Logical Instrumentalism

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree

Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Teresa Kouri

∼6 6

Graduate Program in Philosophy

The Ohio State University

2016

Dissertation Committee:

Professor Stewart Shapiro, Advisor

Professor Chris Pincock

Professor Craige Roberts

Professor Kevin Scharp

Professor Neil Tennant c Teresa Kouri, 2016 Abstract

Logic and reasoning go hand in hand. So often, we hear that someone has reasoned poorly about something if they have not reasoned logically, or that an argument is bad because it is not logically valid. To date, research has been devoted to exactly just what types of logical systems are appropriate for guiding our reasoning. Traditionally, classical logic has been the logic suggested as the ideal for this purpose. More recently, non-classical logics have been suggested as alternatives. Even more recently, it has been suggested that multiple logics are reasoning guiding, or none are. So far, no one has addressed the impact that natural language has on our ability to reason well. My project fills this gap.

I focus on the relationship between the meaning of the connectives (“and”, “or”, “not”, etc.) in natural language and logic. By assessing what these connectives mean in natural language, we can figure out what they must mean in the formal language in order to guide our reasoning. I show that the connectives do not have a single meaning across all contexts in natural language, and thus there can be no single meaning to the connectives in the formal language which guides our reasoning. This means that the right logic to guide our reasoning depends on our context.

My dissertation is divided into five chapters. The first is a literature review. In the second, I examine the classical and non-classical answers to the question “which logic guides our reasoning?” and find them all wanting. In the third chapter, I show that the views that postulate either that there are multiple logics that guide our reasoning or that there are none are flawed for the reason that they do not appropriately account for the relationship between natural language and logic. In the fourth chapter, I show how we can adapt an

ii old view to solve this problem. Finally, in the last chapter, I propose a formal reasoning- guiding system that takes natural language into account and defend it from some potential objections. The appendix begins to address a major problem for this type of project.

Given what my dissertation shows, we can see how the traditional view on logic guiding reasoning was misguided. However, if we replace the notion of “right logic” with the system

I develop, we can reap the same benefits.

iii Dedication

To my parents, Brian and Vikki, and my husband, Andrew

iv Acknowledgements

There are a number of people in my life without whom this dissertation would never have come to be. First and foremost, I owe a debt of gratitude to my advisor, Professor Stewart

Shapiro. Without his generosity, guidance and encouragement and engaging discussions,

I would not be the philosopher I am (and this dissertation would likely not exist). My other dissertation committee members, Professor Chris Pincock, Professor Craige Roberts,

Professor Kevin Scharp and Professor Neil Tennant also provided invaluable feedback and support throughout the development of this work. Professor Ben Caplan and Professor Lisa

Shabel, having been members of my committees for other exams, also helped this project come to fruition.

My colleagues in the graduate student program at Ohio State have been instrumental in my philosophical development. They have always been willing to engage in philosophical discussions on topics of interest to me, and in return to let me hear about their projects and interests. Additionally, colleagues in other departments at Ohio State, and at philosophy departments other than Ohio State have played no small role. Discussions with them, and the resulting feedback about various sections of this project have been unbelievably helpful.

Finally, I need to thank my family. My parents, Brian and Vikki Kouri, have always been supportive of my pursuits, even when I told them that I would be in school until my early thirties, and then *maybe* find a job. My brother, Bradley Kouri, has always kept me grounded by asking questions about reality, rather than the lofty ivory tower I sometimes find myself in. My family-in-law, John, Peg, Emily, Paul, Jimmie and Johnny, has so graciously welcomed me into the fold, and always offered a much needed escape from philosophy with cocktails and dinner on their patio. Most importantly, I need to thank my husband Andrew Kissel. Sometimes marrying someone in the same field as you is a

v mistake, but in this case it made me a better person and better philosopher. Between our philosophical conversations and his ability to keep me sane, I know I could not have found a better partner, nor a person more critical in my philosophical development.

vi Vita

2008 ...... B.A., Queen’s University 2010 ...... M.A., University of Calgary

Publications

“A New Interpretation of Carnap’s Logical Pluralism,” Topoi, online first, August 2016

“Restall’s Proof-Theoretic Pluralism and Relevance Logic,” Erkenntnis, online first, De- cember 2015

“Ante Rem Structuralism and the No-Naming Constraint,” Philosophia Mathematica, 2016, Volume 24, Issue 1, Pages 117-128

“A Reply to Heathcote’s “On the Exhaustion of Mathematical Entities by Structures”,” Axiomathes, 2015, Volume 25, Issue 3, Pages 345-357

Fields of Study

Major Field: Philosophy

vii Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgements ...... v

Vita ...... vii

List of Figures ...... x

1 Introduction ...... 1 1.1 Logical Instrumentalism...... 2 1.2 Dissertation Outline...... 4

2 Literature Review ...... 6 2.1 Monism...... 6 2.2 Nihilism...... 9 2.3 Pluralism...... 9 2.3.1 More than one Logic...... 10 2.3.2 More than one Correct Logic...... 11 2.3.3 More than one Correct Logic in a Single Language...... 15 2.4 Logical Instrumentalism...... 29

3 Logical Monism from a Pluralist’s Perspective ...... 30 3.1 Why Logic Ought Not to Legislate to Properly Functioning Sciences.... 31 3.2 Logic and Natural Language...... 32 3.2.1 The Players...... 32 3.2.2 The Problem...... 36 3.3 Logic and Mathematics and the Sciences...... 38 3.3.1 The Players...... 38 3.3.2 The Problem...... 42 3.4 Our Next Step...... 44

4 Beall and Restall’s Logical Pluralism ...... 46 4.1 Connective Meanings in Beall and Restall’s Logical Pluralism...... 46 4.1.1 Introduction...... 46 4.1.2 Beall and Restall’s Pluralism...... 47 4.1.3 Overlapping Clauses...... 48

viii 4.1.4 The Problem...... 50 4.1.5 Merely Technical Meanings...... 55 4.1.6 A Possible Solution...... 56 4.1.7 Conclusion...... 57 4.1.8 Appendix...... 57 4.2 Title...... 59 4.2.1 Introduction...... 59 4.2.2 Restall’s Proof-Theoretic Pluralism...... 60 4.2.3 No relevance validity...... 63 4.2.4 Other Admissibility Conditions...... 66 4.2.5 Conclusion...... 70

5 Title ...... 72 5.1 Introduction...... 72 5.2 The Traditional Carnapian View...... 73 5.3 Carnap’s Position...... 74 5.4 The External Question...... 76 5.5 Carnap’s Agreement...... 86 5.6 Shapiro’s Position...... 90 5.7 Conclusion...... 92

6 Logical Pluralism from a Pragmatic Perspective ...... 93 6.1 The Intuitions...... 93 6.1.1 Carnap’s and Beall and Restall’s Pluralisms...... 96 6.2 The New System...... 96 6.2.1 QUD framework...... 98 6.2.2 A New Theory of Connective Meaning...... 109 6.3 Conclusion...... 114

References ...... 115

Appendices ...... 121

A Two Normative Issues for for the Carnapian Pluralist ...... 121 A.1 Introduction...... 121 A.2 How to Choose our Goals...... 122 A.3 How to Choose our Logics for those Goals...... 124 A.4 Some Possible Solutions...... 125 A.4.1 Solutions to Which Goals are Good...... 125 A.4.2 Solutions to which logics fit which goals...... 127 A.4.3 Pragmatics and Explication...... 131 A.5 Conclusion...... 133

ix List of Figures

4.1 A is true in the future of s ...... 58

4.2 A is not true in the future of s ...... 58

x Chapter 1

Introduction

Logic is often thought of as a tool for figuring out what follows from what. In this disserta- tion, I explore the consequences of taking logic to be such a tool. This is the position I call

Logical Instrumentalism, but it could equally well be called goal-driven logical pluralism.

In effect, though most people will agree that logic is a tool, they think there is something additional to certain logic(s) that makes them more fundamental, more basic, or “righter”.

The position I explore in this dissertation stops short of this. The claim is that logic is a tool for deductive reasoning, and nothing more. If we take this position seriously, and if we

assume that there might be more than one tool available, one of the most notable fallouts

is that we cannot be logical monists, taking it that there is One True Logic, but must be

logical pluralists, arguing that there is more than one right logic.

What does it mean for logic to be a tool for studying what follows from what?1 It means

that when we reason, when we try to figure out what follows from a given set of premises, we

do so in a logical fashion. That is, reasoning is governed by norms, which tell us when the

reasoning is good, and those norms are given by a logic. The difference between this view,

and the traditional view that there is exactly one right logic, or one right way to reason,

is that it imposes no such restrictions. Logic is a tool because it can be adjusted based on

our purpose; logic can be changed based on what we are reasoning about. In this sense, the

instrumentalist’s tool box is replete. Are you reasoning about classical mathematics? Then

do so with classical logic. Are you trying to deduce syntactic relationships between words

in English? Then maybe linear logic is right for you. Depending on what we are up to, on

1The “what follows from what” terminology is borrowed from Graham Priest.

1 what our goal in reasoning is, we will be able to use different tools. Just what follows from what, then, will depend extensively on what we are doing, and what tool we are using to reason about what we are doing.

1.1 Logical Instrumentalism

I will not directly argue for the instrumentalist thesis here. Rather, I will consider what the consequences of its truth are. However, I think there are at least two good pieces of evidence that ought to suggest it might be true. We can look at some natural language data and work done by linguists to suggest pluralism is true (if we think logic is related to language), and to some recent historical examples to suggest that logic is a tool for deductive reasoning.

With these two pieces of evidence in hand, we can develop the beginning of an argument for Logical Instrumentalism, whereby logic is a tool for reasoning and there is more than one good tool.

First, if we assume that logic somehow flows from meaning, then we are already on a path towards pluralism. The claim is often made that once we identify the meanings of the logical connectives, or the turnstile, or “validity”, the one right logic will simply fall out of this meaning. Several authors who hold this view are considered below, and include at least Michael Dummett, Graham Priest and Alan Ross Anderson & Nuel Belnap. The claim amounts to saying that the logical connectives match up with the connectives in natural language in some appropriate way. However, if we take the connective meanings to be connected to natural language, we can see how this view would lead to pluralism.

Natural language is rife with vagueness, ambiguity and, in general, is not precise enough to fold smoothly into a logical language. In fact, if we take seriously what linguists say, then there may very well be more than one conditional connective in English, let alone in all of natural language. Even logicians agree that the conditional in natural language is probably not the material conditional, and yet in face of this maintain that natural language

2 connectives must be appropriately related to a logic’s connectives in order for that logic to be the right logic. If logic is actually related to natural language in this way, then the right logic (which flows from meaning) will have multiple connectives of each type, with multiple validity relations for each connective. I claim that a better way to make sense of this is to claim that there is a plurality of logics that flow from meaning, and thus if logic flows from meaning, we ought to be pluralists.

A piece of evidence we might consider for in favor of logic being a tool for reasoning would be whether the applications that people were making of logic sometimes necessitated that they change the logic they are using. If we could find this, then we could conclude that sometimes a change in task requires a change in tool, that people have already been thinking about logic as a tool for deductive reasoning. There are at least two examples we can point to in recent history that suggest that, depending on the goal someone is trying to achieve, a change in logic is sometimes required. I will briefly consider two here: the move from classical to constructive mathematics, and some considerations about paradoxes.

In both the literature review and chapter 2, I will consider the positions of some philoso- phers who hold that validity, and logic, ought to be about preserving provability from premises to conclusions, rather than preserving truth. They made this change in their method of thinking about logic because of a change in their way of thinking about mathe- matics. For this group, mathematics is about what we can prove. Whether or not something is true is irrelevant unless we can prove it. Because they conceived of the application of logic differently (as a proof-driven enterprise, rather than a truth-driven one), they changed the logic they used to pursue it.

A second example of this type of change is spearheaded by Graham Priest and other dialetheists. Paradoxes, they claim, like Russell’s paradox and the liar, are problematic if we want to treat them with classical logic. They cause the inconsistent theory to explode, and so if we consider one in a classical system, we wind up being able to prove everything. This is not useful for trying to study paradoxical sentences. Rather, to study these sentences,

3 we must change our logic. They claim that we must change it to a dialetheic logic, a logic where these types of sentences are both true and false, but that does not explode in the same way classical logic does.

Though meaning considerations and examples of past changes in logics together do not make a conclusive argument for Logical Instrumentalism, they give us good reason to believe the thesis might be true.

1.2 Dissertation Outline

Each chapter in this dissertation is written as a separate paper, but they fit together in a simple and coherent way.

The first chapter of the dissertation is a literature review, focusing on explaining the views of those I take to hold one of the three positions which I take to be available in the philosophy of logic. These include logical monism, the thesis that there is exactly one right logic, logical pluralism, the thesis that there is more than one right logic, and logical nihilism, the thesis that there are no right logics.

The first two substantive chapters, chapters 2 and 3, of the dissertation focuses on discussion and critical analysis of positions in the literature. In the first chapter, I suggest that many monistic positions about logic are incredibly close to pluralist positions, and on the assumption that we ought not to legislate to properly functioning sciences, ought to be pluralistic positions. In the second substantive chapter, I show that two of the pluralist positions on the table so far do not allow the logical connectives to be defined in a coherent way.

In the fourth chapter, I turn to a position given by Rudolf Carnap. Carnap is often taken to be a pluralist with a very rigorous view of the meanings of the logical connectives. The view is normally taken to be that logics are sets of inference rules, and connectives are defined by those inference rules. Thus, when we change the logic, we change the rules, and so some

4 of the connective’s meanings must change. I suggest an alternative interpretation of the

Carnapian position. I suggest that we cannot compare two logics without first embedding them into a meta-logic. Once we do this, though, I claim that whether two logics have corresponding connectives which share meanings will depend on which meta-logic we have embedded them into. This, I claim, accords better with the Carnapian picture, and ought to be the way we think about logical pluralism more generally.

In the fifth and final chapter, I spell out a position which takes seriously this Carnapian insight. I use a formal pragmatic framework to spell out the position. At its essence, it is a position which takes seriously what practitioners of logic think they are doing when they use a logic. I use pragmatic conversational indicators present in certain contexts but not others to figure out when two connectives mean the same thing and when they do not.

Further, I suggest that, in line with the Carnapian position, there are certain questions we might want to ask but which cannot be answered unless embedded into certain contexts. I suggest that “which logic is right?” is one such question, along with “what do the logical connectives mean?”.

Finally, in the appendix, I begin to address one potential problem with this type of view.

The problem is with whether Logical Instrumentalism allows logic to retain any normative force. I sketch a conception of logical normativity on which it does, relying heavily on the notion of Carnapian explication.

5 Chapter 2

Literature Review

There are three positions about the number of right logics in logical space available to a philosopher of logic: she may be a monist, pluralist or nihilist. Roughly, the monist holds that there is exactly one right logic, the pluralist holds that there is more than one, and the nihilist holds that there are none. In this chapter, I will focus primarily on explicating the possible pluralist positions, with a brief overview of the monists and the nihilists.

2.1 Monism

Logical monism is the position that there is exactly one right logic. It is often thought that there is one right way to reason, and with this, that an argument is either right or wrong.

Folk intuitions about deductive reasoning normally run in this vein, as do positions in applied science. As logic is often thought to be the science of deductive reasoning patterns, it is not very surprising that many think there is one right account of logical consequence.

According to W.V.O. Quine, “logic is the systematic study of logical truths” (Quine,

1986, p vii), and logical truth can be defined as “a sentence from which we get only truths when we substitute sentences for its simple [atomic] sentences”(Quine, 1986, p 50). It would be impossible to translate a foreign language in such a way that it conflicted with the logical truths, because any language which conflicted with such truths would be close to incoherent. Quine’s favorite example of such a truth is the law of non-contradiction

(¬(P ∧ ¬P ), where P is some sentence). Quine claims that anyone who “tries to deny the

[law of non-contradiction]... only changes the subject” (Quine, 1986, p 81). In this sense,

6 the classical logician and the paraconsistent logician are talking past each other; they are talking about different negations and/or disjunctions.

According to Quine, logical monism is an intuitive position. Those who deny a logical truth are really using different connectives, and so are not denying a logical truth at all. We might think that logical monism must be true, and that, given the meanings Quine attaches to the logical connectives and his insistence that those are the only logical connectives, disagreements over which logic is correct are merely verbal. As we will see, just what the logical connectives are (or can be) will be an important theme for us.

Logical consequence, on this classical account, is truth preservation. A sentence B is a logical consequence of a sentence A just in case whenever A is true, B is true.

There are several defenders of something like classical logic as the right logic. Burgess

(1992) and Resnik(1985) are notable on this front, who both hold that if we conceive of the aims of logic, and the purpose of logic, properly, then classical logic presents itself as the obvious option. Burgess claims that classical logic may be right for mathematics because, in effect, mathematicians (note: not necessarily just mathematical logicians) reason classically.

Resnik holds a broadly Quinean position: all propositions are up for revision, even what we take to be logical tautologies. For Resnik, then, there are no facts of logic. Because of this, whenever possible we ought to minimize the scope of our logic (see Resnik(1996)). This restricted logic he takes to be something like classical logic.

Even if one is a logical monist, there are alternatives to the classical position. I will consider only two here, as they will be important for the remainder of this project.

Intuitionist logic has the same classical deduction rules, but the law of excluded middle

(for all sentences P , P ∨ ¬P is true) (LEM) is not valid. Relevance logic has the same infer- ence rules, but conclusions must be relevant to the premises from which they are deduced.

Sometimes is is cashed out by making ex falso quod libet (from a contradiction, anything follows, hereafter EFQ) invalid.

Intuitionism can be thought of as preserving proof or knowledge rather than truth.

7 Several authors put forward such views. Of note are L.E.J. Brouwer (see his 1983b; 1983a) and Heyting(1956), who hold that mathematics is dependent on human thought, and for this reason we can only have mathematical truths which can be constructed. The best logic with which to accomplish this is one which is restricted in certain ways, namely that those things which cannot be constructed have no truth value. The classical mathematician and logician mistakenly take mathematical propositions to be independently true or false.

Intuitionistic logicians also typically take the connectives to have a particular meaning.

This is evident in Michael Dummett’s claim that the classical logician has assigned an inappropriate meaning to several of her connectives. Dummett has a particular philosophical stance which implies that the only right logic is intuitionistic (see his 1973; 1974; 1991; 2000).

For Dummett, “the meaning of an expression must be exhaustively manifested by the use of that expression” (Dummett, 2000, p 260). It is not only the truth or falsity of a statement that matters, but our ability to know that truth or falsity. If sentences had evidence- transcendent truth-conditions (as in classical logic), knowledge of sentences could never be manifested in our use of those sentences (see (Dummett, 1978, p 107)). In order to satisfy the manifestation requirement, Dummett argues that “the notion of truth, considered as a feature which each mathematical [and logical] statement either determinately possesses or determinately lacks, independently of our means of recognizing its truth-value, cannot be the central notion for a theory of meanings of mathematical [and logical] statements...we must, therefore, replace the notion of truth, as the central notion of the theory of meaning for mathematical [and logical] statements, by the notion of proof ”(Dummett, 1978, p 107).

Thus, by re-conceiving what meaning is, we must re-conceive what the connectives mean, and that re-conception, for Dummett, makes intuitionistic logic the right logic.

The last monistic position I will consider is a relevantist position. Alan Ross Anderson and Nuel Belnap, in their 1975, provide several logics which do not permit use of EFQ, which they think is a fallacy, and in doing so produce several relevant logics. In relevant logics, premises must be relevant to the conclusions. Since from any contradiction we can

8 conclude something completely irrelevant to the contradicted propositions using EFQ (e.g. from “it is raining” and “it is not raining” we can conclude “the moon is made of green cheese”), we must rule it out as a validity-preserving inference.

Anderson and Belnap accomplish this by changing the notion of logical consequence. In their system, B is a logical consequence of A if and only if any time A is true B is true and B is relevant to A. The paradoxes of material implication make it such that material implication does not match up with natural language implication. On the other hand, they claim, relevant implication, which does not suffer from the material paradoxes, matches up with natural language implication in a much more intuitive manner.

There are many ways in which one can change the conditional, or change something else in classical logic, to make the logic relevant. Given this, there is much variation even within relevant logic itself. I will not go into more detail about it here. What is important for my purposes is that part of what makes a logic right for relevance logicians is that the conclusion must be appropriately related to the premises.

2.2 Nihilism

Nihilism is the position that there are no right logics. In some sense, any of the pluralist positions below could be trivially turned into a nihilist position, if we thought that the right logic must be unique. Franks(2015) argues that such a view is in all likelihood true: since we have made use of many different logics historically, and the right logic must be unique, there can be no right logic.

2.3 Pluralism

We now turn to the main focus of this dissertation: logical pluralism. This view is roughly characterized by the fact that there is more than one right logic.

9 In this section, I will mostly follow the layout given in Cook(2010), with one important exception. Cook takes it that a logic is correct (right) exactly when “the logic validates an argument if and only if the natural language statement corresponding to the conclusion is a logical consequence of the natural language statements corresponding to the premises”

(Cook, 2010, p 496). I will not use this definition here. It is not clear that all proponents of pluralism agree to this, nor is it clear that we ought to think about logics being right in this way. I will reserve the word “correct” for this technical notion of what makes a logic right, and use the word “right” more generally.

One thing to point out immediately is that it is not contestable that there is more than one logic (this is Cook’s MLP, mathematical logical pluralism). What we will be interested in is, for the most part, something stronger than this thesis. What the logical pluralist wants is a theory that shows that there is more than one right logic, whatever right amounts to.

It is also important to note that many logical pluralisms are logical relativisms, in a sense. This would mean that the right logic will be relative to some factor. Note that, importantly, I will use “relativism” here in a different manner than many other authors. For my purposes, relativism for a subject will mean that the subject is relative to something external. This external factor does not have to be, as many seem to believe, language, society, a form of life, etc. For our purposes, I will use “A is relative to B” to mean roughly

that the account of A is “a function of a distinct set of facts”, namely those in B (Cook,

2010, p 492).

2.3.1 More than one Logic

Philosophical logical pluralism (PLP) is the position that “there is more than one logic that

can fruitfully be applied to philosophically interesting phenomena” (Cook, 2010, p 494).

Susan Haack argues that this is true. In Haack(1974), she proposes that there are needs

classical logic cannot fill, and thus we must look for different logical consequence relations.

She gives some examples, including a three-valued logic, whose third truth-value has no

10 analogue in the classical system (see Haack(1974, p 22)), and so, in instances where that logic is needed, classical logic will not do. For Haack, “logic is a theory...on par...with other ‘scientific’ theories...[and] choice of logic...is to be made on the basis of [coherence and simplicity]” (Haack, 1974, p 26). Thus, if we can find a domain that calls for a non-classical logic on the basis of “economy, coherence and simplicity of the overall belief set” (Haack,

1974, p 26), we will have good reason to think we need a logical consequence relation that is not strictly classical, at least for certain applications.

PLP can be read either in a strong way, or in a weak way. Both are theories that think that the right logical consequence is relative to a particular application, but the strong

PLP claims that this is all there is to a logic’s being right, while weak PLP (which is significantly less contentious) is consistent with there being a single right logic over and above the applications. Weak PLP, then, is consistent with monism, while strong PLP is not. What strong PLP contends is that the only thing which makes a logic right is that it is appropriately suited for dealing with a particular application. Weak PLP simply holds that sometimes different logics are called for when different applications are considered, but says nothing about which logic is right overall. Thus, someone might hold that weak PLP is true while maintaining that classical logic is the one right logic, because, for example, classical logic is the logic with which we reason about which logic is best for each application.

I believe that weak PLP is not generally contested, but that strong PLP is not held by any monists, and possibly not even many pluralists. I am not even certain that Haack would agree to strong PLP as I have formulated it. It may in fact be equivalent to something like

Cook’s LCP (see section 2.3.2).

2.3.2 More than one Correct Logic

Logical Consequence Pluralism (LCP) is the thesis that there is more than one correct logic (Cook, 2010, p 496). For our purposes, we will read this as “there is more than one right logic”. Strong PLP, above, is a version of LCP. In this section, we will look at three

11 possible positions that hold that at least LCP is true. All three are relativisms in the sense mentioned above, and so I will sort them by what they take the correct logic to be relative to.

Relative to Linguistic Frameworks

Carnap takes the view there is no one right logic, because the question “which logic is right?” is illegitimate in a particular way. Carnap’s position is pluralistic in the sense that, though the question of which is the one right logic is incoherent, it does accept that different linguistic frameworks require different logics. This position is also a position in which the choice of logical consequence relation is relative to the choice of linguistic framework.

Carnap’s position can be summarized by two often-quoted passages:

Our attitude towards...requirements...[of a logic] is given a general formulation in the Principle of Tolerance: It is not our business to set up prohibitions but to arrive at conventions. In logic, there are no morals. Everyone is at liberty to build up his own logic... All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments. (Carnap, 1937, p 51/2)

We find that for Carnap, what is important is not the logical system itself but whether the logical system suits the purpose it was meant to be used for, and whether the logical system can be explained by formal syntactical rules. It is not important that one logic is better than another, or that one is the right logic while the others are simply useful tools.

These are, in effect, not even coherent questions.

Carnap holds that whether these logics are correct outside of a given linguistic frame- work is an incoherent question. Questions of existence of abstract objects can only be asked relative to a given linguistic framework, as with questions about the right logic, or which logical consequence relation is correct. “Logical syntax is the same thing as the construction and manipulation of a calculus” (Carnap, 1937, p 5), and since we can construct different

12 calculi depending on the language, different logics may result. Outside of a linguistic frame- work questions like this are only pseudo-questions. When we introduce new axioms or terms we do so within a linguistic framework. Since we cannot introduce axioms without already having a framework in mind, we do not need to be worried about our axioms being true about the external world, but only being true relative to our chosen framework. That being said, accepting a linguistic framework does not imply accepting that the framework accu- rately matches up with reality: we can accept many conflicting frameworks, we can even accept many conflicting frameworks for the same subject. There is an important pragmatic question about which linguistic framework is appropriate to use for any given need. This question, though it looks external, is not about which framework most accurately describes the actual world, but which framework best suits our purposes, and so it is not external in the same problematic way as metaphysical questions.

For Carnap, the question of which account is the one true account, or of which logic is the one true logic, is incoherent. It asks about something external to a given framework; it asks a question about what is really true. What we must ask instead is whether a logic (or account of logical consequence) is correct relative to a given linguistic framework. Once we do this, we can accept or reject a given framework/logic pair depending on its efficiency as an instrument (Carnap, 1983, p 257) and whether it accomplishes the goals we intended it to.

Relative to the Logical Constants

Cook(2010) hints at another possible way to generate logics that satisfy LCP: “by varying which bits of primitive vocabulary count as logical” (p 496). There are at several authors who propose such a view: Tarski(2002), Sagi(2014), Varzi(2002), Etchemendy(2008) and Shapiro(2014). Tarski(2002), Sagi(2014), Etchemendy(2008) and Shapiro(2014) hold that there is no privileged distinction between logical and non-logical terminology, and so the right logic can be varied by changing which parts of the language are to count as

13 logical. Varzi(2002) agrees on this front, and presents what I take to be the most complete formulation of what such a system would look like. I will restrict my discussion to Varzi

(2002), with the caveat that Sagi(2014), Etchemendy(2008) and Shapiro(2014) would likely agree, at least with the big picture.

For Varzi, there is no way to legitimately divide the logical vocabulary from the non- logical vocabulary. Varzi claims that we cannot take what is logical to be determined by its semantic invariability across models, nor are the logical terms exactly those which are general (where general means something like “makes sense on every domain”), nor can the divide be drawn by how the rules of logical constants are specified. All three of these considerations, Varzi claims, will not be able to draw a line between “is parallel to” (normally thought to be a non-logical term) and “is identical to” (normally thought to be a logical term). Varzi’s first point is that the extension of “is identical to” will vary depending on the domain, thus meaning that it is not in fact semantically invariable, making “is identical to” a non-logical term. His second claim is that what is general must be general for some class of domains, and there will be classes of domains across which “is parallel to” makes perfect sense (for example, those where the domain consists entirely of line segments), which makes

“is parallel to” a logical term. Finally, he claims that both “is identical to” and “is parallel to” can come to have their rules specified in exactly the same way: if V al is a function assigning semantic values, then they both have the form “ ‘a R’s b’ if and only if V al(a)

R’s V al(b)” (Varzi, 2002, p 8). Thus, this condition will either make both “is identical to” and “is parallel to” logical, or both non-logical. Varzi claims, more strongly still, that any term can be made to follow this pattern of semantic value assignment, and so any term can be logical. This, claims Varzi, there is no privileged way to divide logical and non-logical terminology.

