From Philosophy of Mathematics to Philosophy of Language
In The Foundations Frege proposed a definition of number as a kind of concept-extension and outlined how arithmetical truths can be derived from the general laws of logic in terms of this definition. Before the proof can be formally executed in the Basic Laws of Arithemtic, clarification is needed of the notions of concept and concept-extension. In the Begriffsschrift, Frege suggested that a functional analysis of concepts is required in order to explicate their logical function. Unfortunately, while relatively clear about the nature of the argument of such functions, that is, the denotations of non-empty singular terms, Frege was less than explicit about the values of such functions. Until Frege has specified the nature of the values of those functions with which concepts are identified, we have as yet no clear conception of Fregean concepts nor what he takes their extensions to be. It also follows, then, that we have no clear conception of what numbers are if they are to be, as Frege urges, a particular class of extensions, nor do we have a clear conception of the logical principles governing concept-extensions. It is to this project, clarifying the operative notions of concept and concept-extension, that Frege turns in his writings of his middle period. Finding a class of values for concepts will dictate the structure of his mature philosophy of language and logic. As standardly presented, Frege’s philosophy of language is developed more or less independently of his logic and philosophy of mathematics. We are told that Frege was driven to investigate the philosophy of language because of what he took to be the demands of his logical and mathematical research. But we are given no clear conception of the connection. I have already suggested that Frege had to rethink the traditional conception of a statement as a kind of saying because one’s account of valid inference will depend on what one takes to be the logically relevant structure of a statement and because he considered the subject–predicate analysis to be an insurmountable obstacle to a mathematically rigorous treatment of valid inference. That is the connection between his logic and philosophy of language. The connection between Frege’s philosophy of language and philosophy of mathematics is precisely the specification of the fugitive values of those functions with which concepts are identified. Frege’s decision, in conjunction with his commitment to the compositionality principle, will dictate his views about truth, his distinction between sense and reference, his identification of the content of assertions with thoughts, the sense expressed by the sentence asserted, his account of propositional attitude constructions, etc. And make a decision he must, for without an explicit statement of what extensions are and of the logical principles that govern them, he cannot complete the system of proofs outlined in The Foundations. Thus Frege’s essay, “Function and Concept,” is of first importance, perhaps even more important than “Sense and Reference.” It is there that he makes his crucial and, as it turns out, fatal decision, for, not only has he thereby laid a foundation for his philosophy of language and mathematics, but he has also laid the groundwork for Russell’s paradox.
Function and Concept
Frege begins “Function and Concept” by investigating the notion of a function as it originally appeared in higher analysis. Frege claims that the first place that a novel scientific expression receives a clear meaning is where it is required for the statement of a law. And this is the case with respect to functional expressions as they occur in analysis because there was a need to state general laws regarding functions. Throughout the essay Frege will clarify and extend the original meaning of a function until he has arrived at his mature conception of a concept that is a function from objects to truth-values. It is worth noting that Frege explicitly describes the content of his essay as supplementing the logic of the Begriffsschrift with new conceptions “whose necessity has occurred to [Frege] since”. What is new is the account of concepts and concept-extensions and they are necessary for the project of uncovering the covertly logical subject matter of arithmetic.
According to formalism, functions just are functional expressions. So, if asked what is a function in the context of higher analysis, one will likely be confronted with the answer that a mathematical function of the variable x is a formula containing x such as:
Moreover the corresponding formula
would be a function of 2. Frege complains of this answer that no distinction is made between form and content, sign and thing signified. Frege observes that the expression:
denotes the same thing as “18” or “3*6”, and thus we are licensed to assert, for example:
But how could this be if functions are identified with mathematical expressions? What stands on either side of the identity sign are different expressions, are they then equal but mysteriously not the same. Frege is here enforcing the use/mention distinction and urges that a difference in sign is not sufficient for a difference in thing signified. Thus
- sweet smelling violet
- viola odorata
mean the same even though the audible sign is different. Frege further observes that the properties of a sign and thing signified are different. The number one, for instance, has the property that the result of multiplying it by itself is equal to itself. However no chemical or physical investigation of the properties of the written symbol will uncover this property. He further argues that the introduction of a novel numerical notation doesn’t mean that we have thereby introduced new numbers whose properties we have yet to determine.
