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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproduction Further reproduction prohibited without prohibited permission. without permission. FREGE’S ONTOLOGY AND THE PROBLEMS OF THE THIRD REALM by Sylvia Alexis Rolloff submitted to the Faculty of the College of Arts and Sciences of The American University in Partial Fulfillment of the Requirements for the Degree of Master of Arts in Philosophy

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FREGFS ONTOLOGY AND THE PROBLEMS OF THE THIRD REALM

BY

Sylvia Alexis Rolloff

ABSTRACT

In this thesis, I investigate Gottlob Frege's philosophy, particularly his conception of thoughts as existing in a "third realm" separate from the physical world and independent of human minds. I begin with an exegisis of Frege's philosophy, examine the interpretations of Michael Dummett and Joan Weiner, and finally adjudicate between their differing conceptions of Frege's thought. In conclusion, I find that Frege's third realm is a metaphysical entity, yet one which serves a useful purpose of providing an ontological grounding for the expression of assertoric sentences.

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Page INTRODUCTION...... 1

CHAPTER 1 Exegesis Historical Background...... 4 Concept-script...... 6 Foundations of Arithmetic...... 11 Function, Concept and Object...... 20 Sense and Reference...... 35 Thoughts...... 37

CHAPTER 2 Dummett's Interpretation of Frege...... 46

CHAPTER 3 Weiner's Interpretation of Frege...... 73

CONCLUSION...... 90

BIBLIOGRAPHY...... 94

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION

Gottlob Frege started his philosophical work with a simple aim: to establish a secure ontological grounding for the concept of number. This seemingly innocent endeavor has wrought much controversy, not only about the content of his thought, but also over the interpretation of that content. Frege's inquiry into the status of numbers led him on a progressive philosophical path in which the nature of his ontological grounding became more refined and mature as his thought progressed. He begins with a rather unsophisticated realist position and evolves towards a position which cannot accurately be called nominalism, but rather a sparse realism. Frege's original presuppositions involve a rather ontologically naive background of mathematical number and set theory. As his work continues through the years, Frege develops an increasingly metaphysical conception of functions which strongly influences his views of thoughts. Of particular concern to me is Frege's views of thoughts. His conception of thoughts existing in an objective yet non-material "third realm" is where we can see most strongly his progressive embrace of a pared-down ontology. As his thought progresses, the ontological categories in Frege's world become fewer and fewer, until there are only two: concepts and objects. There has been a great deal of debate over whether or not 1

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thoughts are concepts or objects within Frege's thought -- or if his formulation of the third realm is merely philosophical myth-making. It is

important to remember that the mature Frege is concerned withsharpening the ontological boundaries within his thought, rather than performing a metaphysical investigation. The first chapter of this thesis is an exegesis of Frege's views. I explain his works in the order in which they were written. I intend to emphasize the historical progression of his work, and how his evolving philosophy ties together certain conceptions (e.g., sense and reference, concept and object, truth and falsity) into a progressively more coherent whole. The second chapter deals with Frege's most influential interpreter: Michael Dummett. Dummett views Frege primarily as a philosopher of language, and considers his context principle to be the most important element within Frege's philosophy. Although Dummett's interpretation can be very compelling, it overlooks Frege's original (and maintained) distinction between generalized and instantiated functions. In the third chapter, I explore Joan Weiner's interpretation of Frege. Specifically, she is concerned with Frege's intentions and how they influenced his philosophy, rather than with categorizing Frege as a particular type of philosopher. Weiner maintains (convincingly) that Frege was concerned to produce (through his Begriffsscrift) a logical tool to be used for the advancement of science. "Science" is here broadly conceived as the 19th-century classification which included mathematics, chemistry, history, etc.

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Both Weiner and Dummett bring to the foreground interesting (and sometimes conflicting) interpretations of Frege. My attempt is to adjudicate between the views of these two thinkers as well as to explore a particularly obscure aspect of Frege's thought.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 EXEGESIS

Historical Background Gottlob Frege attempted to establish an ontological grounding for the concept of number. This ontology springs from logicism, the doctrine that logical concepts can exhaustively explain arithmetical ones. This ultimately unsuccessful attempt led him to a wide-ranging philosophical investigation of logic, numbers, ordinary language and geometry. Although relatively unknown during his lifetime, Frege's stature has grown over the years, and the scholarship on him still examines and contends with his thought. Frege's university studies consisted of mathematics, philosophy and physics. He went on to teach mathematics and mathematical logic. Although the field of mathematics at that time was in the midst of exciting new discoveries, Frege felt that the primitive basis of mathematical entities -- the natural numbers — had not been sufficiently examined or explained. He was determined to supply the foundations of natural numbers and arithmetic as a remedy to this situation.

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Historical Progression An essential aspect of Frege's work is that it is progressive. His original distinction, in theBegriffsscrift, between generalized and instantiated functions leads his later work through a sharpening of the ontological boundaries between concept and object. Functions (in particular, functions that possess truth-values when instantiated) become subsumed under the category of concept. In conjunction with his progressively more metaphysical conception of functions, Frege's ontological category of "object" grows larger as his thought progresses. Frege's increasing distinction between the sense and the reference of a sentence reflects this progression. Originally defined as the difference between a referent and the way by which it is referred, the sense/reference distinction evolves into a dichotomy between the thought expressed in a sentence and the truth-value of that thought. Frege's thought, which begins with a scientific realism, moves towards a metaphysical nominalism. The ontological categories in Frege's world become fewer and fewer, until there are only two: concepts and objects, both of which he eventually admits are logical simples. This is not to say that Frege become a metaphysician or a nominalist in the conventional understandings of the terms; he simply became more concerned with sharpening his ontological boundaries.

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Concent-scrint Frege began his logicist program in order to avoid mathematics' reliance upon intuition in defining the most basic aspect of mathematics: the concept of the number. In order to do this, Frege invented the concept- script, a system of logical notation, to describe logical arguments without letting any hidden presuppositions slip in. Because many arguments turn on unstated suppositions, a logically valid argument needs to illuminate unstated presuppositions in order to avoid imprecision. Frege holds that many philosophical mistakes (especially the confusion of cause and proof) arise from argumentative imprecision. His concept-script derives the laws of logic from primitive principles, using a specific method of inference. This endeavor differs from past attempts in that logical systems previous to Frege had relied upon Aristotelian syllogistic logic. By using the starting point of primitive principles, rather that propositions, Frege created a much more subtle system of logical analysis. His concept-script allowed him to analyze a sentence in terms of its conceptual content, rather than as a subject/predicate dichotomy. Although this may seem to be a tool specific to logic and mathematics, Frege's goal is to clarify scientific methodology by creating a single form language useful for all disciplines. Concept-script is intended to be a tool for "science," broadly conceived. To this effect, Frege looks at the conceptual content of sentences and arguments in terms of scientific matters. This does not mean that only scientists will have an interest in Frege's work:

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If it is the task of philosophy to break the power of words over the human mind, by uncovering illusions that through the use of language often almost unavoidably arise concerning the relations of concepts, by freeing thought from the taint of ordinary linguistic means of expression, then my Begriffsscrift, further developed for these purposes, can become a useful tool for philosophers.1

The invention of the concept-script brought with it the most important contributions to modern logic: quantification theory. With the concept-script, Frege attempts to generalize mathematical functions using a notation that could apply to every Held which made use of rigorous proofs. The goal was a scientific one: namely, to achieve the highest possible level of certainty of any particular truth. Achieving this certainty is a logical process, as " ... a universal proposition becomes more and more firmly established by being connected with other truths through chains of inference."2 This logical process can be applied, in theory, to any scientific field. Frege attempts to set the foundations for valid chains of inference by paring down ordinary language to what he believes to be its conceptual content, namely, functions and objects and utilizing only one primitive law of inference: modus ponens. From these starting points, Frege set out to reconstruct the laws of logic upon a firm foundation and, consequently, prove numbers and arithmetic principles to follow from logical laws.

^ Frege, Gottlob, Begriffsscrift, in The Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, pp. 50-51. 2ibid., p. 48.

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Because Frege determined to create a system of logical notation that could conceivably apply to all fields of thought that made use of proofs, he recognizes that proofs can be both empirical and logical. Accordingly, he separates all truths that require justification into two categories: proofs which are purely logical and proofs which are grounded in empirical facts. In making this separation, he points out that purely logical truths need not be apprehended by the human senses. For example, we do not need to have experienced two apples plus two apples equaling four apples for the arithmetic proposition 2+2 * 4 to be true. Thus," ... it is not psychological origination but the most perfect method of proof that lies at the basis of the division."3 In order to establish the foundations of arithmetic through the use of logical rules, Frege sets himself to establishing a generalized system of logic adequate to the task. In order to prove that arithmetic (and more specifically the concept of the natural numbers) does not rely upon psychological intuition,

I first had to see how far one could get in arithmetic by inferences alone, supported only by the laws of thought that transcend all particulars. The course I took was first to seek to reduce the concept of ordering in a series to that of logical consequence, in order to progress to the concept of number.4

Frege admits that this investigation of logical consequence is impeded by the imprecision of natural language. Frege was inspired to create the concept-script as a means of circumventing the inadequacies of natural

3ibid. 4ibid.

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language in this regard. By reducing the concept of ordering to that of logical consequence, Frege made possible the translation of mathematical set theory to logical arguments. By making the revolutionary leap of separating concepts from objects, rather than relying upon the traditional division of subject from predicate, Frege envisions concepts as entities that can include or be included in other concepts. Additionally, individual entities (objects) can fall under concepts in much the same way that numbers can belong to various sets. His conceptual notation is modeled upon the ideas, rather than the specifics, of mathematics. As we shall see, numbers are objects which must be treated differently than physical objects such as the moon. The idea behind Frege's invention is an improvement in the scientific methodology specifically for arithmetic:

Arithmetic . . . was the starting point of the train of thought that led me to my Begriffsscrift. I therefore intend to apply it to this science first, seeking to provide further analysis of its concepts and a deeper foundation for its theorems.5

Frege utilizes two sorts of signs: letters and symbols. Letters represent an undetermined (and therefore generalized) number or function, such as the variable x and the function F( ). Symbols, such as +, -, 0,1,2, represent specific numbers and operations. The letters and symbols respectively represent generality or determinacy, an ontological division similar to Frege's separation of concepts and objects.

5ibid., p. 51.

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Frege was the First to axiomatize propositionai logic. This project has its historical roots in Boole's expression of logical relations using algebraic formulae, and utilizes Cantor's set theory. With his genius for generalization, Boole saw that algebra could be developed into an abstract calculus that could be interpreted in various ways. Frege extended this tendency towards that generalization of algebraic applications to apply the mathematical concept of functionality to propositions. He used set theory (the supposition that propositions make statements about classes and sets of classes) as a backdrop for analyzing the functionality of propositions. He attempted to prove logical theorems from a small set of axioms. These axioms, expressed in modern notations, are as follows:

i) (q->p) *> (~P -> ~q) ii) —P -> p iii) p -> ~~p iv) p -> (q->p) v) [r -> (q->p)] -> [(r -> q) -> (r -> p)] vi) [r ~> (q -> p)] -> [q -> (r -> p)] vii) (c = d) -> [(f(c) -> f(d)i viii) c ^ c ix) (x)f(x) -> fc

Axiom nine is the crucial element of Frege's functional calculus in that it expresses the generality of functions which he used as a system for expressing and instantiating arguments. If f(x) holds generally, then it holds for any particular object c. Axiom nine is essential for Frege's version of implication. In order to infer something about a particular object c, we

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must be able to instantiate c into a general function. Obviously, every function has its own particular range and set of objects that fall under it.

Foundations of Arithmetic Frege wrote the Foundations of Arithmetic to bolster his logicist program. Logicism maintains that the foundations of mathematics can be valid using logical structures and techniques. Before Frege, logicians and mathematicians (especially Peano) -had proven that all mathematical concepts and structures were based upon the natural numbers. All that was left, and the most important element in Frege's agenda, was to prove that the natural numbers themselves are derivable from logical concepts. Frege concerns himself with proving the underlying concepts within mathematics: zero, number and succession. In pursuing a program of defining the specific conception of numbers, Frege attempts to provide a firm basis for generalized arguments. If, for instance, the number one is defined simply as a "thing," this begs the question of what sort of "thing” a number is. This description of numbers, namely that one equals a thing a, is too generalized. While it is true thata + a - a - fora a equaling any number, we can instantiate this equationonly with numbers, not with physical entities. If we replace a with a singular entity, such as a cat named Octavia, then the equation a + a = 2a means something entirely different than 1 + 1 = 2. From this, Frege concludes that

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. . . arithmetic cannot get along with a alone, but has to use further letters (b, cand so on), in order to express in general form relations between different numbers.6

In the equation concerning Octavia,a + a = 2a slips in a definite (rather than general) number, the number2. In order to express the equation entirely in terms ofa, we would have to say thea +- a = a + a, which is useless. Because a cannot be equated to any definite particular object,

... a has no properties that can be specified, since whatever can be asserted of a is a common property of all numbers, whereas 11 -1 asserts nothing of the Moon, nothing of the Sun,... for what could be the sense of any such assertion?7

In claiming that a has no specific properties, Frege is not arguing against the use of generalized numbers in philosophical thinking. Instead, he is arguing against a specific conception of numbers, that numbers are things in the same way that the moon or the sun are things. Because the variable a possesses all the common properties of numbers, talk of variables should not be confused with talk of ordinary physical objects “ a confusion made by many philosophers of Frege's day.

6Frege, Gottlob, The Foundations o f Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston, 1994, p. ii. 7ibid.