Though Varzi’s claim that there are many ways to divide the logical and non-logical vocabulary is controversial, once we take it on board, logical pluralism follows. If there are many legitimate ways to divide the vocabulary, then which logic is right will depend on how

14 the division is settled.

Relative to the Normative Demands

Field(2009) presents a unique take on pluralism, in which logic is relative to the set of norms in play at any given moment. For Field, a logic can be right only if it is appropriately epistemically normative. Additionally, for Field, what epistemic norms are appropriate will be relative to our epistemic goals. That is, there is more than one set of applicable and useful epistemic norms. This means that depending on which norms are in play, a different logic may be required to analyze the outcomes of a situation. Field claims this type of pluralism will be very weak: for the most part, logics will agree on the consequences of reasoning within most sets of norms. I characterize this pluralism as LCP, because Field makes no claims about which languages each logic are “in” (more on what it takes for a logic to be “in” a language in the next section). Field holds that a logic cannot be right without being somehow epistemically normative, and that since epistemic norms are relative, so too is logic.

2.3.3 More than one Correct Logic in a Single Language

There are several authors who put forward views about logical pluralism in which two logics can share a language. This corresponds to Cook’s “substantive logical pluralism” (SLP), where a pluralism is substantive if, roughly, there are at least two distinct right logical consequence relations which share all of their logical vocabulary (Cook, 2010, p 496). In effect, the claims about language sharing in the literature seem to be slightly different.

What those in the literature seem to mean by logics sharing languages is that the logical connectives in each system have the same meaning. I will turn to these views now.

15 Relative to Cases

According to Beall and Restall(2006) (and Beall and Restall(2000)), logical pluralism is

“the view that there is more than one genuine deductive consequence relation, and that this plurality arises not merely because there are different languages, but rather arises even within the kinds of claims expressed in one language” (Beall and Restall, 2006, p 3). This is a substantial logical pluralism (SLP).

Beall and Restall propose the Generalized Tarski Thesis (GTT) as a definition of logical consequence:

An argument is validx if and only if in any casex in which the premises are true, so is the conclusion (Beall and Restall, 2006, p 29)

They settle on the definition of logical consequence as a relation that must be necessary,

normative and formal.

Necessity is an essential component of logical consequence, claim Beall and Restall,

because “the truth of the premises of a valid argument necessitates the truth of the con-

clusion” (Beall and Restall, 2006, p 14). A relationship is necessary when “it applies under

any conditions whatsoever.” (p 16). Thus, logical consequence is necessary because, for a

valid argument from A to B, if A were the case, “then we ought to be able to conclude

that...B would be the case as well” (p 16).

Normativity gets an equally succinct characterization. For Beall and Restall it amounts

to something like an ability to “go wrong” if you accept the premises of a valid argument

but not its conclusion (p 16). They also hold, though, that sometimes it can be rational to

violate the norms of logical consequence. Thus, normativity amounts to there being a way

to make a mistake.

Formality is more complicated. Beall and Restall consider four types of formality. The

first is what they call schematic-formality (p 17-18), which is to say that logic is somehow

16 about the forms of the relata, and must in some way be schematic. They seem to think this is not enough, and consider three more types of formality, which they take from MacFarlane

(2000):

• F1: logic provides constitutive norms for thought as such (also characterized as pro-

viding laws applicable to content as such)

• F2: logic is indifferent to the particular identities of objects

• F3: logic abstracts entirely from the semantic content of thought (Beall and Restall,

2006, p 21)

They do not settle which of F1, F2 and F3 are required for being a formal relation of logical consequence, but rather suggest that one of the three must be satisfied in order for a relation to be formal, and perhaps more.

They claim an instance of GTT is admissible if it produces a logical consequence relation that is necessary, normative and formal. Their pluralistic claim amounts to accepting more than one admissible instance of GTT. They take classical, intuitionistic and a relevant logic1

to be results of admissible instances of GTT.

To generate classical logic, Beall and Restall take cases to be either possible worlds or

Tarskian models. On the first account, an argument is valid if and only if, in any world

where the premises are true, so is the conclusion. On the other, an argument is valid just in

case, in any model where the premises are true, so is the conclusion. Both these relations are

necessary. The possible world variant is constructed to meet the necessity constraint. The

Tarski model variant is necessary because if it is possible that the premises of an argument

are true and the conclusion is false, then there is some model on which the argument is

not valid. Both are also normative: on the possible worlds account, our mistake would be

reasoning from truth to untruth, on the Tarski model account, our mistake is reasoning

such that we infer an untruth from a truth. 1It is not clear which of the Anderson and Belnap relevant logics they take their relevant logic to be.

17 Finally, both accounts are in some sense formal. The Tarski model account is constructed to be schematic, though the possible worlds account is not schematic. Both accounts sat- isfy F1-formality. The Tarski model account is also F2 and F3-formal (again, almost by construction). It will be useful to quickly explain why Tarski models satisfy all three of F1,

F2 and F3, since relevant logical consequence and intuitionistic logical consequence are F1,

F2 and F3-formal for similar reasons. The logical consequence relation generated by Tarski models is F1-formal, meaning that it provides norms for thought as such, because “the analysis of forms of statement[s] proffered in the language of predicate logic is applicable to all kind of propositional content” (Beall and Restall, 2006, p 41). Thus, since we can analyze any propositional content, and have laws applicable to content as such, it is F1- formal. Tarski-model-logical-consequence is also F2-formal, meaning that it is indifferent to the identities of objects. This is clear. It is F3-formal, in other words it abstracts from the semantic content of thought, because it cares about the form of the relata of logical consequence, and not about their content.

The possible worlds account, according to Beall and Restall, may not be F2-formal, since possible world semantics are potentially sensitive to the identities of objects. Addi- tionally, the possible world account is not F3-formal, since “the content of a sentence is what determines its truth value at a world” (Beall and Restall, 2006, p 43). Thus, both of the logical consequence relations generated are necessary, normative and formal, meaning that, according to Beall and Restall, we have achieved a position of logical pluralism: two instance of GTT are admissible. This pluralism is strong because different sentences are logical truths in each set of cases: where possible worlds are cases, the sentence “red is a color” is logically true, while where models are cases, it is not.

Next, they consider a relevant logic, where cases are situations in the sense of Barwise and Perry (see Barwise and Perry(1983)). Situations can be thought of like possible worlds, but they do not need to be complete or consistent (the analogy with possible worlds is very weak). That is, they do not need to say, of every proposition, whether it is either true or

18 false. There are some propositions about which they say nothing. At the same time, there can be inconsistent situations. For example, if Professor Smith is talking to his student,

Jones, in Smith’s office, we can have a situation that consists of just the two of them in the office. The situation would say nothing about Jones’s brother, and thus any sentence concerning him would be un-evaluable. The situation may also be the set of beliefs of a particular person, and if that person happens to believe inconsistent things, then the situation would be inconsistent as well. An argument is relevantly valid if and only if, in any situation where the premises are true, so too is the conclusion. This ensures relevance, since if the conclusion is not relevant to the premises, there will be at least one situation where the premises are true and the conclusion is not. This also ensures normativity: we go wrong when we reason from premises and conclude something irrelevant to those premises.

Necessity also comes cheaply: since the arguments which are relevantly valid form a subset of the arguments which are classically valid, they will all be necessary. Relevant consequence also satisfies all four of the formality requirements (Schematic, F1, F2, F3), for the same reasons Tarski models are formal and because relevant validities will be a subset of classical validities. Again, we have a logical consequence relation that is necessary, formal and normative, and so another acceptable instance of GTT.

Finally, they consider how one might reconstruct intuitionistic logics. They take cases to be stages in Kripke models. Though it is in general harder to fit intuitionistic logic into a model-theoretic framework than others, Kripke models do successfully accomplish the task. Thus, we have that “an argument is intuitionistically valid if and only if, for any stage (in any model) at which the premises are satisfied, so is the conclusion” (Beall and Restall, 2006, p 68). Importantly, this is a very restricted version of an intuitionistic logic. The version of intuitionism here will never prove the negation of a classical theorem.

It will prove only a (proper) subset of the classical theorems. This is what makes the valid arguments necessary: since classical validities are necessary, and we have a subset here, so are constructive necessities. Normativity comes because the mistake we make

19 is not reasoning constructively. This logical consequence relation also satisfies all four of the formality relations, for reasons similar to why Tarksi models satisfy these criteria and because intuitionistic validities will be a subset of classical validities. Again, this forms a logical consequence relation that is necessary, normative and formal, so we have a fourth admissible instance of GTT.

What will become very important for this project is that Beall and Restall claim that their pluralism is within one language. For Beall and Restall what this amounts to is that all four logical consequence relations characterized so far have the same connectives.2 Restall, sometimes with Beall, spends some time trying to parse out exactly what it would mean for two logics to have the same connectives. In Restall(2002), he discusses a number of ways the connectives may be defined so that they maintain their meanings across logics. I will say more about these options in chapter4, but for now it suffices to point out that he thinks Quinean assent/dissent conditions, truth values and inferential role are all able to play the role of a meaning of the logical connectives which is preserved across logics.

More specifically, what Beall and Restall take a connective to be defined by is something like a general truth value. A connective is defined by the combination of its classical clause in Tarski models, its constructive clause in Kripke structures, its relevant clause in states, and so on. In Beall and Restall(2006), they state that:

The clauses can both be true of one and the same object simply in virtue of being incomplete claims about the object [the referent of the connective]. What is required is that such incomplete claims do not conflict, but the clauses for negation do not conflict. The classical clause gives us an account of when a negation is true in a model, and the constructive clause gives us an account of when a negation is true in a construction. Each clause picks out a different feature of negation. (Beall and Restall, 2006, p.98)

In Beall and Restall(2001) and Restall(1999), this notion is spelled out more formally.

In effect, what it comes to is that all three of classical, intuitionist and relevance logic can

2This is in some sense weaker than the requirements of Cook’s SLP. Cook’s SLP requires that all logical vocabulary be shared. In order to make SLP come out true for Beall and Restall’s logical pluralism, we have to assume the connectives exhaust the logical vocabulary, which may be controversial.

20 be “formalized” in a situation-like semantics. I will use negation as my example here, but this can purportedly be done with all of the logical connectives. We know the usual truth conditional clauses for negation in Tarski models, Kripke models and situations. They are

• Tarski models: for a model M, M |= ¬A if and only if M 2 A

0 0 0 • Kripke models: for a node K, K |= ¬A if and only if, ∀K such that K ≤ K , K 2 A, where ≤ is the ordering on the nodes in the Kripke model

• Situations: for all situations x, x |= ¬A if and only if, ∀y such that xCy, y 2 A, where C is the compatibility relation on the situations in the model

What Restall(1999) and Beall and Restall(2001) notice is that all three of these clauses

can be thought of as precisifications of the clause: x |= ¬A if and only if, ∀y such that

xRy, y 2 A, where R is some appropriate relationship, and x, y are elements of some suitable domain. The “elements of some suitable domain” can all be found in a situation semantics.

If we would like to find classical logical consequence, we restrict R to identity, and x, y must

be complete and consistent situations. If we care about intuitionist logical consequence,

then we restrict R to the appropriate reflexive and transitive relation, and x, y to explicitly

consistent situations (situations which make ¬(A ∧ ¬A) true for all A). If we want relevant

consequence, we allow x, y to range over all situations in the domain, and read R as the

compatibility relation.

This is how Beall and Restall claim to capture different features of negation. Since all

three clauses for negation can be “read off” of the more general clause, they take this as

evidence that all three negations are part of the same negation, and moreover, since we can

provide models for the general clause, there will be no conflict between the various readings

of said general clause. Thus, for Beall and Restall, all three negations are part of the same

more general negation, and they do not conflict. This is why their pluralism is an instance

of SLP (as long as we restrict what counts as logical vocabulary in appropriate ways).

21 The language in question is the one of the “general” connectives, and the different logical consequence relations are found by focusing on particular features of those connectives.

Though the pluralism in Beall and Restall(2006) is entirely model theoretic, Restall

(2014) gives a proof theoretic counter part. According to Restall, another way we can define the connectives is by their left and right logical rules in a sequent calculus. This means, in particular, that when we change the structural rules in that sequent calculus, we do not affect the meanings of the connectives. If Restall is right, then this means that classical, intuitionistic and relevance logic can share connectives. We start with a classical sequent calculus where a sequent Γ ` ∆ (where Γ and ∆ are finite sets of formulae) is to be read “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true” (Restall, 2014, p 282). The rules for negation are something like:

Γ,A ` ∆ ¬R Γ ` ¬A, ∆ and

Γ ` A, ∆ ¬L Γ, ¬A ` ∆

If we restrict the number of sentences on the right then we have intuitionistic consequence, and if we remove the weakening rules, we have relevant consequence. Neither of these changes, claims Restall (in Restall(2014) and Restall(2000)) change the left and right logical rules, and so both preserve the connective meanings. If Restall is right, then we have another instance of SLP.3

There are several common objections to the Beall and Restall project.

First, as Goddu(2002) discusses, Beall and Restall do not give a clear analysis of what a case is. Just what are these things which we must consider to decide what the logical consequence relation in question is? Goddu’s claim is that, without a proper account of

3Hjortland(2013) presents another proof-theoretic logical pluralism, which he calls intra-theoretic plural- ism. It is much weaker, as it only contains Kleene’s K3 and Priests’s LP, both tri-valent logics, the difference between them being that one value is designated in K3 and two are designated in LP. Essentially, it relies on the fact that these two logics have the same truth tables, but designate different truth values.

22 what a case is, we will never know for sure whether the Beall and Restall position is really pluralism, or collapses into a monism. In effect, if we think of logical monism as “the claim that for any argument there is exactly one correct answer as to whether the argument is deductively valid” (Goddu, 2002, p 225), then we need cases to be such that they provide different answers to validity for one and the same argument. What Beall and Restall respond with comes essentially to the claim that “cases, whatever they are, are ‘things’ in which claims may be true” (Beall and Restall, 2006, p 89). Because of the claims Beall and Restall make about logical vocabulary, we can see how this will lead to a form of pluralism. Since we might have two cases, say a classical world and a relevant situation, assessing one and the same argument (since the logical vocabulary is the same), we may indeed find that both cases rule differently on the validity of that argument. For example, if the argument used disjunctive syllogism, we might find this to be the case. Thus, claim Beall and Restall, they do not need a more detailed analysis of what a case comes to.

The second objection is that Beall and Restall (and similarly Restall(2014)) do not have a robust enough notion of the definitions of the connectives to achieve their goal of having all the logics in question share logical vocabulary. This is essentially a criticism of Beall and Restall’s claims about what the logical connectives can mean. Versions of this criticism are given in Hjortland(2013) 4, Griffiths(2013), Bremer(2013), Field(2009), Priest(1987),

Paseau(2007). The main claim in all cases is that, though Beall and Restall claim that they have defined the connectives in such a way that their meaning is preserved across logics, their definition actually results in a meaning shift when the class of cases in question shifts.

Though not as apparent in the literature, these types of criticisms can also be levied against

Restall’s proof-theoretic pluralism. I will not consider this objection further now, as it will

4As a response to this type of criticism, Hjortland presents a logic which has two distinct entailment relations. His position “combines more than one consequence relation in a single logical theory”(Hjortland, 2013, p 11). If effect, he combines two different non-classical logics, a paraconsistent logic and a three valued logic. Although interesting from a technical point of view, the project of combining two entailment relations in one logic using Hjortland’s method cannot go very far. We cannot, for example, combine classical and intuitionistic logic in this manner. I will not explore this possibility further.

23 occupy most of the discussion in Chapter 3.

Keefe(2014) criticizes Beall and Restall’s notion of endorsing. Beall and Restall adopt this notion in order to demonstrate how a dialethiest might still endorse logical pluralism.

Roughly, one strongly endorses a logical consequence relation when “one takes it to be an instance of GTT...where the actual case is in the domain of its quantifier” (Beall and Restall,

2006, p 82). One weakly endorses a logical consequence relation when “one takes it to be an admissible instance of GTT” (Beall and Restall, 2006, p 83). This means that a dialethiest can only weakly endorse most logical consequence relations on this version of pluralism, since she thinks that the actual case in inconsistent. For Keefe, weakly endorsing a logic comes to merely “recognizing it as a logic” (Keefe, 2014, p 9). Thus, weak endorsement is not really acceptance of GTT-style logical pluralism. For Keefe, strong endorsing fares no better, since you can be said to strongly endorse any logic contained in one you actually think is true. Thus, someone could be said to strongly endorse a three-valued logic if they strongly endorse classical logic, but hold that there are no sentences with non-classical truth values. This, for Keefe, is obscure, and she holds that the notion of endorsing can thus do no work for Beall and Restall. If she is right, and we need to be able to (weakly or strongly) endorse the logical consequence relations generated by instances of GTT to be pluralists, then we cannot be pluralists.

Another criticism lodged against Beall and Restall is that they have an inappropriate notion of what the settled core of logical consequence is. Beall and Restall take it that for a logic to be right, the logical consequence relation needs to be necessary, normative and formal, and satisfy GTT. Several authors disagree. Criticisms of this kind can be found in Griffiths(2013), Bremer(2013),Paseau(2007) Bueno and Shalkowski(2009), Woodward

(2008), and Field(2009).

Finally, Beall and Restall claim that they are not relativists about logical consequence, or about truth. Read(2006b), Burgess(2010), and Priest(1987) all disagree. It seems, at least on one front, they are right. Beall and Restall are at least relativists in one sense:

24 logical consequence is relative to the cases in question. More contentiously, it seems there are other potential relativisms in the offing. If those who claim that Beall and Restall have to allow the meanings of the connectives to change every time the cases in question shift, then there will be a relativism to languages. More pressingly, it is possible that

Beall and Restall must be relativists about truth. They claim that all three of their logical consequence relations are truth preserving. However, in order to be pluralists, they think that relevance and intuitionistic logic preserve less than classical logic. Thus, there might be some argument, say from α to β, which is valid in one way and not in another (this is related to Priest’s charge). If this is the case, then since both are truth preserving, if I know α is true, can I infer β? It seems that the answer might have to be that β is true in one sense (and hence inferable), and false in another sense (and hence not inferable). If this is the case, then Beall and Restall must be relativists about truth. As one might expect, they do not think they are truth-relativists. Beall and Restall claim that this is a pluralism about entitlement to infer. Thus, we might say, not that β is true in different senses, but that we are entitled to infer it in one way, and not in another.

Beall and Restall(2006) reopened this very important debate about logical pluralism.

Though the position developed has several insurmountable problems (as will become clear in chapter 3), it was critical to the reassessment of the debate. For this reason alone,

GTT-pluralism deserves much attention.

Relative to the Context

Stewart Shapiro motivates his conception of logical consequence by considering a Hilbertian perspective, that “consistency is the only formal, mathematical requirement on legitimate theories” (Shapiro, 2014, p 88). Since consistency is defined relative to a logical consequence relation, if we relax what counts as a logical consequence relation from something monistic

(ex: classical consequence) to something pluralistic, we can generate more consistent and

25 legitimate theories. This may be an instance of SLP because sometimes the consistent and legitimate theories can share logical vocabulary.

For Shapiro, logical consequence, at least logical consequence as it is used for mathemat- ics, is relative to structure (where a structure is characterized by any consistent mathemati- cal axiomatization). Because the axioms of both classical and constructive analysis systems are consistent relative to classical and intuitionistic logical consequence respectively, they generate different structures. Each structure calls for a different logic, a logic that is relative to that structure. Thus, more than one account of logical consequence must be correct for

Shapiro, because there are at least two distinct structures calling for distinct logics.

Shapiro makes a similar move with natural language. Shapiro suggests that “a formal language is a mathematical model of natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects, a set of differential equations is a model of the growth of bacteria in lakes and streams, and a Turing machine is a model of an algorithm or computing device” (Shapiro, 2011, p 537). On this conception, each conversational context (where conversational contexts include at least mathematical structures, as described above) determines the correct logic. Logic is relative to context.

Shapiro’s view is a pluralistic view. In order to show this, it suffices to show that there are at least two situations which are best modeled by different logics. This, we can show.

Further, Shapiro claims that

For some purposes...it makes sense to say that the classical connectives and quan- tifiers have different meanings than their counterparts in intuitionistic, paracon- sistent, quantum, etc. systems. In other situations, it makes sense to say that the meaning of the logical terminology is the same in the different systems. (Shapiro, 2014, p 127)

Shapiro’s two examples come from comparing classical analysis and smooth infinitesimal analysis,5 and asking whether the logical systems required for each have connectives which

5Smooth infinitesimal analysis (SIA) is an intuitionistic analysis system, in which all functions are smooth. Importantly, it is such that 0 is not the only nilsquare (elements whose square is zero, i.e. elements x such that x2 = 0). This is because every function is linear on the nilsquares. From this, it follows that 0 is

26 mean the same thing or something different. In the first scenario, we compare the systems when we are interested in differences between the logics themselves. In the second scenario, we compare the systems in terms of their mathematical consequences. For example, in the first case, two people might be comparing axioms of each system; they might discuss whether the existence of nilsquares is possible. In the second, they might be comparing whether both systems prove some theorem; they might discuss whether both systems prove the intermediate value theorem.

This is a version of SLP, since Shapiro claims that in the first case, “[it is] natural to speak of meaning shift” (p 128), while in the second case, “it is more natural to take the logical terminology in the different theories to have the same meaning” (p 130). Thus, we have two scenarios which call for distinct logics, and in which those distinct logics sometimes have the same language and sometimes do not.

Continuum-Many Non-Classical Logics

Cook(2013) takes a logic to be correct exactly when “for any way of interpreting the non- logical vocabulary, the logic validates a particular argument in the formal language if and only if the natural language statement corresponding to the conclusion of that argument is a logical consequence of the natural language statement corresponding to the premises of that argument” (p 6).

Cook also holds that the law of excluded middle is not valid, for broadly Dummettian reasons. However, even having dismissed the law of excluded middle as valid, he still holds that intuitionism is not yet determined as the one correct logic. For Cook, there are not the only nilsquare even though there are no nilsquares distinct from 0. This would be inconsistent in classical logic (because of the validity of LEM), and so intuitionistic logic is required. More formally, in a classical system, the sentence ¬∀x(x = 0 ∨ ¬(x = 0)) is a contradiction. In an intuitionistic logic, since the law of excluded middle is not valid, the sentence can be true. Importantly for us, the SIA system has a very simple and straightforward proof of the fundamental theorem of the calculus (that the area under a curve corresponds to its derivative). Rather than, as usual, taking approximations of the rectangles under a curve as they approach a width of 0, we take a rectangle under the curve which has the width of a nilsquare. No approximations are necessary, and we do not need the concept of “approaching zero”. See Bell(1998) for more details.

27 continuum-many ways to eschew the validity of the law of excluded middle, and many of the logics thereby generated will satisfy his correctness principle. Cook(2013) claims that the usual intuitionistic reasons for thinking rejecting the law of excluded middle implies that intuitionism is the one right logic in fact all show merely that classical logic fails, and not that intuitionism is right. He holds that this is true whether the argument against LEM is motivated by arguments from harmony, arguments from knowability, and/or arguments from determinacy. In effect, what these arguments show is that there must be at least one formula, P , for which P ∨ ¬P is not always valid, and not, as is often claimed, that this is the case for all P .

Interestingly, if Cook is right, this may be our first case of a pluralism which is not relativist. All the logics he considers will be right, no matter what context/language/etc we use.

To Account for Possibility

Bueno and Shalkowski(2009) also have a very particular account of what makes a logic right.

According to them, Beall and Restall erred on their characterization of necessity in their conception of logic. For Bueno and Shalkowski, Beall and Restall’s notion of a case does not exhaust all possibilities. This is an issue for them, and Bueno and Shalkowski suggest solving it by taking possibility, rather than case-hood, as primitive. Thus, “an argument is valid if and only if the conjunction of its premises with the negation of its conclusion is impossible” (Bueno and Shalkowski, 2009, p 307). They call this the modalist account of logic consequence. They claim we can still be pluralist on this account, because rather than logics being relative to cases, the pluralist can “characterize the appropriateness of distinct logics in terms of subject matters” (p 308). The pluralist can, for example, talk about the possibility of contradictions without having to assent to the fact that there are any true contradictions. If Bueno and Shalkowski are correct, then this would be an advantage

28 over Beall and Restall, who have to introduce notions of weak and strong endorsement to accommodate the possibility (but not truth) of contradictions.

2.4 Logical Instrumentalism

The position which I advocate for in this dissertation, that the right logic is relative to the goal at hand, is a logical pluralism. Further, it is a pluralism where we sometimes have more than one correct logic in a single language, and sometimes we do not. In this sense, it can be seen as something of a combination of the Carnapian and Shapiroian positions. In

Cook’s terms, it is either a strong PLP or an SLP. Corresponding connectives in distinct logics can share a meaning, as in SLP, but the only thing that makes a logic right is its suitability to a goal, in the sense of strong PLP. In this sense, the right logic is relative to a goal or purpose.

29 Chapter 3

Logical Monism from a Pluralist’s Perspective

Logical monists all hold that there is exactly one right logic. On this they agree. They do not, however, agree on what the purpose of logic is, what it is that makes a logic right, or which logic is right. There is much disagreement on these fronts.

In this chapter, I will consider two typical purposes that monists claim logic has. I will show that, taking the motto that we ought not to legislate to a proper, functioning, science, these two purposes each lead to a pluralism about logic. That is, if we think that the purpose of logic is one of the two which I consider here, then we ought to be logical pluralists rather than logical monists.

I will start by considering the purposes of regimenting natural language, or formalizing the natural language consequence relation. As examples of this type of monist, I will consider Michael Dummett, Alan Ross Anderson and Nuel Belnap, and Graham Priest.

We will see that arguing that monism is right on this front comes up against the science of linguistics. Second, I will consider the purpose of regimenting mathematics or science.

I will consider Arend Heyting and L.E.J. Brouwer, Hilary Putnam, and Neil Tennant as examples of this type of monism. We will see that requiring monism on this front requires someone to legislate to the mathematics or physics communities.

As a general disclaimer, it ought to be pointed out that the views I present on behalf of the monists considered are rational reconstructions, rather than historical exegeses. I take it the discussion is still of interest, as many monists might hold similar views.

Generally, I take the upshot of this chapter to be the following. If we want to adopt the purposes for logic which monists typically take to be sacrosanct, while at the same time

30 assuming that properly functioning sciences need no legislation from philosophy, we are in a quandary. I conclude by suggesting that we should just bite the bullet and accept a logical pluralism.

3.1 Why Logic Ought Not to Legislate to Properly Function-

ing Sciences

The fundamental assumption I rely on to make most of my arguments in this chapter is that philosophers ought not to legislate to properly functioning sciences. Because of its fundamentality to the project, it bears spelling out in a bit of detail.

The “sciences” I have in mind are a broad group. They include physics, chemistry, biology and the typical sciences grouped with them in Schools of Science in university programs. However, “science” as I am characterizing it here also includes disciplines less traditionally thought of as sciences, like linguistics (as practiced in linguistic departments, at least), and theoretical physics (as practiced in physics departments). In general, I take it that if you can be a philosopher of X, then X is a good candidate for being a science

(with exceptions, of course).1

Legislating to these sciences comes down to telling the science they are wrong about their practices on philosophical grounds. I take it that as a point of humility, we ought not to do this. At the very least, we ought not to do it if we can avoid it, and I will demonstrate in the next sections that for certain logical monists to avoid it, they must be logical pluralists.

1It is not clear whether on this picture philosophy itself comes out as a science, but this will have no effect on the project. If one does consider philosophy as a science, then the claim that we ought not to legislate to properly functioning sciences ought to be rephrased as “those who are not practitioners of science X ought not to legislate to science X”.

31 3.2 Logic and Natural Language

One common purpose attributed to logic is that it is meant to capture, in a formal way, the consequence relation in natural language, or that it is meant to regiment deductive reasoning in natural language. In this section, I will show how the claim that logical monism is true, and that logic’s purpose has something to do with natural language runs afoul of linguistics.

