
This is an argument that our logic should find to be valid. With the obvious substitutions, we can symbolize this argument as:
1. @b o
2. @ p
3. @ Fo
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Sentence (3) clearly follows from (1) and (2), since @ b . But, consider the following putative argument:
1. Ophelia was born on May 1, 2010;
2. Pat was born on May 1, 1980;
3. Quinella was born on May 1, 1950;
4. Therefore, Quinella was born before Pat and Ophelia was born after Pat
In this case, sentences (3) and (4) cannot be symbolized in Pat‟s frame of reference, since there is no ordinal in Pat‟s frame of reference that corresponds to instant at which Quinelle was born. Even with the function, introduced earlier, there is no
correspondence. If we set iO , iP , and iQ to be, respectively, the frames of reference occupied by Ophelia, Pat, and Quinella, we can see, first that, with being the
privileged frame of reference, O , the instant in Pat‟s frame of reference
corresponding to the first instant in Ophelia‟s, is defined, while Q , the instant in
Pat‟s frame of reference corresponding to the first instant in Quinella‟s, is not. Because of this we cannot, as we were able to do in section 7.3.1, translate all sentences and sub- sentences from other frames of reference into the privileged frame of reference. One approach we can clearly take is to shift the privileged frame of reference. Because both the instants of Ophelia‟s birth and Pat‟s birth are contained in Quinella‟s frame of reference, we can use hers as the privileged frame of reference to write:
1. @ o , where is the instant in Quinella‟s frame of reference O
that corresponds to the first instant in Ophelia‟s frame of reference;
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2. @ p , where is the instant in Quinella‟s frame of reference P P
that corresponds to the first instant in Pat‟s frame of reference;
3. @ q , where is the first instant in Ophelia‟s frame of reference;
4. Therefore, since @ and @ , @ Pq Fo , at the O P P P
instant in Ophelia‟s frame of reference that corresponds to the first instant
in Pat‟s, it is the case that Quinella‟s birth is in the past and Ophelia‟s birth
is in the future.
In this instance, because frames iO , iP , and iQ are so similar, and the events o, p, and q all occur within Quinella‟s lifespan, the use of the function is not strictly
necessary. It does, however, carry a syntactical expediency since P and O refer neatly to particular instants.
A similar manoeuvre can be undertaken even when there is no overlap in frames of reference. Consider „Socrates birth occurred before McTaggart‟s birth and Kripke‟s birth occurred after McTaggart‟s.‟ In this example, there is no overlap26, so we cannot use the frame of reference of any of the three. We can, though, use some other frame of reference that does contain instants corresponding to these three events. In cases such as
this, we can use the Earth‟s frame of reference iE . We then have, relative to the Earth‟s frame of reference, @ s ; @ m; @ k ; @@ ; S M K S MKM therefore, @ Ps Fk . M
26 Socrates dies in 399 BC, McTaggart was born in 1866 and died in 1925, and Kripke was born in 1940.
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The accords fairly well with our intuition concerning events that occur within our lifespan as compared with those that occur outside of our lifespan. There is a clear way in which my experience of the „reality‟ of events occurring within my lifespan differs from my experience of the „reality‟ of those that occur prior to my lifespan. An event such as the First World War is less vivid when viewed from my frame of reference than is the Vietnam War. I was not present at either, but I did have indirect experience of the one that occurred during my lifespan. Even if I had no experience of the event that occurred during my lifespan, say the third birthday party of a cousin I have never met, I do have experience of other events that occurred, casually speaking, at the same time.
The time during which the birthday party occurred is part of my experience, while the time during which the First World War occurred is not.
I create a context for, and clarify my notion of, the First World War partly though my grandfather‟s frame of reference, in which the war occurs during his late-adolescence.
More, it is through my own experience with object and descriptive artefacts that remain.
My context for, and notion of, the Métis Rebellion may be slightly informed by its having occurred during my great-grandfather‟s childhood, but only slightly. It is almost entirely made of my own experience with some object artefacts, but mostly descriptive artefacts in the form of reports and stories, of the rebellion. These artefacts belong to other frames of reference than my own. The greater the distance in time, the less I rely on frames of reference of people or objects from my own experience, and the more I rely on frames of reference like those of a country and, ultimately, of the Earth. Next to the Sun or, perhaps, the Milky Way Galaxy, the Earth is the oldest object in my direct experience.