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Reaction Against Psychology In pursuing this work, Frege assumes logic to be prior to both mathematics and philosophy. Hans Sluga writes:

Logic possesses for [Frege] the status of primacy within the body of human knowledge because we must use the laws of logic as well as those of arithmetic in order to gain knowledge from our subjective experiences. Arithmetic, in turn, he believed in those early years, rests on logic, and logic can occupy this place of primacy within our knowledge because of the generality of its laws.8

As a part of his logicist program, Frege is trying to probe the logical underpinning ofa priori objects (specifically numbers), rather than expound psychological statements about the nature of thought. In doing this, Frege is reacting to the influences psychology has on the philosophy of his day, particularly as it was manifested in the empiricist school. Any investigation of the concept of numbers, Frege claims, is common to both mathematics and philosophy. He laments the gulf between mathematicians and philosophers that existed during his career. The gulf, he believed, existed because of the adverse influence of psychology on philosophy, an influence with which mathematicians have no sympathy. Psychology, at least in Frege's day, dealt with empirical experiments that investigated the methods of human understanding and reaction to the world. Many philosophers, especially those of the empiricist school, celebrated psychology as a means of accounting for all human knowledge on the basis of sensory experience. These philosophers, particularly J. S. Mill,

8 "Frege: The Early Years," by Hans Sluga, inPhilosophy and History, Rorty, Scneewind and Skinner, eds., 1984, p. 349.

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held that the human mind holds sense-impressions and mental images. A tenet of the psychology of Frege's day (which focused on the study of these subjective experiences and images) was that mental images are formed from traces of earlier sense-impressions. The psychological account of numbers is that they are the result of aggregative thought. We conceptualize numbers through extrapolations of everyday experiences: one orange added to a basket of two oranges leads to the sense-impression of three oranges in a basket. However, the empirical basis of numbers is problematic: when I add one unit of liquid to another unit of liquid, the result is one unit of liquid, unless we introduce volumetric measure. Numbers become mental images resulting from sense-impressions of liquid units, apples and oranges, and so on. But these mental images rely heavily upon psychological and contextual considerations of what aspect of the empirical world we are trying to count. Adding two units of liquid together may result in either one large glass of liquid or two volumetric units of liquid. As a result of this inconsistency:

Psychology is interested in the causal conditions of our mental processes; mathematics is interested in the proof, or justification, of the thoughts we think. But cause and proof are quite different things.9

Frege disagrees not only with the empiricist conflation of cause and proof, he also objects to the fundamental presupposition that mental processes are inherently subjective. Frege differentiated thought from

9Kenny, Anthony, Frege, Penguin Books, New York, 1995, p. 53.

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feelings, sensations, and emotions. The content of thought concept,is a rather than sensations or images. For him, thought consists of the mind grasping something external to it. In applying this conception of thoughts to arithmetic, Frege concludes that"... sensations arc absolutely no concern of arithmetic. No more are mental pictures, formed from the amalgamated traces of earlier sense-impressions."10 Sense-impressions cannot give us a conception of the number 456,731, or any other significantly large number. It is doubtful that any human has ever had an empirical sensation of 456,731, or can imagine it in a mental picture. Even though the psychological explanation of numbers may work to explain our conceptions of the smaller numbers, its scope is so limited that it does not provide us with a viable theory of numbers. The solution, for Frege, lies in a platonic, rather an empiricist, ontology. Because of Frege's belief in an objective world that exists outside of our subjectivity, to think is to think about something. Therefore, thoughts, in Frege's view, are not independent actions of the mind - they are not contained within the mind but exist independently outside of the mind, to be apprehended by a perceptive individual. These thoughts must maintain some sort of permanence in order for us to acquire knowledge about the world, for

If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know anything about the world and everything would be plunged in confusion.11

^Frege, Gottlob, The Foundations o f Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston, 1994, p. v. 11ibid., p. vii.

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Here, Frege expresses his platonic influences by claiming that true knowledge requires some form of permanence of what is known. Just as Protagoras claimed that "man is the measure of all things," and Heraclitus claimed that the world is in a state of constant flux, psychology's subjectivism and empiricism's shaky foundations were a significant threat to the possibility of a grounding ontology. Frege's response is platonic, yet his platonism does not reify concepts as Forms (e.g., beauty, goodness, etc.), but instead reifies thoughts as entities that exist in a "third" realm, separate from the physical world and subjective mental processes, that happen to possess conceptual content. Frege holds that many, in fact most, things exist outside of human minds and are grasped by us through a mental act. Thoughts are therefore outside of our personal, subjective experience and can thus be classified as objective. Frege begins his writings with the conviction that thoughts, such as mathematical truths, are eternally real. Later on in his writings12, he classifies thoughts as neither subjective ideas nor objects in the external world, but as existing in a "third realm." Concepts, the object of our logical thinking, are not created by the mind, but discovered through mathematics. In maintaining this fairly rigid form of realism, Frege attempts to rid his system of any reliance upon intuition. Philosophers previous to him (e.g. Kant and Mill) had incorporated an element of intuition in explaining human understanding of numbers. Numbers are existing entities in Kant’s theory, but we mere mortals cannot perceive them without a leap of faith.

12Particularly in "Logical Investigations.11 Frege's third realm will be discussed in a later section of this paper.

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In pursuing his goai, Frege holds to three fundamental principles in his work: i) always separate sharply the psychological from the logical, the subjective from the objective; ii) never ask for the meaning of a word in isolation, but only in the context of a proposition; iii) never lose sight of the distinction between concept and object.13 The first principle reiterates Frege's realism and his emphasis upon viewing concepts as objective entities. If we do not maintain a sharp distinction between the subjective/psychological and the objective/logical then the possibility of unacknowledged presuppositions slipping into logical arguments increases greatly. The second principle, the "context principle," allows Frege to investigate the sense/reference distinction in sentences (i.e., the actual meaning) rather than the grammatical structure.14 We should note that Frege views the context principle as essential for maintaining the distinction between the logical and the psychological. He points out that:

If the second principle is not observed, one is almost forced to take as the meanings of words mental pictures or acts of the individual mind, and so to offend against the first principle as well.15 The implication of this quote is that a word or sentence in context is a part of the thought expressed, rather than a verbalization of one's own

1.3 Frege, Gottlob, The Foundations of Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston, 1994, p. x. 14Michael Dummett views the context principle as the most important element in Frege's thought. Other philosophers take issue with this, arguing that an emphasis upon the context principle ignores the important concept of functionality. (See Baker and Hacker, "Dummett's Purge," inThe Philosophical Quarterly, April 1983, p. 115.) 15Frege, Gottlob, The Foundations of Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston, 1994, p. x.

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(subjective) idea. This explanation of the context principle makes no mention of the use commonly ascribed to it: as a tool allowing for flexibility in logical descriptions of the world. The third principle reduces Frege's concerns to two entities; namely, concepts and objects. Everything within Frege’s logical system is either a concept (in the form of a function) or an object falling under a concept (in the form of an instantiating variable). This separation becomes sharper as Frege's thought evolves. Using the first and third principles, Frege hits upon an important method of determining the logical nature of numbers:

The crux of this solution is the correct recognition of the logical status of natural numbers: they are logical attributes which belong not to things, but to concepts.16

By making this leap, Frege is able to rid the field of logic of many inconsistencies, such as viewing numbers as simple adjectives. Instead, he wishes to establish the existence of structures through explicit definitions, rather than using axioms as the sole foundation. However, from the fact that he was using this technique, we should not conclude that Frege saw numbers only as products of definitions. For him, numbers are non-physical objects. If numbers are only definitions, then the concept of number can only be expressed by using language. We can use definitions to arrive at concepts only through linguistic definitions. A modern philosopher might conclude that, because we acquire our understanding of the world through linguistic

16Camap, Rudolf, "The Logicist Foundations of Mathematics," inPhilosophy of Mathematics, Bennacerraf and Putman, eds., p. 32.

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definitions, all concepts are dependent upon language for their expression and understanding. However, this seems to lead us in a circle. For, how can we grasp a concept through a definition without first understanding the concepts behind the definition? Infinite regress results. Frege’s system allows a way out of this regress in that he posited concepts as objectively existing outside of human understanding. We understand them through the mental act of "grasping" them, rather than requiring an explicit linguistic definition of them. The platonic conception of concepts allows us to grasp generalizations and Frege's theory of naming as an instantiation of a generalized function helps us grasp and isolate individual particulars. For Frege, thoughts are objective and to think is to think of something. The way in which we grasp concepts is the same for both mathematical and non-mathematical propositions. That is, the principles of logic apply to many other areas than those of just mathematics. This does not mean that Frege embarked upon an investigation of how we understand language. He explicitly states that the purpose of logic is not to investigate how we learn, but what it means for something to be true.17 Instead, Frege thought that an investigation into how we learn, without the proper foundations of explicit and agreed upon basic meanings, was an intellectual waste of time. Our knowledge of the basic meanings of terms is communicated only through linguistic expressions, in that the expression of knowledge is the mode of presentation. Frege equates the mode of presentation with the sense of a proposition, which, we shall see

^7Frege, Gottlob, The Foundations o f Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston, 1994, p 3.

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later, is a primary ingredient in the meaning of a sentence. lie also thought that we can discuss what we know and how we come to know it without analyzing the expression of the knowledge. To opt for the alternative is to put the cart before the horse. The standard saying is that Frege displaced epistemology as the starting point of knowledge and moved that point to logic. However, the logic with which Frege is concerned can be divided into two parts: the strict laws of inference used for mathematical statements, and the more general justifications of judgments. The justifications of judgments cannot rely on a chain of inference, but must examine the leaps of faith which underlie the whole system.

Function. Concent and Object Frege, using a mathematical paradigm as a starting point, divides the world up into objects and functions, with concepts being a specialized form of function. As everything about which we philosophize must be dealt with in language, we must name everything we discuss. This act of naming is viewed by Frege as the application of a mapping function.18 That is, functions serve to map objects onto concepts. By labeling an object, we are placing that object under a particular concept. Concepts act to group

18There is an equivocation within Frege's thought concerning the application of any mapping function in language, it is unclear as to whether the act of naming creates a (mapping) function, or if the groupings of objects under concepts is simply elucidated by naming. If the latter, the function must exist prior to the naming, and the act of naming is simply an instantiation of proper names within the function.

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various individual objects together. For instance, the concept "horse" groups all the particular objects called "horse" under it.

Function Frege uses the function as a starting point to express not only the difference between sense and reference, but also between object and concept. He gives a very narrow and specific definition of what a function is in order to prevent many of the equivocations between form and content that commonly occur in philosophy. Frege is careful to stipulate that functions arenot simply expressions in which a number is indicated indefinitely. For, to define functions as expressions containing "x" is to confuse the indefinite number, x, with the function itself. Instead, the function is the formula surrounding x, indicating what arithmetic operations should be performed on x. For the series of expressions: 22 - 5 2^-5 27-5;

" ... it is the common element of these expressions that contains the essential peculiarity of a function .. ”19 All three functions have the form 2X - 5, and this general form is the function. The difference in expressions for the same object, e.g. 2 + 5 and 3 + 4 as expressions for the object 7, has been taken to mean that the two expressions are equal but not identical. Frege strongly objects to this view,

l^Frege, Gottlob, "Function and Concept," trans by Peter GeachThe in Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 133

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claiming that it confuses signs with what they signify. Instead, the two expressions have the same referent (e.g., 7) but different senses. The sign is the mode of expression, and is used to refer to a particular object. This confusion of sign and content had adverse effects upon the conception of numbers in Frege's day:

There is at present a very widespread tendency not to recognize as an object anything that cannot be perceived by means of the senses; this leads here to numerals being taken to be numbers, the proper objects of our discussion.20

In order to have a discussion about numbers, we need to get to the meaning behind the various numerals. Numerals are a notational system each standing for a number. In other words, we need to concern ourselves with the content of the numerical signs, in order to discuss that for which they stand, namely numbers. One way of doing this is to focus on the indefinite indication of a number through the variable x. Frege refers to the variable as the argument of a function. Through the conception of a generalized or indefinite variable, we are able to see the same function in different numerical expressions, e.g.

2( 1)2 + 1 2(3)2 +3 2(5)2+5 All of these are instances of the same generalized function but with different arguments (1, 3, and 5, respectively). By using a variable, we can

20ibid., p. 132

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write the function as 2(x)2 + x. It is important to note that variables (or arguments, as Frege calls them) do not fall «inder the scope of the functions:

. . . the argument does not belong with a function, but goes together with the function to make up a complete whole; for a function by itself must be called incomplete, in need of supplementation, or unsaturated.21

Frege calls the generalized (rather than instantiated) function F( ) " unsaturated.'' The argument, "c," instantiates the function F( ) with an object c. The augmented function F(c) Frege calls "saturated." Because of the inherent unsaturation of functions, they must be strictly separated from numbers. The ontological distance between numbers (which Frege considers to be objects) and functions explains why we are able to see the same function in two expressions that designate two different numbers. For example, we can see the expressions 2(3)2 + 3 and 2(5)2 + 5 contain the same general function, yet stand for different numbers. Frege thus splits a mathematical expression into two parts: the function and the argument. The function by itself is unsaturated; it needs to be instantiated with an argument. The argument, in this case a number, is saturated in and of itself. By completing the function with the argument, we reach the "value" of the expression. Thus the function 2(x)2 + x has the value 3 for the argument 1. Frege shows his brilliance by extending the scope of his theory of functions beyond mathematics and on to expressions in language. Specifically, he applies his theory of functions to propositions. The

21ibid., p. 133.

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expression "The Epson Stylus Plus is a wretchedly slow printer" can be seen as the function "x is a wretchedly slow printer." Within this example, "x" serves as a placeholder for names for various objects, and a certain object, namely the Epson Stylus Plus, falls under the concept of a wretchedly slow printer. The printer does not belong to the concept of slow printers, but its name serves as an argument to accurately instantiate the function with this printer as object.