If one holds this position, one also has to hold that there is exactly one right meaning for the equivalents of the logical connectives in natural language, and linguistics offers much evidence to the contrary.

3.2.1 The Players

Here, I consider Dummett, Anderson and Belnap, and Priest.

Dummett

Michael Dummett claims that the principle of bivalence is illegitimate.

It is when the principle of bivalence is applied to undecidable statements that we find ourselves in the position of being unable to equate an ability to recognize when a statement has been established as true or as false with a knowledge of its truth-condition, since it may be true or false in cases when we lack the means to recognize it as true or false. (Dummett, 1976, p 101)

Dummett’s arguments against bivalence stem from three key assumptions (the following is from Tennant(1997, chapter 6)):

(A) The meaning of a declarative sentence is its truth-conditions (B) To under- stand a sentence is to know its meaning (C) Understanding is fully manifestable in the public exercise of recognitional skills (Tennant, 1997, p 176-7)

The ability to recognize and understand a sentence plays a critical role in whether it is actually truth-evaluable. Unrecognizable/non-understandable sentences are not candidates for truth values. Thus, Dummett says

32 Precisely because a realistic theory forces so large a gap between what makes a statement true and that on the basis of which we are able to recognize it as true, the theory has difficulty in explaining how we derive our grasp of the latter from a knowledge of the former. (Tennant(1997, p 178), originally from The Interpretation of Frege’s Philosophy, p 71)

These three features of Dummett’s view require that all truths are knowable, and so realistic theories of truth, those theories which allow some truths to be unknowable, are not proper theories of truth. Thus, realistic theories are illegitimate.2 The One Right Logic, then,

cannot be bivalent. Since classical logic, under suitable assumptions (for example, assuming

Tarski’s T-scheme), is bivalent, classical logic cannot be right.

How do we determine which logic is right, then? In summation of this position, Dummett

states that

The intuitionistic explanations of the logical constants provide a prototype for a theory of meaning in which truth and falsity are not the central notions. The fundamental idea is that a grasp of the meaning of a mathematical statement consists, not in a knowledge of what has to be the case, independently of our means of knowing whether it is so, for the statement to be true, but in an ability to recognize, for any mathematical construction, whether or not it constitutes a proof of the statement... Such a theory of meaning generalizes readily to the non-mathematical case. Proof is the sole means which exists in mathematics for establishing a statement as true: the required general notion is, therefore, that of verification. On this account, an understanding of a statement consists in a capacity to recognize whatever is counted as verifying it, i.e. as conclusively establishing it as true. (Dummett, 1976, p 70)

Our theory of use, which is generated by looking to natural language to see how native

speakers use words, tells us that we need to look to how logical constants are used in order

to know what they mean, and that the meanings of these connectives determine which logic

is the one right logic.

If we substitute verification (or knowable truth) for truth, according to Dummett, then

we need only to look to the meanings of the logical connectives to find the one right logic.

2As shown in Tennant(1997), Dummett’s arguments here do not prevent a certain type of realism: that of the G¨odelianoptimist, who is a classical monist, but holds that any statement will be decidable in some system (importantly, the optimist does not hold that there is one system which will decide all statements). See (Tennant, 1997, chapter 7) for a solution to this.

33 The meanings of the connectives are given by their use, and their use for Dummett is given by their inferential rules. Thus, the one right logic will be that which corresponds exactly with the inferential rules for the connectives. This, for Dummett, is intuitionistic logic.

In summation, Dummett holds that a logical theory flows from a theory of meaning, and that a theory of meaning flows from a theory of use. We know how to use a word when we know how it contributes to the assertability conditions of a sentence in which it occurs. Thus, a word’s meaning is given by its rules of use. Importantly, this applies to the logical constants as well. We simply identify the constants, identify their rules of use, and collect those to build our logic. In this way, logic “falls out of” a theory of meaning: it is the formal theory generated by whatever meanings the logical connectives get on the theory of meaning in question. The right theory of meaning is inferential, and thus the right connectives are defined inferentially. According to Dummett, the best logic to study the inferential connectives is intuitionistic.

Anderson and Belnap (and Friends)

For Alan Ross Anderson and Nuel Belnap (and later, J Michael Dunn and others) the heart of logic is the notion of “if...then...” as it occurs in natural language, and the relationship between that notion and the formal connective “→”(Anderson and Belnap, 1975, p 3).

Once we have identified the correct meaning for “if...then...”, which will also give us the correct meaning for “entails” (see quote below) they claim we will have the right logic. For

Anderson and Belnap, the purpose of logic is the study of “if..then...” and the corresponding

→ operator.

Anderson and Belnap wish to develop a logic with a conditional operator (→) which does not fall prey to what they call the “paradoxes of material implication”. For example, the conditional connective in classical logic is symmetric, that is (A → B) ∨ (B → A) is

logically true. This, they take it, is paradoxical, and does not latch onto the conditional we

34 actually use in natural language. In their words, “[material implication] is no more a kind of implication than a blunderbuss is a kind of buss” (p 5).

They claim that they “wish to interpret A → B as A entails B” (p 7), and moreover that

“valid inferences require both necessity and relevance” (p 23). This implies that the right logic must be relevant. That is, the conclusions we infer must be related in the right kinds of ways to the premises. This allows them to avoid the paradoxes of material implication, which they think gives them a notion of implication closer to the actual one. We can no longer derive (A → B) ∨ (B → A), since it may not be the case that A and B are related in the appropriate ways. In this way, they seem to think they are onto a notion of natural language implication. The claim seems to be that we never assert any of the paradoxes of material implication in natural language (certainly, I have never heard anyone make use of the fact that the natural language conditional might be symmetric), and so the material implication conditional cannot be the right one. They take themselves to have constructed a much better approximation. They develop several logical systems from these principles.

They claim some relevance system is right.

Priest

Graham Priest holds that the right logic is a paraconsistent logic. There are two steps in his argument.

First, Priest claims that “there are numerous pure logics” (Priest, 1987, p 164). He is a pluralist about pure logics, which are simply formal systems. However, he holds that

“the canonical application of logic is to reasoning” (Priest, 1987, p 165). He acknowledges that there are several applications to which a logical system can be put, but the most important one is the application to reasoning. That is, the ultimate purpose of logic is to provide us with norms for good reasoning. He claims that the canonical application of logic is “what follows from what” (Eastern APA 2015 comments at Shapiro’s Author Meets

Critics session). When we reason, then, we try to figure out what follows from what. The

35 best theory of this will be the one true logic. Moreover, he claims that the best logic for reasoning is a paraconsistent logic, since it appropriately accommodates paradoxes, amongst other things.

When it comes to the meanings of the connectives, Priest holds that there is exactly one natural language meaning for each connective, and that the formal meaning of each connective is given by its (unique) natural language meaning (see Priest(1987, p 199)).

3.2.2 The Problem

Here is the pluralist rub: what we have for all three positions is a single logic which relies on each connective in natural language having a single meaning, and thus relies entirely on a particular theory of meaning, or on there being exactly one meaning attributed to the connectives by any such theory. At the very least, they must say that even though there may be different kinds of meaning, there is only one such kind which is relevant to logic.

The monists must insist either that there is exactly one type of meaning period, or that there is some relevant difference between the meanings attributed to the natural language connectives by their logical system and the other candidates for natural language connective meaning.

As a first step, it is relatively simple to show that the first horn, that there is exactly one type of connective meaning, legislates to sciences in a problematic way. According to our best science of meaning, linguistics, there may be many other theories of meaning, and many other meanings which are attributed to the connectives. For example, when considering

Dummett’s theory, we have to rule out Montague semantics, which many linguists take to be the truth, since Montague semantics is inherently truth-conditional. Montague semantics does not proceed from an inferentialist theory of meaning, and since Montague semantics is a theory used by the experts in meaning (the semanticists in linguistics departments), it ought at least be taken seriously.

36 Moreover, there are many uses of the logical connectives which require more than one meaning of a connective of any given type. For example, there are some uses of conjunction where conjunction does not commute. These can be found in dynamic semantics, where

“Tom and Mary got married and had a baby” is not equivalent to “Tom and Mary had a baby and got married”. Assuming that there is exactly one conjunction, and that it is the usual one, rules out dynamic semantics as a logical system. It rules out using conjunction as it is modeled in dynamic semantics, because it is not really logic. This, I claim, is illegitimate: if we want to find the right theory of meaning, we ought to take seriously what

the linguists do. Thus, on the first horn, Dummett, Anderson and Belnap and Priest are all

in a quandary. Priest thinks there will be exactly one right meaning to each connective in

natural language, and Anderson and Belanp think that at least the arrow has one natural

language meaning. Dummett thinks that Montague semantics is not a theory of meaning.

If we take linguistics seriously, though, this cannot be the case.

Thus, if we take linguistics seriously, we cannot be logic monists and think that logic

latches onto the single meaning of the connectives in natural language.

The second horn must be dealt with differently: if there is something pertinent which

distinguished logical meaning from ordinary, run of the mill, linguistic meaning, then what

is it? It cannot depend on how we use the connectives, since this would default to the

results in linguistics, so Dummett and Anderson and Belnap will be at a loss. Perhaps it

is about how we ought to use the connectives. This is something Priest can accept, and

something Dummett and Anderson and Belnap can adjust their theories to accommodate.

Here we have something more promising. Since linguistics is a mostly descriptive science,

this normative claim will not be stepping on their toes. It will, however, require insisting

that there is exactly one right set of norms which we follow when we reason.

37 3.3 Logic and Mathematics and the Sciences

A second common claim is that logic is for regimenting mathematical or scientific reasoning.

As in the case with logic and natural language, what we find here is that assuming that logic is for regimenting deductive reasoning in mathematics or science and that there is exactly one right logic runs afoul of the mathematical or scientific deductive reasoning to be regimented. The regimentation by exactly one logic will rule out certain useful mathematical or scientific tools as legitimate, which is a clear act of legislating to a properly functioning science. Here, I take the claim that logic regiments science or mathematics to be something like the claim that logic provides us with norms for reasoning about the subject.

3.3.1 The Players

Here, I consider Heyting and Brouwer, a time-slice of Putnam, and Tennant.3

Heyting/Brouwer

For Arend Heyting and L.E.J Brouwer, the question of which logic is right arises only in the context of analyzing what the right approach to mathematics is. Thus, the purpose of logic is that it be applicable to mathematics, where that mathematics is practiced in the right kind of way.

The logic Heyting and Brouwer hold is correct is intuitionistic logic. This is because

“mathematical objects are by their very nature dependent on thought” (Heyting, 1983, p

52). This leads them to reject the law of excluded middle (LEM) as a valid logical law.

3One might wonder why I do not consider any classical logicians here. In effect, I have run into a bit of a problem. It seems that most of the prominent figures in the literature who take logic to be related to science and mathematics advocate for classical logic as the right logic without much argument. The claim seems to be something like “classical logic is the logic we use to produce the most mathematics, so it must be the right logic.” See, for example, Burgess(2015) as an example of such an argument. This position, though it does claim that classical logic is right, does so on a primarily descriptive basis, rather than a normative one, and so I will not consider it here. Notice, though, that it falls prey to a similar legislation problem: it requires claiming that constructive mathematics is illegitimate, or lesser. For similar reasons, then, we must either dismiss the view, or assume it generates a pluralism.

38 Since, they hold, we can only assert LEM for a particular proposition if it “has been proved or reduced to a contradiction” (Heyting, 1983, p 59), and we cannot prove or reduce to a contradiction any arbitrary sentence (specifically, the undecidable ones), we must reject

LEM as a valid law. This follows from an observation that “the point of view that there are no non-experienced truths and that logic is not an absolutely reliable instrument to discover truth has found acceptance with regard to mathematics much later than with regard to practical life and science” (Brouwer, 1983a, p 90). This, I take it is an early anti-realism: mathematics, as opposed to being about some sort of third realm of abstract objects, is rather about mental entities. As such, we cannot ask whether what we cannot know is true or false. For Brouwer and Heyting, classical logic is not an absolutely reliable instrument for discovering truth, since there are no unverifiable truths, and if we adopt classical logic it follows that there must be (all propositions are either true or false).4 Thus,

Brouwer and Heyting claim we must adopt intuitionistic logic, where LEM is not a valid law, but which is nonetheless strong.

Putnam circa 1968

Hilary Putnam(1968) likens logic to geometry. For Putnam, a change in the right logic is something like a change between using Euclidean and non-Euclidean geometry. Geometry is used to study space, and once we realized that the theory we were using (Euclidean) was not sufficient, we revised it (to a non-Euclidean one) in order to better track the way the world is. Logic is the same: logic is, in some sense, for studying the world (at least, for studying correct inferences in the world), and we revise our logical theory as we learn more about how the world is and about how we ought to think about it.

The purpose of logic is to regiment our best science. Putnam holds that logic needs to be applicable to quantum physics (mechanics in particular), since this is our best science.

4Except, again, for the G¨odelianoptimist. See footnote2.

39 To this end, he proposes that the one right logic is quantum logic. The distributivity laws for conjunction and disjunction do not hold.5

There is one thing to take note of here: the right logical theory is open to revision.

Since logic, like geometry, is used to study the world (or, in this case, regiment the best science), when we discover new things about the world, or when we learn that our science needs to be revised, our logic and geometry may need to be revised as well. This time slice of Putnam thinks that quantum logic is the one right logic, and likely holds that we ought to revise our logic so that our uses of the distributivity principles in ordinary contexts are non-logical.

There are two ways to interpret what revision means here. Lets consider as an example revising the logic so that the distributivity principles no longer hold. On the first way of spelling out the meaning of “revision”, it was always the case that the distributivity principles were invalid, and now we know. On the second way of spelling out the meaning of “revision”, something has changed about the world: the distributivity principles were once valid, but now are not. On the first, we have learned something about the world, and on the second, we have changed something about the world (on the condition that logic is a part of the world). It is not entirely clear which position Putnam holds, though it is likely it is the first, since the first corresponds more appropriately to the revision we see in the sciences. Since logic is for studying the world, when we learn something new about the world we may learn that the logic we had been using was wrong all along, and not that the validities had changed.

5One of the distribution rules is as follows: A ∧ (B ∨ C) ⇒ (A ∧ B) ∨ (A ∧ C). Putnam claims in quantum physics, this cannot hold. Take an example: it is known that every electron is either spin up or spin down along any given axis. It is also known that you cannot, at one and the same time, measure an electron along two distinct axes. Thus, we let A be the proposition that a given electron, say e, has been measured as spin up along the x axis. Let proposition B be that e is measured as spin down along the y axis, and proposition C be that e is measured as spin up along the y axis. Then A ∧ (B ∨ C) is true, since we have that A is true because e is spin up on the x axis, and B ∨ C is true because we know that for any axis, e will be either spin up or spin down. However, neither A ∧ B nor A ∧ C is true, because we know that we cannot measure one electron along two axises. Thus, the disjunction is false, and the distributivity law does not hold.

40 Ultimately, for Putnam (at least at this stage in his career) the purpose of logic is to regiment our best science, and the right logic is quantum logic.

Tennant and Core Logic

For Neil Tennant, the purpose of logic is regimenting mathematical proofs and deductions.

The best logic suited for constructive mathematics, Tennant claims, is his own Core logic, an intuitionistic-relevance system. The best logic suited for regimenting classical mathematics, he claims, is his Classical Core logic, or Core logic with the addition of dilemma or Classical

Reductio. In this way, Tennant is already a (limited) logical pluralist, and on board with our project: both Core logic and Classical Core logic are right, depending on the application.

Core logic is proposed as an alternative to classical, intuitionistic and Anderson & Bel- nap relevant logics in Tennant(1997) and Tennant(2005). Core logic has introduction and elimination rules similar to those found in Gentzen (but disjunction elimination and conditional introduction altered) but none of the strictly classical rules (classical reductio, double negation elimination, etc.), and without the rule EFQ. In effect, core logic is both intuitionistic and relevant. The system validates disjunctive syllogism, unlike that of Ander- son and Belnap, but is not unrestrictedly transitive in the object language (a meta-theorem is proven to get some instances of transitivity, see below).6 The goal of core logic is to pare

6Tennant notes (personal correspondence) that he gets “all the needed instances” of transitivity. He writes (1) Core Logic proves every intuitionistic theorem; reveals every intuitionistic inconsistency; and secures every intuitionistic consequence of any intuitionistically consistent set of premises (Tennant(2012)); and that (2) Classical Core Logic proves every classical theorem; reveals ev- ery classical inconsistency; and secures every classical consequence of any classically consistent set of premises (Tennant(2015)). This, as you know I argue, makes the Core systems com- pletely adequate unto all legitimate methodological demands in science and in mathematics! In the case of (1) and (2), he is certainly correct. I have used “some” here, because he does not get all of the classical instances. The procedure given by Tennant takes as input the premises in the proofs to be “cut”, and produces as output either the original sentence, or a contradiction. In particular, the procedure, when applied to a set of contradictory premises, produces a contradiction rather then what we might have concluded from the original classical proof. If one thought that EFQ was made use of in mathematics (which Tennant holds is not the case), then this procedure would in fact not be adequate to the methodological demands of mathematics. I will not pass judgment on whether EFQ is ever actually used in mathematics here. For this reason, I will withhold judgment about whether Tennant gets all of the necessary instances of cut.

41 down the premises and conclusion as much as possible to avoid irrelevancies and to make epistemic gains.

Epistemic gain is achieved, claims Tennant, because we either learn from a Core de- duction that a consequence that follows from a subset of a given set of premises does so relevantly, or we learn that our original premises were inconsistent. In this way, Tennant claims that “our canon of deductive reasoning has been relevantized; but our powers of reasoning have been left intact” (Tennant, 2005, p 725). This is essentially what the meta- theorem about transitivity proves: the theorem states that for each step in a proof where we would normally apply cut (but cannot, because the Core system is not unrestrictedly transitive at the object-level), there is an effectively determinable proof of our desired con- clusion from a subset of the premises involved, or there is an effectively determinable proof that the premises involved are inconsistent. In this way, Tennant claims we make epistemic gains: we are better off than we were before, because we either know something follows rel- evantly from a (possibly smaller) premise set, or we know we started with a contradictory set of premises.

3.3.2 The Problem

The problem with the claim that logic regiments science or mathematics is that there seems to be more than one way to go about regimenting science and mathematics.

The purpose of logic for Tennant, for example, must be more than merely regimenting mathematical proofs and deductions, as these are typically thought to make use of cut and EFQ; the purpose of logic must be regimenting mathematical proofs and deductions in an optimal way. Here, there are two ways to spell out “optimal”. The first means something like “with as few premises as possible, while achieving epistemic gain”. The second means something like “as close to the way the mathematician actually proceeds as possible.” What Tennant has to assume is that there are only two ways to regiment mathematical practice. However, if we take the claims that there might be more than one

42 set of norms we can subscribe to (see, for example, Field(2009)), then we could very well have myriad ways to do so. If the set of norms in play is different, for example, if the epistemic goals do in fact change the strength of the normative demands, then there may be a different set of norms which regiments reasoning in those cases. Consider, for example, a mathematics professor who is trying to impart to her introduction to calculus students how to reason about integrals. She is permitted to utter certain falsehoods, and ignore certain counter-examples to get her students to come to understand. However, if this same professor writes a paper for publication in which she ignores those same counterexamples, this will be a mistake. Thus, if regimenting mathematics comes down to providing norms for reasoning about mathematics, and the norms for reasoning about mathematics are sensitive to epistemic goals, then there is no reason to think there are only two ways to regiment mathematics. Tennant has to assume that there are exactly two ways to regiment mathematical practice, which closes the door to other options.

Putnam, too, suffers from cutting off certain branches of science which we would have thought would be legitimate enterprises. Should it result that there are two, equally good, scientific theories (perhaps one for the really small things, and one for the really large things), and should it also turn out that these two theories are not best regimented by the same logic, then Putnam will have to claim that one of them is the best science, and one of them is somehow secondary. But this is odd: imagine the scene when a logician walks into a string theorist’s office and announces that her science is not the science of the world, because it is not regimentable by the logic used to regiment some other scientist’s work.

This would be absurd. Additionally, given the complexity of our world, it does not seem far fetched to assume that we really could wind up with two scientific theories, each describing properly different parts of the world, and each requiring different logics. As examples, consider quantum mechanics and economics. It seems they both describe the world, but that neither can be accounted for in terms of the other. If they are each best regimented by different logics, and we wanted to follow through on the Putnam position that logic is for

43 regimenting sciences, then we would be required to institute a logical pluralism. If Putnam decides to staunchly remain a monist, he must admit that this cannot happen. This, I take it, is good reason to think that his position is open to pluralism, so long as scientific inquiry is.

The Heyting and Brouwer case is a bit different, but follows a pattern in that it requires legislating to mathematics. In the Heyting and Brouwer case we must claim that unverifi- able truths are anathemas to mathematics. However, as mathematics is practiced, this is certainly not how it looks. Certainly, mathematicians seem to reason from negated univer- sals to existentials, and seem to make use of the law of excluded middle. If mathematicians do not think that unverified truths are problematic, then we ought at least to take that position seriously.

In general, if we take seriously the work that mathematicians and scientists do, then we ought to take seriously that both constructive and classical mathematics are legitimate enterprises, or that both quantum physics and special relativity are legitimate sciences. If we assume that there is exactly one way to regiment mathematics or the sciences, then we have to assume that there is only one legitimate way to practice each. This is illegitimate legislating to mathematics and the sciences.

3.4 Our Next Step

There is a simple and practical solution to the tension between the monistic purposes and our desire not to legislate to the sciences: we ought to stop privileging one (logical) purpose over the others! If we acknowledge that many applications and purposes are legitimate, then the debates amongst the monists reduce to one much simpler one: whether there is exactly one right logic for the purpose they find most interesting. Additionally, if we allow that for any purpose there may be more than one logic which is applicable, we reduce the debates amongst the monists to nothing. What we will wind up with, then, is something

44 like a mosaic rather than a melting pot. We will no longer force various purposes to be squashed into a logical framework in which they do not fit, but at the same time we will also have to give up on there being exactly one universally applicable logic.

It is interesting to note that there is still something to the monistic debate: it might reduce to a debate about which logic is best for a given purpose. The Priest debate, for example, might become a debate about which logic is right for a particular set of meanings of the logical connectives. The Tennant debate might reduce to a debate about how it is best to optimize mathematicians’ reasoning on the assumption that one of Tennant’s two ways to optimize is correct. Thus, there is still something to the monistic claims, they are just not as universal as we might have thought.

45 Chapter 4

Beall and Restall’s Logical Pluralism

4.1 Connective Meanings in Beall and Restall’s Logical Plu-

ralism

4.1.1 Introduction

JC Beall and Greg Restall 2006 spell out and defend a version of logical pluralism. A feature of their particular pluralism is that corresponding connectives in admissible logics mean the same thing. This means, for example, that negations in classical, intuitionistic and relevant logic are the same. This is in contrast to a “Carnapian” view, on which change in logic is a change in the meanings of the logical terms.1

Unfortunately, Beall and Restall have not provided a conception of the meaning of a logical connectives on which this is true. I will show that, as they describe it, negation in intuitionistic logic and negation in relevant logic cannot mean the same thing on any reasonable interpretation of their account of meaning. The fact of the matter is that they have not settled just what “same meaning” amounts to, and any reasonable way of spelling out what they say results in connectives which mean different things.

I will start by describing their pluralism, and what they take the logical connectives to mean. I will then show that, as defined, the negation connective in intuitionistic logic does not mean the same thing as the negation connective in relevance logic. Finally, I suggest

1In chapter5, I argue that this is not the best way to understand the Carnapian picture. For the purposes of this chapter, though, it suffices.

46 that, as described, they have no way around this problem. I close with some suggestions about what the truth of the matter might be.

4.1.2 Beall and Restall’s Pluralism

Beall and Restall characterize their logical pluralism as “the view that there is more than one genuine deductive consequence relation, and that this plurality arises not merely because there are different languages, but rather arises even within the kinds of claims expressed in one language” (Beall and Restall, 2006, p 3). Since they hold that the logical connectives are part of the language of a logic, they claim that logics which share connectives are “in one language” (or at least this is not ruled out by connective meanings). For a more detailed explanation of their view, see section 2.3.3 in chapter2.

What we need now, to answer the question of whether the connectives in these logics do indeed share meanings, is to know what they take the identity conditions of the meanings of the connectives to be. Unfortunately, they do not say much on this front. What they do say is

The clauses can [all] be true of one and the same [connective] simply in virtue of being incomplete claims about... [the connective]. What is required is that such incomplete claims do not conflict, but the clauses for negation do not conflict. The classical clause gives us an account of when a negation is true in a model, and the constructive clause gives us an account of when a negation is true in a construction. Each clause picks out a different feature of negation. (Beall and Restall, 2006, p.98)

This means that, somehow, there is one thing which is (for example) negation, and the various negation clauses given above all express different aspects of that one thing.

To make sense of this claim, we need to fill in more details. I take it that Beall and

Restall (or, at least Restall) want to take the meanings of the connectives on this picture to be maximal truth conditions, as described in Restall(2002). The maximal truth condition for a connective is something like the disjunction of all the clauses for the connectives in

47 the admissible cases. That is, the maximal truth condition for negation (assuming the only admissible logics are classical, intuitionist and relevance) would be something like

(M is a Tarski model and M, s |= ¬A if and only if it is not the case that M, s |= A) OR (w is a node in a Kripke structure and w |= ¬A if and only if, for all u such that w ≤ u, u 2 A) OR (s is a situation and s |= ¬A if and only if, ∀t such that sCt, t 2 A)

The meaning of negation is then the disjunction of all three negation clauses given above.2

If this is acceptable, then we need to ensure, as Restall claims, that there is no conflict amongst clauses where they overlap. Restall(2002) holds that “the two accounts [classical and relevant] agree where they overlap” (p 438). There are two questions we need to ask now: what, exactly, does it mean for two clauses to overlap? And do the intuitionist and relevant clauses agree where they overlap? I think the answer to the second question is no, but it depends in large part on our answer to the first. It is to this which I now turn.

4.1.3 Overlapping Clauses

The method that Beall and Restall use to explain overlap involves describing all three consequence relations in a situation semantics (Restall(1999) and Beall and Restall(2001)).

They state

We... provide a model of how these things could be... to show how our three different claims about the behavior of negation can be true together... A world [classical model] is a complete, consistent situation... [We] take all constructions to be situations... [that are] not only consistent, but explicitly so. (Beall and Restall, 2001, p 10-11)

Though this is ultimately the downfall of the view, we need the details to explain the problem. 2There is another way we can view this (given that Beall and Restall’s meta-language is probably classical, these will be equivalent). It might be rather than a disjunction, we have a conjunction of conditionals. The maximal truth condition for negation would then be (if M is a Tarski model then M, s |= ¬A if and only if it is not the case that M, s |= A) AND (if w is a node in a Kripke structure then w |= A ∧ B if and only if w |= A and w |= B) AND (if s is a situation thens |= ¬A if and only if, ∀t such that sCt, t 2 A) Since I take it these are equivalent in their classical meta-language, I will focus on the disjunctive form.

48 This method of explaining overlap involves restricting the clauses for the connectives in the situation semantics in certain ways to generate the clauses for the connectives in the other logics. At the very least, the road is simple enough for finding a classical consequence relation in such a semantics. We simply restrict ourselves to considering complete and consistent situations, and restrict compatibility to be the identity relation.

The case for finding an intuitionistic consequence relation in such a semantics is much harder. Beall and Restall(2001) claim we need to restrict ourselves to considering explicitly consistent situations, where each explicitly consistent situation makes ¬(A ∧ ¬A) true, for all A (Beall and Restall, 2001, p 11). Complete consistent situations (those which generate classical logic) are also explicitly consistent. Presumably, Beall and Restall hold that this generates the intuitionistic clauses for the connectives. In section 4.1.4, I will show that this is not correct.