During my lifespan, its frame of reference is similar to my own, so it is a useful frame of
197 reference for me to shift to when considering temporally-distant terrestrial events. Of course, I would have to look beyond the Earth for a frame to reference when considering cosmic events that predate the Earth.
To review, experienced time is built on the basic temporal hybrid logic with the
addition of the sets FoR i1,,, i 2 i 3 [frames of reference] and corresponding set of sets
j j j j W w1,,, w 2 w 3 [instants for each element of frame of reference] and the function
:WWj . We also include the function :WWP to the semantics and symbol
„ ‟ with the forcing condition If A , then M,, g w A iff w: w w and
M,, g w B . The function is such that, for any instant w, all the elements, w, of
PW will be such that w Rw ; that is, all elements assigned by to w will hold the
„earlier than‟ relation to w. This captures Reid‟s notion of the specious present. In addition, experienced time contains the following theorems:
1. x x y@@ x Py y Fy [tensed] yx
2. x @@@@x y x y z x z z y [non-dense]
3. y x@@yy x , there is a first instant and the nominal is true at
and only at that instant; and
4. y x@@xy y , there is a first instant and the nominal is true at
and only at that instant.
Sentences can be evaluated only at instants that are elements of the set W of instants in the privileged frame of reference. A sentence containing a nominal a, such
198 that neither a nor a is an element of the set W, must be evaluated in an alternate frame of reference whose set of instants does contain a or .
7.3.5.2 Physical Time
Our temporal hybrid logic can model a variety of assumptions concerning physical time. For example, consider a physical time that is non-tensed, relativized, dense, and four-dimensional. Under these assumptions, a logic of physical time can be built from the relativized temporal hybrid logic, as follows.
The relativized temporal hybrid logic, itself, contains the basic temporal hybrid logic with the addition of the set FoR and the corresponding sets W i and the function
:WWj . Four axioms are added to the relativized temporal hybrid logic to create the experienced time hybrid logic.
Because, under these assumptions, physical time is not tensed, the sentences
xx and x x y@@ x Py y Fy will not be theorems in any yx logic of physical time. With no tense, there is no need to include apparatus for the specious present.
On this theory, physical time is dense, so the logic will include as a theorem the
sentence x @@@@x y x y z x z z y . If space-time is continuous, then the time series will be continuous. In this case, the set, W, of instants will not be an uncountable infinite set. Consequently, our completeness, soundness and decidability results will not hold. This does not render the logic useless, but does compromise the assurance of its results.
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Physical time may have a first instant, and last instant, or be infinite in both
directions. If there is a first instant, the set W,,, w12 w will include a first
element, α, and the sentence xx@@x will be a theorem. If there is a final
instant, the set W ,,, wnn1 w will include a last element, ω, and the sentence
x@@xx will be a theorem. If time is infinite in both directions, the set
W ,,,, wn11 w n w n will have no first element or last element and the sentence
x y@@xz y z x is a theorem.
We can use this temporal hybrid logic to model various assumptions with respect to physical time and then use the logic to explore implications of those assumptions.
Without a definitive way of determining a definitive theory of physical time, this logic provides a tool for „test driving‟ various theories. In doing so, we can develop a sense of which characteristics of a theory of time most consistently reflect the way in which the world seems to behave.
7.4 Summary
In this chapter, I have attempted to develop a logic that allows a formalization of the language with which we express our understanding of temporal aspects of the world.
This temporal hybridized logic is a somewhat open structure. Tenets of particular theories of time can be expressed as semantic and syntactic additions to the basic logic.
In this way, the logic was seen to model experienced time; that is, time that is tensed, includes a specious-present, and has a finite time series. Finally, various elements that
200 might be included in a theory of physical time were discussed and an example of a logic based on a particular set of assumptions was produced.
The next step, which is beyond the scope of this work, is to apply various forms of this logic both to prosaic and to problematic examples of temporal situations and temporal reasoning. The logic will serve to help us avoid spurious reasoning and to illustrate some of the consequences of a particular theory of time.