Concept Frege defines a concept as a "function whose value is always a truth- value." In later writings, he reevaluates this statement as an explanation, rather than a proper definition. His reasons for this are that:

One cannot require that everything be defined, any more than one can require that a chemist decompose every substance. What is simple cannot be decomposed, and what is logically simple cannot have a proper definition.22

Instead, Frege tries to show his readers what he means through the use of examples. The function "x2 = 1" has always one of the two possible truth values — it is always either true or false. If we instantiate the function with a definite argument, e.g. -1, then the value of the function will be true at all times, in that (-1)2 = 1. To the number -1 we can assign the property of being the square root of 1. Another way of stating this is saying that the

22Frege, Gottlob, "On Concept and Object," trans. by Peter GeachThe in Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 182.

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number -1 falls under the concept: square root of 1. If we instantiate the function with another argument, say "3," then the value of the expression (3)2 = 1 will be false at all times. The function x2 = 1 is also always true however often we instantiate it with the argument "1." In this way, Frege claims that x2 = 1 is always true for the value range of [1,-1], and untrue for all other values. This value- range is an extension of the concept: square root of 1. Hence," ... we can designate as an extension the value-range of a function whose value for every argument is a truth-value."23 The extension of concepts, when the functions operate in the realm of the natural numbers, are the numbers themselves. Frege therefore defines a number as the extension of a concept. The notion of the "extension of concepts" can also be applied in the analysis of linguistic statements. Every sentence contains a thought as its sense, and this thought is either true or false. The sense of the sentence, when instantiated with the referents, expresses either a true statement or a false statement. In this way, the sense of every sentence, that is, the thought it expresses, must be either true or false. Although Frege's theory of functionality can be applied universally to all logical propositions, he adds a sharpness requirement in order to circumvent nonsensical propositions. As Noam Chomsky has shown, it is possible to create grammatically correct but nonsensical sentences by instantiating functions with inappropriate objects. For instance, the "x" in "x is a lovely shade of blue" cannot be replaced by an object which cannot sensibly be labeled blue, such as a ballet. In order to prevent this from

23Frege, "Function and Concept," trans. by Peter GeachThe in Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 139.

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occurring,"... we have a requirement of sharp delimitation; if this were not satisfied it would be impossible to set forth logical laws about them."24 It is important to remember that Frege began his investigations with a logicist program to try to provide an exclusively logical foundation for arithmetic. To this end, Frege uses numbers as a route to discussing concepts. As shown earlier, Frege extends the values of functions (limited to the scope of the natural numbers) in order to reach a working explanation of numbers in and of themselves. Specifically, he shows that numbers are the extension of concepts, and are therefore objective:

I have said that to assign a number involves saying something about a concept; I speak of properties ascribed to a concept, and I allow that a concept may fall under a higher one.25

As an example, Frege uses "there is at least one square root of four." He claims that we are not saying anything about the numbers 2 or -2, but about the concept square root of Here, 4. the concept has certain properties we can assign to it, namely that it always has a positive and negative pair of numbers falling under it. The concept,square root of can 4 fall under the more general concept ofsquare root. This example shows the difference between a concept, which can be assigned properties, and objects, which cannot. A better explanation of this is that concepts are predicative, whereas objects (such as the number 2) cannot be used as grammatical predicates.

24ibid., p. 141. 25Frege, Gottlob, "On Concept and Object," trans. by Peter GeachThe in Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 187.

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Objects Objects serve as the logical simples of Frege's ontology. They perform the important act of instantiating functions, yet can only be defined negatively, as "anything that is not a function, so that an expression for it does not contain an empty place.”26 This negative definition gives us an idea of what purpose objects serve in Frege's paradigm of functionality. Everything in Frege's mature ontology, as mentioned earlier, is either a function or an object, with names of objects filling in the empty places of functions to create saturated arguments. A second negative definition - one more applicable to language - separates objects from concepts in terms of how they function in language: "A concept. . . is predicative. On the other hand, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate."27 For example, the function " hates tomatoes" serves as a description of any object that fill in the blank. In contrast, the name of an object, e.g., Felicia Nestor, predicates nothing in the saturated argument "Felicia Nestor hates tomatoes." It is important to note that Frege claims that it is the name of an object, rather than the object itself, that is incapable of acting as a grammatical predicate. This builds on his earlier distinction between sense and reference: the name "Felicia Nestor” refers to an object — and object to which we can refer by other names, e.g., "Fifi." Frege's point here is that

26Frege, Gottlob, "Function and Concept," trans. by Peter Geach Thein Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 140. 27Frege, Gottlob, "On Concept and Object," trans. by Peter Geach Thein Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 182.

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even though we can only refer to objects through the use of proper names, neither these proper names nor the objects to which they refer can be used predicatively. In his discussion of "objects," Frege refers to both physical entities as well as thoughts. For example, "objects," within Frege's universe, include all persons, thoughts, and any abstract entity that can be named. If we can name any abstract entity, then presumably we can name concepts. Frege avoids this potential confusion between objects and concepts by maintaining a difference between sense and reference, thereby separating a concept from its name, the same way in which he separates objects from their names. All of this relies upon the action of naming. Frege seems to be multiplying entities through his prolific naming. His distinction between the meaning of a proper name and what it indicates circumvents this seeming explosion of entities. There are, however, some problems within Frege's system of meaning and indication. The distinction between the two, Bertrand Russell claims, leads to some ontological confusion:

The indication of a proper name is the object which it indicates; the presentation which goes along with it is quite subjective; between the two lies the meaning, which is not subjective and yet not objective.28

To Russell, this implies that every proper name has two sides. Assuming that functions can be named, then the naming of a function can

28Russell, Bertrand,The Principles of Mathematics, Norton & Company, New York, 1996, p. 502.

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be contextual, whereas the indication of the proper name, its reference, is distinctly separate from personal ideas. Frege is notoriously vague in giving a definition of proper names — as well as in giving a definition of objects, which are designated by proper names. He simply used the criterion " ... an expression constituted a substantival phrase in the singular, governed by the definite article."29 Frege's use of "proper name" is that of an object (that which is not a function) determined by its sense. It is interesting to note that both objects and their proper names are used extensively within Frege's thought, yet not given explicit definitions. This is where the context principle can play an important role. By asking after the meaning of a proper name in the context of a proposition, although this seems extremely intuitive, Frege maintains that this actually serves to preserve the separation between the subjective and the objective. If one does not observe the context principle, this can lead to perceiving the meanings of proper names as "mental pictures' or "acts of the individual mind."

Mapping Frege uses the notion of mapping to explain how functions work. If we have two classes of entities (or, to use Fregean terminology, two concepts with objects falling under them), a function can map each member of one of the two classes onto a member of the other.

^^Dummett, Michael, Frege: Philosophy of Language, Second Edition, Harvard University Press, Cambridge, 1981, p. 54.

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For example, the function f(x) = x3 maps each number onto another number; the second number being the cube of the first. Frege extends the concept of mapping from numbers to linguistic functions: he uses predicates as a means of mapping meaning onto a subject. For example, "hates tomatoes" can serve as a means of mapping the concept of hating tomatoes onto a specific subject, e.g., Felicia Nestor. Using the mapping paradigm, both fix) = x3 and Felicia Nestor = hates tomatoes are mappings. The former maps "3" onto "27" by fi ). The latter maps a person onto a mental state by ( ) hates tomatoes. It is important to realize that the "=" sign does not represent the mapping, rather the functions fi ) and ( ) hates tomatoes do. One can object that Frege’s move from numbers to language relies upon an equivocation of the "equals" sign. That is, fix) equals x3 in a very different way than Felicia Nestor equals "hates tomatoes." The first is a relationship of equivalence, the second a description. Frege seems to be conflating descriptions with equivalence. To say that Felicia Nestor equals "hates tomatoes" is a logical conclusion from Frege's statements that language functions are equivalent to mathematical functions. However, Frege would not be able to conflate language and mathematics into the same logical system if there were a different rule for language (description) than for mathematics (equality). In fact, he firmly believes that logical rules must be the same for both language and mathematics:

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Thought is in essentials the same everywhere: it is not true that there are different kinds of laws of thought to suit different kinds of objects thought about.30

Although there may not be different sorts of rules for different sorts of objects, this does not entail that the copula "is" can be reduced to one and only one meaning: equality. Other meanings of "is," especially the predicative form, do not map in the same way as statements of equality. Take, for example, the proposition "Octavia is a cat." The "is" serves as a copula connecting the proper name "Octavia" with the concept "cat." A concept functions to express sense (what it means to be a cat), whereas an object (or, more accurately, its name) functions to rigidly designate (the ball of fur that is denoted "Octavia.") This distinction between the object and the concept is not symmetrical: concepts can talk about objects, but not vice- versa. What it means to be a cat can be exemplified by Octavia. However, the rigid designation of Octavia does not imply anything about what it means to be a cat. This, at first, appears to conflict with Frege's mathematical paradigm. Let us compare, for example, the propositions "Octavia is a cat" and "2 + 2 = 4." According to Frege's conception of the functionality of language, Octavia serves to instantiate a generalized (descriptive) function " is a cat." However,"___= 4" is a mapping of numerical values. The proposition concerning Octavia does not, in this description, appear to map a value to another value, but to correlate an object to a concept.

30prege, Gottlob, Foundations o f Arithmetic, Second Edition, trans. by J. L Austin, Northwestern University Press, Evanston,1994, p. iii.

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Some philosophers, most notably Gustav Bergmann, argue that these two functions are not as similar as Frege would like them to be. In the linguistic example, "Octavia" and "Octavia is a cat" are both saturated expressions, whereas" is a cat" is unsaturated. In the mathematical example, the analogy would hold that "2 + 2" and "2 + 2 = 4" are saturated, but "___= 4" is unsaturated. However, there is a sort of type-level tension between these two examples. Octavia falls under the concept "cat," but it is not the case that 2 + 2 falls under the concept "4." If we were to argue that 2 + 2 falls under the concept of "4," we must admit that 3 + 1 and 4 + 0 also fall under the concept of 4. Doing this confuses ontological type levels, for placing 2 + 2 under the concept 4 entails that "2" is a different sort of entity than "4" -- leading us into a vicious circle. There is a type-level resolution to the objection Bergmann raises. Frege freely admits that some concepts can be included in other concepts, but that the important ontological distinction lies between concepts and objects, not between concepts and other concepts. In this way, 2 + 2 and 3 + 1 are both different concepts, but this does not entail that they are a different ontological type than any other number. This objection to Frege's mapping theory presupposes that Frege holds mapping to move in only one direction: from objects to concepts. It seems entirely plausible that Frege thought it possible to map a certain kind of object to another, namely, that it is possible to map numbers (which he considers objects) to other numbers. If we remember that numbers are defined as the extension of a concept, then we can map objects falling under a certain concept to objects falling under a different concept.

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Concept Correlates There is another tension within Frege's theory of mapping, and that has to do with his category of concept correlates. Frege creates the entity of concept-correlates to avoid phrases such as "the square root of 4" being classified as objects rather than concepts. Frege uses classes (and set theory) to define numbers. All objects (as well as numbers) can be viewed as members of a particular set. However, Cantor's theorem proves that every set has 2n subsets. For the example of the class of all people who hate tomatoes, this class would have 2al1 people who hate tomatoes subsets. This leads to the strange situation where there are more entities in the subsets than there are people who hate tomatoes. The class of all people who hate tomatoes has, as its members, people who hate tomatoes, and includes every single person who hates tomatoes. For the sake of example, let us suppose that the class of all people who hate tomatoes has 11,497 members. According to Cantor's theorem, there are 211*497 subsets in this class. In other words, there are more subsets of people who hate tomatoes than there are people who hate tomatoes. Continuing this line of reasoning leads to the paradox that there are more existents in the world than there are existents in the world, if we admit sets as objects alongside their members. However, concepts and functions, although objective, are not objects in Frege's thought. The contention that functions and objects are objective without being objects leads to another objection: if we can give a name to concepts and functions (e.g., the derivative) then why are these names not

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proper names? And if we can label functions and concepts with proper names, then, according to Frege's theory, they are objects. In this way, the distinction between objects and concepts disappears. Frege avoids this contradiction in his theory by claiming that words for concepts are different than the concepts themselves, thus maintaining a type-level distinction between concepts and their names. The words for concepts he simply calls concept-words,31 and are to be distinguished from the concepts themselves. One can re-phrase this to say that concept- correlates are signs which stand for a concept, yet somehow do not designate an object. Because concept-correlates do not rigidly designate an object, but act as a description for a concept, one can argue that concept-correlates do not have sufficiently sharp boundaries to allow their incorporation into set theory. Therefore, any discussion of concepts is a game with signs. With this objection, second level quantification becomes impossible. Since a discussion of concepts is not a discussion of entities rigidly designated, then quantification over fluidly designated entities (not objects) becomes suspect. Frege himself hints at this when he claims that objects fall under concepts, but that concepts fall within other concepts. Within his ontology, he distinguishes between differing kinds of subordination: "An object falls under a first-level concept; a concept falls within a second-level concept."32 Second-level quantification would have to quantify within concepts, rather than over them. How this would happen is a mystery to me.

31 Frege, Gottlob, "On Concept and Object," trans by Peter GeachThe in Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 184. 32ibid., p. 189..

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Final Comments Frege's theory of functionality serves to reduce the number of ontological entities within his system, specifically, everything is either an object or a function. In contrast, his theory that each concept has a concept correlate serves to multiply entities. This ontological tension is not as damning as it first appears if we keep in mind the distinction between form and content that Frege holds so dear. Concept correlates act as names, which are objects. What they name are the concepts themselves, which are functions. Thus, the seeming multiplication of entities that comes with naming each concept is really a multiplication of linguistic signs for concepts. To claim that concept-correlates increase the ontological playing field is the same as claiming that the morning star and the evening star designate separate entities.