To figure out whether two clauses for a connective overlap, we first, we need to assess what all the negation clauses have in common; what is it that makes them disjuncts in the maximal truth condition?3 As noticed in Hjortland(2013), all three negation clauses can be made to “fit” the following clause

for x ∈ D, x |= ¬A if and only if, ∀y such that xRy, y 2 A

(*)

Let’s look at precisely how this will work. First, we take the default to be that D is the entire

situation space, and that R is compatibility (we will be forced to loosen these requirements

later). This means that the default of (*) generates a relevant negation clause. Classical

3This is in part necessary to rule out “junky” connectives, formed by disjoining the “wrong” clauses to the maximal truth condition. For example, it is required to prevent having to claim that the disjunction of the clause for conjunction in Tarski models with the clause for the arrow in situations is a legitimate connective meaning. Without this, the maximal truth condition is not a good candidate condition for the meaning of any connective, and we can dismiss Beall and Restall on those grounds.

49 negation can then be recovered by letting D range over complete, consistent situations, and letting R be the identity relation. So far so good.

Recall that Beall and Restall(2001) claim that the intuitionist clause can be recovered by restricting our attending to explicitly consistent situations. In the language of (*), this means letting D range over explicitly consistent situations. They make no claims about changes in R.4

We are now in a position to articulate a definition for when two clauses overlap. Two clauses overlap when the intersection of their domains is non-empty. I think Beall and

Restall would agree up to this point. We can also see now that all three clauses overlap, since the intersection of situations with complete and consistent situations and/or with explicitly consistent situations will not be empty. This means what is left to determine whether all three negations have the same meaning is to assess whether or not they conflict where such overlap occurs.5

4.1.4 The Problem

I will focus on showing that the intuitionistic clause conflicts with the relevant clause for negation. I will demonstrate this by showing the addition of the intuitionistic clause to a maximal truth condition which already has the relevant clause as one of its disjuncts makes

4This method will be harder to make work for the clauses for the conditional in the three logics. Though there is an available clause which could be said to be the common core of anything called a negation, which I have called (*), it is not at all clear what such a clause would look like for a conditional. See Hjortland (2013) for more details about this line of argument. 5There is an immediate response available on behalf of Beall and Restall here (thanks to Alexandru Radelscu for the suggestion). Beall and Restall claim that what they are doing is providing a precisification of the vague notion of logical consequence. So why not also assume that they are giving a precisification of a vague account of negation? Then, we would not need such a precise definition of negation, as I have suggested with maximal truth conditions and (*), and the examples I give in section 4.1.4 can all be dismissed as borderline cases or outliers. However, I think there will still be a problem here, though it is different from any I will addressed in this chapter. There will be a tension between their firm claim that the connectives must mean the same thing, and the vague meaning of those connectives. In order to make the first claim, we must assume that there is something precise about the nature of the connectives, while to make the claim about vagueness, we must dismiss this preciseness. In effect, I think that if we want to pursue the claim that the connective meanings are vague, we are better to adopt something like the tentative conclusion I give in section 4.1.6.

50 negation arbitrary. If we add the intuitionist clause to the maximal truth condition which already has the relevant clause, then we will be able to add clauses to the maximal truth condition which we ought not to consider part of the meaning of negation. I take it, showing that the relevant negation becomes arbitrary upon the addition of the intuitionist clause shows that the clauses conflict. So, both clauses be cannot part of the meaning of negation.

In order to assess whether the intuitionistic clause for negation conflicts with the relevant clause, we need to settle just what stages-as-situations are. Beall and Restall claim that they are merely explicitly consistent situations. If stages are merely explicitly consistent situations, then changes in the relationship R in (*) need to be made. I call this horn

“Option 1”. If they are not explicitly consistent situations, then there will be no overlap between the intuitionist negation clause and the relevant negation clause. I call this horn

“Option 2”. I will consider each option in turn, and show that both options make negation arbitrary, and thus no matter how Beall and Restall go, they will not be able to succeed in their connective meaning project.

Suppose first that stages are merely explicitly consistent situations. Then on the face of it, we have an immediate problem. Part of being a stage is being a node in a Kripke struc- ture. One and the same explicitly consistent situation can occur in two different Kripke structures. If the future of that situation in each structure is different, then this makes different complex sentences true at that very situation (if A occurs in the situation’s future in one structure, but not in the other, for example). Nothing about being an explicitly consistent situation restricts our “stages as situations” in a way which prevents such situa- tions from occurring in two distinct Kripke structures (see Appendix for more details). In order to prevent this, we not only need to ensure that D is restricted to explicitly consistent

situations, but also that R is restricted so that “stages as situations” are only related to

other “stages as situations” which are intuitively in the same Kripke structure.

To do this, we need to have R parallel the ≤ relationship. We need to restrict R in (*) to be some form of a reflexive transitive relation between the right situations. However,

51 having the parallel between R and ≤ means that R cannot be symmetric. Were the parallel relation to be symmetric, we would lose any notion of constructibility. The ≤ relation is supposed to be something like a “constructed-from” or “built from” relation, so that we have w ≤ v just in case v is “built from” w. Suppose that the relationship between explicitly consistent situations (the “built-from” relationship) were indeed symmetric. Then anytime v was “built from” w, w would also be “built from” v. We could never construct anything new, so to speak, because each explicitly consistent situation would be built from those before and after it. Everything would be true at the first situation in the ordering. So, for

(*) to properly capture the intuitionistic clause, R cannot be symmetric. If we want the negation clause for intuitionistic cases to properly fit (*), we need to restrict D to explicitly consistent situations, and R to some subrelation of compatibility which is not symmetric.

If we allow these two restrictions, though, we have a problem. If all it takes to be part of the meaning of negation is to be “moldable” into (*) (where R can be any sub-relation of

C) then lots of things will fit this mold.6 Not all of them will be things we want to include as part of the meaning of negation.

For example, let D be anything we like, and let R be empty.7 The empty relation is certainly a subrelation of compatibility. The clause here is ∀d ∈ D, d |= ¬A if and only if,

∀t such that dRt, t 2 A. Since R is empty, this means the right hand side of this conditional is always vacuously satisfied. The “empty-R” negation clause becomes d |= ¬A for any A and any d ∈ D. This means that negated sentences are always true. Once again, if Beall and Restall are right, then this operator, which when appended to anything produces a

6Additionally, of course, the logical consequence relationship will have to satisfy GTT and be necessary, normative and formal. For Beall and Restall, necessity is truth preservation in all cases, normativity is the ability to go wrong if you accept the premises and reject the conclusion of a valid argument, and formality is either schematicity or one of providing norms for thought as such, indifference to identities of objects, or contentless-ness (Beall and Restall, 2006, chapter 2). 7This logical consequence relationship is also necessary, normative and formal, and satisfies the GTT. It satisfies the GTT since the cases in question are just whichever situations are “blind”. The empty-R relation is just a sublogic of relevance logic, which we had already assumed was necessary and formal, and so these characteristics are preserved. Additionally, it is normative, since we “go wrong” by assuming that the situations are compatible with something.

52 true sentence, must be part of the meaning of negation. This makes negation arbitrary. An operator which, when appended to any sentence, makes that sentence true ought not be part

of the meaning of negation. Thus, if intuitionistic stages are explicitly consistent situations

ordered by a subrelation of the compatibility relation, then we can see that intuitionistic

negation and relevant negation cannot share a meaning.

This suggests quite strongly that intuitionistic stages simply cannot be situations in

the way Beall and Restall describe, at least not if we want the relevance and intuitionistic

negations to mean the same thing. If they were, they would need to be ordered by a proper

subrelation of compatibility, and that would allow us to add arbitrary clauses to negation.

This rules out Option 1, and so we must now turn to Option 2.

Even though intuitionistic stages are not situations, we can still make the intuitionistic

clause for negation “fit” the (*) mold. We simply let D be nodes in Kripke structures, and

let R be the ≤ relation. Here, we avoid the previous problem: nodes will only be related

to the nodes in the future of their own Kripke structure, and the relationship will not be

symmetric.

There will be no overlap between the clause for negation in situations and the clause for negation in nodes of Kripke structures. The intersection of the domains will be empty, since one contains only situations, and the other contains no situations.

Again, though, we have a similar issue. If all it takes to be part of the meaning of negation is to be “moldable” into (*) (with no requirements on the domain) then there is a problem. Again, there are lots of clauses which would fit this mold, and it is not the case that we would like to allow all of them to be part of the meaning of negation.

Consider the following. Let us assume that the situation semantics is rich enough such that every sentence is not true at some situation. Then, for any list of negated sentences, we can construct a situation-in-a-model at which they will all be true.8 We do

so simply by restricting our domain in certain ways. Suppose, for example, that the list

8Thanks to Graham Leach Krausse for suggested an example of this type.

53 we were given contained ¬A1,...,¬An. Then, we simply restrict our domain to the set/class of situations which make none of A1,...,An true. This will be non-empty, since because of our assumption, there is at least one situation which does not make true the conjunction

9 of A1,...,An. Further, at any situation in this domain, the negations of A1,...An are true, since the compatibility relationship will be restricted to our domain, and no situation in the domain makes A1,...,An true. In this case, negation is arbitrary. We can make the negation of any sentences we would like true, just by restricting the domain in certain ways. Thus, allowing restrictions to the domain in order to get the intuitionistic clause for negation requires allowing restrictions that gets arbitrary clauses for negation. I claim that Beall and Restall cannot allow this and still claim that they have succeeded in pinning down the meaning of negation.

Summing up, we see that if stages are either explicitly consistent situations with the accessibility relation being a subrelation of compatibility, or if they cannot be situations at all, negation as described by Beall and Restall will be arbitrary if it includes both the intuitionistic and classical clause. Given the way Beall and Restall discuss stages, and intuitionistic cases, these seem to be the only options. So, either negation is arbitrary, or intuitionistic negation cannot be part of the “disjunction of all negation clauses” meaning.

This means that either negation is arbitrary, or Beall and Restall have not produced a pluralism without language change after all.10

9These restricted domains of situations still abide by the necessary-normative-formal requirements on being an admissible logic (see definition in footnote7, and satisfy the GTT since the cases are simply situations of a specific type. These situations produce a consequence relation which is formal (since it is in a model) and normative, since we “go wrong” by assuming that the situations we are considering are not restricted to not making true a certain set of sentences. Finally, it is necessary. Beall and Restall define necessity as “the truth of the premises necessitates the truth of the conclusion” (p 14), in other words, as long as it is not possible for the premises to be true and the conclusion to be false (p 40). But in this case, this is not possible, since the possibilities in play are a subset of the original domain, so there are no possible cases where things are different. 10 One might think at this point that we might be able to accomplish the Beall and Restall project if we framed everything in terms of nodes in Kripke Structures rather than situations. After all, terminal nodes in Kripke structures are classical models (see (Beall and Restall, 2006, p 98)). However, we find we have the same problem once again. The default instantiation of (*) would then be D=nodes in Kripke structures, R =≤. We would have to expand ≤ so that it could be symmetric, since compatibility is symmetric.

54 4.1.5 Merely Technical Meanings

It is open to Beall and Restall at this point to simply state that their definition of the connectives is merely technical, and that it does not matter that the meaning of negation really is arbitrary. This would allow them to say that it does not matter that negation is arbitrary, and maintain that their pluralism really is one in which the connectives across logics are pairwise synonymous.

However, on top of being unsatisfying for more general reasons, this move is not open to Beall and Restall because they motivate their pluralism in part by the ability of each logic to capture deductive reasoning in various applications. In particular, this means that the connective meanings have to have something in common with the connectives speakers actually use. If they want their theory to be applicable, then the connectives cannot be merely stipulative. So they cannot just define negation technically, it has to match up with something in the world.

Further, what Beall and Restall are doing seems to be providing a descriptive theory of logical consequence. They gives us examples, and develop their thesis from what we actually do when we use logics. Thus, if the stipulative definitions do not match the connectives we actually use in some appropriate respect, they will not be adequate to this task. So, Beall and Restall’s connectives must match something in the actual world, and cannot be merely technical.11 However, Kripke structures give us no clues as to how to expand ≤ to appropriately capture compatibility, and so we can proceed as we wish, and expand it to any relationship. Then we simply take R to be the universal relation, and re-run the previous counter-example. Another option would be to consider cases as classes of pointed frames (thanks to Shay Logan for this suggestion). Then, we could capture the various logics by restricting what gets admitted to the class of pointed frames in question. Moreover, the two counter examples presented above would be less odd, since one ought to expect that odd frames come equipped with off negations. However, we will still have a problem here (thanks to Beau Madison Mount for this suggestion). It is still the case that we will have “overlap” (for example, pointed Kripke frames for propositional logic with the null signature will overlap with classical frames with a unary relation), and thus we will still be able to develop sentences which are true when the frame is considered as one kind of model, and not true when considered as the other. 11There is another option here, and that is to go the way of Dunn(1993) (thanks to Aaron Cotnoir for suggesting this option.). There, he presents a method by which we might “fit” both relevant and intuitionistic negations into one model.

55 4.1.6 A Possible Solution

There is an interesting moral to draw here. There does seem to be something fundamen- tally different about intuitionistic negation and relevant negation. I strongly suspect this has something to do with the fact that intuitionistic and relevant consequence are both restrictions of classical consequence in different ways. Intuitionistic logic allows explosion but not double negation elimination, while relevance logic allows double negation elimi- nation but not explosion. However, Beall and Restall have provided a meaning for the connectives on which classical and relevant negation mean the same thing. We can also do this for classical and intuitionistic negation. Here, we would use nodes in Kripke models as intuitionistic cases, and recover classical consequence, and the classical connective clauses, by restricting ourselves to one-node Kripke structures (see (Beall and Restall, 2006, p 98) and footnote 10). If both of these systems are legitimate, then we have a pluralism which is pluralistic in both logic and language. We have two languages, that of the situations semantics and that of the Kripke structures, and two logics in each, classical and relevance in the first, and classical and intuitionistic in the second. On the one hand, we have what

Beall and Restall want: two consequence relations with connectives which share a meaning.

This system has two relations, C and v, so there is no need to make one “fit” the other, and we do not have the problem we saw above with trying to make the Kripke structure “in-the-future-of” relation fit with our compatibility relation. Each relation corresponds to one “type” of negation: ¬∗A (defined by x |= ¬∗A ∗ ⊥ ⊥ if and only if x 2 A) and ¬ A (defined by x |= ¬ A if and only if ∀y(y |= A implies y⊥x)) (p 331). Roughly, ∗ can be thought of as the operator from the Routley-Meyer relevant logic semantics, and so we use it to generate v, which can be thought of as A v B ↔ everything A makes true, B makes true. ⊥ can be thought of as an orthogonality relation (see, for example, Goldblatt’s semantics for orthologic), which we can use to interpret this C as a compatibility relation. By varying restrictions on the relations C and v, and we can get classical, intuitionistic and relevant negation in this model. Moreover, it works even when we add ∧ and ∨. One might think now that this is all well and good, but that we still have two negations in play: ¬∗ and ¬⊥. Here is where Dunn’s system shines: one can show, under suitable circumstances, that these two negations really come to the same thing. So Beall and Restall may be saved after all! There is a problem here though. It is not at all clear how to extend this method to the conditional. The conditional is quite a complicated beast, and what we have here is access to two different relations to try to make it work. It seems like it is plausible that we could generate a classical and intuitionistic conditional, since both of these can be generated in models where there are only binary relations. However, most models which generate a relevance logic use a three-valued relation to do so. We would need much more work to see if such a relation could be generated on Dunn’s model, and if so whether it would work. Thus, though it may work for negation, it is distinctly possible it will not work for the conditional, and hence will not be enough to make Beall and Restall’s claims about all connectives sharing a meaning true.

56 On the other hand, we have what our earlier characterization of Carnap wants: sometimes, logic change does require connective meaning change. In a sense, this type of system might be thought of as a synthesis between Carnap and Beall & Restall.

4.1.7 Conclusion

Beall and Restall(2006) propose a logical pluralism where the connectives for each logic are pairwise synonymous. Due to the manner in which connectives are given their mean- ing, relevant negation and intuitionist negation cannot mean the same thing. Thus, their pluralism is a pluralism of languages and logics, not just logics as desired.

4.1.8 Appendix

The following demonstrates why the compatibility relation needs to be restricted to prevent the same situation occurring in different Kripke structures. As I will show, if we do not prevent it, a single sentence may be both true and not true at a situation (i.e. not merely that it will be true and its negation will be true, but that one and the same sentence will hold and not hold at a situation).

Consider a situation which does not make the (possibly complex) sentence A true. This

situation might occur in more than one Kripke structure. For example, it might occur in

one Kripke structure where A becomes true in its future, and one where it does not. For

simplicity, assume both structures have two nodes: our situation is the base node, call it

s, and there is a single node in the situation’s future, call it f. In the first instance, A is

true at the single node in our situation’s future (figure 1), in the second, A is not true there

(figure 2). In the first case, ¬A will not be true at the situation, while in the second case it

will not be. Now we have one and the same situation, in two Kripke structures, and because

of the different futures, different sentences are true at that very same situation. ¬A is both

true and not true. This is a bad outcome.

57 Figure 4.1: A is true in the future of s f : A O

s

Figure 4.2: A is not true in the future of s f O

s

Notice that this will only be an issue if we individuate situations in a particular way.

If situations are individuated by all of the sentences true at them, then differences in the

“future” of each situation will affect what is true in the situation, and so affect the identity of the situation. However, this method of individuating situations makes the compatibility relation strange. The compatibility relationship is supposed to be what governs which negated sentences are true at any situation. If situations are identified in part by which negations are true at them, then which situations they are compatible with will be trivial

(presumably, it will be all and only those situations which do not make true the negation of anything true at the first situation). However, Restall does not talk about compatibility as though it is an afterthought, but rather as though its role is to generate more true sentences at any given situation. It does real work for this theory. Since compatibility cannot be an afterthought in this way, situations must be individuated by something different.

This means that we must individuate each situation by the atomic and negated atomic sentences true at it. But if this is the case, and if A happens to be complex, then the truth or falsity of A in s’s future will not affect the atomic and negated atomic sentences true at A, and so the situations in both figures will be the same. If we are in the figure 1 case,

¬A will not be true at s, while if we are in the figure 2 case, ¬A will be true at s. ¬A will

both be true and not true, a very bad situation indeed (pun intended). We will be put in

a position where a situation being in two Kripke structures will be problematic.

58 4.2 Restall’s Proof-Theoretic Pluralism and Relevance Logic12

4.2.1 Introduction

Restall(2014) proposes a new, proof-theoretic, logical pluralism. It is based on a sequent calculus, where the connectives are implicitly defined (or constituted) by their left and right logical rules. Defining the logical constants by their left and right logical rules is far from new. We see Dummett, Gentzen and even Prawitz suggesting similar moves in their respective systems, though they are not logical pluralists. What is unusual is that Restall takes these rules to implicitly define the same set of connectives across different logics, all of which are taken to be “right”. Not all pluralists hold that connective meanings are the same across all admissible logics, and so this is part of what makes his system’s approach to his chosen three interesting. He holds that because the change in logics is affected by differences in things other than the left and right logical rules for a connective, the connectives mean the same thing in all three logics.

I will show that Restall’s new proof-theoretic pluralism is such that it cannot include a relevance logic. For the purposes of this chapter, I take a relevance logic to be one which minimally avoids the positive and negative forms of Lewis’s paradox.13 In this chapter I will demonstrate why Restall’s system cannot accommodate such a logic. I will suggest that this is because Restall has not provided adequate conditions on which logics are admissible, and that the reasonable ways of spelling out such conditions do not allow us to include a relevance system in this framework. I hold that this provides a basis for dismissing this framework for pluralism in general, as Restall ought to want his pluralism to be able to accommodate a relevance logic.

There are at least two reasons to think Restall should want his pluralism to include a relevance logic. First, in his work with Beall (see Beall and Restall(2006)), they include

12This section is published in Erkenntnis, and is kept exactly as it appears in print, with the exception of footnote 23, which has been added for the dissertation. Much of section 4.2.2 is repeated from Chapter2. 13The positive form is that from a contradiction, one can infer anything. The negative for is that from a contradiction, one can infer any negated sentence.

59 a (particular) relevance logic as a legitimate logic. This gives us at least reason to suspect that Restall thinks relevance logics are legitimate, and that at least one should be included as a legitimate logic in a pluralism. Second, a pluralism which includes only three logics

– none of which are relevance logics – seems unnecessarily restrictive, especially since the logics which are admitted into the pluralism are generated by altering structural rules, and one can generate a relevance logic by altering structural rules (though, I will show, not without changing the meanings of the connectives in Restall’s system). Together, these two reasons suggest that if Restall’s proof-theoretic pluralism cannot include a relevance logic, then it ought to be dismissed. 14

4.2.2 Restall’s Proof-Theoretic Pluralism

Restall(2014) develops his pluralism on the basis of a sequent calculus. This pluralism explicitly encompasses classical, intuitionistic and dual intuitionistic logic.15

Restall’s calculus encompasses at least the following rules:

identity Γ,A ` ∆ Γ,A ` A, ∆ ∧L Γ,A ∧ B ` ∆

Γ ` AA ` ∆ Γ,B ` ∆ cut ∧L Γ ` ∆ Γ,A ∧ B ` ∆

Γ ` A Γ ` ∆,A Γ ` ∆,B weakening R ∧R Γ ` A, B Γ ` ∆,A ∧ B

weakening L Γ ` A Γ ` A, ∆ ∨R Γ,B ` A Γ ` A ∨ B, ∆

Γ ` A, ∆ Γ ` B, ∆ ¬L ∨R Γ, ¬A ` ∆ Γ ` A ∨ B, ∆

Γ,A ` ∆ Γ,A ` ∆ Γ,B ` ∆ ¬R ∨L Γ ` ∆, ¬A Γ,A ∨ B ` ∆ 14The lessons drawn for Restall here will certainly have an impact on the debate on the possibility of defining logical connectives inferentially. See, for example, Hjortland(2014) and Paoli(2014), who discuss the possible meaning-constituting laws of the logical connectives. 15Dual intuitionistic logic is a paraconsistent logic. It has the same sentential theorems as classical logic, but not the same counter-theorems. In fact, it does not disprove all contradictions. See Urbas(1996) for more details. 60 For any connective, ∗, Restall claims that ∗L and ∗R constitute the meaning of, or implicitly define, the connective ∗. The ∗L and ∗R are the logical rules for the connective

∗. For example, ¬ is implicitly defined by ¬L and ¬R. Any rule which does not explicitly involve a connective is a structural rule. Identity, cut, weakening L and weakening R are such structural rules.16 Though the meanings of the connectives are constituted by the rules given above, it will become important for our purposes that, according to Restall, the connective meanings are constituted by those rules read in a particular way. The rules must

fit with reading the sequent as “every evaluation which makes everything on the left true makes something on the right true”. I will discuss this requirement in more detail below.

Now we are in a position to show how Restall’s pluralism works. It is well known that for this calculus, restricting the number of sentences on the right (in all sequents) to at most one results in a calculus where the law of excluded middle is no longer valid. This restriction, in fact, results in intuitionistic logic. This is how Restall, like Gentzen originally, captures intuitionistic consequence with this classical sequent calculus. The original sequent calculus proves all and only the classical truths, and if we restrict ourselves to proofs in which each sequent has at most one sentence on the right, we can prove all and only the intuitionistic consequences. Moreover, if we restrict ourselves to sequents with at most one formula on the left, we get what Restall calls dual-intuitionistic logic. Restall claims that since he has not changed the left- and right-logical rules, he has not changed the meanings of the connectives. Thus, he holds that this is a framework for a pluralism where the connectives are the same across logical consequence relations.

In a sense, what we have is the following: one object-language, one meta-language, and three “notions” of validity.17 That is, we have a meta-language which contains three

16The two weakening rules are not given explicitly in Restall(2014). They have been adapted from Restall (2000). 17I am not quite sure what term to use here, and have settled on “notion” (with thanks to a blind reviewer for the suggestion). It is something like a reading, a sharpening, an interpretation or a precisification. I will use the word “notion” in order to remain neutral on this this topic, as the results I present only require that any “notions”/readings/sharpenings/etc. of the same term have something in common, which all of these ways to spell out what a “notion” is.

61 meta-relations which correspond to three “notions” of validity. I will call these relations

`C , `I and `DI , for the relations which correspond to classical validity, intuitionistic va- lidity and dual intuitionistic validity. On “notion” one, we have classical validity, which is characterized by the meta-relation `C , and is given by the full complement of structural and logical rules. On “notion” two, we have intuitionistic validity, which is characterized by the meta-relation `I , and is given by the full complement of logical rules with a restric- tion on succedents in all the rules. On “notion” three, we have dual-intuitionistic validity, which is characterized by the meta-relation `DI , and is also given by the full complement of logical rules but with a different restriction, this time on the antecedents in the rules.

For i = C,I,DI,Γ `i ∆ can be read “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true” (Restall, 2014, p 282). All three “notions” of validity have the same logical rules, and so all three associated object-languages have the same connectives.

It is of critical importance that all three “notions” result in validity relations, i.e. that they are all “notions” of the concept of validity. In their book, Beall and Restall propose that a logical consequence18 relation must be necessary, normative and formal. A major problem with Restall(2014) is that it does not present such criteria. However, there needs to be one. We would like, for example, to rule out a calculus with the structural rules that there can only ever be exactly zero formulae on the left and exactly zero formulae on the right. This structural restriction rules out making any inferences at all. Additionally, if we chose instead to rule out the identity rule, we could not even infer that A ` A, and

given that this is the only unconditional rule, we would never be able to prove anything

unconditionally. Since, I take it, part of what it is to be a right logic is to be capable of

licensing unconditional inferences, these types of restrictions cannot be permitted.19 We

18Though Beall and Restall(2006) speak of logical consequence relations, and I am here speaking of validities, I take these two notions to be similar enough. A logical consequence relation can, for example, be thought of as a set/class of validities satisfying certain closure properties. 19I take it that this requirement on a logic is inherently plausible, and hold that Beall and Restall(2006) would agree. As they note, their system does not license logics which are not reflexive; logics which do

62 need some way to rule them out, and giving criteria of what counts as a “notion” of validity proper is one way to do this. Moreover, it is not simply a matter of ruling out just one or two odd systems. We can generate these odd systems fairly simply: we just need to restrict the number of formulae on the left- and right-hand side of the sequent to an exact number.

The calculus generated by the structural restriction of exactly three sequents on the left and exactly four on the right, for example, is also as odd, and it is not clear that it ought to be included in any calculus claiming to give us the “right” logics.

The question that we have to ask now is whether there is a fourth “notion” of validity, with a fourth meta-relation, call it `R, which corresponds to some type of relevance validity. The rest of this chapter is devoted to exploring some possibilities for such admissibility criteria of logics into the proof-theoretic pluralism, and assessing whether they admit a relevance logic without admitting the undesirable calculi mentioned above. What I will show is that no reasonable criteria provide us with a framework which admits classical, intuitionist and relevance logics, and which preserves connective meanings across all such logics. In the next section, I will claim that the most natural way Restall has to give a criterion to rule out undesirable “logics” as above is to claim that real “notions” of validity can be read “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true”. This, I will show, causes problems when attempting to include a relevance logic. Following this, I will consider two other options, both of which fail as well.

4.2.3 No relevance validity

Now, one might think that some relevance logics can be added to this pluralism quite simply. It is known that removing the weakening rules given above results in a sequent calculus where it is not possible to prove any irrelevancies, and can reasonably be called not license the inference from A to itself. They refer to such systems as “logics by courtesy and by family resemblance” (Beall and Restall, 2006, p 91). I take it that their dismissal of non-reflexive systems suggests that they would agree that the logic developed by removing the identity rule from the sequent calculus ought not to be one of the right logics, and that they hold a right logic needs to license at least some unconditional inferences.

63 a relevance logic (see, for example, Restall(2000)). 20 Since the structural rule changes do not change the logical rules, Restall could reasonably claim that the connectives in this new system still mean the same thing as those in the original system, and hence that this change allows a relevance logic into his proof-theoretic pluralism. What I will show in this section is that removing the weakening rules requires a change in the sequent calculus which is not acceptable in Restall’s proof-theoretic pluralism, and thus relevance logic cannot be added to his system.