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Chapter Eight: Conclusion
8.1 Experienced Time
We experience the world from a particular perspective. We are a certain size, much larger than atoms and much smaller than galaxies. We perceive the external world through a small collection of senses, each of which is sensitive to a narrow band of sensation. Our brains create a model of the external world from this sense-data, and this model informs our actions. This model has proved through natural selection to be a very successful model in determining our survival. But this model is just that: a model. It is not a description or blue-print of the external world, and there is no assurance that an element of the model will bear any direct resemblance to a corresponding element of the external world. Just as a mathematical model of a storm will not make us wet, the green of the leaf may correspond in an important way to a certain electromagnetic wavelength, but is still a different kind of thing. The model of the storm is successful to the extent that it allows us to predict accurately the storm‟s behaviour. Similarly, our models of the external world are successful to the extent that they allow us to make useful predictions of how the external world behaves; more accurately, appears to behave. A tensed model of time is successful to the extent that it allows us to make useful predictions, whether physical time is tensed or not.
Tense is an undeniable and uneliminable element of our experience of time. This is seen in our language, through simple introspection, and in the different ontological status we grant to objects based on whether their existence is past, present or future. This
202 ontological discrimination based on tense, combined with the direction of causality that we perceive, supports our experience that the past is fixed and the future is open, and that we have some degree of agency in determining what the future will hold. Experienced time is A-time. Now is the experienced instant. The privileged instant is Now because it is the experienced instant; it is not the experienced instant because it is Now.
Besides being tensed, experienced time is three-dimensional and finite. We perceive events as occurring in time. We perceive objects as existing in time. When we speak of Now, we speak of the absolute Now, not our own personal Now. One instant follows the next. Instants are almost inconceivably short, but not infinitely short. There are countless instants from one point in time to another, but there are not infinitely many.
8.2 Physical Time
Four-dimensionalism is consistent with some of the most basic theories of contemporary science. Objects persist as fusions of temporal parts. Minkowski space is a four-dimensional manifold comprised of spatiotemporal points in which there is no absolute Now. There may be individual Nows in each of the frames of reference, but there is no one Now for them all. We speak in terms of objects‟ world lines in relativistic physics to describe their histories, or paths through space-time. Each point in an object‟s world line has a temporal coordinate, and can be considered as a temporal part of the object.
While we can never categorically confirm any conclusion concerning the nature of physical time, or any conclusion concerning the nature of anything in the external world, it is always seductive to look to a theory that accords with our science. But we need to remember that science consists of models of the external world, not naked
203 descriptions. As Prior reminds us, one foundational assumption of general relativity is that time is tenseless. It should come as no more a surprise, then, that a B-theory of time arises from general relativity than that a theory that time is unreal arises from
McTaggart‟s idealism. Borrowing Thompson‟s phrase, our theory of physical time is inextricably linked to our choice among the various „crazy metaphysics.‟
With respect to the basic question of whether physical time is tensed or tenseless, we are left with the choice between an A-theory, with its cost of being unable to satisfactorily explicate the physical Now, and a B-theory, with its cost of being unable to satisfactorily account for the state of affairs that leads to the expression of „Thank goodness, that‟s over!‟ Either theory of physical time leads to an explanation that ends too soon.
Where are we left in our search for a theory of physical time? A B-theory that supports general relativity will prove successful in its explanations and predictions of events in the external world. An A-theory will prove successful in its explanations and predictions of actions in which agency plays a role. Other, more general, metaphysical positions will determine which flavour of theory we choose.
8.3 Logics of Time
I believe that it is Prior‟s position that results in logic provide direct insight into the nature of the real world. My position is that results in logic provide direct insight into our understanding of the real world, and only indirectly apply beyond our thinking. Prior found that the reduction of his B-theory U-calculus to his A-theory tense logic supports for his view that tensed time is metaphysically prior to tenseless time. While I do not
204 find that the metaphysical claim is justified by this reduction, I think it is clear that it does establish the primacy of tense in experienced time.
A sound logic allows us to extend our intuition and to test the consequences of our theories. It helps to ensure the consistency of the tenets of our theories. The temporal hybrid logics developed in Chapter Seven, with the appropriate assumptions, can model both tensed and tenseless theories. It can help us to clarify our theory of experienced time and ensure its consistency, and can also help us to see the consequences of various combinations of assumptions in a theory of physical time.