Sense and Reference Frege distinguishes between what he calls the sense of a sentence and its reference. He first made this distinction in "Function and Concept," and gives an expanded explanation in his famous essay "Sense and Reference." Every sentence is, presumably, about something. For example, the sentence "Venus is the morning star" is about a particular object, namely, Venus. What this sentence is about, Frege calls the reference. The particular object to which we are referring is the referent.

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The sense of a sentence is how the referent is presented to the listening or reading audience. This mode of presentation is important for making meaningful statements of identity. If a = a and a = b, the first is a tautology, whereas the second can meaningfully convey information. In this case, a = b implies that "a" and "b" refer to the same entity, but this referent is presented in different ways. In drawing a distinction between sense and reference, Frege's aim is to uncover how the meaning of propositions is expressed. As shown by the sentence "ThePrinciples of Mathematics is a very difficult book," the hearer understands what it means for this sentence to be true, according to Frege, although he or she may not knowThe Principles of Mathematics. One can say that the sense is actually the precursor to the reference, in that it is how we are led to the referent. In the example, the hearer, not having read Russell's great tome, can learn (be led to) a description of the referent. By the very way in whichThe Principles of Mathematics is presented in the sentence, we equate it with the concept "very difficult book." As mentioned earlier, the referent of a proposition is picked out by a proper name. Thus, it becomes obvious why proper names are so foundational for Frege's philosophy: they serve as vehicles of both sense and reference. A proper name stands for its referent but also, by its very nature, it is how the referent is presented. The very act of naming serves to create a mode of presentation. In Frege's mature thought, he concludes that the reference of a sentence is actually a truth-value. This is a logical outgrowth of his conception of sense. The sense of a word is what we grasp when we understand a word. Similarly, the sense of a sentences is the thought we

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grasp upon hearing or reading a sentence. Keeping in mind Frege's conception of functionality, the thought of a sentence is a function instantiated with referents to make up an argument. This argument is either true or false. Therefore, the saturated function that makes up a thought points out one of two truth values.

Thoughts Frege links the study of thought to the study of language, arguing that the two are intertwined. One cannot discuss thought without using language, and one cannot divorce language from thought. Therefore, any investigation of Frege's conception of truth must look at the relation of language to thought. Frege wrote his essay "Thought" as an attempt to reason out many of the ideas he perceived in Wittgenstein'sTractatus. It is not, however, either an explication or a refutation of Wittgenstein's work. Instead, this essay uses many linguistic examples to show how thoughts are expressed through sentences in an effort to expand Frege's philosophy. Closely associated with the expression of thought is the expression of truth. However, it would be a mistake to conclude that Frege took the simplistic approach of viewing the truth as an ephemeral entity reflected in language. Instead, truth serves as a foundational principle:

Just as "beautiful" points the way for aesthetics and "good" for ethics, so do words like "true" for logic. All sciences have truth as their goal; but logic is also concerned with it in quite a different way: logic has

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much the same relation to truth as physics has to weight or heat. To discover truth is the task of all sciences; it falls to logic to discern the laws of truth.33

Frege is careful to distinguish between the laws of thinking and the laws of thought. An investigation into the laws of thinking is an investigation of taking things to be true, whereas Frege correlates the laws of thought with the laws of truth. This correlation implies that thought must be truth, a correlation that causes Frege some difficulties, to be discussed later. Frege does not paint himself as an epistemologist, but considers himself as only a logician. This is most apparent in an apologetic footnote. In discussing how different thoughts may be obtained from the same sentence, Frege notes that:

I am not here in the happy position of a mineralogist who shows his audience a rock-crystal: I cannot put a thought in the hands of my readers with the request that they should examine it from all sides. Something in itself not perceptible by sense, the thought, is presented to the reader - and I must be content with that - wrapped up in a perceptible linguistic form. The pictorial aspect of language presents difficulties. The sensible always breaks in and makes expressions pictorial and so improper. So one fights against language, and I am compelled to occupy myself with language, although it is not my proper concern here.34

33Frege, Gottlob, "Thought," trans. by Peter Geach and R. H. StootoffinThe Frege Reader, ed. by Michael Beaney, Blackwell Publishers, Oxford, 1997, p. 325. 34ibid., p. 333-334f.

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Language can obscure thought as much as it can illuminate its listeners. Because thoughts are mental, rather than physical, objects, they can only be represented through the use of language. Because of this need to express thoughts through language, Frege is claiming that the physical (sensible) world adversely influences the transmission of thoughts through language by making expressions "pictorial." That is, truth can be seen as the corresponding of a picture to what it represents. Frege views this as an error: correspondence is a relation, and there is nothing relational about the truth. To advocate the pictorial theory of truth effectively confuses mental with physical objects -- a serious error when trying to map objects to concepts.

Relationship of Truth to Thought Frege offers his readers a working definition of "thought" as something for which the question of truth can arise. This not only links truth to thought, but implies that thought is more or less the same as judgment. If judging something (in a logical context) is judging something as either true or false, then what gives rise to the question of truth or falsity — a thought - is the only thing that can be judged. On a first reading, this linkage of truth to thought seems to lend itself to an empiricist epistemology. We can judge some thought (e.g. the apple is red) to be either true or false. However, "truth is not a quality that answers to a particular kind of sense impressions."35 Even though we may see that

35ibid., p. 328.

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the apple is red, and judge it to be true, we are able to judge things by the absence of sense impressions. For example, I can make the true statement: "I do not see the red apple." Just as truth is not perceptible to the human senses, neither is thought. While acknowledging this imperceptibility, Frege asserts that "[fjrom the laws of truth there follow prescriptions about asserting, thinking, judging, inferring."36 The laws of truth spell out how we can validly think of a proposition, assert its truth, and manipulate it according to logical laws. But how can logic investigate the laws of truth if both the truth and our thoughts about it are imperceptible? Even though the meaning of the word "true" may be indefinable, we can still see how truth and falsity operate in thought and language. Frege distinguished three aspects to the apparently seamless conjoining of truth, thought and language in uttering a true sentence:

i) the grasp of a thought -- thinking ii) the acknowledgment of the truth of a thought — the act of judgment iii) the manifestation of this judgment -- assertion.37

Thoughts are joined with truth through the act of judging. This implies that human reasoning and faculties are required to realize the truth behind any thought, much like scientific conjectures require human reasoning to ascertain their truth. One can object that the thought "The tree outside my window has budding leaves" can be either true of false depending upon the time at

36ibid., p. 325. 37ibid., p. 329.

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which it is uttered. Frege mitigates this objection by claimingthat"... the time of utterance is part of the expression of thought."38 The sentence "The tree outside my window has budding leaves" has, contained within the thought it expresses, the time of year at which I say it. This can be tied to Frege's context principle, in that this statement must be analyzed within the (temporal) context in which it is uttered. But is this temporal context a part of the thought? Frege implies that it is. Frege's contention that there is an inherent temporality in thought leads to some difficulties in his conception of thought. Frege began his philosophical work with the presupposition that certain (mathematical) truths are eternally true, and therefore not based upon our personal psychology or the actual state of development of mathematics. For instance, the truth of the Pythagorean theorem does not depend upon our subjective ideas about triangles and/or about number theory. However, the Pythagorean theorem is true without reference to the time at which it is uttered. Fermat's Last theorem, of which it is a special case, has always been true regardless of our ability to prove it true. Thus we are lead to conclude that there are two types of thought: those which are eternally true, and those whose truth is tied to time.

Third Realm Frege differentiates thoughts from feelings, sensations and emotions. For him, thought consists of the mind grasping something external to it. An

38ibid.t p. 332.

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important element in Frege’s philosophy is his insistence upon a sharp delineation between the objective and the subjective. Because of Frege's belief in an objective world that exists outside of our subjectivity, to think is to think about something in the objective world. Thoughts, in Frege's view, are not independent actions of the mind. They are not contained within the mind, but exist independendy outside of the mind, to be apprehended by a perceptive individual. Thoughts, therefore, are outside of one's subjective experience and private cognitions. In "Thought," Frege attempts to prove that thoughts are neither (subjective) ideas nor objects in the physical world, but must exist in an objective, third realm. He posits this third realm in order to differentiate thought from subjective ideas, which have an owner, and physical objects in the external world. Instead of placing thoughts in either one of these ontological categories, Frege points out that, like physical objects, thoughts have no owner but, like ideas, thoughts are intangible. In this way, we must have a third ontological category whose entities are intangible and objective. Humans are able to perceive things in this third realm:

A person sees a thing, has an idea, grasps or thinks a thought. When he grasps or thinks a thought he does not create it but only comes to stand in a certain relationship to what already existed - a different relationship from seeing a thing or having an idea.39

The grasping of a thought, as well as seeing a thing or having an idea, presupposes someone who grasps them. This seems to imply that, despite

39ibid., p. 337f.

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Frege's insistence upon objectivity, personal sense experience and subjective mental processes are necessary for expressing thoughts. Frege does not deny that each person's sense experience can be different from everyone else's, but argues that thought, unlike sense impressions, must not have owners. If every thought required an owner, then we could have no science. For, in order for science to advance, various thoughts (and the truth-values attached) must be disputed in order to move science closer towards the truth.

The Possibility of False Thoughts Frege strongly suggests at various points in his article that thoughts are true, either eternally or at a specific moment in time. Other aspects of his work contradict this. But what about false thoughts? If they are thoughts, then they exist in the intangible yet objective third realm. If there are false thoughts, then these thoughts must be, qua thoughts, objective, so we are lead to believe in entities that are objectively false. We can hesitate to call a thing false; judgments may be either true or false, but to call a thought false is to imply that judgments are objective things -- an implication that Frege admits. An important, if understated, element in Frege’s analysis of thought is the distinction between thought and thinking;

To the grasping of thought there must then correspond a special mental capacity, the power of thinking. In thinking we do not produce thoughts, we grasp them. For what I have called thoughts

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stand in closest connection with truth. What I acknowledge as true, I judge to be true quite apart from my acknowledging its truth or even thinking about it... What is a fact? A fact is a thought that is true.4^

Here, Frege's assertion that thoughts stand in closest connection with the truth implies that anything false, especially thoughts, cannot be close to the truth. If truth is determined by thoughts, then one can conclude that truth is neither subjective nor objective (in the standard sense), but occupies the same third universal realm as thoughts. However, the truth is not simply "determined" by thoughts. If we remember Frege's distinction (distinction #2) between the acknowledgment of the truth of a thought and the act of judgment, we can acknowledge a thought in the same way that we grasp a thought - without extensive logical analysis. Once we begin to evaluate the thought we have grasped through the process of thinking, then thinking is associated with the act of judgment. Thus it becomes clear why Frege has a need to distinguish thoughts from thinking: his scientific endeavors require that separating conjectures into the true (facts) and the false (mistakes) necessitates judgment. One can make the additional claim that this necessitates false thoughts: if thoughts were all eternally true, then any mistakes humans have made in the sciences resulted from psychological error or mistakes in logical reasoning. Frege seems to conceive of science as a process of refining conjectures with truth as the ultimate goal. The refining of conjectures requires that we make true judgments. Because thoughts are things for which the question of truth can arise, Frege counts "what is false among

40ibid., pp. 341-342.

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thoughts no less than what is true."41 So, it would seem, thoughts can be either true or false, what brings them in connection with the truth is the act of judgment. The distinction between thoughts and thinking points out the difference between Platonic Forms and Frege's third realm entities. The Forms are, by definition, eternally true and objective, but thoughts can be false. Thoughts do not reside in the third realm because of their inherent truth, but because of the dual qualities of objectivity and intangibility.

41ibid., p. 328.

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Philosophy and Language vs. Philosophy and Logic Michael Dummett interprets Frege's work toward a logically pure language to mean that he is a philosopher of language — a view which has sparked controversy within the community of analytic philosophers. There are several gaps within Dummett's analysis of Frege. For example, he does not address the theoretical distinction between language as a universal medium and language as a functional system similar to mathematics, other than to claim that there is an inherent conflict between Frege's context principle and his goal of producing a coherent logical system. In addition, he does not address the sticky issue of whether of not thought can precede language. Frege originally started his logicist program with the goal of proving arithmetic from logical principles. Through his logicist program, Frege wants to be able to describe things (particularly arithmetic) in an unambiguous way, and he repeatedly complains about the lack of clarity in natural language. One can argue that his attempts to create a logically pure language produce a tool to deal with both language and mathematics. 46

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Dummett argues that Frege is not limiting himself to mathematics, claiming that:

A crucial tenet of his philosophy of mathematics was that number theory and analysis are reducible to and derivable from a logic that is not specific to mathematics.42

It is true that Frege's system is applicable to much more than arithmetic, for logic, so he believes, is able to incorporate, state, and utilize all valid forms of inference, no matter what the particular subject may be. However, the broad applicability of logic does not, by itself, prove that Frege was primarily concerned with the philosophy of language. There are alternative interpretations of Frege that view him as an epistemologist or even as an idealist of some stripe. Dummett holds that interpreting Frege as a philosopher of language is necessary in order to make his doctrines coherent. He goes so far as to claim that:

Whether or not Frege is to be regarded as an analytical philosopher, he was without question a philosopher of language; construing him otherwise will make nonsense of his doctrines. 43

Dummett's fundamental reason for thinking of Frege as a philosopher of language has to do with Frege's celebrated and, to some, infamous context principle. The context principle — that a word has no meaning in isolation,

42 Dummett, Michael, The Interpretation o f Frege's Philosophy, Harvard University Press, Cambridge, 1981, p. 36. 43ibid., p. 55.