The fact that Restall’s 2014 pluralism cannot take relevance logics seriously stems from two simple, easily derivable and well known, equivalences between the weakening rules and the following two intuitively true sequents:

A ∨ B ` A, B

(4.1) and

A, B ` A ∧ B

(4.2)

Recall that we are meant to read the sequent Γ ` ∆ as “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true” (Restall, 2014, p 282). Thus, the rules (4.1) and (4.2) ought to be at least intuitively valid. (4.1) “says” that if A ∨ B is true, then one of A or B is true. (4.2) “says” that if both A and B are true, then A ∧ B is true. So far so good. The following proofs show that weakening R and weakening L are

20To my knowledge, this is the only method that has taken hold for this particular proof theory. At the very least, Restall(2000) claims this is a method which is natural to use. It is due to Tennant(1984, 1992).

64 respectively inter-derivable with (4.1) and (4.2):

A ` A A ` A B ` B ∨R WR WR A ` A ∨ BA ∨ B ` A, B A ` A, B B ` A, B CUT ∨L A ` A, B A ∨ B ` A, B

A ` A A ` A B ` B ∧L WL WL A, B ` A ∧ B A ∧ B ` A A, B ` A A, B ` B CUT ∧R A, B ` A A, B ` A ∧ B

This is a simple result which most people familiar with sequent calculus could prove, and Restall is no exception. The fact is, though, that Restall does not seem to have realized the impact it has on his pluralism. What causes the problem for Restall is that invalidating

(4.1) or (4.2), as we would need to in order to add a relevance consequence to his system, is problematic.

The combination of weakening R and ¬L actually proves a form of Lewis’s paradox, which violates our minimal criteria of a relevance logic:

A ` A WR A ` A, B ¬L A, ¬A ` B Thus, we need to “get rid of” either weakening R or ¬L. In this instance, it is weakening which must go, since we need to preserve connective meanings, and to do so we must keep all of the logical rules.

To invalidate weakening L and weakening R, we must make (4.1) and (4.2) invalid.21 If we want to claim (4.1) and (4.2) are invalid, we have to claim that there is some evaluation on which the formulae on the left hand side of the sequents are true, but no formula on the right hand side is true (this is just the negation of what it means for a sequent to be valid).22 In the case of (4.1), this means we have to claim that there is some evaluation

21There are other options if we remove the CUT rule, see Tennant(forthcoming) for a relevance logic without CUT. Restall, however, has made claims that logics must be unrestrictedly transitive (see Beall and Restall(2006, p 91)), and so I will not consider the possibility here. 22This will work if the meta-language is classical. It is plausible to think that Beall and Restall(2006) take it to be classical, and they are criticized for doing so by Read(2006a).

65 on which A ∨ B is true, but neither A nor B is true. In the case of (4.2), this means we

have to claim there is some evaluation on which both A and B are true, but A ∧ B is not

true. Neither of these is possible; there is no evaluation which satisfies these claims. Thus,

I claim, (4.1) and (4.2) must be valid if valid sequents are read as Restall suggests.23

There is another way to run this argument: one can run it directly against the weakening

rule itself. If the sequent, Γ ` ∆ is meant to be read “every evaluation which makes

everything on the left true must make something on the right true”, then if an evaluation

which makes all of Γ true must make at least one member of ∆ true, then an evaluation

which makes all of Γ true and also makes A true must still make some member of ∆ true.

Adding a sentence to the premise set, on this conception of the reading of the sequent, must

not change the validity of the sequent.24

This has a drastic consequence for Restall’s proof-theoretic pluralism. On this strict

reading of the sequents, relevance cannot fit. That is, if the sequents must be read “every

evaluation which makes everything on the left true must make something on the right true”,

then adding a relevance consequence obtained by removing the weakening rules will require

evaluations which are not possible. Thus, relevance consequence cannot share the common

core shared by the other “notions” of validity, and thus it cannot be a “notion” of validity

in Restall’s system.

4.2.4 Other Admissibility Conditions

We might be able to make some changes to Restall’s program to make relevance fit. We

might say that the common core of validity is that all validity relations are given by a

23 There is an option available here to read this paragraph by replacing “invalidate” with “avoid”. Thus, rather than claiming we need to invalidate weakening, (4.1) and (4.2), we would claim that we need to avoid them. To do so, we would need to eschew (4.1) and (4.2) rather than invalidating them. I take it the result would be essentially the same, as we would still need a method of reading the sequent which would not require embracing (4.1) and (4.2). Thanks to Neil Tennant for pointing this out. 24Thanks to Marcus Rossberg and Nathan Kellen for pointing this out to me. Restall’s response to this version of the argument would presumably be to read the sequent intentionally, and thus the same reasoning as in section 4.2.4 will apply here as well.

66 set of sequents generated by a particular set of structural and logical rules, rather than the “reading” given by Restall. This, though, is not satisfactory. It seems that this lets more things in as validity relations than we think there are. What, for example, about the calculus mentioned earlier, with the structural rule that there cannot be any formulae on the left or right hand side of the sequent? This is not something we would like to call a logic, since we cannot make inferences in the system. Thus, the structural restriction “exactly one on the left and none on the right” should not result in a proper notion of validity. We need some way to weed out relations which are not validities, and this manner of defining validity does not provide one. This type of change will still not allow Restall to include relevance logic, on pain of including too much, in his pluralism.

There are more drastic changes we might make to include relevance logic in an extension of Restall’s proof-theoretic pluralism. Perhaps if we do not literally read the sequents as

Restall suggests, then the addition of a relevance logic would not be problematic. In order to add a relevance consequence to his 2014 pluralism, then, Restall will need to provide us with a new way to read the sequent. This – though, strictly speaking, it is not his original proof-theoretic pluralism – might make relevance logic admissible.

One option would be to consider the intensional sequent as presented in Dunn and

Restall(2002) (see p 100). One important difference with this sequent compared to the sequents discussed in Restall(2014) is that, rather than using quantifiers (i.e. rather than

“every” and “some” in “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true”), they use “and” and “or”. As Restall(2014) claims that the sets of sentences on the right and left hand sides of the sequents are finite (see p 282 of

Restall(2014)), this difference is easy enough to overlook. 25

The definition of the intensional sequent is constructed as follows. The intensional conjunction of A and B can be defined as ¬(A → ¬B) in Anderson and Belnap’s R, but

25The move to intensional connectives actually precludes the meta-language being classical. It would require a much broader change to Restall’s position than I will address here.

67 can be taken as primitive in many relevance logics as well. Intensional disjunction is similar

(though not discussed in Dunn and Restall(2002)), and can either be primitive or defined as

¬A → B. I will follow Dunn and Restall and use ◦ for intensional conjunction, and follow

Read(1982) and use + for intensional disjunction. The intensional sequent, A; B ` C, given by Dunn and Restall(2002) is then read as “if A ◦ B then C”. We can expand this notion to the sequents we are currently considering. Now, rather than reading the sequent

“every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true,”, we might read it “every evaluation which makes the intensional conjunction of everything in Γ true makes the intensional disjunction of everything in ∆ true.”

It is shown by Dunn and Restall(2002) that this sequent calculus, when we restrict ∆ to at most one member, is a relevance calculus, and similar to Anderson and Belnap’s R.

There are then two ways to proceed with this intensional approach. Either the sequents were meant to be read intensionally all along, or we have sequents which are meant to be read “every evaluation which takes every element of [Γ] to be true takes some element of

[∆] to be true”, with different “notions” of “every” and “some” as we had with validity, so that sometimes “every” and “some” mean the usual quantifiers, and sometimes they mean

“intensional conjunction” and “intensional disjunction”.

The problem with the first option is that there is nothing we can add to this purely intensional sequent calculus to make it classical and preserve connective meaning. It is known that our ∨R and ∨L rules fail for intensional disjunction (or, rather, our +R and

+L rules). For, if we let either +R or +L be the same as ∨R and ∨L, we could derive irrelevancies: ` A +R ` A + B Def of + ` ¬A → B ` ¬A MP ` B This means that the meaning of + can never be constituted by the left- and right-logical rules for ∨ and thus that that Restall will need two disjunctions to make this strategy work:

+ and ∨ (the latter of which is constituted by the usual ∨R and ∨L rules). This, though,

68 implies that the logical connectives will not be the same across logics, which Restall cannot allow. If we went down this path, we would need to have two disjunctions in play, rather than just one defined by the logical rules. Thus, even reinterpreting what Restall says in his original presentation as intensional rather than extensional sequents will not allow us to add a relevance logic to the pluralistic framework while preserving connective meanings.

The problem with the second option is that it also results in different connective mean- ings. As with the first option, this technique will still require two disjunctions. Our usual

∨ will be required for when we are using the usual “notions” of “every” and “some” in the reading of validity. However, in order to remain consistent, when using the “intensional connective” “notions” of “every” and “some” in the reading of validity, we will need +, which again cannot be defined by the left- and right-logic rules for ∨.

I strongly suspect that this type of result will be common with any non-ad hoc rein-

terpretation of how to read the sequents Restall uses. There will be no way to read the

sequent in such a way that classical, intuitionistic and relevance logics are included in the

proof-theoretic pluralism. It will require some sort of reinterpretation of the proof-theoretic

pluralism, and I think this interpretation will be very hard to come by.

This difficulty seems to come about because the types of concerns presented above

about the difficulty of adding a relevance logic to the sequent calculus given in Restall

(2014) arise in part because that sequent calculus does not allow us to track premise use

in any reasonable way.26 Most proof theories for relevance logic (which include negation)

either have two conjunctions and/or disjunction (e.g. + and ∨ as above, or Belnap(1982))

or have a tool which explicitly tracks premise use (see the natural deduction system for R

in Anderson and Belnap(1975)). The system provided by Restall provides us with neither

of these tools. The first is explicitly prohibited by his pluralism, since it would require

more than one conjunction or disjunction, thereby preventing connectives from meaning

the same thing across all admissible logics. The second is a more viable option, but again is

26Thanks to a blind reviewer for suggesting this issue.

69 not available in Restall’s system without changing the meaning of the sequent: if we have to track premises, we will need a more robust meaning than “every evaluation which takes every element of [Γ] to be true takes some element of [∆] to be true”. More generally, I expect this inability to combine all three of classical, intuitionistic and relevance negation has something to do with the fact that intuitionistic and relevant consequence are both restrictions of classical consequence in different ways. Intuitionistic logic allows explosion but not double negation elimination, while relevance logic allows double negation elimination but not explosion. I hope to pursue this in further work.

4.2.5 Conclusion

Restall(2014) presents a proof-theoretic pluralism which cannot sanction a relevance con- sequence relation. This is because Restall does not provide clear enough conditions on what counts as an admissible logic in this framework (as opposed to the Beall and Restall(2006) framework, which comes with detailed admissibility conditions). Further, there seems to be no straightforward way of providing such conditions so that both a relevance consequence relation is admissible, and connective meanings are preserved across all admissible logics.

I have suggested that this points to a problem more generally with finding a proof theory which accommodates exactly the logics we think are correct, which I have argued should include a relevance logic if they include both classical and intuitionistic logics.

One might also ask the following question: just what is “validity”? It seems that there are many ways to spell out what, exactly, “validity” means. Restall has happened upon at least two: the one presented with Beall in their model-theoretic pluralism, and the one he gives in his more recent proof-theoretic pluralism. Though these two types of validity do not account for the same logics, they at least give us two options to choose from when we are discussing what validity really is. The issues of what validity amounts to, and how the connective meanings influence validity are essential here. 27

27I owe a debt of gratitude to Stewart Shapiro very carefully read and commented on several earlier drafts. I am grateful to comments from several people, including three blind reviewers, Scott Brown, Nathan

70 Kellen, Chris Pincock, Marcus Rossberg, Kevin Scharp, Matt Souba and Neil Tennant. I am also grateful to audiences at the Dubrovnik Conference on Pluralism (2015), the Pacific APA (2015), PhiloSTEM (2014), the Society for Exact Philosophy (2013) and The University of Western Ontario Philosophy of Logic, Math and Physics Graduate Conference (2013).

71 Chapter 5

A New Interpretation of Carnap’s Logical Pluralism1

5.1 Introduction

There is one slogan that many people who hold that Carnap was a logical pluralist agree

upon: Carnap’s pluralism is one in which a change in logic can occur only when there is a

corresponding change in connective meaning. In this way, it is claimed that a Carnapian

pluralism requires language change any time there is logical change. Logical change can

only occur because of language change.

What I will show in this chapter is that there is a different interpretation of Carnap

(1983) and Carnap(1937), where he is better accounted for as a Shapiro(2014)-style plu-

ralist. First, I will show that, for Carnap, the ordinary question of whether two connectives

in different languages share a meaning is meaningless and has no answer. Second, I will

suggest that the only way to make this question meaningful is to assess it with respect

to a meta-linguistic framework, a framework which we use to talk about other linguistic

frameworks. I give some evidence for thinking that Carnap would agree. Further, this

will allow us to claim that, with respect to some meta-frameworks, connectives in different

logics will share meanings, while with respect to others, they will not. This final step is

what allows for logical change without language change. Finally, I show how this new inter-

pretation puts the Carnapian position much more in line with that of Shapiro(2014). On

this interpretation, obtained by extending what Carnap says about linguistic frameworks

to meta-linguistic frameworks, when two linguistic frameworks are embedded in distinct

1This section is published in Topoi, and is kept exactly as it appears in print, with the exception of footnote5. Much of section 4.2.2 is repeated from Chapter2.

72 meta-linguistic frameworks, the connective meanings in the two original frameworks may or may not be the same.

5.2 The Traditional Carnapian View

It will be useful to explain carefully why the typical slogan about Carnap’s view takes him to be claiming that change in logic requires a change in connective meanings. Authors who often seem to be making use of such a slogan to describe Carnap’s position include

Cook(2010), Hellman(2006) and Restall(2002). Shapiro(2014) makes this claim more prominently. He states “the crucial Carnapian conclusion is that these [relevant and classical negations] are different negations”(Shapiro, 2014, p 107). Rarely does anyone explicitly assert that the slogan applies directly to Carnap’s position, but much of what is said can be thought to imply it. In a way, this chapter can be read as a cautionary note against taking the slogan seriously in interpreting Carnapian himself.

The slogan readily makes sense of passages like the following, discussed in chapter2:

Our attitude towards...requirements...[of a logic] is given a general formulation in the Principle of Tolerance: It is not our business to set up prohibitions but to arrive at conventions.... In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e. his own language. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical [as opposed to scientific] arguments. (Carnap, 1937, p 51/2) Let any postulates and any rules of inference be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols. (Carnap, 1937, p xv)

These passages are typically taken to entail two things. The first is that we ought to be tolerant of different logics. As long as we can build a logic (which is given by its syntactic rules), and provide applications for it, that logic is legitimate. If we have two applications which call for different logics, then we end up with logical pluralism. The second passage is taken to imply that the meanings of the logical connectives are given by their inference

73 rules. In this way, it is very easy to (mistakenly) make the inference that changing the logic requires changing the connective meanings. If connectives are defined by their inference rules, and changing the logic just amounts to changing the rules, then changing the logic seems to amount to changing the meanings of the connectives.

What I will claim in the next two sections is that this “sloganed” position conflates two notions: that of building a language from rules, and that of building a language from rules which the builder knows to be distinct (at a fundamental level) from the rules of other languages. As I will show, Carnap meant the former, as the question which corresponds to the latter cannot be answered (“Do these two languages have connectives which mean the same thing?”). A builder can build any language she wants, as expressed in the “i.e., his own language” phrase in the quote above. In this respect we have to be tolerant. However, a builder cannot claim that her language is different from any other without first making some other assumptions about a meta-framework in which she is making that claim.

5.3 Carnap’s Position

As we saw above, Carnap says several things which make it reasonable to assume that any change in the rules of a logic is necessarily a change in connective meanings. In order to show that this is not the case, we need details on three aspects of his view: we need to know what linguistic frameworks are, how they relate to connective meanings, and what a pseudo-question is.

Carnap claims that a linguistic framework is “a system of... ways of speaking, subject to... rules” (Carnap, 1983, p 242). We will assume here that the “ways of speaking” which Carnap mentions really are just the syntactic rules for a logical system.2 Recall

2These syntactic rules may have later been expanded to include model-theoretic rules. Early quotes from Carnap suggest a purely proof-theoretic view of logic. Though he held that proof theory was the best way to “do” logic early in his career, his position about logic being a purely proof-theoretic endeavor changed after he met and spoke with Tarski, who convinced him that model-theory was a legitimate enterprise (see, for example, Carnap(1947a)).

74 that the connectives are defined by “[letting] any postulates and any rules of inference be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols” (Carnap, 1937, p xv). Thus, on this interpretation, the meanings of the connectives are determined by the rules and postulates of the linguistic framework.3

For Carnap, a theoretical question is one asked relative to a linguistic framework. It is internal to a linguistic framework, and asked assuming the rules of that framework. A non-theoretical question is one asked about reality itself, without a linguistic framework in mind. It is external to any given framework. There are two types of external questions: pragmatic questions and pseudo-questions. A pragmatic question is a question about which framework is best for a given purpose, and can be answered.4 Pseudo-questions (non-

pragmatic external questions) cannot be answered, on Carnap’s view, and this is the sense

in which they are illegitimate. For Carnap, a pseudo-question is “one disguised in the form

of a theoretical question while in fact it is a non-theoretical [question]” (Carnap, 1983, p

245). Examples of pseudo-questions include questions of existence of abstract objects, which

can only be answered relative to a given linguistic framework, and questions about the right

logic, or which logical consequence relation is correct. Outside of a linguistic framework

questions like this are only pseudo-questions.

Thus, linguistic frameworks are (for our purposes) syntactic logical systems, and their

rules define the connectives. If we ask a question with respect to no linguistic framework,

then it is external, and it is a pseudo-question unless it is pragmatic.

3In particular, they will be given by the L-rules of the framework, which are the rules that govern transformations of logically true sentences into logically true sentences. We will not here concern ourselves with P-rules, which govern transformations of descriptively true sentences into descriptively true sentences. For more information, see (Carnap, 1937, p 133-5). 4Pragmatic questions are in principle answerable. There is some question as to whether they are actually external, though. See Steinberger(2015) for an interesting discussion about how pragmatically to select the appropriate linguistic framework.

75 5.4 The External Question

We can now ask the following question: when do corresponding connectives in distinct linguistic frameworks have the same meaning? According to the slogan from section 5.2, the answer to this question on Carnap’s behalf ought to be “never”. A different, perhaps less drastic version of the slogan suggests that the answer ought to be something like “con- nectives are defined by their rules, and so if we change the rules, we necessarily change the meanings of the connectives.” According to this version of the slogan, two linguistic frameworks share some connective meanings then they share some of the same logical rules.

Each of these ways of answering the question (“never”, and “only when the rules are the same”) is flawed. This is because the question of when two linguistic frameworks have cor- responding connectives with the same meaning is unanswerable; it is an external question which is not pragmatic.5 It is not pragmatic, since it is not about framework selection, but it is external, because it is not asked with respect to some given linguistic framework. It is a pseudo-question. Even if two linguistic frameworks have postulates and rules which “look exactly the same” (i.e. are typographically the same) we still cannot ask the question, since we still have no postulated meta-linguistic framework in which to compare the rules and postulates in question. The answer to the question, then, cannot be “never”.

In addition to the fact that questions about the meaning of logical connectives are asked outside of a linguistic framework and are not pragmatic, there is at least one other reason why we ought to think they are pseudo-questions: they have a similar characteristic to the metaphysical questions which Carnap also dismisses as pseudo-questions. I will show that,

first, there is no evidence acceptable by all parties that would settle the question one way or another, and second they are trivially analytic once asked in the appropriate way.

5 There is an immediate possible response here that we might be able to produce two distinct logical systems with generate exactly the same deducibilities, and thus must have the same connective meanings. For example, Frege-Hilbert systems, which have many axioms, and only a small number of rules, produce a deducibility relation which is co-extensive with several natural deduction system, which have very few axioms and many rules. Thanks to Neil Tennant for pointing this out. Though this is a possibility, it will run afoul of one of the possible “mistakes” I discuss at the end of section this section, namely that of assuming that since rules look the same, they must in fact be the same.

76 In Carnap(1983), he considers two philosophers debating the status of numbers. One thinks they are real entities, and the other does not. Of the debate, Carnap claims “I cannot think of any possible evidence that would be regarded as relevant by both philosophers, and therefore, if actually found, would decide the controversy or at least make one of the opposite theses more probable than the other” (Carnap, 1983, p 254). Because of this Carnap claims he feels “compelled to regard the external question as a pseudo-question, until both parties to the controversy offer a common interpretation of the question as a cognitive question; this would involve an indication of possible evidence regarded as relevant by both sides”

(Carnap, 1983, p 255). Thus, if there is no evidence which would decide the issue, external questions are pseudo-questions. But notice that we can reconstruct the debate with meaning questions. Suppose we have a debate between a philosopher who holds that connective meanings are given by truth conditions, and another who holds that connective meanings are given by inference rules. Again, it seems unlikely that there would be evidence which both philosophers would agree is relevant. Natural language data seems to support neither

(or both) as connectives in natural language do not behave like they do in formal language.6

Additionally, with the exception of uncontroversial argument patterns which support both

positions, no argument pattern will be accepted as relevant evidence by both philosophers.

In general, in a piece of evidence supports one position, the other philosopher will find

reason to dismiss it as irrelevant.

Further, Carnap writes that

[The statement “there are numbers”] follows from [an] analytic statement and is therefore itself analytic. Moreover, it is rather trivial, because it does not say more than that the new system is non-empty... Therefore, nobody who meant the question “Are there numbers?” in the internal sense would either assert or even seriously consider a negative answer. This makes it plausible to assume that those philosophers who treat the question of the existence of numbers as a serious philosophical problem and offer lengthy arguments on either side, do not have in mind the internal question. And, indeed, if we were to ask them:

6For good examples of the complexity of natural language connectives, see Horn(1989) on negation or Jennings(1994) on disjunction.

77 “Do you mean the question as to whether the framework of numbers, if we were to accept it, would be found to be empty or not?”, they would probably reply: “Not at all; we mean a question prior to the acceptance of the new framework”... Unfortunately, these philosophers have so far not given a formulation of their question in terms of the common scientific language. Therefore our judgment must be that they have not succeeded in giving to the external question and to the possible answers any cognitive content. Unless and until they supply a clear cognitive interpretation, we are justified in our suspicion that their question is a pseudo-question. (Carnap, 1983, p 245)

Thus, questions which are trivially analytic once embedded into the appropriate framework

(where trivially analytic just means that they follow “easily” from the empty set of premises and the rules and postulates of the framework in question, see Ebbs(2016) for more details) and which have not otherwise been formulated rigorously are pseudo-question if they are not meant to be trivial. However, meaning questions, in addition to number questions, are like this. Once we properly specify a meta-linguistic framework, which includes statements to the effect that for any two terms, they either share a meaning or do not, meaning questions become trivial. Further, we can only answer then once they are rigorously formulated, and once we have specific rules for what counts as sameness of meaning. But this is just to say that without such rules, they are external and illegitimate questions, in the same way the number questions are. Meaning questions, like metaphysical questions, cannot be resolved by evidence accepted by both parties as relevant and are trivially analytic when asked internally, and thus we have an extra reason to think of them as pseudo-questions.

What, though, if we postulated the existence of such a meta-linguistic framework, one which was capable of talking about the two linguistic frameworks in question? This would,

I will show, give us an opportunity to answer the question of when two distinct frameworks have the same logical terminology. Additionally, I will show that the answer to whether two object-frameworks have corresponding connectives with the same meanings in a meta- framework will vary depending on our pragmatic goals, and hence the meta-framework we select.

78 First, when both frameworks in question are considered from the vantage point of a meta- linguistic framework, we could answer the question “when do two linguistic frameworks have logical terminology which means the same thing?”. The question is now being asked, not about what is really true, but about what is true with respect to the meta-framework. This makes it a theoretical question, and so it is no longer a non-pragmatic external question.

So far so good. The new question is answerable, and not a pseudo-question.

Second, I hold that the same-meaning question will have different answers depending on our pragmatic goals, and hence meta-linguistic framework selection. There will be no single meta-framework which will do the trick for us here, there might be a whole spectrum of them.

As with object-level frameworks, we pick one meta-framework which suits our pragmatic aims and operate with it. Given this choice, and presumably given that each meta-linguistic framework will come equipped with some rules and postulates for determining when two terms have the same meaning, we can infer the following. With respect to some meta- framework embeddings, corresponding connectives will share a meaning. With respect to others, though, they will not be the same.

As an example, let us consider two meta-linguistic frameworks and two object level frameworks. One object level framework is classical and the other intuitionistic.7 The only thing that will concern us about the meta-linguistic frameworks are which rules they have for determining sameness of meaning. In this example, each is equipped with a different version of a double negation translation, and two terms are synonymous if they are inter- translatable. Here, I take it that these translations map synonyms to synonyms, as they do in natural language, and that synonyms share a meaning. Thus, translations preserve meaning.8

7Thanks to Roy Cook for suggesting this example. 8There are important relations between synonymy and analyticity on Carnap’s view. In effect, if Quine (1951) and subsequent authors are right, then Carnap cannot get his notion of analyticity, or his project, off the ground. As this chapter is only meant to present an interpretation of Carnap’s views, and not assess whether they are viable, I will not address this further here.

79 The first meta-framework has the typical G¨odel-Gentzen translation from classical to intuitionistic logic, call it T1, and is defined inductively as follows:

1 if φ is atomic, then T1(φ) = ¬¬φ

2 T1(φ ∧ ψ) = T1(φ) ∧ T1(ψ)

3 T1(φ ∨ ψ) = ¬(¬T1(φ) ∧ ¬T1(ψ))

4 T1(φ → ψ) = T1(φ) → T1(ψ)

5 T1(¬φ) = ¬T1(φ)

It is a known result that φ is provable classically if and only if T1(φ) is provable construc- tively. In this sense, we have a translation between classical and intuitionistic logic. Now,

consider a second translation, call it T2, and assume that this is the translation available in

the second meta-framework. T2 is the same as T1 for all clauses expect conjunction. The

conjunction clause for T2 is

∗ 2 T2(φ ∧ ψ) = ¬¬(T2(φ) ∧ T2(ψ))

T2 also has the desired property that φ is provable classically if and only if T2(φ) is provable constructively (the proof proceeds by simply replacing the inductive steps for the conjunc-

tion clause in the G¨odel-Gentzen proof). Here, though, we can ask the following question:

do intuitionists and classicists mean the same thing by “∧”? Well, if the translations pro-

vide us with a relation of synonymy, then because conjunction is translated as conjunction

via T1, while it is translated as a double negation of conjunction by T2, we have sameness of

meaning via T1, but difference in meaning via T2. In essence, it seems that if we are using

T1 the conjunctions mean the same thing, while if we are using T2 they do not. Depending on our theoretical purposes, or on which meta-framework we are using, we will have access

to different translations, and different logical terms will be synonymous.

On the face of it, if we think there is such a thing as real meaning, and that all transla-

tions must preserve real meaning, it might seem unlikely that we will be able to use different

80 translations depending on our purposes and aims. However, consider the difference between translating between intuitionism and classicism for the purposes of a logic class, where the

G¨odel-Gentzen translation might be sufficient and simplest, and translating between the two for the purposes of writing computer programs, where the Kolmogorov translation might be more successful (the Kolmogorov translation of φ is generated by affixing a double negation to every subformula of φ). Here, depending on our purposes, one translation will be better than another, and so we ensure that we select a meta-framework which can make sense of that translation, and thereby may generate differences in the “same meaning” relation.

Take another example, from (Field, 2009, p 346-7). He considers a translation between three logics: classical logic, intuitionist logic and some paraconsistent logic. In particular, he focuses on translating between the negations of the three logics. The classical negation is the typical boolean negation, the intuitionist negation is defined as A → ⊥, and the paraconsistent logic has two non-equivalent negation-like operators: one which obeys the

De Morgan laws and double negation elimination, and another which is defined as A → ⊥.

When considering a translation between classical and intuitionistic logic, Field claims it is best to translate the negations homophonically. When considering a translation between classical logic and the paraconsistent logic, Field claims it is best to translate the classical negation as the first of the negation-like operators (the one obeying double negation elim- ination and the De Morgan rules). One would think, then, that if translation preserved meaning, we ought to follow through and translate the intuitionist negation as the De Mor- gan negation in the paraconsistent logic as well. But there is a method for translating between the intuitionist logic and the paraconsistent logic which will preserve more of the behaviour of the intuitionist negation: we ought to translate the intuitionist negation as the negation-like operator which is defined as A → ⊥.