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References
Carlos Areces and Patrick Blackburn, “Bringing Them All Together,” Journal of Logic and Computation, Volume 11, (2001), 657-669.
Carlos Areces, Patrick Blackburn and Maarten Marx, “Hybrid Logics: Characterization,
Interpolation, and Complexity,” The Journal of Symbolic Logic, Volume 66, (2001), 977-
1010.
Frank Arntzenius, “Time Travel: Double Your Fun” Philosophy Compass, Vol. 1, No. 6,
(October, 2006), 599 – 619.
Patrick Blackburn and Miroslava Tzakova, “Hybridizing Concept Languages,” Annals of
Mathematics and Artificial Intelligence, Volume 24, (1998), 23-49.
Patrick Blackburn, “Arthur Prior and Hybrid Logic,” Synthese, Vol. 150, (2006), 329 –
372.
Patrick Blackburn and B. ten Cate, “Pure Extensions, Proof Rules, and Hybrid
Axiomatics,” Studia Logica, Volume 84, (2006), 277-322.
Arthur W. Burks, “Icon, Index and Symbol” Philosophy and Phenomenological
Research. Vol. 9, No. 4 (June, 1949), 673 – 689.
Jonas Dagys, “J. McTaggart and H. Mellor on Time,” Problemos, Vol. 73 (2008), 115 –
121.
206
Barry Dainton, “The Experience of Time and Change” Philosophy Compass, Vol. 3, No.
4, (June, 2008), 619 – 638.
Robert DiSalle, “Space and Time: Inertial Frames,” The Stanford Encyclopaedia of
Philosophy, (Winter 2010 Edition).
Michael Dummett, “A Defense of McTaggart‟s Proof of the Unreality of Time” The
Philosophical Review, Vol. 69, No. 4, (October 1960), 497 – 504.
Stephen W. Hawking, A Brief History of Time, (Bantam Doubleday, 1988).
Katherine Hawley, “Theodore Sider. Four Dimensionalism [Review]” Nous, Vol. 40, No.
2, (2006), 380 – 393.
William James, The Principles of Psychology, Volume One, (Dover, 1950).
Ali Akhtar Kazmi, “Parthood and Persistence,” Canadian Journal of Philosophy, Sup.
Vol. 16, (1990), 227 – 250.
Genevieve Lloyd, “Tense and Predication: A Response to Dummett” Mind, Vol. 86, No.
343, (July 1977), 433 – 438.
E. J. Lowe, “The Indexical Fallacy in McTaggart‟s Proof of the Unreality of Time,”
Mind, Vol. 96, No. 381, (January 1987), 62 – 70.
Detlef Laugwitz, “Definite Values of Infinite Sums: Aspects of the Foundations of
Infinitesimal Analysis Around 1820,” Archive for History of Exact Science, Vol. 39, No.
7, (September 1989), 195 – 245.
207
Henry S. Leonard and Nelson Goodman, “The Calculus of Individuals and Its Uses,” The
Journal of Symbolic Logic, Vol. 5, No. 2, (June 1940), 45 – 55.
Murray Macbeath, “Dummett‟s Second Order Indexicals,” Mind, Vol. 97, No. 385,
(January 1988), 113 – 116.
Matthew McGrath, “Temporal Parts” Philosophy Compass, Vol. 2, No. 5, (August,
2007), 730 – 748.
John Ellis McTaggart, “The Unreality of Time,” Mind, Vol. 17, No. 68, (October 1908),
457 – 474.
John Ellis McTaggart, “The Relation of Time and Eternity,” Mind, Vol. 18, No. 71, (July
1909), 343 – 362.
John Ellis McTaggart, The Nature of Existence, Volume 2, (Cambridge at the University
Press, 1927).
D. H. Mellor, “The Self from Time to Time,” Analysis, Vol. 40, No. 1, (January 1980),
59 – 62.
D. H. Mellor, Real Time II, (Taylor & Francis e-Library, 2002).
John D. Norton, “The Hole Argument,” The Stanford Encyclopaedia of Philosophy,
(Winter 2010 Edition).
L. Nathan Oaklander, The Ontology of Time, (Prometheus Books, 2004).
208
John Perry, “Indexicals and Demonstratives” in A Companion to the Philosophy of
Language, Bob Hale and Crispin Wright, editors. Blackwell Publishers (1999), 586 –
612.