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but must be part of a sentence in order to make sense -- is readily applicable to an analysis of language. As we saw earlier, the context principle plays a key role in the temporal expression of a thought, as well as separating the subjective from the objective. The context principle is intertwined with another of Frege's intellectual achievements; the distinction between sense and reference.

Sense versus Reference Frege defines the sense of a sign as where the "mode of presentation is contained."44 Frege stresses the importance of the "mode of presentation" in order to differentiate between two sorts of statements of equivalence: the identity expressed by a = a and the identity expressed a = b. The statements of "a cat is a cat" and "Octavia is a cat" both mean something very different, yet are both statements of identity. Frege points out that what we say when we claim that "a = b" is that the signs "a" and "b" both designate the same object.45 From this, Frege concludes that differences in signs indicating the same object corresponds to different ways in which that particular object is designated. The association of any particular object with its sign must take place in a certain way. For example, there is a certain cat who was referred to by a code number in the animal shelter when she was a wee kitten, yet is now

44Frege, Gottlob, "On Sense and Meaning," Translationsin from the Philosophical Writings of Gottlob Frege, Third Edition, Geach and Black, eds., trans. by Geach, Black, Jourdain and Stachelroth, Basil Blackwell, Oxford, 1980, p. 351. 45 Frege, Gottlob, "On Sense and Meaning," Thein Frege Reader, ed. by Michael Beaney, trans. by Max Black, Blackwell Publishers, Oxford, 1997, p. 151.

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referred by the name "Octavia." Each of these terms, a numerical code and "Octavia" serves as a method of presenting the same object. As we operate within language, we must refer to objects in certain ways within that language. Sense is therefore a relational concept. That is, sense is the relation of a term to the object it denotes. A better example of Frege's distinction between sense and reference is the over-used example of "the morning star" and "the evening star." Both phrases designate the same object, but present the referent in different ways. To use Fregean terminology, they have the same reference, but different sense. Since we may never know that Venus is both the morning star and the evening star, what is important to Frege's philosophy is how we present our propositions, rather than the exact nature of the referents. Like thoughts, Frege holds senses to be objective and eternal. This seemingly bizarre idea becomes more believable when we take Frege's mathematical background into consideration. The sentence "two plus two equals four" will always be true (in base 10), and therefore, Frege holds, always have a true sense. If I state "three plus one equals four" it will also be eternally true. However, its sense, although true, will be different. The support for this theory is based upon what Frege perceives to be an intimate connection between the sense of a sentence and the thought expressed. This will be discussed at length in a later section. The supposition that senses can be true or false introduces problems into Frege’s sense and reference distinction. Sense, as mentioned earlier, is a relational concept. At first glance, it seems nonsensical for a relation to be true or false. Relations can either hold or not hold between two entities, but this is not the same as a relation being true or false. For instance, the

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relationship of the phrases "morning star" and "evening star," to Venus cannot sensibly be called true or false. However, I think Frege has in mind something different than the relationship in and of itself. Rather, I think that Frege calls the sense of a sentence true in order to locate truth in the objective realm of thoughts rather than the physical world of the referents. As we saw earlier, the sense of a sentence is the mode of presentation of a thought. Just as thoughts may be true or false, so too may be the senses. He states that:

And when we call a sentence true we really mean that its sense is true. And hence the only thing that raises the question of truth at all is the sense of sentences.46

The meaning of a sentence, Frege claims, is the sense conjoined with its truth-value. In this way, the sense of a sentence is more important than reference for determining truth-values. It is not plausible to claim that a referent, such as the morning star, is true or false. What is either true or false is a statement about the morning star. Frege must separate truth from meaning in order to differentiate between sense and reference. As shown in the "evening star" example, both phrases refer to the same object, although in different ways. If we are given the phrase "the morning star is Venus," then the truth of this sentence depends upon our determining that the two objects mentioned are equivalent. The meaning of the phrase "the evening star is Venus" is not the same as "the morning star is Venus," yet

their truth values are the same. If Frege equated truth with meaning,then

46 Frege, Gottlob, Logical Investigations, in Beitrage zur Philosophie des deutschen Idealismus, p. 355.

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either all meanings would be reduced down to two (true or false) or we would have a plethora of different truth values, each meaning something slightly different. Obviously, then, Frege cannot let meaning and truth-values be equivalent. As Frege's thought progresses, he draws a more streamlined ontology, wherein the senses of sentences, when instantiated with the referents, point to one of two truth-values. In this way, the ultimate reference of sentences is either the True or the False. Frege's distinction between sense and reference is the crucial philosophical move he makes in order to refine his ontology. If the exact nature of the referents is not of terrible importance to the meaning of a sentence, as Dummett suggests, one might ask why Frege puts so much emphasis on the distinction between sense and reference. In drawing this distinction, Frege's aim is to uncover the path our minds take towards determining the meaning of propositions. As shown by the example "A. J. Ayer has a dog," the hearer of this sentence understands what it means for this sentence to be true, according to Frege, although he or she may not know "A. J. Ayer." One might say that the sense is actually the precursor to the reference, in that it is how we are led to the referent. Dummett, certainly, believes in the primacy of sense in determining the meaning of a sentence. In the example, the hearer, not knowing "A. J. Ayer," can be lead to a description of the referent through the sense of the proposition. Of course, this presupposes that the hearer can understand the sense without knowledge of the referent. More particularly, we must draw a distinction between the referent and the reference of a sentence. This is because there are some sentences

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that have a sense but lack a referent. The classic example of this is the phrase "the present King of France is bald." Here, we have a phrase which serves, semantically, as the reference (the present King of France), but there is no referent in the outside world. Frege does not seem particularly concerned about the ontological ramifications of referring to non-existent objects. He concedes that because we are not in a position to determine whether of not the present King of France is bald, the truth-value of any sentence is not necessarily connected to the meaning of that sentence. However, he denies that sentences about the present King of France or the center of mass of the universe indicate that we should completely separate truth from meaning, rather, "One has the right to conclude only that the meaning of a sentence is notalways its truth-value." (emphasis in the original)47 Reference can therefore be split into two aspects: a semantic role and a name/bearer prototype. The semantic role of "the present King of France" is what serves as the subject of a sentence, whereas the name/bearer prototype acts as a proper name to denote an entity. In terms of distinguishing the truth-value, the name/bearer prototype has, obviously, the most important role to play within a realistic conception of the world. Dummett does not think that the semantic role of reference is necessary to discovering the truth-value of a sentence:

47Frege, Gottlob, "On Sense and Meaning," Thein Frege Reader, ed. by Michael Beaney, trans. by Max Black, Blackwell Publishers, Oxford, 1997, p. 161.

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The conception of reference as semantic role is, in itself, purely programmatic: it does not tell us what the semantic roles of expression of the various logical types are taken to be.48

One can argue that, because Frege is not concerned about the semantic role of referents, but rather with name/bearer prototypes (especially in the form of proper names), that his work with language is more concerned with a realist description of the world, rather than an investigation of semantics. The truth of any description of the world does not (except in extraordinary circumstances) depend on what is being described, but how it is described. But, as mentioned earlier, how we go about determining the truth-value of a proposition depends upon the circumstances in which it is uttered. Frege, through using his context principle, necessitates that the truth is dependent upon context, i.e., truth must be found within the implications of a proposition, rather than in the linguistic constructs:"... a mere expression, the form for a content, cannot be the heart of the matter; only the content itself can be that."49 This is where Frege's philosophy differs radically from previous methods of logic. Previous logic systems had considered meaning as given, and focused only on what made a proposition true. The truth of logic is demonstrated (for Frege) within the overall system of judgment and content. However, his context principle necessitates that his system be proved within the context of propositions.

48Dummett, Michael, Frege: Philosophy o f Language, Second Edition, Harvard University Press, Cambridge, 1973, p. 458. 49Frege, Gottlob, "Function and Concept," Translationsin from the Philosophical Writings o f Gottlob Frege, Third Edition, Geach and Black, eds., trans. by Geach, Black, Jourdain and Stachelroth, Basil Blackwell, Oxford, 1980, p.22

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Dummett claims that Frege does not provide us with a system of what it means to mean something in language -- he just relies on context to let us know what to do. The claim that there is no "meaning" model for language is controversial. Dummett is here laying modern concepts from the philosophy of language over Frege's form of platonism. Frege's use of the name/bearer prototype assumes that the semantic unit playing the role of reference should always relate to something in the real world, rather than a linguistic construction. If language describes an objective reality, there must be an intelligible meaning in our statements about the world. What Dummett is reacting to is the lack of a definition of meaning outside of context. Because meaning cannot be expressed outside of a context, Dummett views sense as a necessary ingredient for meaning:

The sense of a word . . . constitutes the contribution which it makes to determining the truth-conditions of sentences in which it occurs precisely by associating a certain reference to it.50

The truth-conditions of sentences are arrived at through an understanding of its meaning -- meaning which can only be understood in context. In order to know the meaning of a sentence, we must know what conditions are necessary for the truth of that sentence. Because the sense is how an object is presented, the sense aids in understanding the truth- conditions of a sentence by leading us to a certain object of reference.

SODummett, Michael, Frege: Philosophy o f Language, Second Edition, Harvard University Press, Cambridge, 1973, p. 93.

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Universal Language versus Language as Calculus The principle source of debate over Frege's view of language concerns whether his context principle or his view of language as a system of functions should be given primacy in his philosophy. These views are often seen as opposite: if language were a series of functions, then the context principle would not be terribly important in any logical analysis of language. However, if the context principle were the central idea in Frege’s analysis of sentences and propositions, then the calculus of language is merely a supporting structure for a theory of semantics. Dummett claims that, for Frege, logic is the study of truth. Truth in his thought is intertwined with meaning: truth attaches to the meaning of a proposition. Conceived thusly, philosophy of meaning takes precedence over any other philosophical inquiry. Dummett argues for linking meaning to truth as a means of circumventing problems that arise with the classical theories of truth such as the correspondence and coherence theories. The fault with classical theories of truth is that they attempt to give a description of truth while simply assuming the meaning of any given proposition. Within a philosophy of language, Dummett's arguments are quite persuasive. Any description of the truth must be given in a series of propositions. The meaning of the propositions, assuming Frege's context principle to be correct, must be recognized by the listener through the context of the argument. Therefore, the primacy of a philosophy of

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meaning in Frege's thought is the primacy of the context principle in the determination of truth. The implementation of the context principle ends up both linking truth and meaning as well as separating them. Dummett's arguments are a bit unclear in this (very important) area. We cannot examine the truth of propositions, and therefore reach a general description of truth, without looking after the meaning of the proposition. However, any investigation of meaning requires us to examine the context of a proposition, which in turn separates what it means for a proposition to be true from the truth of that particular proposition. We can conceive of situations where almost all propositions could be true, but in order to delineate between what propositions could be true and what propositions are true, we must understand what meansit for the proposition to be true. We are able to do this through an understanding of the context. For example, if the proposition "my cat Octavia likes to eat broccoli" is to be apprehended as true, the listener must understand all of the ideas within the proposition, as well as all of them put together. There is a step between understanding the meaning of a proposition and judging it to be true. This distinction between understanding and judgment separates meaning and truth. On the other hand, it is impossible to arrive at a truth- value of a proposition without first comprehending the meaning. Many contemporary philosophers subscribe to the belief that language creates a web of meaning to describe the physical world. Set in this philosophical context, Dummett's linkage of truth and meaning is quite compelling. The thought that underlies any particular proposition can only

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be expressed through language and, some would argue, is made possible only by language. This interpretation of Frege, that he presupposes language to be a universal medium, does not take into account Frege's mathematical roots. Some philosophers, most notably G. P. Baker and P. M. S. Hacker, claim that Frege's conception of a function is the zenith of his philosophy: "A generalization of function theory over non-mathematical entities is visibly the support of the core of his thinking about logic and mathematics."51 Therefore, to view the context principle as the primary paradigm of Frege's analysis of language removes the central pillar of function theory from Frege's thought. Hacker and Baker ascribe this error to Dummett, and as a consequence, argue that Dummett's view that a semantic analysis of sentences is the foundation of philosophy is in serious error.

51Bakerand Hacker, "Dummett's Purge," in The Philosophical Quarterly, Vol. 33, April 1983, p.116.

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Frege's Conception of Thought as the Bearer o f Truth Any philosophy of language derived from Frege's thought will have to examine his conception of thought. Dummett argues that Frege's platonism requires us to consider two sides of thought: the thought behind any particular proposition and the platonic character of thought. In its first aspect, thought is expressed through a sentence that has a particular context and sense. In the second aspect, thoughts are eternal and immutable entities whose existence is independent of our grasping them. This is essentially a distinction between a thought's state of existence and the uses a thought is put to. Frege calls thoughts objective in order to distinguish thoughts from individual, subjective, mental functions:

By a thought I understand not the subjective performance of thinking but its objective context, which is capable of being the common property of several thinkers.52

Frege wants to deal with the thought behind, for example, the proposition "this apple is green" — a proposition whose sense can be understood by a variety of people. This is in contrast to private feelings and emotions. As mentioned earlier, an objective thought, is, for Frege, an eternal which can be expressed through propositions. In this way, Dummett argues, thoughts have a dual character: both eternal and context-dependent.

52Frege, Gottlob, "On Sense and Meaning," Thein Frege Reader, ed. by Michael Beaney, trans. by Max Black, Blackwell Publishers, Oxford, 1997, p. 156, footnote.

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Regardless of how we interpret his view of thoughts -- either embedded in a proposition or as plaronic forms -- they function as the bearers of truth:

Frege was perfectly clear that to which truth attaches, to which the predicate "is true" is applied: it is a thought which is the primary bearer of truth or falsity . . . a sentence can be said to be true or false only in a derivative sense, as expressing a true or false thought.53

Frege thought that the truth of a proposition stood outside of the speaker, in much the same way the truth of mathematical propositions does not depend upon the personal experience of those who express them. Despite Frege's strong realist streak, Dummett shies away from labeling Frege as a philosopher of thought. Instead, he argues that an account of meaning necessarily employs the notion of truth in giving that account.54 Logic, for Dummett, is the "theory of meaning" analogous to epistemology rather than the theory of truth.