Functions are not functional expressions. Functions are what functional expressions express. Though Frege rejects the formalist account of function that identifies functions with functional expressions he does acknowledge that it is correct to emphasize the distinctive role of variables. Reflection on the fact that the role of a variable in analysis is to indicate numbers in an indefinite manner, leads us to a correct understanding of function. The variable “x” is recognized as marking the place of an argument of a function and the same function is recognized in:
only with different arguments, that is 1, 4 and 5. What is common to the content of these formulas is what is expressed by
In order to avoid the confusion that this is a function whose argument is an indefinite number, Frege suggests that the functional expression could alternatively be written:
(While Frege claims that a variable indicates a number indefinitely, he rejects the claim that a variable indicates an indefinite number.)
It is at this point that Frege introduces the distinction between saturated and unsaturated entities, and complete and incomplete expressions. (Frege tends to use the pairs “saturated”/”complete” “unsaturated”/”incomplete” interchangeably. It is useful, however, to regiment his usage and restrict the saturated–unsaturated distinction to entities and the complete–incomplete distinction to expressions that denote these entities. Far from falisfying Frege’s thought, this regimentation makes explicit what is merely implicit in Frege’s writing.) As indicated by the empty brackets of the functional expression, the function itself, the content of the expression, is in need of supplementation before it can determine a value. It must be fed a number as its argument before it can yield a number as its value. It is in this sense that functions are said to be unsaturated. Frege sometimes glosses the notion of an object or saturated entity as whatever is denoted by a non-functional expression. The characterization is fair enough once his system is set up an understood, but it is important to notice that when he introduces the distinction between saturated and unsaturated entities the notion of an unsaturated entity is explained in terms of the antecedently understood notion of saturation. In what does this antecedent understanding consist? A saturated entity is whatever is denoted by a complete expression. That characterization is only as good as our understanding of what a complete expression is. If the notion of a saturated entity is explained in terms of the denotation of a complete expression, our grasp of that notion is only as clear as our grasp of the notion of a complete expression. So what is a complete expression? One mark of a complete expression is that its denotation is determined without supplementation. Here we are coming dangerously close to explicating the notion of a complete expression in terms of the content of an incomplete, or functional, expression, that is, in terms of an unsaturated entity. Is there any way to break out of this circle? We will return to this issue. Frege held that there were a number of important logical and grammatical analogies that unite the class of complete expressions. The notion of a complete expression can be understood as the summation of these analogies. If the Fregean analogies are persuasive, then we would have a non-circular characterization of a complete expression. We could then explain saturated entities as whatever are denoted by complete expressions, and unsaturated entities in contrast to these. Frege is explicit that there is no question of defining the correlative notions of saturated entity and complete expression as they are, in the end, logically simple. So the logico-syntactic analogies among complete expressions should not be expected to form the basis of anything like a definition, rather, they are a system of hints—a heuristic to aid us in grasping the relevant, logically primitive concepts.
In the context in which the notion of a function was first introduced, functional expressions were constructed out of the arithmetical operations such as addition, multiplication, exponentiation, etc. and the arguments and values of these functions were taken to be the rational numbers. As mathematics developed, however, the notion of a function was extended along both these dimensions—the class of signs used to construct functional expressions was extended as was the range of things that served as their arguments and values. Thus in addition to the arithmetical operations, various means of transition to a limit were used to construct functional expressions, and the domain and range of functions were extended to include complex numbers. Frege, confronted by a situation in which the original notion of a function has been extended to meet the demands of mathematical research, felt licensed to himself extend the notion of a function even further given the demands of his logical research. Moreover his extension fit the pattern in which the notion of a function had been extended within mathematical practice:
Frege allowed novel signs to be the constituents of functional expressions, and
Frege extended the class of things that could be the arguments and values of functions.