Robin Le Poidevin and D. H. Mellor, “Time, Change and the „Indexical Fallacy‟,” Mind,
Vol. 96, No. 384, (October 1987), 534 – 538.
Huw Price, “Time Symmetry in Microphysics,” Philosophy of Science, Vol. 64 (1997),
S235 – S244.
Arthur Prior, “Three-Valued Logic and Future Contingents,” The Philosophical
Quarterly, Vol. 3, No. 13, (October, 1953), 317 – 326.
Arthur Prior, “Diodoran Modalities,” The Philosophical Quarterly, Vol. 5, No. 20, (July,
1955), 205 – 213.
Arthur Prior, “Thank Goodness That‟s Over,” The Journal of the Royal Institute of
Philosophy, Vol. 34 (1959), 12 – 17.
Arthur Prior, “Tense-Logic and the Continuity of Time,” Studia Logica, Vol. 13, (1962),
133 – 151.
Arthur Prior, Past, Present, and Future, (Oxford University Press, 1967).
Arthur Prior, Papers on Time and Tense, (Oxford University Press, 1968a).
Arthur Prior, “Fugitive Truth,” Analysis, Vol. 29, No. 1, (October, 1968b), 5 – 8.
209
Arthur Prior, “A Statement in Temporal Realism,” in B. Jack Copeland, Logic and
Reality, (Clarendon Press, 1996), 45 – 46.
Hillary Putnam, “Time and Physical Geometry,” The Journal of Philosophy, Vol. 64, No.
8, (April 1967), 240 – 247.
W. V. O. Quine, The Ways of Paradox and Other Essays, (Harvard University Press,
1976).
Thomas Reid, Essays in the Intellectual Powers of Man, (Edinburgh: printed for John
Bell, and G. G. J. & J. Robinson, London, 1785).
Bertrand Russell, “Our Experience of Time,” The Monist, (1915), pp. 212 – 233.
Bertrand Russell, Our Knowledge of the External World, (Allen and Unwin, 1926).
Bertrand Russell, The Principles of Mathematics, Second Edition, (Cambridge University
Press, 1937).
Steve F. Savitt, “The Direction of Time,” The British Journal for the Philosophy of
Science, Vol. 47, No. 3, (September 1996), 347 – 370.
Steve F. Savitt, “Being and Becoming in Modern Physics,” The Stanford Encyclopaedia of Philosophy, (Winter 2010 Edition).
Yaroslav Sergeyev, “A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities,” Informatica, Vol. 19, No. 4, (2008), 567 – 596.
210
Sydney Shoemaker, “Time Without Change,” The Journal of Philosophy, Vol. 46, No.
12, (June 1969), 363 – 381.
Theodore Sider, Four-Dimensionalism - An Ontology of Persistence and Time, Oxford:
Oxford University Press, 2001. (Oxford Scholarship Online. Oxford University Press. 31
December 2009).
Howard Stein, “On Einstein-Minkowski Space-Time,” The Journal of Philosophy, Vol.
65, No. 1, (January 1968), 5 – 23.
Judith Jarvis Thomson, “Parthood and Identity Across Time,” Journal of Philosophy,
Vol. 80, No. 4 (April, 1983), 201 – 220.
Judith Jarvis Thomson, “McTaggart on Time,” Nous Supplement: Philosophical
Perspectives, Vol 15. (2001), pp. 229 – 252.
Michael Tooley, Time, Tense, and Causation, Oxford, Oxford University Press, 2000.
(Oxford Scholarship Online. Oxford University Press. 11 April 2010).
Peter van Inwagan, “Temporal Parts and Identity Across Time,” Monist, Vol. 83, No. 3,
(July 2000), 437 – 459.
Clifford Williams, “The Metaphysics of A- and B- Time,” The Philosophical Quarterly,
Vol. 46, No. 184, (July 1996), 371 – 381.
N. L. Wilson, “Space, Time, and Individuals,” Journal of Philosophy, Vol. 52, (October
1955), 589 – 598.
211
N. L. Wilson, “The Indestructibility and Immutability of Substances,” Philosophical
Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 7,
(1956), 46 – 48.