The Context Principle and the Role of Structures Dummett holds this view as a result of his emphasis upon the context principle. If truth (and therefore meaning) can only be expressed within the context of a proposition, then it makes little sense to view thoughts as eternal and immutable. Here Dummett may be placing too much emphasis upon the context principle in Frege's thought. It is possible to view the

53Dummett, Michael, Frege Philosophy of Language, Second Edition, Harvard University Press, Cambridge, 1973, p. 38 54ibid.

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contextual expressions of eternal thoughts as derivative in the same way that sentences are true or false in a derivative fashion. The truth is out there, but can only be expressed through the flawed lens of language. Dummett's claim that logic is the study of meaning is based on his premise that the philosophy of thought and the philosophy of meaning analyze isomorphic structures. Unfortunately for his readers, Dummett does not spell out which structures the philosophies of thought and meaning analyze. One would presume, given the general tenor of Dummett's arguments, that the "structure" is the structure of language. He presents his readers with two possibilities. The first possibility is that language mirrors the structure of thought. In other words, humans are able to grasp objective thoughts and express them through language. Because of Frege's footnote, cited above, we can conclude that Frege held this view. Unfortunately for Frege, this view presupposes that humans are able to think without language and also that thoughts can have a structure independent of language. The second possibility Dummett presents is where the complexity of a sentence corresponds to the total, systematic structure of thoughts. In this possibility, there is a grand structure to all thoughts, and these thoughts can be expressed through languages in sentences that preserve some aspect of the total structure. This view presupposes that there is, in fact, a systematic structure to thoughts, that these thoughts can be expressed, and these expressions possess a structure that corresponds to the grand system. Dummett argues for the first option, calling the second option the "map- reference view of language," meaning that any particular sentence refers

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to a thought. In other words, sentences have thoughts as their immediate referents. Dummeit refutes the map-reference view of language by pointing out that it holds that a sentence merely maps its thoughts onto the realm of sense. Because the sense of a sentence is, on Frege's view, the thought expressed, the map-reference view of language leads us to conclude that the immediate reference of sentences is the sense of the sentence. That is, instead of language mapping statements about the world onto the objective world, language maps statements to meaning. When applied to Frege's theory of sense and reference, the map-reference view of language becomes incoherent: the immediate reference of any sentence is its sense. Because of this conflation of sense and reference, Dummett argues that the map theory of language has no place in it for the expression of thoughts; it simply tells us where a certain feature is, without telling us what it is.55 An element of a sentence must correspond to a constituent of the thought in a sentence, rather than a thought located in the sense-realm. What I find flawed in the map theory of language is that it does not allow for someone to learn either the complexities of a sentence or the big map of language as a whole. Dummett's contention that language mirrors thought grants the possibility of someone learning a language without anpriori a knowledge of language.

5 5 Dummett, Michael, The Interpretation o f Frege's Philosophy, Harvard University Press, Cambridge, 1981, pp. 42-43.

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Thought as Language Dummett's contention that language mirrors the structure of thought entails that thought is prior to language — either as a human capacity or as the necessary precursor to uttering a sentence. The concept that thought is prior to language is not a very appealing one. Dummett argues against this entailment by claiming that thoughts are constructed out of the building blocks of sense. He argues that we can understand new sentences because we are able to grasp the senses of words in a sentence (through the context principle) and thereby grasp a previously unconceived thought:

. . . it is because the sense of a word must be conceived as a contribution to the sense of asentence that Frege represents it as a building-block of thoughts.56

This is a plausible explanation of how a listener might be able to grasp the sense of a new sentence, but it does not explain how we are able to articulate new thoughts. Through this "building-block" interpretation of how we grasp a thought, Dummett holds to the primacy of sentences in the order of explanation of its thought. He argues this against Hans Sluga's contention that thoughts are grasped in unarticulated wholes.57 Although Sluga’s theory also posits thoughts as prior to language (albeit in stronger terms than Dummett), it does make sense coming from Frege’s platonic conception of thoughts as existing in an objective realm. However, Sluga's theory leaves no room for the context principle: if thoughts are grasped in

SGibid., p. 374. 57ibid., p. 373.

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unarticulated wholes, then it would seem to make no difference when and where they are verbalized. Frege seems to be arguing for a progressive, rather than all- inclusive, method of comprehension. If we comprehend sense from the context of a sentence, and the context of the sentence depends upon the context of our language as a whole, then he leaves open the possibility that we can have an evolving knowledge of context. Although we do not "create" the objective thoughts that are reflected in our language, we can come closer to a comprehension of the truth through an analytic system of reasoning. Dummett views the context principle as concerning sense. It singles out sentences as having a unique role in any account of the senses of expressions. By assigning primacy to the context principle in an explanation of Frege's philosophy, Dummett is leaning towards the linguistic, rather than platonic, interpretation of thoughts.

The Ineffability of Truth A central problem of Dummett's analysis of thought and truth as they relate to language is his failure to distinguish between language as a system of rules of operation and language as a (and the only) method of communication. These mutually exclusive views of language have been competing ontological paradigms in the field of analytic philosophy for the past century. Both views can be supported by various aspects of Frege's work, yet the adherence to either one of these two conflicting paradigms

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leads to radically different interpretations of Frege's philosophy. I am unable to determine if Dummett favors one view of language over the other, foi his writings give the impression that he adheres to both. Because Dummett views Frege as primarily a philosopher of language, rather than a philosopher of mathematics, it is plausible that Dummett favors viewing language as a universal medium. Upon reading Frege's works, it is tempting to try to combine the two views: one could view language as our only means of describing the world, but that language can operate as a series of functions within the universal medium. However, these two views of language, when carried to their logical conclusion, leave no room for a "middle ground" theory. The conception of language as a universal medium holds that its symbolic structure would mirror our conceptual structure of the world. In short, our conceptual understanding of the world can only take place through the medium of language. On the other hand, conceiving of language as something similar to calculus serves to mirror the processes of human reasoning. It implies that we are able to step outside of language in order to analyze its operations, and that there are a multitude of possible interpretations or models of any contextually embedded sentence. This is a tension within Frege's own works, in that he portrays language as both a series of functions as well as something dependent upon context for comprehension. Some philosophers, most notably Jaakko Hintikka, claim that Frege definitely views language as a universal medium, and this presupposition underlies his work to form a logically pure language. The proof for this interpretation of Frege is that Frege does not attempt to operate outside of language, something which is possible in

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viewing language as a calculus, but unfathomable when viewing language as a universal medium. As mentioned earlier, Frege's writings can support both views — that is, he can support either one by contradicting himself. For instance, in his essay simply entitled "Logic" Frege claims boththat"... logic is the science of the most general laws of truth"58 and that:

. . . it would be futile to employ a definition in order to make it clearer what is to be understood by 'true.' If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality.59

Because any definition of truth would presuppose a contextual judgment, Frege concludes that we cannot give an overall definition of truth. That is, every time we judge a sentence as either true or false, we must rely upon the context in which the sentence is uttered in order to make a sound judgment. In this way, true judgments are performed in a piecemeal fashion, rather than falling out of a grand theory of truth. This inability to define truth is a consequence of viewing language as a universal medium. As a result of the context principle, Hintikka argues that Frege's Begriffsschrift serves as not merely a tool for better argumentation, but as a universal language in which everything can be expressed. In conceiving of his concept-script, Frege did not attempt tocreate a logically pure language, rather, his formalization

58Frege, Gottlob, "Logic," in The Frege Reader, ed. by Michael Beaney, trans. by Peter Long and Roger White, Blackwell Publishers, Oxford, 1997, p. 228. 59ibid.

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... is not a particular development beyond our ordinary language; it is a purified and streamlined version of the entire ordinary language itself. It is calculated to replace ordinary language, at least in its mathematical uses, not to extend it. The meanings of the expressions of this language cannot be defined, for they would have to be presupposed as a precondition of any such attempt to explain its semantics.60

This view of Hintikka's supports (partly) Dummett's interpretation of Frege as a philosopher of language, in that ordinary language is a necessary presupposition to Frege's logical work. However, if Frege was only interested in the mathematical uses of language, as Hintikka implies, then it would make more sense to call Frege a philosopher of mathematics.

Frege's Ontology and Epistemology At the heart of this controversy lie different conceptions about Frege's ontology and epistemology. Dummett seems to waiver on calling Frege a realist. Frege is often referred to as a realist because of his adherence to a platonic ideal of thoughts: they are grasped by our minds, rather than created. Dummett does not entirely agree that Frege is a simple platonist. He argues that, because Frege has a sophisticated conception of objects, he is not a true platonist. Dummett also goes on to say that, although Frege holds that mathematical statements are objective, he does not support

60Hintikka, Jaakko, Lingua Universalis vs. Calculus Ratiocinator, Kluwer Academic Publishers, Boston, 1997, p. x.

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this claim is any substantial way,61 thereby implying that Frege did not pursue an in-depth investigation into the ontology of mathematics - a daring claim, given Frege's views about the independent existence of numbers. If Frege is primarily concerned with determining sense (and therefore truth) through the context of a proposition, then his philosophy concerns the intuitive elements of language. Any expression of a thought relies upon an unexplicated intuition of meaning for transmission. On the other hand, if Frege views language as a flawed expression of analytic functions, then the context in which any particular sentence is uttered is of less importance. Dummett views there to be a conflict between Frege's mathematical means of describing sentences and the context principle. Other philosophers, namely Wolfgang Carl, believe that the conflict is not inherent in Frege's philosophy, mainly because he does not view Frege as a philosopher of language, and places much less importance on the context principle. Instead, Carl views Frege's context principle as an epistemological, rather than a linguistic, doctrine. To view the context principle as a part of epistemology leaves open the interpretation of Frege's logical work as grounded in epistemology, rather than linguistics. Carl claims that:

. .. the context principle and the claim that a word has meaning only in the context of a sentence do not belong to a particular semantic

61 Dummett, Michael, The Interpretation o f Frege's Philosophy, Harvard University Press, Cambridge, 1981,p. xxxviii.

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theory but to an epistemological theory that draws a fundamental distinction between thinking and having representations.62

This conception of the context principle's role can be easily linked to Frege's separation of thoughts from thinking. Thinking takes place within a temporal context, but the thoughts behind it are objectively real and non­ temporal. Those who are inclined to emphasize Frege's mathematical, rather than linguistic, venture rely upon Frege's realist conception of mathematical objects (and objects in general) to support their claims.

The Correspondence Theory of Truth Perhaps one of the reasons why Dummett hesitates to ascribe a blatant realism to Frege is because of Frege's disbelief in the correspondence theory of truth. Correspondence theory is often conflated with realism in that the correspondence theory claims that "Truth is a relation -- that of correspondence - between what is said or thought and a fact or state of affairs in the world."63 The realist position holds that the state of affairs in the world actually exists independent of our knowledge. Of course, one does not need to be a realist to hold that truth corresponds to the state of affairs in the world. The state of affairs may be construed as the world as we know it, rather than as it actually is.

62Carl, Wolfgang, Frege's Theory o f Sense and Reference, Cambridge University Press, Cambridge, 1994, pp. 43-44. 63Pitcher, George, "Introduction," inTruth, George Pitcher, ed., Prentice-Hall, Englewood Cliffs, 1964, p. 4

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Frege justified his disbelief in the correspondence theory by claiming that "true" is not a relative term. He uses the example of truth consisting of a correspondence of a picture to what it depicts. If we are given a picture of the Cologne Cathedral, but do not know that the picture is meant to represent that particular cathedral, then we have no means of comparing the picture to something in order to determining its truth-value.64 This points out the same fault that the correspondence theory has when linking language to truth as the map theory has when linking language to thought: it presupposes knowledge of the state of affairs in the world independent of the linguistic means of expressing it -- which would beg the question. Frege goes on to point out that a correspondence is perfect only if the corresponding things coincide. For instance, we can see if a dollar bill is genuine by comparing it to another, genuine dollar bill. However, we cannot compare ideas (including ideas of truth and falsity) in the same way as we can physical objects.65 If I claim that one dollar bill is the same as four quarters, then there is no way of determining an exact correspondence of one dollar bill and four quarters. In this way, Frege disproves the correspondence theory through a subtle use of epistemology. We could say that an idea corresponds in a certain respect to another idea, but then we would have the difficulty of determining in what sort of respect this correspondence takes place. Frege claims that this leads to an infinite regress in that we would have to specify the respect of correspondence between an idea and a physical entity, and

64Frege, Gottlob, "Thoughts," in Translations from the Philosophical Writings o f Gottlob Frege, Third Edition, Geach and Black, eds., trans. by Geach, Black, Jourdain and Stachelroth, Basil Blackwell, Oxford, 1980, p.352. 65ibid., p. 353.

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then specify the correspondence between the first correspondence and the idea, and so on. Dummett takes his readers through a careful examination of Frege's infinite regress argument, using the example of sentences. The correspondence of a sentence with an object requires that yet another sentence, namely, "This sentence corresponds with this object," be true. Perhaps a more definite example would help us at this point. The truth of the sentence "Sylvia owns a cat" would correspond to a particular cat, say X. However, we would then have to determine the correspondence between the sentence and the object X with another sentence: "This sentence (Sylvia owns a cat) corresponds to X." The second sentence thus will correspond to (what I suppose to be a thought) X*. We would then have to inquire into the truth of the sentence: "This sentence 'this sentence corresponds to X' corresponds to X*. And so on. In this way

... an infinite regress is generated. The same reasoning shows that truth is absolutely indefinable: For, if the truth of a sentence were to be defined as its possessing such-and-such characteristic, we should have, in order to determine whether the sentence is true, to inquire into the truth of the sentence which ascribed those characteristics to the first sentence; and again we should be launched on an infinite regress.66

Because of this infinite regress, Frege claims that truth is indefinable. Dummett blocks this regress by pointing out that once we determine a particular statement to be true, we simultaneously determine an infinite

66Dununett, Michael, Frege: Philosophy o f Language, Second Edition, Harvard University Press, Cambridge, 1973, p. 442.