Frege’s first innovation is to extend the class of functional expressions to include predicate expressions such as the identity sign and the greater than sign familiar from arithmetic. Thus not only does:
count as a functional expression, but now so does:
Recall the analogy that Frege draws between predicates and functional expressions traditionally conceived in the Begriffsschrift. Frege observes that just as we can substitute the numeral “3” in for “2” in the expression:
to arrive at the well-formed expression:
so can we substitute “Nero” in for “Caesar” in the sentence
Caesar conquered Gaul
to arrive at the well-formed expression:
Nero conquered Gaul
That part of the complex expression that remained invariant under the substitution of number words is a functional expression as originally conceived. Frege’s suggestion in the Begiffsschrift was that that part of the complex expression that remained invariant under the substitution of “Nero” for “Caesar” was similarly a functional expression albeit broadly conceived. Notice that number words and ordinary proper names aren’t themselves functional expressions but are, in Frege’s mature vocabulary complete expressions as are the original complex expressions and the ones arrived at by means of the substitutions. We can understand Frege’s syntactic proposal in the Begriffsschrift as something like the following principle:
Let A[…e…] be a complex complete expression whether a singular term or a sentence and let e be a complete expression that is a grammatical constituent of A[…e…]. Let e’ be a complete expression that is of the same grammatical category as e. That part of the complex expression that remains invariant under the substitution of e’ for e in A[…e…] is a functional expression, A[…( )…].
Once again, the Fregean characterization of incomplete or functional expressions as that which remains invariant under the substitution of complete expressions while preserving well-formedness is only an adequate grammatical characterization of the relevant class of expressions given an antecedent understanding of complete expressions. The adequacy of Frege’s syntactic characterization of functional expressions depends on the persuasiveness of the logico-syntactic analogies that unite the complete expressions as a class.
Having extended the class of functional expressions to include predicates, Frege immediately faces the question, what are the values of these novel functional expressions? Just as syntactic analogies guided Frege’s decision to extend the class of functional expressions to include predicates, semantic analogies between functional expressions as traditionally conceived and predicate expressions guide Frege in his decision as to the values of these novel functional expressions. Consider the result of substituting “3” in for “2” in the complex expression:
to arrive at:
Since the number words aren’t codesignative—they differ in denotation—the denotation of the complex expressions also differ (given the function denoted by that part of the complex expression that remained invariant under the substitution). Notice that just as the number words “3” and “2” aren’t coreferential neither are the proper names, “Nero” and “Caesar”. And just as the denotation shifts as a result of the substation in the arithmetical example, so the truth-value shifts as the result of substituting “Nero” in for “Caesar” in the sentence:
Caesar conquered Gaul.
While the original sentence is true, the sentence resulting from the substitution:
Nero conquered Gaul
is false. This semantic analogy suggests that truth-values are the values of these novel functional expressions. We thus arrive at one of the clarifications that we have sought—just as predicates form a subclass of the class of functional expressions, the denotations of predicates, that is, concepts, form a subclass of the class of functions broadly conceived. In particular, concepts are functions from objects to truth-values.
Let’s pause to consider two commitments of Frege’s identification of truth-values as the values of a concept understood as special kind of function:
If truth-values are the values of concepts, then truth-values are the denotations of sentences
Since the values of functions are saturated entities, that is, objects, truth-values are themselves special kinds of objects
Both of these are controversial commitments, and the second is especially surprising—so much so that an entire genre of philosophical humor has subsequently arisen around the Fregean expression “naming the True”. How can semantic commitments as controversial and counterintuitive as these be made merely on the basis of the semantic analogy that Frege puts forth in “Function and Concept”?
Recall Frege has just suggested that just as “2*2” denotes 4, “2*2 = 1” denotes the False. Similarly, all of the following sentences all denote the same thing, namely the True:
The problem is this: despite the fact that each of these sentences denote the same thing, the True, intuitively they seem to say different things and hence express different thoughts. So how can they really be codesignative?
The problem can be put in the form of a puzzle, an inconsistent set of claims:
The thought expressed by a sentence is its denotation.