It seems like an available step at this point to say something like the following. Some- times, translating the connectives by the transitive translation above will serve our theoret- ical purposes, as in the case of translation loosely between classical and intuitionist logic.

81 Sometimes, however, it will not, and the best available translation will not map two things we thought might mean the same thing onto each other, as in the case of classical and intuitionistic logic together being translated into the paraconsistent logic. In those cases, translation will preserve something else, something which is not captured by similarity of

“shape” (perhaps it preserves the inferences associated with each connective).

What our translation needs to do (i.e. what it needs to preserve) will depend in part on what our aims are, which is cashed out by what we choose our third language to be.

We pick a third language to suit our pragmatic purposes, and then we see whether one of the available translations maps homophones to homophones. Sometimes it will be avail- able, and sometimes it will not. In other words, sometimes the intuitionist negation will be synonymous with the De Morgan negation (when we are using the meaning-preserving transitive translation and taking classical logic into account), and sometimes the intuitionist negation will be synonymous with the A → ⊥ paraconsistent negation (when we are trans- lating via the “closest in inference” translation). Depending on our theoretical purposes, different translations will be appropriate, and hence different connectives will be said to be synonymous.

In the remainder of this section, I will explain what two types of mistakes we might make in trying to address the original external question without being careful about which meta-linguistic framework we are using, and I will also address an immediate potential objection from Friedman(1988).

First, then, let us consider two examples of where this type of confusion might be a problem. In the first case, consider two linguistic frameworks, each defined by the Gentzen- style left and right logical rules for a sequent calculus, but one which has the full complement of structural rules, and one which does not have weakening. The connectives in both frameworks are defined by their logical rules on the Carnapian picture, since these are the

“postulates and rules” to which the earlier quote was referring. In this case, one might this that the only way to embed these two object-frameworks into a meta-framework is to use

82 one where the identity map between the two object frameworks counts as a good translation.

However, in this case even though the logical rules look the same in both logics, it might not be best to translate them as rules which have the same meaning. Perhaps, for example, if we are trying to deduce what effect the structural rule of weakening had on the behaviour of the connectives. In this case, assuming that the connectives must mean the same thing might obscure the results of adding or removing the weakening rules. Additionally, the logics will be different (the first classical, the second relevant), and so if we think that the traditional view of Carnap’s pluralism holds here, the connective meanings ought to be different. This is an example where rules which look the same may not be the same once considered from the perspective of a meta-linguistic framework, the framework which we use to compare two linguistic frameworks.9

In the second case, consider an adaptation of a case from (Shapiro, 2014, p 127-133).

Here, we have two mathematicians, one classical and one constructive, discussing some form of analysis. The classicist normally defines her connectives via truth conditions, and the constructive mathematician normally uses proof conditions. One might think that the only way to embed these two object-frameworks into a meta-framework is to do so in such a way that none of the acceptable translations translate the connectives homophonically. However, during their exchange, and for the purpose at hand, they never discuss the connectives, nor do they consider any results which are not acceptable to both of them. In this situation then, it makes sense to talk about them as though they are “speaking the same language”, that is, as though they are both using connectives which mean the same. This is an example where rules which do not look the same may in fact imbue corresponding connectives with

9There is some question as to whether Carnap would accept this type of suggestion. It seems that Carnap would have thought that structural rules are meaning determining as well. This type of holism seems to be part of what is expressed in the quote above, when he claims that the postulates and rules of inference determine the meaning of the “fundamental logical symbols”. The point I am trying to make still stands, though. It would be a mistake to think that the connectives in question shared a meaning, since even on the traditional view, they were never candidates for having the same meaning in the first place. Thanks to a referee for suggesting this possibility.

83 the same once embedded into a meta-framework.10

Here, we see that if we are not careful about our meta-framework selection, we will be liable to make two types of mistakes. First, we may think that rules which look the same must mean the same thing. Second, we may think that rules which do not look the same cannot mean the same thing. Neither of these is good.

There is an immediate potential objection here.11 I have given no method by which we

individuate meta-frameworks. I have only made vague claims that somehow which transla-

tions the meta-frameworks contain individuate them. However, it is distinctly possible that

there is only one meta-framework, perhaps one which contains all of the translations we

might ever need. For example, primitive recursive arithmetic (PRA) might play this role.

This is the position advocated for in Friedman(1988) (see also Friedman(2001)). This is

a system which can encode all of the translations I mention above, but then there would

only be one meta-framework. Friedman claims

Carnap takes the idea that primitive recursive arithmetic constitutes a privileged and relatively neutral “core” to mathematics and, moreover, that this neutral “core” can be used as a “metalogic” for investigating much richer and more controversial theories. (Friedman, 1988, p 87)

In other words, for Carnap, the only meta-language available is PRA, since in this language

we can execute all of our logical and scientific investigation. What becomes of my claim that

whether two object-level frameworks have connectives which share meanings is dependent on

which meta-framework we embed them into? It seems, if there is only one meta-framework,

that answer will never change. For any two frameworks, either their corresponding connec-

tives will always share meanings, or they will always have distinct meanings, since we can

10There is another option for interpretation here: it is possible to claim that the mathematicians are simply talking past each other. If we assume this, though, we would have to assume that any dispute they had would be a merely verbal dispute. However, it seems more charitable to assume that sometimes the mathematicians in question can have substantive disagreements, as in the case, for example, where they discuss the status of the intermediate value theorem, which the classicist is provable in the classical system, but not in the intuitionist system. See Shapiro(2014) for more details. Thanks to a referee for pushing me on this issue. 11Thanks to Neil Tennant for bringing it to my attention and to a referee for doing so as well and suggesting much of the literature discussed here.

84 only ever embed the object-level frameworks into a single meta-framework. There are two possible responses here.

First, we can side with Devidi and Solomon(1995), and point to textual evidence that

Carnap would not hold that PRA was the only legitimate meta-framework. If Carnap thought that there was more than one available meta-framework, then there is no reason for us to assume there was only one. Devidi and Solomon cite section 45 of Carnap(1937) where he claims that

Our attitude toward the question of indefinite terms conforms to the principle of tolerance; in constructing a language we can either exclude such terms (as we have done in Language I) or admit them (as in Language 1I). It is a matter decided by convention .... Now this holds equally for the terms of syntax. If we use a definite language in our formalization of a syntax... then only defi- nite syntactical terms may be defined. Some important terms of the syntax of transformations are, however, indefinite (in general); as, for instance, ‘derivable’ ‘demonstrable’, and a fortiori ‘analytic’, ‘contradictory’,... and so on. If we wish to introduce these terms also, we must employ an indefinite syntax-language (such as Language II).

In this passage, Carnap suggests that just as the object language (here just “language”) can include or exclude indefinite terms, so to can the meta-language (here, “syntax”). If this is right, then there are at least two legitimate meta-languages, and hence two legitimate meta- frameworks. Further, since the second meta-framework has more terms in the language than the first (“derivable, “contradictory”, etc.) there may be a different class of translations available in the second language. Hence, depending whether we embed our original object language into a meta-language with or without indefinite terms, we may find different results about our meaning question.12

Second, even if it does turn out to be the case that Friedman is correct, we have another avenue available to us. What Friedman presents is a single meta-language rich enough to

12There is much literature on the topic of meta-language on Carnap’s view. See, for example, Tennant (2007) on a different response to Friedman’s PRA proposal. There, Tennant argues that “Friedman is demanding too much in thinking that the resources of such combinatorial analysis as Carnap requires should not exceed those of primitive recursive arithmetic.” (p 103).

85 address all of the mathematical and logical questions about the Carnapian picture. A single meta-language, and PRA in particular, cannot answer the meaning question we are trying to resolve for a simple reason: it does not contain a notion of meaning, or of same meaning.

We need these notions to answer the questions with which we are concerned. Thus, though

Friedman may be right that there is a unique meta-language for all mathematical pursuits on the Carnapian picture, in order to resolve meaning disputes, we need a meta-language with the appropriate terminology. Thus, PRA just won’t do for us in this situation. So again, we find ourselves with at least two available meta-languages: PRA, and one with a notion to express “same meaning”. Presumably, again, we will have access to different translations in both.

5.5 Carnap’s Agreement

There are passages where Carnap says things which support the interpretation of section

5.4. I will discuss several here.

First, let us reconsider the quotes of Carnap given above to support the traditional view.

Our attitude towards...requirements...[of a logic] is given a general formulation in the Principle of Tolerance: It is not our business to set up prohibitions but to arrive at conventions.... In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e. his own language. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical [as opposed to scientific] arguments. (Carnap, 1937, p 51/2) Let any postulates and any rules of inference be chosen arbitrarily; then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols. (Carnap, 1937, p xv)

Here, it is important to note that we never explicitly have the claim that we can tell when two languages are distinct. Everyone might still be at liberty to build up their own language, but nothing here explicitly prevents that in order to compare any two languages, we must

first embed them in a meta-language. In fact, given that whether two terms share a meaning

86 is a theoretical question, we are required to address meaning questions from within a meta- linguistic framework, and so are required to embed any framework into a meta-framework to answer the question.

There is some sense, when interpreted as speaking loosely, that Carnap may be thought to be saying exactly the opposite here. On the face of it, what he seems to be saying here is something like “we recognize rules and postulates without needing any theoretical appa- ratus, and can tell when they are the same just by looking at them”. I think this is reading of the Carnapian position is too loose. First, in order to interpret Carnap consistently, we ought to interpret him as not putting forward this loose interpretation as a serious compo- nent of his view, as it would require suggesting that some external questions are legitimate

(the meaning ones). Second, Carnap insisted that we need a theoretical tool to engage in rigorous and meaningful discussion of logic. Reading Carnap as claiming the above would vitiate this. As I suggested above, meaning questions are external questions on his view, and can only be answered if we make them internal by asking them with respect to some framework. Thus, I claim, the loose interpretation just will not do here.

Taken as not speaking loosely, what Carnap can be thought to have said in these pas- sages is that, though the rules do determine a connective’s meaning, they only determine a meaning within a particular framework. Without considering the two frameworks from the perspective of a meta-framework, there is no way to know whether the meanings of the connectives in the two different frameworks are the same. There are two reasons why we are prevented from knowing this: first, because sometimes rules with the same shape will wind up meaning different things under the embedding into a meta-linguistic framework, and sec- ond because sometimes rules which intuitively look different will end up meaning the same under the embedding into a meta-linguistic framework (see the two possible “mistakes” in section 5.4). Since rules determine connective meaning, this implies the connectives will sometimes share a meaning and sometimes not. The passage above, then, can only be re- ferring to rules determining a meaning for the connectives within a given framework. The

87 passage makes no claim about considerations from outside of a framework (nor should it).

There are other passages where Carnap makes similar sounding claims. For example, in section 14, when he claims that “if we wish to determine what a sentence...means...we must find out what sentences are consequences of that sentence” (p 41). Similar claims can be found in section 16, 50 and 61.13 Here, I take it, we must interpret him in the same way as we have above. We must interpret him as claim that if we wish to determine what a sentence means in a particular linguistic framework, then we must find out what sentences are consequences of it in that framework. The same would go for the other passages that are similar in nature.

Second, it seems as though Carnap explicitly agrees that language comparison must be done in a meta-language (here, languages just are linguistic frameworks). Consider the following from (Carnap, 1937, section 62). There, he discusses translations from one language into another. These translations must occur within a meta-language:

The interpretation of the expressions of a language S1 is thus given by means of a translation into a language S2, the statement of the translation being effected in a syntax-language S3... (Carnap, 1937, p 228)

If all translations are effected in a meta-language, then there is reason to suspect that

when using two distinct meta-languages, sometimes a term in S1 will be translated into a

particular term in S2 and sometimes to a different term in S2. Third, Carnap holds that just what a translation must do depends on our theoretical

goals (seeCarnap(1937), p 228, where he discusses the special conditions which can be

imposed on translations). If the above is correct, then it matters what S3 is. However, if

there were only one type of translation, then we might expect the relationship between S1

and S2 to always look the same no matter what we select as S3. Given the examples from section 5.4, we can see that there will not always only be one type of translation available,

nor will that translation always be required to do the same things. Carnap himself suggests

13Thanks to a referee for pointing me to these passages.

88 that what a translation is required to do is variable; depending on our theoretical aims, a successful translation will be different. Sometimes it “must depend upon a reversible transformance, or it must be equipollent in respect of a particular language, and so on”

(Carnap, 1937, p 228).14 But it need not always be as such. It might not be reversible, and it might not be equipollent (bijective). Depending on what we are wanting our translation to do, we impose different adequacy conditions on it. So what a translation must do, and what it must preserve is a pragmatic choice.

Given that the original quotes taken to support the traditional view can be reinterpreted, and that Carnap explicitly says that two languages must be discussed in a meta-language and what translations have to preserve is variable, it looks like he would have supported the interpretation given in section 5.4.

Finally, it pays to keep in mind what Carnap says directly about intuitionism as a philosophical position. In section 43 of Logical Syntax, Carnap criticizes the intuitionists for ruling out languages which contain indefinite terms for all purposes. He says

In any case, the material reasons so far brought forward for the rejection... of indefinite... terms are not sound. We are at liberty to admit or reject such definitions without giving any reason. But if we wish to justify either procedure, we must first exhibit its formal consequences. (p 165)

What he is saying here about intuitionism is that it is not always permissible to rule out indefinite terms.15 However, if the formal consequences of a language which contains indefinite terms do not suit the purpose to which it is being applied, then I take it he would allow a ruling out of indefinite terms for that purpose. Thus, in some contexts, intuitionism is good, while in others it is not. Similarly, in Carnap(1939), he states

14Here, a transformance is a map between two languages such that “the consequence relation in [the first] is transformed into the consequence relation in [the second].” A reversible transformance is a transformance such that the reverse relation is also a transformance. Being equipollent in respect of a language amounts to the requirement that any sentences which are mapped to each other are consequences of each other in the language we are respecting. The details here are not as critical to the view as the fact that these are different requirements. 15There are serious questions about whether the intuitionists Carnap took himself to be addressing would agree with these criticisms. See Koellner(Koellner) for more details.

89 If we compare the systems of classical mathematics and intuitionistic mathemat- ics, we find that the first is much simpler and technically more efficient, while the second is more safe from surprising consequences. (p 50-1)

Here, we see Carnap suggesting that depending on our aims (either simplicity or safety when we do mathematics) one of classical or intuitionistic logic may fare better.

In these passages we see that Carnap both thought that intuitionism was good for some- thing, and that the intuitionists were too restrictive in thinking it was good for everything.

Take together with the passages above about translation and language comparison, we can thus infer that Carnap would agree with the proposal in section 5.4 and that classical logic and intuitionistic logic are two candidates for the L-rules of object-level linguistic frame- works.

5.6 Shapiro’s Position

Finally, I would like to suggest that the section 5.4 interpretation puts Carnap in a position much closer to that of Stewart Shapiro than one might have thought. This is odd, given the way Shapiro interprets Carnap (see section 5.2). Carnap should be interpreted as a friend of the Shapiro position, rather than an opponent. I will look at two scenarios from Shapiro

(2014), which are also discussed in chapter6. Shapiro claims that

For some purposes...it makes sense to say that the classical connectives and quan- tifiers have different meanings than their counterparts in intuitionistic, paracon- sistent, quantum, etc. systems. In other situations, it makes sense to say that the meaning of the logical terminology is the same in the different systems. (Shapiro, 2014, p 127)

The purposes/situations in question here can be thought of as something like the S3 dis- cussed above. Shapiro’s two examples come from comparing classical analysis and smooth infinitesimal analysis, and asking whether the logical systems required for each have corre- sponding connectives which share a meaning. In the first scenario, we compare the systems when we are interested in differences between the logics themselves. In the second scenario,

90 we compare the systems in terms of their mathematical consequences. For example, in the first case, two people might be comparing axioms of each system; they might discuss whether the existence of nilsquares is possible. In the second, they might be asking whether both systems prove some theorem; they might discuss whether both systems prove the intermediate value theorem.

Here is the rub: Shapiro claims that in the first case, “[it is] natural to speak of meaning shift” (p 128), while in the second case, “it is more natural to take the logical terminology in the different theories to have the same meaning” (p 130). In our language, in the first case, the logical connectives are different in each system, while in the second they are the same.

To put this in Carnapian terms, let classical analysis and smooth infinitesimal analysis be the first two linguistic frameworks, or S1 and S2. Then, when we are discussing the logics of the systems, as in the first scenario above, we will find ourselves in a meta-linguistic framework where the maps between S1 and S2 which are translations will not map A ∗ B

0 0 0 0 in S1 to A ∗ B (where ∗ is any connective and A and B are the translations of A and B)

0 0 in S2. Rather, they will map A ∗ B to some formula not equivalent to A ∗ B in S2. This will be akin to the case when we considered T2 above, where conjunction was mapped to a double negation of a conjunction, and so the meaning of conjunction was not the same across the two logics. In the second case, discussing the mathematical implications of each system, we ought to find ourselves in a meta-linguistic framework where we will be able to produce a trivial translation between CA and SIA. I suspect in the second case, the translation given by the identity map will be salient, where A ∗ B just gets mapped to

A0 ∗ B0. In the first case, however, because our meta-language is so interested in the details of each system, such a translation will not be particular enough about what gets translated to what. This accords very well with the way Carnap seems to think about translation and cross-framework meanings as discussed in section 5.5.

91 This upshot of this that Shapiro might be a Carnapian logical pluralist (especially given what Carnap says about what a translation needs to preserve above), which would make the traditional slogan about Carnap’s view mistaken. Additionally, by making this comparison, we can address the worry that there is something quite anachronistic about the interpretation of Carnap’s project which I have just presented. There are several differences between the way that Carnap thought of logic and the way we do. Notably, the version of

Carnap I am considering here only used syntactic rules, and his languages were not purely logical by our lights, as they included arithmetical terms. Both of these make it harder to assess what Carnap would say about our current state of affairs. By comparing him to a modern logical pluralist, I take it that we can at least learn something about what he might say, and can certainly learn something about what he would not say. If I am right, no version of Carnap would claim that “change in language necessitates a change in logic.”

5.7 Conclusion

It is often claimed that Carnap must hold that there is language change whenever there is logical change. However, this assumes that we can ask questions about meanings outside of any linguistic framework. This assumes that we can answer the question “does logical change require language change really?”; it is illegitimate on Carnap’s view. What we should be asking is “does logical change require language change in framework X?”. By doing so, we embed the external question into a linguistic framework, and thus make it answerable.

Additionally, when we embed the question into an additional linguistic framework, we see that sometimes there is meaning change when there is logical change, and sometimes there is not. In a sense, this means that Carnap must be Carnapian about the meta-theory.

92 Chapter 6

Logical Pluralism from a Pragmatic Perspective

There are several options for logical pluralism on the table. Rudolf Carnap puts forward a position which is taken to imply that no two distinct logics have connectives which are pairwise synonymous, and Beall & Restall put forward a position in which they typically do.

However, neither Carnap nor Beall & Restall can make sense of the fact that there are some contexts in which distinct logics seem to have logical terms which mean the same thing, and some contexts in which distinct logics have logical terms which are not pairwise synonymous.

In this chapter, I will present a view which can account for both types of contexts. I will demonstrate what factor affects the meanings of the logical connectives, and then provide an account of logical connective meanings which works in light of this factor. Ultimately,

I will suggest that we cannot ask about the meanings of the logical connectives outside of a context, as doing so is asking an external question, in the Carnapian sense, and is illegitimate.

6.1 The Intuitions

There are two particular conversational thought experiments we will focus on throughout this chapter. In both conversations, two participants (either philosophically minded math- ematicians or mathematically minded philosophers) are discussing two analysis systems: classical analysis (CA) and smooth infinitesimal analysis (SIA).

Briefly, SIA is an intuitionistic analysis system, in which all functions are smooth. Im- portantly, it is such that 0 is not the only nilsquare (a nilsquare is an element whose square

93 is zero, i.e. elements x such that x2 = 0). This is because every function is linear on the nilsquares. From this, it is provable that 0 is not the only nilsquare even though “there are nilsquares distinct from 0” leads to a contradiction. This would be inconsistent in classical logic (because of the validity of LEM), and so intuitionistic logic is required. More formally, in a classical system, the sentences ¬∀x(x2 = 0 → x = 0) and ∀x(x2 = 0 → ¬¬(x = 0))

are contradictory. In an intuitionistic logic, since double negation elimination is not valid,

the sentences can both be true. Importantly for us, the SIA system has a very simple and

straightforward proof of the fundamental theorem of the calculus (that is, a proof of “if

F (x) = R f(x)dx then F 0(x) = f(x)”). Rather than, as usual, taking approximations of the

rectangles under a curve as they approach a width of 0, we take rectangles under the curve

which have the width of a nilsquare. No approximations are necessary, and we do not need

the concept of “approaching zero”. See Bell(1998) for more details.

The conversations we are interested in are as follows.

Conversation 1: Here, the participants are discussing the two analysis systems in a

classroom setting.

Rudolf: In my system, I can prove the fundamental theorem of the calculus.

Does your system prove it?

Geoffrey1: I can prove a version of the fundamental theorem, too. I use nil-

squares, numbers whose square is 0. They are not distinct from 0, but it is also

the case that 0 is not the only one!

Rudolf: What? Such nilsquares cannot exist. There is only one nilsquare and

it is 0.

Geoffrey: Well, using your negation, sure.

Rudolf: Oh, I see. Your negation must behave differently from mine.

1Named after Geoffrey Hellman, a staunch modern Carnapian and a proponent of the view that different logics cannot have connectives with the same meanings. See Hellman(2006).

94 Conversation 2: Again, the participants are discussing the two analysis systems in a classroom setting.

JC: In my system the fundamental theorem of the calculus holds. Does it hold

in yours?

Greg: In mine too. I use non-trivial nilsquares to prove it.

JC: There are no non-trivial nilsquares in my system. Interesting. We both

prove the same thing differently.

Intuitively, in both conversations, two logics are in play: the logic of CA, classical logic, and the logic of SIA, intuitionistic logic. Additionally, in the first conversation, between Rudolf and Geoffrey, the logical connectives in each logic in question seem to mean something different (the participants even suggest as much, stating that the negations must be different). In the second conversation, they seem to mean the same thing. At the very least, they seem to be talking about one and the same fundamental theorem. If the theorems are the same, then the logical terminology in the formal statements of the fundamental theorems must have the same meaning.

Carnap can make sense of the first conversation, and Beall & Restall can make sense of the second.2 However, neither can accommodate both of them. In this chapter, I will present a system which accommodates both intuitions. This conclusion in very much in line with the underlying theme of Shapiro(2014), but is put in a theoretical framework (Shapiro presents no such system).

In the next section, I will explain in more detail why neither Carnap nor Beall & Restall can account for both of the two situations above. In some sense, I will show that Carnap is a pluralist about the connectives and thus about logic, and so on the traditional reading will not be able to easily account for conversation 2, while Beall & Restall are pluralists

2It is possible that, depending on one’s interpretation of Carnap, he can actually accommodate both intuitions. See chapter5 for details on one way of spelling this out.

95 about logical consequence but monists about the connective and so cannot easily account for conversation 1.3 I will then develop a method for accounting for both conversations, using the Question-Under-Discussion framework of Craige Roberts(2012), and show how we can account for the change in connective meaning by a factor picked out by the framework.

Finally, I will develop an account of meaning which accords well with this shift.

6.1.1 Carnap’s and Beall and Restall’s Pluralisms

If we take seriously the traditional positions attributed to Carnap and Beall & Restall in sections 2.3.2 and 2.3.3 in chapter2, then we can see how neither of them can account for both conversations. Carnap cannot account for conversation 2, because his view requires that any time a new logic is introduced, a new language is introduced, and so logics cannot share languages. Beall & Restall cannot account for conversation 1, because any admissible logic on their view will share a language with all of the other admissible logics, and so they cannot have a situation with two logics in distinct languages.

6.2 The New System

In this section, I will show how we can use Roberts’s Question Under Discussion (QUD) framework to analyze these conversations, and how whether we can maintain a certain default assumption in the common ground is the factor which affects connective meanings.4

In section 6.2.1, I will present the framework, and show how it models the two conversations.

This section, I claim, shows that the factor which affects whether two connectives mean the

same thing is whether a certain default assumption can be preserved in the common ground

(what all conversational participants agree is true). This assumption is that if two words

sound alike, are spelled the same way, and are generally used in the same sentences in the

same way then they mean the same thing. I will refer to this as the “correlation as identity

3Thanks to Giorgio Sbardolini for the suggestion to put it this way. 4Thanks to Brian McLean and Chris Pincock for much help in refining the structure of this section.

96 proposition”, or CIP for short. I claim that CIP is a default for most conversations (at least the deductive ones I consider here), and like any default, it can be given up. Conversations where CIP is part of the common ground are conversations where the logical connectives in question mean the same thing. Conversations where CIP cannot be part of the common ground are conversations where the logical connectives can mean something different. In section 6.2.2, I will provide an account of the meanings of the logical connectives which accord with whether CIP ∈ CG.

It is important to note that the conversations I will discuss, and those to which I think

my framework applies, have three important features. The conversations I consider are all

what I will call deductive conversations. These are conversations, broadly construed, about

mathematics or where the participants are doing or discussing mathematics. For example,

lectures in a typical mathematics class count. As does a mathematician working out a proof

by herself on a blackboard. Anything that can be construed as the exchange of mathematical

or deductive information between a speaker and an audience, even including those cases in

which the sole speaker is the sole audience member, is a deductive conversation on this

picture.

Additionally, I will assume that the conversational contexts are sufficiently regimented

to carry the appropriate logics with them. This will help give us a transcription between

the natural language conversation and a formal language counterpart for each deductive

conversation. I will proceed for the time being by assuming that there is an intuitive

transcription from assertions to sentences in formal languages. In most cases, I think this

will be the simple enough.5

5It is not, however, clear that this will always work. Field(2009) provides an example where translating negation into formal language is quite complicated (p 346-7). Additionally, for the purposes of this chapter, since we are only considering mathematical conversations which are easily transcribable into formal lan- guage, we can ignore one other potential problem: the problem of assessing whether people “speak” formal languages. This is a hard question, and I will not answer it here. It suffices for my purposes that people “control” the formal language. That is, they must know “what constitutes a well formed formula” (Roberts, personal correspondence) and know how to use it model theoretically and/or proof theoretically. They must be familiar enough with the system to have some sense of what follows from what, and be able to give a rough gloss on what that means.

97 Finally, I will assume that something like a principle of charity and accommodation is in play. Each conversational participant is meant to be as charitable as possible to each other participant. For example, if it looks like a participant has uttered a falsehood, or a contradiction, the remaining participants are obliged to do everything they can to rationalize the utterance. Participants must accommodate as much as possible to ensure they do not have to believe their fellow participants are irrational. For more details, see Grice(1969) and Lewis(1979). 6

Interestingly, the requirement of charity is very much in line with the Hilbertian per- spective of Shapiro(2014). The Hilbertian perspective, roughly, is the position that most mathematical theories, so far as they are coherent, are legitimate. Here, we share a similar point of view: so long as a deductive utterance can be accommodated, it can be seen to be coming from the perspective of a legitimate logical theory.

6.2.1 QUD framework

The framework I will consider is a direct adaptation of the system provided by Roberts

(2012) for analyzing natural language conversations.7 The system is particularly useful when dealing with presuppositions and for explaining why conversational participants participate

6I think this framework can be extended to cover more situations that just those discussed here as long as there is a strong principle of charity and accommodation in play. 7One immediate objection here is that the following framework necessitates a contextualist definition of the logical connectives, and leaves very little for the semanticists to study. Though a polysemous reading of the connectives is ultimately what I argue for in section 6.2.2, it does not need to be the case, and moreover, even if it is contextual, this still leaves room for semantics. First, the framework I propose here is a pragmatic framework. Even if this framework is accepted, one is still at liberty to assume that there is an invariant semantics for the connectives underlying the surface/pragmatic variance. For example, one would be at liberty to say that either the “different logics, same connectives” camp or the “different logics, different connectives” camp was right on the semantic level, and that what this system is pointing to is pragmatic variation for Gricean and Lewisian purposes which does not affect the semantics. In this sense, though there might be different usages of one and the same sentence which convey different things, a single sentence would have a single invariant conventional meaning. Though I do not think this is ultimately correct, it is a position available. Thanks to Kevin Scharp for pushing me on this issue. Second, the most common position in semantics today is that meanings are dependent on context. There is no reason why the work in linguistics departments done on context sensitive terms cannot be extended here. The claim that this type of project leaves no work for semanticists is thus mitigated.