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series of statements to be true. For instance, if we determine that Sylvia does indeed have a cat, then the statements "the statement 'Sylvia owns a cat' is true” and "the statement 'the statement "Sylvia owns a cat" is true' is true" and so on. So long as the determination of the original statement does not depend upon the determination of its predecessor, we can effectively end the infinite regress. Dummett concludes that "This objection succeeds in showing that Frege's argument does not sustain the strong conclusion that he draws, namely, that truth is absolutely indefinable."67 However, Dummett himself does not give his readers a definition of truth. There is one area of confusion within Dummett's deconstruction of Frege's argument. He bases his objection on the premise that Frege assumes the statement "Sylvia owns a cat" to rely upon a correspondence between the statement and an object X (in this case a cat). However, the truth of the statement does not depend upon any particular cat, but rather the relationship between this cat and Sylvia. In other words, the truth of this sort of statement is determined by establishing that the relationship stated actually exists, rather than the existence of any particular object named in the statement. To be sure, for the statement "Sylvia owns a cat" to be true, there must be a Sylvia and a cat, but it also requires that the two objects are connected through the relationship of possession. This does not mean that the correspondence theory of truth is correct, for Frege effectively demolishes it. Frege shows his readers that we cannot formulate a theory of truth in language without leading to a vicious circle. However, there is a correspondence between language and the physical world. Obviously, if there were no correspondence between language and

67ibid., p. 443.

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the world, Frege would not be so interested in distinguishing between sense and reference. What the correspondence theory does show is that we need certain conditions for our definitions of truth-values. All of these arguments are heavily dependent upon the context principle and the primacy of sense in determining the meaning and truth-value of a sentence. However, if the context principle is not of such great importance, as other philosophers have argued, then Dummett's interpretation needs to expand to include Frege's conception of functions. Dummett's contention that the context is the most important element of Frege's philosophy relies upon a reading of Frege that overlooks the importance of functionality in Frege's thought. The context principle goes hand-in-hand with functionality, as Frege originally portrays Theit (in Foundations of Arithmetic) as important for maintaining a distinction between the objective and the subjective. This distinction is important in the creation and operation of a logically pure language — which is tailored after mathematical set theory.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 WEINER’S INTERPRETATION OF FREGE

In her article "Has Frege a Philosophy of Language?"68 Joan Weiner answers in the affirmative -- but only as ordinary language can aid the development of a different, more precise language to be used as a scientific tool. Weiner claims that Frege is pursuing the goals of science through the creation of his logical system. Frege's concept-script and his subsequent logical writings serve as a tool to magnify and clarify certain aspects of the logical process. However, we cannot, Weiner claims, create a logical language without relying upon a commonly understood ordinary language in the first place. Weiner has Michael Dummett in her sights, who views Frege as primarily a philosopher of language. The crucial move, according to Weiner, that Dummett makes to support the contention that Frege is a philosopher of language is that of aligning "truth" with "meaning." This alignment extends any theory of meaning beyond logical terms. The implication in Dummett's move is that meaning cannot be expressed solely through logical symbols and their operations, but must rely upon ordinary language for its expression.

68Weiner, Joan, "Has Frege a Philosophy of Language?" inEarly Analytic Philosophy, ed. by William Tait, Open Court, Chicago, 1997. 73

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Weiner argues instead that Frege's philosophical treatment of truth is allied, not with meaning, but with science. She supports this theory by Frege's writings in which he compares ordinary language to the naked eye and his concept-script to a microscope. As our scientific investigations began with imperfect instruments yet evolved through the creation of more sensitive and limited instruments , so too our investigations of truth must begin with ordinary language and work towards creating a more perfect tool. As a part of this scientific venture, Weiner discounts any theories that Frege is interested in metaphysics or metaphysical entities.

Frege's Theory o f Meaning — of Lack Thereof Any contribution to a theory of meaning that Frege might have made is tied to his concept-script. Frege viewed his concept-script as one of the most important, if not the most important, results of his work. Dummett claims that the concept-script". .. serves as a base for the semantic theory embodied in his theory of reference."69 However, Weiner points out that Frege did not have a theory of reference until he wrote the celebrated and infamous article "On Sense and Meaning" several years later. In fact, Weiner claims that Frege did not set out to create a theory of reference. Instead,

When Frege first attempted to set out this syntactic analysis in 1879, he characterized it as a means for expressing all content that is of

69ibid., p. 250.

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significance for inference and as a means of preventing presuppositions from sneaking into inferences unnoticed.70

Frege wished to find a means of expressing the content and form of arguments free from presuppositions. There are, as we shall see later, certain presuppositions that do slip in, with important consequences for Frege's philosophy. Dummett identifies two distinct projects in Frege's work: the creation of a system of logic and the creation of a theory of meaning. Both of these projects require a theory of reference. Unfortunately for Dummett, any evidence of a theory of meaning in Frege's works is indirect at best. Weiner's intent is to show that Frege was not interested in studying the semantics of everyday language, nor was he interested in creating a theory of meaning. The primary example for Weiner that shows Frege's lack of interest in creating a theory of meaning is that Frege did not give a definition of proper names. Dummett views this as a serious error in Frege's theory, as any theory of meaning requires a definition of the function and meaning of proper names. Another interpretation, put forth by Weiner, is that "No definition of 'proper name' is required for purposes of setting up a logical notation and logical laws."71 If Frege were only concerned with the first half of Dummett's perceived dual project, that is, if Frege were only concerned with setting up a theory of logic, then he would have no need to offer a definition of proper names.

70ibid. 71 ibid., p. 251

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Another error that Dummett sees in Frege's theory of meaning is Frege's contention that a sentence serves as a proper name for either the True or the False. As sentences express an objective thought, the thought must be either true of false. Due to this expressive function, sentences serve to refer to both the referents and the sentence's own truth-value. Because sentences refer to truth-values in much the same way that proper names refer to that which they denote, Frege concludes that sentences act as proper names, and that which they denote are the True and the False. This conclusion, although quite odd, springs from the logical outcome of Frege's own concept-script:

It is an essential part of Frege's logical notation to show how the truth-value of a complex sentence is dependent on its composition . . . a sentence's contribution to the truth-value of a sentence in which it appears can only be explained if sentences are names of truth- values.72

If, as Dummett holds, any theory of meaning must move beyond logical expressions and into natural language, then it makes no sense for sentences to mean, ultimately, one of two things. If Frege were indeed trying to establish a theory of meaning, then reducing sentences to either true or false does not give us much explanation behind themeaning aof sentence. If Frege has only two referents in his theory of reference, then we can conclude that he was not terribly interested in producing a subtle theory of reference or a theory of meaning. However, Dummett contends

72ibid., p. 252.

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that a theory of meaning must underlie a theory of logic. This in itself makes perfect sense. To claim that we can have a coherent logical system that is somehow disconnected from meaning is implausible, for we cannot successfully pursue the truth through logic if we cannot understand what it means for something to be true. Dummett goes on to claim that any theory of meaning should not be an accurate account of how natural language functions, but that it is meant to apply to natural languages "in so far as it does function properly."73 Dummett wants Frege's logic to be responsible for a systematic theory of meaning that correlates to correct usage of natural language. Here there is an inherent circularity in this argument that Weiner does not point out: we can only adjudicate "proper" functions of language through a logical system of validity and invalidity. However, Weiner herself claims that ordinary language serves as an aid to Frege's logic. Weiner responds to Dummett's linkage of truth to meaning by claiming that there is no indication that logical laws must answer to a theory of meaning. If anything, a theory of meaning must answer to the laws of logic. Dummett places meaning as prior to logic, whereas Weiner wants logic to be prior to meaning. The primary source of their disagreement springs not from Frege's concept-script, but from his context principle. Dummett places great emphasis on the context principle, while Weiner does not think it important enough to be mentioned in her article. If meaning, which is understood in context, is adjudicated through logic (as Dummett claims) then any investigation into Frege’s thought requires an investigation of the "meanings” that are prior to logic. Because meaning

73ibid., p. 253.

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precedes logic, Frege must, Dummett argues, use a theory of semantics that allows logic to adjudicate the meanings of sentences we speak. Weiner, on the contrary, emphasizes the utilitarian aspect of logic that allows it to serve as a set of rules for conveying information. On this view, the logic itself takes the place of a semantic theory of meaning for Dummett. The consequence of this disagreement is that Dummett is interested in examining Frege's underlying semantic theory and metaphysical assumptions, whereas Weiner does not think either exists in Frege's thought. Weiner's emphasis upon the utilitarian aspect of concept-script has some far-reaching philosophical consequences for her interpretation of Frege. Not only does her contention that logical analysis is the crux of Frege's product influence her reading of the context principle, it also entails that any metaphysical project ascribed to Frege is an error.74 Despite Weiner's arguments to the contrary, Frege does seem to give his readers a theory of meaning in "On Sense and Meaning." Dummett thinks that the point of a theory of sense is its contribution to a theory of meaning. That is, the standard view of the sense/reference distinction is that the sense of a sentence communicates the meaning. Weiner does not agree with the standard view. She points out some difficulties in the relationship between sense and logic. Frege claimed that we can only determine two senses to be equivalent if their contents are logically equivalent. In a letter to Husserl, he writes "equipollent propositions have something common in their content, and this is what I

74See especially "Burge's Literal Interpretation of Frege", Mind, vol. 104, July, 1995. In this article, she argues against Burge's interpretation of Frege as a staunch platonist, claiming that Frege never gives his readers enough textual information for us to label him a platonist without hesitation.

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call the thought they express."75 The implication of this statement is that two different senses can have the same logical content. If this is so, then there must be a difference between sense and logical content — if there were not, then the two senses would be the same. This implies that there is something to sense which cannot be reduced to its logical content. However, if sense had a content beyond what could be expressed in concept-script, then Frege would need an additional system of logical notation. This is the crux of her argument against Dummett's interpretation of Frege: if the concept-script is a complete system for the logical expression of sentences, then there is no need for a theory of meaning beyond Frege's logic. However, Weiner fails to take into account Frege's context principle. Frege states explicitly that sense is grasped through an understanding of context. The context principle is another reason, and, one can argue, the main reason why Dummett views Frege as a philosopher of language. If the meaning of a sentence can only be understood by taking into account the context in which it is uttered, then any judgment concerning the truth or falsity of a particular proposition must consider the context. Context, of course, cannot be expressed through logical notation, as it is dependent upon the situation in which a sentence is uttered, rather than upon logical laws. Frege's admission that the context of a proposition is necessary for determining its truth or falsity implies that there is something beyond logical laws that goes into a logical analysis of propositions. I find the context principle to be the most compelling evidence of any theory of

75Weiner, Joan, "Has Frege a Philosophy of Language?" inEarly Analytic Philosophy, ed. by William Tait, Open Court, Chicago, 1997, p. 254.

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meaningFrege may have pursued. However, Frege himself does not probe what it means for a sentence to have context; rather, he simply states this as a method of separating the logical from the psychological (inFoundations) and later as a means of avoiding the "antinomies of reason" by ensuring a temporal aspect to the expression of sentences.

Logic as a Scientific Tool Weiner concedes that the laws of logic cannot be independent of meaning, but argues that it does not follow that Frege neededtheory a of meaning. Frege famously declared that the laws of logic are the laws of truth. Therefore, the laws of logic are meant to be applicable whenever our interest is in establishing the truth. Weiner then claims that truths are the ultimate basis of science - implying that Frege is in pursuit of a tool for the philosophy of science, broadly conceived. If Frege's logic serves as a system for deducing truths, then the concept-script serves as a useful tool. However, it is important not to confuse the system with the truths themselves. Weiner responds to the work of Thomas Ricketts who claims that the roots of Frege's scientific enterprise lie in everyday agreements and disagreements, rather than scientific enterprises. Because of this, Frege adopts ordinary language as a starting point. But, if thoughts are objective, as Frege held them to be, then there can be noreal disagreements about the status of the external world. Because of Frege's realism, there is a true

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description of the world, and disagreements are not competing theories, but correct or incorrect descriptions of the world. Ricketts argues that Frege's notion of meaning originates in an attempt to explain generality and its quantifier representation in his concept-script. Frege's concept-script involves a presupposition that there are no empty denoting phrases, to use Russellian terminology. However, the status of this presupposition changes after Frege drew the sense/reference distinction. Ricketts claims that Frege drew this distinction after recognizing the existence of meaningless names. However, Frege thought "meaningless" names were literary, poetic, psychological, but not logical. Ricketts seems to be imposing a Russellian concept -- that of denoting phrases -- over Fregean philosophy. Because of this, Weiner does not subscribe to Rickett's emphasis and interpretation of proper names. Specifically, she does not agree that the purpose of the concept is to create a semantic science. Rather, the scientific purpose of the concept-script is to serve as a tool to examine the workings of logic. In order to avoid the existential fallacy, (moving from naming objects to assuming that for every name there must be an object) Frege must presuppose that every logically valid expression picks out a definite object. Weiner claims this need to determine what any given name or sense means entails that Frege utilizes semantics in his logical analysis. In separating reference from sense, Frege attempts to differentiate between linguistic descriptions and the objects to which they refer. However, this does not entail that Frege works out, or needs to work out, a theory of semantics. Weiner, obviously, allies herself with Ricketts rather than Dummett even though her interpretation of "general science" is different than

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Ricketts'. However, she determines that Frege's concept-script is more closely aligned with natural language that had previously been thought. According to Weiner, Frege does not need a semantic theory and

. . . the predominant reading of "On Sense and Meaning" is a misreading. My difference, both with Dummett and with Ricketts, has to do with the relation of logic, as Frege conceives it, to ordinary language and ordinary reasoning.76

Dummett, for instance, claims that a logically perfect language should apply to ordinary language. The main difference between concept-script and natural language is that natural language contains content that is not relevant to the logical expression of the thought. However, any thought that can be expressed in a logically perfect language can also, if done correctly, be expressed in natural language. Logic, for Dummett, serves as a clarifying mechanism for ordinary language. Weiner bases her interpretation of Frege on a passage from the preface to Begriffsschrift comparing his concept-script to a microscope. If natural language is comparable to the human eye, then concept-script serves as a means of examining small imperfections not normally seen by the naked eye. Both methods of perception have their uses, but

% . . . as soon as scientific goals demand great sharpness of resolution, the eye proves to be insufficient. The microscope, on the other hand, is perfectly suited to precisely such goals, but that is just why it is useless for all others. This ideography [Begriffsschrift], likewise, is a

76ibid., pp. 257-258

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device invented for certain scientific purposes, and one must not condemn it because it is not suited to others.77

The image of the microscope, with its limited but specific purpose, suggests that a logically pure language can express something that ordinary language is unable to express. Accordingly, natural language and concept- script have different uses. However, the microscope can be used to illuminate various aspects of ordinary language.