The denotation of a sentence is its truth-value.
The sentences “2*2 = 4” and “2 > 1” each denote the True.
The sentences “2*2 = 4” and “2 > 1” express different thoughts.
If the thought expressed by a sentence is its denotation, and the denotation of a sentence is its truth-value, then the thought expressed by a sentence simply consists in the truth-value it denotes. It follows that since the sentences “2*2 = 4” and “2 > 1” each denote the True, they each express the same thought. This however conflicts with the final claim that the sentences “2*2 = 4” and “2 > 1” express different thoughts. We have a contradiction and so claims 1-4 cannot be rationally maintained together. We must reject at least one of these claims on pain of inconsistency. Call this The Puzzle about Truth-Value: How can sentences with the same truth-value differ in cognitive value.
The objection that Frege is presently considering consists in the suggestion that the best way out of this puzzle is to reject the second claim, that the denotation of a sentence is its truth-value. Frege, in effect, responds by suggesting that the inconsistency is just as easily avoided by rejecting the first claim, the claim that the denotation of a sentence is the thought expressed. From a formal perspective, each are equally good ways of avoiding the charge of inconsistency. So why should we prefer Frege’s way out of this puzzle over the objectors’?
Frege goes on to suggest that there is independent reason to distinguish the denotation of a sentence from the thought expressed. The independent reason is provided by a related puzzle, call it The Puzzle about Subsitution: How can sentences that differ only in the substitution of codesignative names differ in cognitive value? (We will return to exactly how The Puzzle about Truth-Value and The Puzzle about Substitution are related when we discuss “Sense and Reference”) The Puzzle about Substitution gives rise to the following argument:
Compositionality of denotation Let e and e’ be expressions belonging to the same syntactic category, and let S[…e’…] be the result of substituting e’ in for at least one occurrence of e in S[…e…]. S[…e…] and S[…e’…] denote the same thing just in case e and e’ denote the same thing.
The names “Morning Star” and “Evening Star” denote the same thing, the planet Venus.
The sentences:
The Morning Star is a planet with a shorter period of revolution than the Earth. (a)
The Evening Star is a planet with a shorter period of revolution than the Earth. (b)
differ only in the substitution of codesignative terms and so denote the same thing (from 1 and 2).
If it is possible to regard one sentence S as true while regarding another sentence S’ as false then S and S’ express different thoughts.
It is possible to regard (a) as true and (b) as false.
(a) and (b) express different thoughts. (from 4 and 5)
Since (a) and (b) denote the same thing and yet express different thoughts, the denotation of a sentence cannot be the thought expressed.
Notice, Frege’s argument does not rely on the controversial claim that the denotation of a sentence is its truth-value. Let the denotation of a sentence be some as of yet determined value. Frege argues that given the compositionality principle, and given the principle that a difference in cognitive value make for a difference in the thought expressed, we have reason to distinguish the denotation of a sentence from the thought expressed—a reason that does not rely on truth values being the values of concepts and hence the denotations of sentences.
How persuasive you think Frege’s supporting argument is, depends of course on how plausible you think it is that sentences have denotations. This point too will be taken up in “Sense and Reference”.
Extensions
So far, we have discussed Frege’s defense of the claim that concepts are functions from objects to truth values. This was one of the major clarrifications needed in order to formally execute the definitions and proof sketches of the Foundations. In addition, Frege needs to clarrify the notion of a concept-extension and the logical principles that govern that notion. Here too, “Function and Concept” makes progress.
Frege begins with an anlogy from analytic geometry:
The method of analytic geometry supplies us with the means of inutitively representing the values of a function for different arguments. If we regard the argument as the numerical value of an abscissa, and the corresponding value of the function as the numerical value of the ordinate of a point, we obtain a set of points that present itself to inutition (in ordinary cases) as a curve. Any point on the curve corresponds to an argument together with the associated value of the function.
Thus, e.g.,
yields a parabola; here “y” indicates the value of the function and the numerical value of the ordinate, and “x” similarly indcates that argument and the numerical value of the abscissa.