98 in conversations at all. I give the details briefly here - it is a technical system and I will only be interested in certain aspects, which I will highlight in the next sections. The system is roughly an expansion and formalization of the system in Lewis(1979).

1. I: the set of interlocutors at time t

2. G: a function from pairs of individuals in I and times t to sets of goals in effect at t

such that for each i ∈ I and each t, there is a set G(i, t) which is i’s set of goals at t

3. Gcomm: the set of common goals at t; i.e. {g|∀i ∈ I, g ∈ G(i.t)}

4. M: the set of moves made by interlocutors up to t with the following distinguished

subsets: A, the set of assertions, Q, the set of questions, R, the set of requests, and

Acc, the set of accepted moves

5. ≤: a total order on M that reflects the chronological order of moves

6. CG: the common ground; i.e. the set of shared presupposed propositions at t8

7. QUD: the set of questions under discussion at t; i.e. a subset of Q ∩ Acc such that

for all q ∈ QUD, CG does not entail an answer to q and the goal of answering q is a

common goal

The conversational score is updated as follows:

1. Assertion: if an assertion is accepted by all interlocutors, then the proposition asserted

is added to CG and Acc

2. Question: if a question is accepted by all interlocutors, then the set of propositions

associated with the question is added to QUD. A question is removed from QUD if

and only if either its answer is entailed by CG or it is determined to be unanswerable

8Here, “presupposed” is meant to mean something like “treated as true” rather than simply “true”.

99 3. Request: if a request is accepted by an interlocutor, i, then the goal associated with

the request is added to G(i, t) and the proposition that i intends to comply with the

request is added to CG

We need to make some small changes to this framework to make it suit our purpose.

First, I will use the term “proposition” only roughly, for something like a formalized sen- tence, in order to avoid having to make substantial (controversial) claims about just what propositions are. This view can be made to fit many views about the nature of propositions.

Second, we will not assume that CG cannot entail an answer to a question q for q to be added to QUD. If it did, when our common ground includes the axioms for a theory (as will often be the case for us), we will not be able to ask any question about what that theory entails. Rather, we will assume that if CG entails an answer to such a question in a simple, straightforward and clear way, then that question cannot be added to QUD.9 This will, in most cases, be a context sensitive matter. Finally, I will ignore the sets R and DR for ease of exposition.

Our primary concern is the notion of a question under discussion. QUD is a set of ques- tions that all of the conversational participants are trying to answer. The conversational goal of answering each question in QUD is shared by all participants. Answering the ques- tions under discussion is an element of Gcomm. Thus, the goals of answering the questions in QUD are some of the (shared) conversational goals.

I will show that conversations as described above fall into one of two categories: conver- sations where CIP ∈ CG and conversations where the interlocutors are forced to remove

CIP from CG.10 In the first case, the logics in play in the conversation will have connec-

9One might substitute Stalnaker’s term “available” for what I am calling “simple/straightforward/clear” inference. See also (Stalnaker, 2014, p 24). 10Whether or not CIP ∈ CG does not affect the status of the proposition expressed by CIP as true or false, but only whether the conversation participants are treating it as true for the purposes of the conversation. See footnote8. In effect, this means that the theory I’m giving here is a pragmatic theory. In section 6.2.2, I will suggest that there is no further story to tell. Thanks to a blind referee for pushing me on this issue.

100 tives which mean the same thing. In the second case, the logics in play in the conversation may have connectives which do not share meanings. This will roughly line up with the framework in Shapiro(2014). I suspect that his “logical” contexts will correspond to con- texts in which we have to remove CIP from CG, and that his “mathematical” contexts will correspond to the contexts in which we can leave CIP in place.11

Test Case

The first conversation we will consider is a simple conversation. The context is such that

a student and a professor in a typical North American analysis class, discussing classical

analysis:

Teacher: Is the fundamental theorem of the calculus true?

Student: Yes.

Teacher: How do we prove it?

Student: We use a limit proof by Riemann sums.

Here, the conversational participants (Teacher and Student) are simply proving a theo-

rem in a system. The questions are something like “is the fundamental theorem a theorem?”

and “how do we prove it?”. Since the system is implicitly agreed upon, and requires a cer-

tain logic, this is the right logic for the conversation. In other words, this conversation also

has exactly one salient logic: classical logic.

Assessing the connective meanings is another matter. The only connectives “used” are

those in the formalization of the fundamental theorem. For the participants, the issue is

never raised, and does not matter. As interpreters, we can spell out the connective meanings

11This may not always happen. For example, though CA proves the intermediate value theorem, SIA does not. It would, on my view, be possible for a conversation between two people about the intermediate value theorem to be one in which we have to remove CIP from CG, which would result in a conversation where the corresponding connectives did not share a meaning, and thus where the intermediate value theorems in each system were not the same. This is not possible on Shapiro’s view.

101 in a number of ways (truth conditions, inferential role, etc.), as long as the meanings accord with the manner in which the conversational participants use the connectives.

Formalization in the Roberts framework

We start by assuming, for the purposes of simplicity, that the teacher’s first utterance takes place at time t1 and the student’s last utterance takes place at time t4, with t0 being the time before the conversation starts (in the class, I assume). The set I of interlocutors

at each of the time in question is {Teacher, Student}.

The common ground, CG, at t0 contains the axioms of classical analysis and some other

information about the teacher/student relationship, the classroom, etc. The sets Gcomm, M, A, Q, QUD, and Acc are empty.

At t1, Q contains the question “Is the fundamental theorem of the calculus true?”, as does Acc and QUD. The first move in M, then, is the accepting of the teacher’s question.

Answering this question is added to the set Gcomm. Notice that the question gets added to QUD even though the CG entails its answer. It is not the case here that the CG simply,

straightforwardly and clearly entails the answer here. In this context, the student may be

unaware of this entailment, or still working out the answer. Thus, on our adapted system,

the question “Is the fundamental theorem of the calculus true?” is a legitimate question.

At t2, the student provides us with an answer to this question in the affirmative. This means that the student has realized that the CG entails the answer to the question. The

answer is “promoted” in the CG to one of the simple, straightforward and clear entailments.

A now includes the assertion that the fundamental theorem is true, and the question is

removed from QUD. The proposition “The fundamental theorem is true” is also added to

Acc.

At t3 and t4 a similar thing occurs. The teacher’s question of how the proof works is added to Acc, Q and QUD. The student, in answering “We use a limit proof by Rie-

mann sums” successfully removes the question from QUD and adds the proposition that

the proof is conducted by a limit proof of Riemann sums to A and Acc. CG now con-

102 tains the original information, plus the assertion the fundamental theorem is true and simply/straightforwardly/clearly inferable, and that the proof is done via a limit proof by

Riemann sums.

G is the function which can be reconstructed from this conversation. Presumably, at all times, the teacher has the general goal of helping the student learn, and the student has the general goal of learning. In addition, the student’s goals at t1 include answering the question, and the teacher’s goals include having the student answer the question. The same goes for t3. At t2 and t4, respectively, these goals have been achieved.

This is a simple example of a conversation and its formalization in the Robert’s frame- work. I now move onto the two conversations we are primarily concerned with.

Conversation 1

Recall conversation 1. Here, the participants are discussing two analysis systems: classical analysis (CA) and smooth infinitesimal analysis (SIA).

Rudolf: In my system, I can prove the fundamental theorem of the calculus.

Does your system prove it?

Geoffrey: I can prove a version of the fundamental theorem, too. I use nilsquares,

numbers whose square is 0. They are not distinct from 0, but it is also the case

that 0 is not the only one!

Rudolf: What? Such nilsquares cannot exist. There is only one nilsquare and

it is 0.

Geoffrey: Well, using your negation, sure.

Rudolf: Oh, I see. Your negation must behave differently from mine.

One of the questions in this conversation is something like “in each system, do there exist types of things which are not distinct from 0, but such that 0 is not the only one?”,

103 and another is something like “what is consistent in your system given the meanings of your negation?” (in addition to something like “what does your system prove?”).

Intuitively, if we assumed that both participants would interpret themselves as using the same connectives, we would have a problem with charity and accommodation. Geoffrey has uttered something which can be taken to imply the negation of a classical truth. His utterance implies that ¬∃(x 6= 0) ∧ ∃(x 6= 0). Since the double negation cancels in classical logic, this has the form ∀x(x = 0)∧¬∀x(x = 0) for the classical analyst, a clear contradiction.

Hence, adding the existence of these types of nilsquares to a classical system will result in

a contradiction. Thus, Rudolf cannot accept the existence of these objects and maintain

a consistent theory. If the conversation continues after the existence of such nilsquares is

posited (and not just by a simple “no” from Rudolf), then we must assume that Rudolf

is assuming that Geoffrey is using different connectives. Geoffrey must mean something

different by the use of his connectives for the conversation to continue, and so the most

charitable interpretation of this conversation is one in which the negation connectives have

different meanings.

If we assumed that both participants were using the same negation connective, then

when Rudolf accepts Geoffrey’s claims about nilsquares, he would have to accept that he

had an inconsistent theory. This is not something we can attribute to Rudolf’s reasoning.

This is in line with our principles of charity and accommodation. The best interpretation

of this conversation is one where the participants are “speaking different logical languages”;

the best interpretation is one in which there are two negations (say ¬CA and ¬SIA), and no connective means the same thing as any other. If we follow our intuitions, then the Robert’s

framework ought to generate a Carnapian style pluralism: “different logics, different con-

nectives”.

The story in the Roberts’s system works out the same way.12 What we wind up showing

is that, in effect, the presupposition that the connective meanings were shared is what

12Thanks are due to Craige Roberts for suggesting that the original common ground was defective.

104 makes the original context defective (because the CIP default needs to be changed). The conversation proceeds as normal (in fact, the first two utterances in both conversations are incredibly similar) until Rudolf challenges Geoffrey with another question: a question expressing skepticism about the possibility of nilsquares, perhaps something like “how is it that your nilsquares exist without contradiction?”. Geoffrey, in order to answer this, suggests that the negations in question must be different, which Rudolf accepts with his

final contribution to the conversation. Here, we have to make a change to the common ground. If we assumed that CIP ∈ CG from the beginning, then this is what we have to change. In this sense, the default is that these words have the same meaning, until we are put into a position where the only way to accommodate is to assume that they do not. The formalization of the conversation, then, will show that the way Rudolf and Geoffrey have to accommodate each other is by adding to the common ground that the connectives in question actually mean something different, even though they sound/look the same.

What this shows is that the original common ground was defective. When the original question “does your system prove the fundamental theorem?” is posed, Geoffrey and Rudolf are assuming that they are talking about the same fundamental theorem, and are therefore both looking for an answer to a single question. This, though, cannot be the case, as we see by the manner in which each participant had to accommodate the other by assuming that the logical connectives in question were different. They must, in effect, both prove a very closely related thing, though not exactly the same thing. This is an effect of the common ground being updated to include Geoffrey’s implication that the negations in each system are different.13 13One possibility not addressed here is that “in my system” is something more like “in my book”. In this way, we might liken the debate between Geoffrey and Rudolf to a debate between two historians wondering about the nature of Cleopatra (this example and issue are due to a blind referee. Certainly, if two historians were disagreeing about Cleopatra, we would not want to interpret them as discussing two distinct historical figures. In this sense, there is a non-removable proposition in the common ground which expresses the fact that there is exactly one Cleopatra. On the other hand, we have no such certainty in the logical case. The evidence available in the logical cases is quite different. We cannot point to historical records or relics, but only to the behaviors of the conversational participants and the rules and/or truth conditions they profess to associate with each connective. Thus, in the historical case, it is more charitable to assume that there is

105 Formalization in the Roberts framework

I assume that t0 is the time before the first utterance, t1 the time of the first utterance

and t5 the time of the last. I is {Rudolf, Geoffrey}. At t0, CG includes the axioms of classical analysis and smooth infinitesimal analysis, and the fact that each analysis system

is coupled with a different logic, and the CIP . Gcomm, M, A, Q, QUD, and Acc are empty.

At t1, Rudolf promotes the proposition that his system proves the fundamental theorem to a simple, clear and straightforward implication in CG, and A and Acc. At the same time,

he adds to Q the question of whether the other system proves the theorem. It also gets

added to QUD, since in this context, even though CG implies it, it is not yet a simple, clear

and straightforward implication. Rudolf’s goals at t1 include finding out whether Geoffrey’s system proves the theorem.

At t2, Geoffrey answers the question by claiming that the other system proves a version of the fundamental theorem. Additionally, Geoffrey attempts to add new information to

the CG: that his system uses nilsquares to prove this theorem. He also proposes to add to

CG and A that nilsquares are not distinct from 0, but it is also the case that 0 is not the

only one.

At t3, Rudolf tries to block these two additions from entering Acc and CG by claiming that the utterance is incoherent. “Non-trivial nilsquares do not exist” is added to A, but

not Acc. This is the moment where he challenges Geoffrey’s assumption that one can posit

the existence of non-trivial nilsquares without thereby engendering a contradiction.

At t4, Geoffrey implies that because of his distinct negation meaning, the existence of such nilsquares is not contradictory. He promotes to a simple/clear/straightforward

implication in CG that his logic, and connectives, are different from Rudolf’s. He also adds

this assertion to A. Here we see that Geoffrey and Rudolf have to accommodate by changing

a fundamental assumption in the common ground, namely CIP . This is a radical change, exactly one thing, Cleopatra, which the historians are discussing. While, in the second, it makes more sense to let how many connectives are (pragmatically) in play be guided by the behaviour of the participants.

106 but is necessary for the conversation to continue charitably. CIP is removed from CG.

At t5, Rudolf accepts this. Propositions about the difference in connective meanings, logics, and existence of nilsquares in are added to CG. Geoffrey’s assertions that non- trivial nilsquares exist in his system and that the negation connective does not entail that these are contradictory are added to Acc and CG, and the proposition that SIA proves a fundamental theorem is promoted to a simple/clear/straightforward implication in CG. All is well. We can make sense of this conversation if we assume that the initial common ground was defective, in that it contained CIP . By the end, Rudolf and Geoffrey have rectified this, and have discovered that they both prove a theorem which looks like the fundamental theorem, but because of the addition to the common ground that words which satisfy the antecedent of the CIP do not have to mean the same thing, might not be the same theorem in both systems.

Conversation 2

Recall conversation 2. Two participants, JC and Greg, are seeing what is common to their two theories.

JC: In my system the fundamental theorem of the calculus holds. Does it hold

in yours?

Greg: In mine too. I use non-trivial nilsquares to prove it.

JC: There are no non-trivial nilsquares in my system. Interesting. We both

prove the same thing differently.

Here, one of the questions is something like “what tools does each system use to prove the FTC?”. JC and Greg are behaving as though there is one fundamental theorem which they both prove. If we think that the statements of the fundamental theorem mean the same thing, then because those statements contain logical operators, we must consider those as meaning the same thing as well. This means that we would have to treat the connectives

107 as meaning the same thing. However, it still requires two distinct logical systems: one which licenses non-trivial nilsquares and one which does not. This generates something like a Beall-Restall pluralism: there are two logics, but both have connectives which have the same meanings.14

Importantly, nothing that is said flags the common ground as defective. We are given no reason to change the assumption that CIP ∈ CG, and so there is no need to do so.15

Formalization in the Roberts framework

I assume that t0 is the time before the first utterance, t1 the time of the first utterance and t3 the time of the last. I is {JC, Greg}. At t0, CG includes some basic information about analysis and the proposition that both participants use different analysis systems and different logics, and Gcomm, M, A, Q, QUD, and Acc are empty.

At t1, JC adds the proposition that his system proves the fundamental theorem to A and it is also added to Acc, and promoted in CG to a simple clear and straightforward implication. At the same time, he adds to Q the question of whether the other system proves the same theorem. It also gets added to QUD. JC’s goals at t1 include finding out whether Greg’s system proves the theorem.

At t2, Greg answers the question affirmatively. The question posed by JC is removed from QUD and the proposition that Greg’s system proved the fundamental theorem is promoted in CG to a simple, clear and straightforward theorem, and added to A and Acc.

JC’s goal of finding out whether this new system proved the theorem has been satisfied.

Additionally, Greg adds new information to the CG: that his system uses nilsquares to

14Interestingly, Beall and Restall would not accept this style of pluralism, since SIA proves negations of classical theorems, and they claim that all admissible logics are either classical or classical subsystems. It is also not clear that Carnap would accept the previous case, as it may involve one linguistic framework with more than one logic. This is an interesting result, and I will explore it in further work. 15One thing to notice here additionally is that if this conversation continued, it is distinctly possible that Greg’s next utterance would be something like “You must mean something different by “proof” than I do.” In this case, it would be possible that we would have to remove CIP from the common ground, but that it would not affect the meanings of the logical connectives, but rather the meaning of “proof”. This is interesting, since it shows two things. First, it makes explicit the fact that there are two logics in play in the conversation. Second, it shows that CIP does not need to be restricted to just the logical connectives, but can affect much of what is going on in the conversation.

108 prove this theorem. That is, he adds the propositions that his system licenses non-trivial nilsquares, and that non-trivial nilsquares are used in the proof of the fundamental theorem.

He adds no new questions.

At t3, JC adds to the common ground that his system has no non-trivial nilsquares. This proposition is promoted in CG, and added to A and Acc. Again, nothing said suggests the common ground was defective, and so we can maintain that the logical connectives mean the same thing in both logics. At the end of the day, CIP ∈ CG.

The Factor Affecting Connective Meanings

So far, so good. What is important for this project is the following: whether CIP ∈ CG can directly affect the meanings of the logical connectives in question.

Roughly, the idea is that, when CIP ∈ CG the logical connectives mean the same thing across different logics, while if the interlocutors are forced to remove CIP from CG, the assumption that the connectives must mean the same thing is given up. The factor responsible for the intuitions in section 6.1 is the presence or absence of CIP from the CG at the time we are assessing whether the connectives share a meaning.16

6.2.2 A New Theory of Connective Meaning

We need an account of the meaning of negation (and the rest of the connectives) which can be treated in such a loose way. I sketch a possibility in this section to show that there is logical space for such a theory of meaning. I also motivate why we should only consider

16I need to give a story about what happens in “mixed” conversation. That is, what happens in conver- sation where CIP cannot be part of the common ground at a certain point, but can be part of the common ground before and after that point. It seems that we ought to maintain that if CIP is removed from the common ground and then “added back” that the meanings of the connectives or “proves” might change throughout the conversation. Though on the face of it, this is weird, there is a reason to think this might be the case. The purpose of the conversation can be cashed out in terms of which conversational goals are in play at any given moment (i.e. by what is in Gcomm). Any time a new goal becomes salient (or the participants are trying to answer a new question), the status of CIP might change. But this is to be expected. We have been maintaining that the right logic and meanings of the connectives are relative to a purpose and a context, so we ought to accept that when out purpose changes, sometimes the meanings will change.

109 meaning relative to contexts context by suggesting that what the connectives really mean is something of an external question. This is very much along Carnapian lines.

Connective Meanings as Polysemous

There is something it seems everyone is after when they claim to have provided a new formal definition of negation. This is true not only for the classical logical, the intuitionist, the relevance logician, but also the linear logician, the dialetheic logician, etc. The claim

I want to make in the section is that they are all, roughly, after the same thing, which we might call “pre-theoretic negation”, and further that the concepts they wind up arriving at are related to each other in a polysemous way. We might also call this a “fluid concept,” in the sense of Lynch(1998). There, he states that “the search for the essence or common property expressed by a [fluid] concept is futile” (p 62). In a sense, this is what I will spell out here. We have some idea of what “pre-theoretic negation” is, but giving any more than broad-strokes descriptions of it is impossible.

I will make this argument in two steps. First, I will show that at least sometimes, the meaning of negation is contextual. That is, sometimes it depends on a parameter in a definition of negation being spelled out in a particular way. Second, I will suggest that even the meanings which cannot be spelled out in such a way are relevantly similar, and thus can be thought of as polysemous with the contextual concept. Once we have that the formal versions of negation are polysemous, we can immediately see that what they are aiming at,

“pre-theoretic negation” cannot be very robust at all. If it were, then many of the things called “negation” would not be negations at all. This is a consequence we do not want to accept.

First, there are at least two formal negation connectives which can be thought of as contextual variants of each other. Consider negation in a Kripke structure, for example. A

Kripke structure is a model for intuitionistic logic, on which we have a set of nodes, say

W, and an accessibility relation, say ≤, with the relation intuitively being equivalent to

110 ‘constructed out of’. ≤ is reflexive and transitive. The clause for negation (adapted from

Hjortland(2013)) is

For all w, u ∈ W, w |= ¬A if and only if, for all u such that w ≤ u, u 2 A

Here, we ought to notice that we can make a change to have this clause generate a classical clause for negation: if we restrict the w, u not to everything in W, but rather to only those nodes in W which are terminal, i.e. which have nothing “constructed out of them” except

(trivially) themselves. We might then construct a contextualist clause like the following

For all w, u ∈ X, w |= ¬A if and only if, for all u such that w ≤ u, u 2 A where X varies over some contextually salient domain. If this is a candidate for the mean- ing of negation, then at least sometimes, the meaning of negation is dependent on some contextual factor. I suspect that in cases which the negations in play vary only in that some parameter in their definition is filled out different by the context are those in which CIP is reasonably maintained as a default.

It should be clear, though, that not all negation clauses will fit this mold. What of those which can only be defined relative to a proof theory? Or the clause from a multi-valued logic, like Priest’s LP? These types of negations will not fit the contextual clause given above.

However, they are in some sense “close”. There is something common to the contextualist clause and the clauses given by the proof theorist and multi-valent logician. This might be something like a family resemblance. In this case, I claim it is close enough that these formal negations are polysemous with the contextual ones. Further, the questions under discussion will help resolve this polysemy. The polysemy of the connectives only needs to be resolved when the questions under discussion make CIP salient, or require that it be challenged. Once we are in a situation where the questions make the CIP salient, then we can decide how to resolve the polysemy based on the conversational goals. Depending on what we are after, we either resolve it by letting the connectives in question mean the same thing, or resolve it by forcing the connectives in question to have distinct meanings.

111 There is something about being a conjunction, disjunction, negation, etc. that we seem to intuitively recognize. In the words of Potter Stewart, “I know it when I see it”. We might say that each connective has a core meaning. There is something that draws us to class one thing as negation and another as disjunction. They share a family resemblance.17

In the terms we used earlier, they are logical terms which usually abide by the antecedent of CIP . This is the sense in which they are polysemous.

The interesting feature of the system presented here is that we can use the conversa- tional goals and questions under discussion to resolve much of the polysemy. In effect, questions under discussion are excellent tools for helping to identify the topic of any given conversation. Once we identify just what it is we are talking about, we can identify which negation we are using from the polysemous bundle of them. For example, if we are talking about or doing classical mathematics, then we can say safely that we are using classical negation. However, if we are talking about or doing non-exploding naive set theory, we can safely say we are using a paraconsistent negation.18

Further, this distinction between those negations which are merely polysemous, and those which can sometimes be contextual variants of each other allows us to make sense of some intuitions. In the two conversations above, it was easy for the conversations to continue either after CIP was removed from CG, or assuming that the negations in question were the same. However, imagine a similar conversation, but this time between a classical logician and a dialetheist. It seems in this conversation, when the classicist discovers that her interlocutor is using a negation with which both A and ¬A can be true, she is allowed to claim that she does not know whether that negation means the same thing as hers. It could be that she can make such a claim because it would be very hard to find a definition of negation on which the paraconsistent negation was a contextual variant of the classical

17One should also note that this is another place where this view diverges from Shapiro(2014), who holds that there no such meaning is likely to be articulable. The considerations in section 6.2.1 can be seen roughly as a formalization and extension of his views, and by and large agree with the underlying theme. 18This will also work for the other connectives, although some, like conditionals, may have more versions related by polysemy rather than contextualism.

112 one. Thus, the mere polysemy (rather than contextualism) of the two negations in question might explain why sometimes uses of negation are hard to reconcile with each other, either by assuming they are the same, or assuming they are distinct. Sometimes it is just not clear which camp they fall into.

Real Meaning as an External Question

Here, then, we have a system which governs the pragmatic story of when two connectives share a meaning. It does not address the question of whether any two sentences involving negation really mean the same thing, as this will boil down to whether the “pre-theoretic” notion of the connectives in play is the same. This is not a robust enough notion of “same meaning” to do any significant work; no logic will flow from the pre-theoretic meaning, it is simply to thin to stand in any logic relation to the pre-theoretic meaning of the other connectives.

We can ask instead about logical meanings: when do two connectives, as they are spelled out in a context, share a meaning robust enough to be logical? I think this is something like a pragmatic external question, in the Carnapian sense. Two sentences involving negation cannot be said to share a logical meaning simpliciter, because then we would have to ask whether they share a logical meaning without some conversational goal guiding our inquiry.

But without such a goal, or a determination of the status of CIP , we do not know how to partition the sentences. Without such a partition, we cannot answer these robust meaning questions.

More generally, I think that what I have proposed is that logical connectives simply have no meaning outside of their meaning in a context. This means that the question of the meaning of the logical connectives, when asked without a purpose or application of the logic in mind, is unanswerable, in the same sense that external question on the Carnapian picture are unanswerable. In this sense, we have extended the Carnapian picture: not only is it illegitimate to ask for the right logic outside of a pragmatic structure, it is also

113 illegitimate to ask for the meanings of the logical connectives for any particular logic outside of a pragmatic structure. Where there was once only one external question, we now have two. Without a context, without a goal, we can have no logical meaning. Once we fill out these, details, though, we are at liberty to “build up [our] own logic”, to preach Carnapian tolerance, and moreover, we can build that logic so that it shares connectives with another, or so that it does not. We might change his second slogan to “in logic and connective meanings, there are no morals”.

6.3 Conclusion

I have shown one can produce a logical pluralism which makes sense of both conversations in question. Carnap was able to make sense of one, where the connectives in question intuitively mean something different. Beall & Restall are able to make sense of the other, where the connectives in question mean the same thing. Following the Hilbertian intuition in Shapiro(2014), I have shown how one can construct a logical pluralism which relies on the linguistic system presented in Roberts(2012) which makes sense of both conversations.

That is, sometimes the logical connectives in such a conversation have the same meaning, and sometimes they have pairwise distinct meanings. Questions about the meanings of the logical connectives cannot be asked outside of a context of use. Such questions are illegitimate; they are Carnapian pseudo-questions.

114 References

Anderson, A. R. and N. D. Belnap (1975). Entailment: The Logic of Relevance and Nec- cessity, Vol. I. Princeton University Press.

Arthur, P. (1963). The Philosophy of Rudolf Carnap, Volume XI. Open Court.

Barwise, J. and J. Perry (1983). Situations and Attitudes. Mit Press.

Beall, J. and G. Restall (2000). Logical Pluralism. Australian Journal of Philosophy 78 (4), 475–493.

Beall, J. and G. Restall (2006). Logical Pluralism. Clarendon Press Oxford.

Beall, J. C. and G. Restall (2001). Defending Logical Pluralism. In J. H. Woods, B. Brown, and S. of Exact Philosophy (Eds.), Logical Consequences: Rival Approaches, pp. 1–21. Hermes Science.

Bell, J. L. (1998). A Primer of Infinitesimal Analysis. Cambridge University Press.

Belnap, N. (1982). Display Logic. Journal of Philosophical Logic 11, 375–417.

Bremer, M. (2013). Restall and Beall on Logical Pluralism: A Critique. Erkennt- nis 79 (Supplement 2), 293–299.

Brouwer, L. (1983a). Consciousness, Philosophy and Mathematics. In P. Benacerraf and H. Putnam (Eds.), Philosophy of Mathematics, pp. 90–96. Cambridge University Press.

Brouwer, L. (1983b). Intuitionism and Formalism. In P. Benacerraf and H. Putnam (Eds.), Philosophy of Mathematics, pp. 77–89. Cambridge University Press.

Bueno, O. and S. Shalkowski (2009). Modalism and Logical Pluralism. Mind 118 (470), 295–321.