Weiner and Dummett's Interpretations of the Third Realm Disagreements over the status of Frege's third realm often boil down to disagreements over whether of not Frege is a Platonist. If Frege were a Platonist, then his third realm is a realm of the Forms, or some such type of entities. As I pointed out earlier, the inhabitants of Frege's third realm - thoughts - are not Forms in any traditional sense, in that they can be either true or false, and do not express any sort of metaphysical absolute (e.g. Truth, Beauty, etc.) That Frege's thoughts are non-physical, mind-independent entities implies an antecedent metaphysical assumption that there are some entities in the world which are neither created by the human mind nor physical objects. Weiner points out that there are many epistemological problems concerning how we are able to come to know these entities.78 This is a

77ibid., p. 258. 78See especially "Platonism, Fregean and UnFregean"Frege in in Perspective, by Joan Weiner, Cornell University Press, Ithaca, 1990. In this article, Weiner primarily examines Frege's concept of number as a Platonic entity, and argues that "Frege's mere

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serious question, for thoughts themselves, which serve (in the Fregean worldview) as the primary ingredient in any analysis of propositions, belong to this third realm of non-physical, mind-independent objects. Frege posits a logical faculty that allows humans to grasp thoughts. However, Weiner claims that because the passages in which Frege talks about the ontological status of concepts and functions are notoriously paradoxical, we cannot make any definite claim about Frege's metaphysical assumptions, if indeed he made any in a meaningful fashion. 79 Dummett, of course, has a completely different take on the implications of thoughts residing in a third realm. Because of his insistence upon a semantic theory of meaning underlying Frege's philosophy, he interprets thoughts as always beingabout something, that is, they have a strong connection to the physical world and are effectively unarticulated propositions. Consequently, he argues that it is a mistake for Frege to place thoughts in a third realm separate from the world of everyday linguistic practices. If thoughts must always be about something, then the third realm is directly related to our everyday expressions. On this account, the separation of thoughts into a third realm seems unnecessary. However, as we have seen, the creation of a third realm falls out of Frege's ontological separation of the logical from the psychological:

assertion that something is a non-physical, non-subjective object does not commit him to any of the contemporary epistemological or metaphysical views customarily attached to such an assertion." p. 216. 79Weiner, Joan, "Burge's Literal Interpretation of Frege," in Mind, vol. 104, July 1995, p. 586.

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Given the strict dichotomy between the radically subjective and the wholly objective . . . it follows from the fact that the senses of the words and of sentences can be grasped by different individuals that they exist eternally and immutably in complete independence of us: but is this not a textbook example of philosophical mythology?80

So, Dummett claims, Frege's third realm is philosophical mythologizing. In a sense, he is correct, as the third realm can neither be conclusively proven nor disproven, but must be taken as a matter of faith. However, the fact that Frege creates this mythical realm can be evidence of metaphysical assumptions he makes concerning the nature of thoughts. One central assumption that Dummett makes in his analysis of Frege's conception of thoughts is that they are objects of the mind. Because the mind can grasp thoughts and thereby make them objects of our personal cognitions, thoughts are objects. This ties in with Dummett's contention that thoughts must always be about something. Although this may seem trivially true at first glance, we must be careful in determining in whatfashion thoughts about something. Dummett makes the crucial mistake of not accounting for Frege's conception of sentences (and therefore the thoughts behind them) as instantiated functions. In describing thoughts as objects of our minds, Dummett uses the example of chess moves. There are many different chess moves, some practiced often, some no longer used, and some that have never been used. In keeping with his contention that thoughts are always about something,

80Dummett, Michael, "Frege's Myth of the Third Realm," inFrege and Other Philosophers, Clarendon Press, Oxford, 1991, p. 249.

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Dummett points out that chess moves have a definite scope: the scope of all chess pieces:

It is harmless to say that "there are" such moves; but it would be insane to deny that moves areof (actual or possible) chess pieces. This "of' of logical dependence is not properly expressed by saying that a certain move exists only if there is a piece that has that move, since, as just noted, we can speak of moves that have never been assigned to any piece. It means, rather, that to conceive of any move is to conceive of a piece as having that move.81

The linking of a chess piece to a chess move can be easily accommodated within Frege's philosophy through his context principle which ensures the proper scope of objects falling under a thought. However, Dummett makes two crucial and linked mistakes: he conceives of thoughts as objects rather than functions and confuses the scope of objects for the function with a relation of dependence between two sets of objects. Frege's remark that "to conceive of any move is to conceive of a piece as having that move" is indicative of this ontological type-level confusion. The mistake has to do with Frege's rather vague definition of what it means to be an object, rather than a concept of function: "Chess moves are objects by Frege's criteria, for they can be named and have predicates applied to them."82 However, it matters a great dealwhich criteria for the definition of objects. It is true that Frege gives a vague definition of objects as that which can be named and predicated, but he also defines objects as everything which is not a concept. I think that a closer examination of

81 ibid. 82ibid.

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Frege's thought will show that Dummett's classification of chess moves as objects is inaccurate. Frege defines functions as expressions of the rules of operation for objects such as numbers. Under this conception, chess moves are functions, in that they are the expression of the rules of operation for chess pieces. If we take x3 as an analogy, this function can be named (e.g. "x cubed") as well as have predicates applied to it (e.g. "exponential"). However, x3 describes a rule of operation for numbers, rather than an object. Dummett also maintains that chess moves are objects in that they are objects of our minds, as are thoughts in general. However, this turns on an equivocation of "object." The distinction to draw is between thinking a thought and thinkingabout a thought. We are all perfectly capable of thinking about x3 , that is, have that function as an object of our consciousness, with x3 remaining an eternal, mind-independent entity. The relation between chess moves and chess pieces is not, as Dummett would have it, a logical relation between two sets of objects, but a relation of a set of objects falling under a set of functions, which in turn fall under the concept of "chess moves." The context principle maintains the boundary of all objects on which chess moves can be performs (all objects which are chess pieces, or, as in the famous Indian movie, acting as chess pieces.)

The Sharpness of Boundaries As mentioned earlier, Frege supposed that proper names pick out definite objects. Weiner points out that we cannot simply ask if a proper

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name picks out a certain object, because not all objects have definite boundaries. The issue of meaningful reference has to do not just with proper names, but with definite descriptions of the objects we mean as referents. The constituents of definite descriptions (which must exist in Frege's theory) are concept expressions. And "a concept expression is meaningful only if it has a sharp boundary, that is, only if for each object it determinately holds or fails to hold."83 Because of this need for sharp boundaries, the terms we use in everyday discourse will only be meaningful if we replace our vague concept-expressions with definite terms and that we only use definite descriptions that pick out unique objects. In other words, Frege's emphasis on the distinctness of terms and reference to existing objects requires us to replace natural language with a logically perfect language. From this, Weiner concludesthat"... Frege has no need for a semantic theory for natural language."84 Frege's sharp boundary requirement is not very plausible. Weiner tries to show that the solution to this problem is that systematic scientific investigation provides Frege with his model of how to use the concept-script. The solution, in short, is to draw distinctions between categories in much the same way that modern science separates various phenomena into categories, even though the boundaries, when pressed, can dissolve. Because of Frege's scientific, rather than metaphysical or linguistic concerns, Weiner does not take Frege to be the strict realist and Platonist most have thought him. Weiner's cautious reading of Frege concludes that

83Weiner, Joan, "Has Frege a Philosophy of Language?" inEarly Analytic Philosophy, ed. by William Tait, Open Court, Chicago, 1997, p. 261. 84ibid., p. 262.

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we cannot say many things that have been said because it would involve reading between the lines. However, Frege implies so much in his works that, despite his meticulousness, many previous ideas are implied in his passages, especially in his later works.

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The evolution of Frege's thought involves a streamlining of his ontology and an emphasis upon functionality. Frege's crucial distinctions between sense and reference, concept and object, and the True and the False are connected through a functional mapping paradigm. Frege's methodology of instantiating variables into functions to arrive at a value plays out in all three distinctions. This methodology is his paradigm of mapping. Within the sense/reference distinction, the referent of a proposition is presented (instantiated) in a particular way that is the sense of the proposition. This instantiation leads us to the reference. The referents and the sense of propositions correlate to Frege's separation of concepts and objects, with the sense of a proposition being an expression of the underlying concept within the thought expressed, and the referents serving as objects which fall under the concept. Within the distinction of concept and object, the methodology is much more apparent. Frege views objects a members of sets that can fall under concepts. As this must be expressed through language, concepts and objects are verbalized through sense and referring expressions. Originally defined as the difference between a referent and the way to which it is referred, the sense/reference distinction evolves into a dichotomy between the thought expressed in a sentence and the truth-value of that thought. 90

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It is Frege's conception of reference that leads him to streamline his ontology. The instantiation of an object into a concept points the way to a value. In the case of mathematical functions, instantiating the function x3 with the object 3 leads us to the value of 27. In other words, x3 with x = 3 refers to 27. In much the same way, the instantiation of a unsaturated proposition with a referent leads to the reference. An important factor to remember when investigating Frege's work is his conception of propositions. For instance, "x3" is not a proposition for any x, but rather "x3 = ___" is. In the same way, "The morning star" is the name of an object, whereas "The morning star is Venus" is a proposition referring to a particular object through the use of a function.

How Frege's Conception of Functionality Relates to Thought Thought is closely related to concepts within Frege's ontology. The content of a thought is a concept, rather than a sensation or image. As sentences are an expression of the thoughts behind them, thought is isomorphic to the saturated function a sentence verbalizes. Because, within Frege's ontology, functions are a particular type of concepts, thoughts serve as an ontological basis for the verbal expression of (instantiated) concepts. Ultimately, functions, when satisfied, are either true or false. Using an earlier example, x3 = 27 is true for the number 3, and false for all other numbers. Not only is it false for every number except 3, it is false for any other object, e.g., the moon, instantiated within the function. This conception of truth and falsity relies strongly on the context principle: the

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context of any proposition or function determines the range of objects to be instantiated within that function. In other words, the context principle serves as a criterion for sharpness in determining the ontological boundaries of various functions. Although Frege's theory of functionality can be applied universally to all logical propositions, he must include a sharpness requirement to ensure that the conceptual content of thought can ultimately produce meaningful statements about the world. With the context principle in place, all propositions ultimately refer to one of two truth values; the True or the False. There is a sort of type-level differentiation between the reference of the proposition "The morning star is Venus" to a particular physical object and its reference to the truth. The instantiation of objects into the sentence leads us to a description of Venus. On a different level, this instantiated function provides us with a description of the world that is either true or false. In this particular case, it is true. Frege does not spell out this type-level difference in his writing — that job was left to Bertrand Russell.

Thoughts as Third Realm Entities Because instantiated functions, either mathematical or verbal, ultimately refer to either the True or the False, this implies that thoughts themselves can be either true or false. Frege's modified platonism conceptualizes thoughts as permanent -- a requirement for meaningful knowledge and, therefore meaningful statements about the world.

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Frege's correlation of the laws of thought with the laws of truth seems to conflate thought and truth -- a tenuous position. However, we should note that it is the laws that are conflated, not thought and truth themselves. These laws are, presumably, logical laws. It is in his essay "Thought" that Frege can be labeled a metaphysician. In positing a third realm, separate from subjective ideas and physical objects, Frege is indeed positing a transcendental ontological category. However, the entities in this category refer to objects in the world. Thoughts, as precursors to saturated propositions, refer both to the object and to the truth-value of the proposition. However, we are never able to get to the thought itself; it always must be referred to through the use of language. In this way all assertoric sentences serve as expressions of thought. One can argue that Frege's third realm is a metaphor, rather than a reified concept. Because Frege correlated thoughts with mathematical functions, one can argue that this correlation is metaphorical: the thoughts in my head and the questions is my algebra textbook seem to be disparate entities. However, Frege is concerned with theconcepts behind mathematics and thought, rather than simply the expression of them. Because thoughts cannot be examined like a rock, they must always be referred to in an elliptical fashion, such as metaphysical statements about

their subsistence in a thirdrealm . I find that Frege's third realm is a metaphysical entity, yet one which serves a useful purpose of providing an ontological grounding for the expression of assertoric sentences.

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