Burgess, J. (1992). Proofs about Proofs: A Defense of Classical Logic. In M. Detlefsen (Ed.), Proofs, Logic and Formalization, pp. 8–23. Routledge.

Burgess, J. (2010). Review of JC Beall and Greg Restall, Logical Pluralism. Philosophy and phenomenological research 81 (2), 519–523.

115 Burgess, J. P. (2015). Rigor and Structure. Oxford University Press.

Carnap, R. (1937). The Logical Syntax of Language. New York: Harcourt, Brace and Company.

Carnap, R. (1939). Foundations of logic and mathematics. International encyclopedia of unified science. The University of Chicago press.

Carnap, R. (1947a). Meaning and Necessity - A Study in Semantics and Modal Logic. University of Chicago.

Carnap, R. (1947b). Meaning and Necessity - A Study in Semantics and Modal Logic. University of Chicago Press.

Carnap, R. (1950). On Explication. In Logical Foundations of Probability, pp. 1–18. Uni- versity of Chicago Press.

Carnap, R. (1983). Empiricism, Semantics and Ontology. In P. Benacerraf and H. Putnam (Eds.), Philosophy of Mathematics, pp. 241–257. Cambridge University Press.

Carnap, R. and W. Quine (1995). Dear Carnap, Dear Van: The Quine-Carnap Correspon- dance and Related Work. Berkeley, CA: University of California Press.

Cook, R. T. (2010). Let a Thousand Flowers Bloom: A Tour of Logical Pluralism. Philos- ophy Compass 5/6, 492–504.

Cook, R. T. (2013). Should Anti-Realists be Anti-Realists about Anti-Realism? Erkennt- nis 79, 233–258.

Devidi, D. and G. Solomon (1995). Tolerance and Metalanguages in Carnap’s Logical SYntax of Language. Synthese 103, 123–139.

Dummett, M. (1973). The Philosophical Basis of Intuitionistic Logic. In Truth and Other Enigmas, pp. 215–247. Duckworth.

Dummett, M. (1976). What is a theory of meaning? (ii). In G. Evans and J. McDowell (Eds.), Truth and Meaning: Essays in Semantics. Oxford: Clarendon Press.

Dummett, M. (2000). Elements of Intuitionism. Oxford: Clarendon Press.

Dummett, M. A. E. (1974). The Justification of Deduction. In Truth and Other Enigmas. Oxford.

116 Dummett, M. A. E. (1978). Truth and Other Enigmas. London: Duckworth.

Dummett, M. A. E. (1991). The Logical Basis of Metaphysics. Harvard University Press.

Dunn, J. M. (1993). Star and Perp: Two Treatments of Negation. Philosophical Perspec- tives 7 (May), 331–357.

Dunn, J. M. and G. Restall (2002). Relevance logic. In D. G. Guenther and Franz (Eds.), Handbook of Philosophical Logic Vol 6, pp. 1–1136. Kluwer.

Ebbs, G. (2016). Carnap on Ontology. In Carnap, Quine and Putnam on Methods of Inquiry. Cambridge Univeristy Press.

Etchemendy, J. (2008). Reflections on consequence. pp. 1–39.

Field, H. (2009). Pluralism in Logic. The Review of Symbolic Logic 2 (2), 342–359.

Franks, C. (2015). Logical Nihilism. In A. Villaveces, R. Kossak, J. Kontinen, and A. Hirvo- nen (Eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, pp. 147–166. De Gruyter.

Friedman, M. (1988). Logical Truth and Analyticity in Carnap’s ”Logical Syntax of Lan- guage”. Essays in the History and Philosophy of Mathematics, 82–94.

Friedman, M. (2001). Tolerance and Analyticity in Carnap’s Philosophy. In J. Floyd and S. Shieh (Eds.), Future Pasts: The Analytic Tradition in Twentieth Century Philosophy, Number August 2012, pp. 223–256.

Friedman, M. and R. Creath (2007). The Cambridge Companion to Carnap. Cambridge Companions to Philosophy. Cambridge University Press.

Goddu, G. C. (2002). What Exactly is Logical Pluralism? Australasian Journal of Philos- ophy 80 (2), 218–230.

Grice, H. P. (1969). Utterer’s Meaning and Intention. The Philosophical Review 78 (2), pp. 147–177.

Griffiths, O. (2013, May). Problems for Logical Pluralism. History and Philosophy of Logic 34 (2), 170–182.

Haack, S. (1974). Deviant Logic. London: Cambridge University Press.

117 Hellman, G. (2006). Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis. Journal of Philosophical Logic 35 (6), 621–651.

Heyting, A. (1956). Intuitionism. Amsterdam, North-Holland Pub. Co.

Heyting, A. (1983). The Intuitionist Foundations of Mathematics. In P. Benacerraf and H. Putnam (Eds.), Philosophy of Mathematics, pp. 52–59. Cambridge.

Hjortland, O. T. (2013). Logical Pluralism, Meaning-Variance, and Verbal Disputes. Aus- tralasian Journal of Philosophy 91 (2), 355–373.

Hjortland, O. T. (2014). Verbal Disputes in Logic: Against Minimalism for Logical Con- nectives. Logique & Analyse 227, 463–486.

Horn, L. R. (1989). A Natural History of Negation. Center for the Study of Language and Information - Lecture Notes Series. Center for the study of language and information.

Jennings, R. (1994). The Genealogy of Disjunction. Oxford University Press, USA.

Keefe, R. (2014). What Logical Pluralism Cannot Be. Synthese 191 (7), 1375–1390.

Koellner, P. Truth in Mathematics : The Question of. pp. 1–44.

Lewis, D. (1979). Scorekeeping in a Language Game. Journal of Philosophical Logic 8 (1), 339–359.

Lynch, M. (1998). Truth in Context: An Essay on Pluralism and Objectivity. A Bradford Book. MIT Press.

MacFarlane, J. (2000). What Does It Mean to Say That Logic is Formal? Ph. D. thesis, University of Pittsburgh.

Paoli, F. (2014). Semantic Minimalism for Logical Constants. Logique & Analyse 227, 439–461.

Paseau, A. (2007). Review: Logical Pluralism. Mind 116 (462), 391–396.

Priest, G. (1987). Doubt Truth to be a Liar. New York: Oxford University Press.

Putnam, H. (1968). Is logic empirical? Boston Studies in the Philosophy of Science 5.

Quine, W. V. (1986). Philosophy of Logic. Cambridge, Massachusetts: Harvard University Press.

118 Quine, W. V. O. (1951). Two Dogmas of Empiricism. Philosophical Review 60 (1), 20–43.

Read, S. (1982). Disjunction. Journal of Semantics 1 (3), 275–285.

Read, S. (2006a). Monism: The one true logic. In D. de Vidi and T. Kenyon (Eds.), A Logical Approach to Philosophy: Essays in Memory of Graham Solomon, pp. 193–209. Springer.

Read, S. (2006b). Review of J.C.Beall, Greg Restall,Logical Pluralism. Notre Dame Philo- sophical Reviews 2006 (5).

Resnik, M. (1985). Normative or Descriptive? The Ethics of Belief or a Branch of Psychol- ogy? Philosophy of Science 52 (2), 221–238.

Resnik, M. (1996). Ought There to Be but One Logic. In Logic and Reality: Essays on the Legacy of Arthur Prior, pp. 489–517. Oxford University Press.

Restall, G. (1999). Negation in Relevant Logics (How I stopped worrying and learned to love the Routley star). In H. Wansing and D. Gabbay (Eds.), What is Negation?, Volume 13 of Applied Logic Series, pp. 53–76. Kluwer Academic Publishers.

Restall, G. (2000). An Introduction to Substructural Logic. New York: Routledge.

Restall, G. (2002). Carnap’s Tolerance, Language Change and Logical Pluralism. Journal of Philosophy 99, 426–443.

Restall, G. (2014). Pluralism and Proofs. Erkenntnis 79 (2), 279–291.

Roberts, C. (2012). Information Structure in Discourse: Towards an Integrated Formal Theory of Pragmatics. Semantics and Pragmatics 5 (6), 1–69.

Sagi, G. (2014). Formality in Logic: From Logical Terms to Semantic Constraints. Logique Et Analyse 57 (227), 259–276.

Shapiro, S. (2006). Vagueness in Context. Oxford: Clarendon Press.

Shapiro, S. (2011). Varieties of Pluralism and Relativism for Logic. In S. Hales (Ed.), A Companion to Relativism, Blackwell Companions to Philosophy, pp. 526–552. Wiley.

Shapiro, S. (2014). Varieties of Logic. OUP.

Stalnaker, R. (2014). Context. Oxford University Press.

119 Steinberger, F. (2015). How Tolerant Can You Be? Carnap on Rationality. Philosophy and Phenomenological Research 92 (2), 645–688.

Steinberger, F. (forthcoming). Frege and Carnap on the Normativity of Logic. Synthese, 1–20.

Tarski, A. (2002). On the Concept of Following Logically. History and Philosophy of Logic 23 (3), 155–196.

Tennant, N. (1984). Perfect Validity, Entailment and Paraconsistency. Studia Logica 43 (1- 2), 181–200.

Tennant, N. (1992). Autologic. Edinburgh information technology series. Edinburgh Uni- versity Press.

Tennant, N. (1997). The Taming of the True. New York: Oxford University Press.

Tennant, N. (2005). Relevence in Reasoning. In S. Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics, pp. 696–726. New York: Oxford University Press.

Tennant, N. (2007, September). Carnap, Godel, and the Analyticity of Arithmetic. Philosophia Mathematica 16 (1), 100–112.

Tennant, N. (2012). Cut for core logic. Review of Symbolic Logic 5 (3), 450–479.

Tennant, N. (2015). Cut for classical core logic. Review of Symbolic Logic 8 (2), 236–256.

Tennant, N. (forthcoming). Core Logic.

Urbas, I. (1996). Dual-Intuitionistic Logic. Notre Dame Journal of Formal Logic 37 (3), 440–451.

Varzi, A. (2002). On Logical Relativity. Noˆu{}s 36 (s1), 197–219.

Wagner, P. (2009). Carnap’s Logical Syntax of Language. History of analytic philosophy. Palgrave Macmillan.

Woodward, R. (2008). Logical Pluralism, by J. C. Beall and Greg Restall. European Journal of Philosophy 16 (2), 336–339.

120 Appendix A

Two Normative Issues for for the Carnapian Pluralist

A.1 Introduction

A Carnapian pluralism is one in which any logic is admissible, so long as there is an appro- priate purpose to which you can apply it. This leads to an immediate worry should we want logic to remain normative: norms cannot be based on correct reasoning across the board, since then we ought to conclude that the one correct logic is given by a single set of norms.

This would not appeal to a Carnapian.

Rather, they must be based on something like “correct reasoning for purpose X”. Norms, then, are purpose or goal dependent. When we decide to pursue a goal with a certain logic, we “volunteer” to be subject to the normative demands of that logic for that goal.

Steinberger(forthcoming) has an excellent defense of such a position. Once we take this type of normativity on board, though, we an ask the following: how does a purpose relate to its norms? And what are the good purposes?

This is an immediate problem for positions like those of Shapiro(2014) and logical instrumentalism, as presented in this thesis. Additionally, it makes the recent Carnap revival much more challenging then we may have first thought. If we want logic to be normative in any sense, then we need to be able to answer the above two questions. We need to be able to tell a story about why it is normative. Otherwise, we would have to insist, on Carnap’s behalf, that logic is not normative at all. This would be a big blow for the Carnapian revival.

121 I develop both of the normative problems for the Carnapian pluralist in this chapter.

Using the view attributed to Carnap in section 2.3.2 of chapter2, I will discuss the two problems, and explain just how far reaching and problematic they are. Finally, I look at some possible solutions, and conclude that none is adequate.

A.2 How to Choose our Goals

There are two things we need to discuss with respect to goal and purpose selection. Carnap’s position seems to be that any goal is a good goal, as long as we frame it in the appropriate linguistic framework. At the very least, this is what would seem to follow from the Principle of Tolerance. There are, however, two immediate types of aims/goals which are problematic here. The first is scientific, and the second we might call a pseudo-goal.

In the first case, we might think that in asking questions like “what is heat?” or “what is the shape of the Earth’s orbit around the Sun?”, we are asking something about reality.

These questions are in a significant way different from questions like “what are numbers?” and “do propositions exist?”. When we ask them, we do not expect the answer to be “it depends on the framework you are using” or “we cannot ask these question about capital-

R-Reality”. These are exactly the types of questions we think lead us to knowledge about what the world is really like. However, on the Carnapian picture sketched above, we can only ask these questions relative to some appropriate linguistic framework, and not about reality itself. This is a problem for scientific inquiry generally. Science is, or claims to be, about reality. Carnap would have to deny this.

An immediate response here might be that scientific questions are pragmatic questions, and so do not need to be embedded in a linguistic framework to be answerable. This would be odd, as Carnap holds that the only pragmatic questions there are are the questions about framework selection. As these types of scientific questions are not about framework selection, they are not pragmatic. If the Carnapian picture requires that scientific inquiry is

122 not about reality itself but only about the consequences of some framework-or-other, then we ought to dismiss it. The Carnapian needs to provide an account of appropriate goals for study within a linguistic framework which excuses scientific inquiry from the system.

Second, a Carnapian will need to provide an account of goals which dismisses as illegit- imate (rather than excuses from being a part of the system to begin with) a certain class of pseudo-goals. Imagine, for a moment, that you are teaching an introduction to logic class, and a group of obnoxious students start making poor inferences for the purposes of showing off to classmates. As you have taught them to be good Carnapians, when you ask that they stop, they respond that their goal is to make inferences to be impressive. There is something illegitimate about this. The Carnapian need not license every whim and fancy as a legitimate goal, even when we can impose a logic on it. The Carnapian needs a way to rule these types of illegitimate purposes out. At the very least, he or she needs to be able to develop a method by which we can say that some goals are better than others.

There is yet a third type of purpose which may be thought to be problematic, but actually is not. One might wonder why we cannot pursue the goal of discovering the truth about the One True logical consequence relation. This would fit under the “any goal goes” claim that we can assume tolerance allows us to make, but would seem to contradict the

Carnapian project. There is one problem here, though: the question “which logic is the one true notion of logical consequence?” is external. It seems problematic only because it is after the “one true logic” of reality, rather than the one logic of any given framework.

We can pursue the goal of trying to figure out which logic any given framework is equipped with, but not which logic is the Logic of reality. Though this might initially seem like a problematic goal, it only seems that way because we interpret it externally, which we cannot do on the Carnapian picture.

123 A.3 How to Choose our Logics for those Goals

There is another normative concern which will rear its ugly head for any Carnapian. This comes from an attempt to answer questions such as “which logical principles do we use to select a logic for our legitimate purpose?”, “is the selection of logics subject to rational norms?”, etc. We need to be capable of being wrong about purpose/logic pairs; we need to be able to claim that certain logics are worse for certain purposes than others. This does not need to amount to saying that each purpose has a right logic, but merely that some purposes are not appropriately pursued with certain logics.

This will help us prevent, for example, snarky undergraduates in their first logic class from asserting logically false statements and claiming they are true “in their logic”. It will also prevent more subtle issues: it will give us grounds to reject the claims of a mathemati- cian who takes himself to be doing constructive mathematics with a classical logic. We need to be able to say that some logics should not be used for some purposes.

One way to get at this might be to have a meta-framework in which to make the object- level logical selection for our preferred framework. If we have this, we will be able to accept or dismiss logic/framework pairs based on the normative guidance of the meta-framework.

However, we would still need to answer the question “in which meta-framework does our desired purpose “select” a logic, and which logic is right for that meta-framework?”. There is a threat of an immediate regress here: we can also ask in which meta-meta-framework we select the meta-framework, in which meta-meta-meta-framework we select the meta-meta- framework, etc. We need a way of fending off these types of worries. We need a method to pair logics with linguistic frameworks in a way which does not lead to regress, i.e. a method to select meta-frameworks which does not require a meta-meta-framework.

To solve this problem, then, we need not only to provide a framework for our inquiries in which we can make claims about being right or wrong about whether a logic is suited to study our desired purpose, but also to do so in such a way that we do not wind up with an

124 infinite regress.

A.4 Some Possible Solutions

The previous two sections outline two normative problems any Carnapian view of pluralism will face. I now turn to some possible solutions, some which answer only one of the issues, and some which answer both.

A.4.1 Solutions to Which Goals are Good

I first consider two possible solutions to the first problem, that of eliminating bad purposes.

No Bad Purposes

One story that can be told about purpose selection, or goal selection, is that really and truly, anything goes. On this picture, we can play fast and loose with the purposes themselves. In a certain sense, every purpose is legitimate. In another, certain purposes are not pragmatically advisable.

The sense in which every purpose is legitimate might be said to be the following. As

Carnap states, “it is not our business to set up prohibitions but to arrive at conventions.”

There is a distinct sense in which in the two examples in section A.2, about heat and inferring-to-impress, there is nothing strictly speaking wrong with applying logic and lin- guistic frameworks to the purpose. The people putting forward those applications as good applications for logic are only doing something wrong in that they are not being pragmati- cally practical. However, there is nothing wrong with pursing those purposes from a purely theoretical perspective. They are bad to pursue in the Carnapian framework at best because they are goofy and are hard to make precise, or thought to be about reality itself. But as it stands, we do want to maintain the position that at a fundamental level anything goes.

125 The sense in which some purposes are not pragmatically advisable is the following.

According to the Carnapian characterization, one is at liberty to do what one wishes, as long as he or she is able to explain it to others. If a given purpose is so obscure, mysterious and nebulous that it cannot be explained to others, then it will not be pragmatically advisable to apply a logic to it. The obscurity of the purpose is a distinct sign that it is not rigorous enough to be studied logically. This pragmatic advisory, however, does not make the purpose illegitimate, but merely ill-advised. As noticed above, it is actually the case that anything goes, but some things are more motivated than others.

There is an initial problem here, though. This method seems fine for the people sug- gesting “bogus” goals, like those goals had by the person who is trying to infer a bunch of things to impress a professor. However, this solution will still make most of scientific in- quiry illegitimate. What we need is a way to rule out scientific question from the framework in general: we need an account of the goals which can be asked in a linguistic framework which excludes the scientific ones. Otherwise, we will have to require that scientists only ask question with respect to a certain framework, and not about reality itself. Since it can be assumed that what (most) scientists study is reality, this makes this solution a bad one.

Testability

Perhaps, then, what we need to do is rule out the scientific goals in the Carnapian sense and then impose the pragmatic structure suggested above. One way to do this would be to suggest that questions which can be answered experimentally would not fit this mold.

So, we rule out experimental questions, and then are left with the “any goal goes” proposal above.

There is also an initial problem with this proposal: I have argued elsewhere that ques- tions about meaning are theoretical questions, and can only be answered within the bounds of a linguistic framework (see Chapter 5). However, if experimental questions cannot be

126 part of the Carnapian framework to begin with, this would mean that experimental at- tempts to get at what meaning is cannot be legitimate. This would rule out linguistics as a legitimate science, and that is not a consequence we should be will to bear. So testabil- ity/experimentability won’t get us the answer we want here.

There is an additional problem here, in that this picture would not accord with the

Carnapian position. Earlier, we distinguished between L-rules and P-rules. The P-rules were meant to govern the transformation of descriptively true sentences into descriptively true sentences. In this way, the natural laws are encoded into a linguistic framework, and so even questions that purport to be about reality ought to be asked and answered from within a given linguistic framework with the set of P-rules that we think govern the laws of nature. Thus, on a strictly Carnapian interpretation, even scientific questions ought not to be excused from the picture.

A.4.2 Solutions to which logics fit which goals

We now turn to solutions to the second problem, which require providing a tool by which we can tell whether a logic is suited to its purpose.

Theoretical virtues

One way we might rule out certain logics for certain goals is to appeal to something like the theoretical virtues. A logic is good for a goal when it provides us with some appropri- ate combination of simple, clear, fruitful results which are easy enough to get. This is a straightforward answer, but it fails for several reasons.

First and foremost, the theoretical virtues are normative, so we will likely wind up with a regress in the style we considered in section A.3. The theoretical virtues include things like simplicity, fruitfulness, clarity, etc. They get spelled out in norms like “all things being equal, prefer a simpler theory to a more complicated one” or “all things being equal, prefer a more fruitful theory to a less fruitful one.” These are norms, if anything is. So we can

127 ask, why do these norms get to decide which goals are best studied by which logics? If we pick certain theoretical virtues over others, we ought to be able to tell a story about why those were more important. On the face of it, we can do this by appealing to a framework which embeds the theoretical virtues, and use that framework to actually deduce which virtues are most important for out original goal. But then, we begin a regress: why is this third framework, the one which embeds the theoretical virtues, is giving us the right answer to the question of which virtues we ought to care about most when pursuing our original goal? Why not use a different third framework. Again, we can appeal to the theoretical virtues at this level, but again, we will have to embed into yet another framework to check to see that we have focused on the right virtues. Thus, we find ourselves in a regress. The question here boils down to the following. How are we going to be able to tell how to maximize the theoretical virtues for any given purpose? Or which theoretical virtues we ought to maximize? We would have to appeal to a meta-framework (and then a meta-meta- framework, and then a meta-meta-meta-framework, etc.), and figure out which theoretical virtues are best suited for our goals there. This would be bad.

Steinberger(2015) considers two additional objections to this type of view. The first criticism Steinberger makes is that allowing theoretical virtues to, at a certain level, guide the framework and logic selection, is not a move Carnap could accept. He states “the position that the norms guiding framework choice should somehow resist formalization and so be essentially informal sits very uneasily with both Carnaps philosophical methodology and his overarching philosophical aims” (Steinberger, 2015, p 9). Carnap’s problem relies heavily on questions being legitimate if they can be formalized. So, having legitimate questions which could not be formalized goes against the Carnapian spirit.

The second criticism Steinberger levies is that there is something deeply philosophi- cally unsatisfying to having to claim that framework/logic pairs are good or bad based on something mysterious, in that they cannot be spelled out in terms of norms.

[T]here just is something deeply unsatisfactory about claiming that it is not

128 possible to offer a rational reconstruction of the process by which scientists choose linguistic frameworks without offering any explanation as to why that should be. If framework choice really is central to scientific practice, it would be all the more worrisome if it were an unanalyzable know-how comparable to chicken sexing.

What I take it Steinberger means here is that in order to block the infinite regress, we have to assume that scientists somehow just know, or intuit, which theoretical virtues to maximize, and how to do that. But if we cannot embed their reasoning about which theoretical virtues to maximize into another framework, then their reasoning cannot be normative, then there can be no right or wrong results to achieve by their reasoning. But if this is the case, then we cannot really say that one logic is better than another for some goal, since there will be no good or bad way to reason in order to pursue that goal. However, since scientists

(we think) mostly get this right, this would mean that they have somehow latched onto an intuition, and this mysterious intuition normally gets things right (hence the comparison to chicken sexing). If we really could not formalize the questions about theoretical virtues and framework selection, then they could not be normative on the Carnapian picture. But if the theoretical virtues cannot be normative, then it is not clear what the theoretical virtues are. I take it that this is what Steinberger is getting at here.

Thus, not only would the appeal to theoretical virtues require an infinite regress, but

Carnap could not allow it, and it would be philosophically unsatisfying.

Steinberger’s Selection Framework

Steinberger’s solution to all of this is to postulate something he calls a “selection framework”, which is “the linguistic framework that articulates the standards against which framework choices are to be appraised” (Steinberger, 2015, p 10). Steinberger claims that framework selection is done with decision theory, emulating the work that Carnap did on probability theory. This mitigates both of his additional concerns about merely appealing to theoretical

129 virtues: it would agree with the Carnapian picture, and it would not be not philosophically mysterious.

What Steinberger essentially proposes is that we select the logics for our goals in a selection framework which is equipped with decision theoretic utility calculation tools. We then plug into this framework the utility we place on each expected outcome (the results of applying each logic to our purpose), and it returns the best possible logic for our goal.

The problem here is that, though two of the three problems with the theoretical virtues are dealt with, it does not solve the regress problem. We would still in this case have to choose a decision theory, and there would be a question about which decision theory was right for our chosen goal. But then, we would need a decision theory to answer that question, and so on. We have the regress worry all over again.

The Quinean Answer

There is a more or less Quinean option we might consider here as well. One might simply say “start from where you are”. In this case, we would take whatever logic came most naturally for our goal, or take the logic we were already working with, see what results obtained by applying those logics to our desired purpose, and update our logic as needed.

The problem here is that this move likely has to appeal to theoretical virtues in order to tell whether we need to update our logic. As before, this would lead to an infinite regress.

The reason we likely need an appeal to theoretical virtues is that we need some way of tracking when and how we need to shift from our starting point. It is all well and good to say that we start from where we are, but we need to provide a system by which to tell if we need to switch. Given that the Quinean answer suggests that we decide whether we need to change logics depending on the results of applying our original logic to the problem, it can be assumed that if the results do not satisfy the right theoretical virtues, or are not virtuous enough, then we ought to change. But this is essentially an appeal to theoretical virtues,

130 and so we are back to the problems faced in section A.4.2: infinite regress or philosophical mystery.

A.4.3 Pragmatics and Explication

Here, I sketch my preferred solution to both problems.

One solution to the first problem would be to develop a more robust theory of pragmatic external questions to answer the first normative concern. If we had a more robust theory of which questions were pragmatic, external questions, then we could tell exactly when questions were about framework selection (or needed to be about framework selection).

One would hope that this development would rule out scientific questions as those types of questions which needed to be about framework selection, or allow the frameworks used by the scientists to correspond somehow to reality. On the other hand, one might just need to bite the bullet here and claim that even scientific questions do need to be asked in a framework, and cannot be answered about reality itself. In some sense, this is appealing to the first proposed solution to the problem: there are no bad goals. Carnap essentially does not need to have views about which goals are good or bad, as whether a goal is a good one will be a pragmatic question. The work comes primarily from assessing whether a logic is suited to a goal.

Once we have picked our goal, we still need a method by which we can determine which logics are suited to be applied to our goal. Here, we can appeal to Carnap’s notion of explication. I claim that the logical system and linguistic framework provide a potential explication of the goal. They provide one way to consider it in a formal, rigorous system.

In the sense of Cook(2010) and Shapiro(2006), they model the original task. But they provide only one such model. Other linguistic frameworks and logics will provide other models, other explications. If we think of linguistic framework and goal pair as explication and explicated, then we need to answer the question, not of when a logic is suited for a goal, but of what make an explication a good one.

131 Carnap characterizes explication as

The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication for, the earlier concept. (Carnap, 1947b, p 7-8)

Here, we can read “concept” as being something more general. We are making precise and exact the task we have in mind, and the deductive relation appropriate for reasoning about that task. An explication of a task, then, is a replacement of our ordinary methods to pursue the task by a formal structure providing methods to pursue the task. How do we tell whether an explication is good or not? In other words, if logics are seen as explications of goals, how do we tell whether a particular logic is a good explication? Carnap has an answer here. He proposes four requirements on when an explication is adequate

(1) Similarity to the explicandum, (2) exactness, (3) fruitfulness, (4) simplicity (Carnap, 1950, p 5)

Rightfully, he claims that these criteria will not settle uniquely which explication is best.

There may very well be a number of explications which suit the purpose at hand. But what we needed here was a way to rule out certain logics for certain goals, not a way to select a unique logic for a goal. And it seems we can do this here. The answer to the question of “when is a logic bad for a given goal” is when it does not conform to the four criteria

Carnap gives here.

One thing to notice is that these are essentially theoretical virtues. But we can avoid the regress problem we had above in this situation. One thing we do not get from this solution is any sort of ranking of which logics are better than others for a goal. A logic is good for a goal if it can be conceived of an explication of that goal which meets the four criteria, and bad otherwise. We do not need to know when one is better or worse at meeting the criteria, and so have no need to appeal to a meta-framework to check this.

132 A.5 Conclusion

There are two normativity problems for a Carnapian pluralist. Such a pluralist needs to be able to rule out bad purposes, and rule out bad logics for good purposes. As I have shown, several immediate solutions to this problem do not work. The normativity problem for the Carnapian is hard and potentially leads to infinite regress. This makes the current

Carnapian revival trend challenging. I have suggested a new type of solution, inspired by

Carnapian explication, which might just do the job for us. But there is much more work to be done before we can settle the question of whether the explication-style solution is an adequate one!

133