So functions of integers cast objectual shadows—corresponding to every such function there is a saturated entity, the graph of a function, that represents that function.
Frege’s notion of a course of values closely follows this pattern. Courses of values are objectual shadows cast by predicative contents—corresponding to every concept there is a course of values that represents that concept. Just as the graph of a function pairs every argument of the function with the value it yields, a course of values pairs every argument of the concept witht the value it yields, in this case, the True and the False. Frege’s notion of a course of values is meant to be a partial explication of the notion of a concept-extension. Intutitively, the extension of a concept includes all and only objects that fall under that concept. Notice, however, this is equivalent to the notion of a course of values. The objects that fall under a concept are the arguments that the concept maps onto the True.
We might be tempted to further explicate the graph of a function as a set of ordered pairs, where the first element of the pair is the value of the abscissa and the second element is the corresponding value of the ordinate. The graph in its totality would be represented by the set of all such pairs. This procedure would have the advantage of generality—it is not limited in its scope to pairs of integers as is the analogy from analytic geometry. In this way we could generalize the notion of a graph of a function to arrive at a conception of Fregean courses of values. But it is important to recognize that such a conception would be in tension with Frege’s thought:
Frege took a dim view of the emerging conception of set as represented in the work of Cantor and Dedekind. Indeed by modern standards their informal exposition of the notions of set or system strike us, as it did Frege, as vague and inconsistent. The notion of set, if at all legitimate, must undergo the same kind of conceptual clarification that Frege undertakes to subject the notion of number. This would involve logical analysis and further mathematical investigation. As such the notion of set, and the set-theoretic representation of ordered pairs, can play no role in the logical analysis of number. The more fundamental point is this. The logical analysis of number required the existence of concept-extensions and logical laws governing them. The notion of set, unsettled as it was by mathematical usage, could play no role in the statement of a logical law. Logical laws are self-evident while the principles governing sets were far from self-evident at least during the time Frege was writing.
The emerging conception of set that resulted in the now standard iterative conception of set, is motivated by intuitions distinct from those motivating Frege’s talk of extensions. There is some evidence that by his Middle Period Frege had adopted an extensional view of concepts: Two concepts were the same just in case they had the same extension. The concept F is the same as the concept G just in case everything that falls under the concept F also falls under the concept G and vice versa. In contrast the intensional logician held that identity of extension is insufficient to determine sameness of concept. Nevertheless, Frege continued to hold with the intensional logician that concepts determine extensions. Hence the Fregean metaphor of the relation between an item in an extension and the concept as that item falling under that concept. Indeed Frege sometimes writes (“Boole’s Logical Calculus and the concept script”) that what unites the items in an extension as a class is their falling under a concept. Frege conceived of extensions as the objectual correlates of predication. Extensions were the objectual shadows cast by possible predicates. It is the contents of predicates, that is, the designated concepts, that determine their objectual correlates. And so it is concepts that determine extensions, even if they are extensionally individuated. In contrast, one of the motivating intuitions of the iterative conception of set is that the elements of a set, those objects that fall under it, determine that set. There is then an important difference between Frege’s purely logical conception of extension and the emerging conception of set.
There are important mathematical differences between sets and extensions as Frege conceived of them. Infamously, Frege’s conception of extension allows for the possibility of extensions that are members of themselves. Given that Frege is conceiving of extensions as objectual correlates of predicative contents, this was no more mysterious to him than a predicate applying to itself. (Consider the predicate “is a predicate”—that predicate truly applies to itself.) However, the iterative conception of set has restrictions on set formation that explicitly disallow this possibility. No set can be a member of itself—not so for Fregean extensions.
What are the logical principles governing courses of values or concept-extensions? Again, Frege is moved by the analogy with analytic geometry. Consider the following equation:
This equation is true quite generally—no matter what value is assigned to “x”, the identity statement is true. Frege takes it that it is undemonstrable fundamental law of logic that if this is so, the functions in the equation will produce identical graphs. If we extend this fundamental law of logic to the case of concepts and courses of values, we get a version of Basic Law V:
Basic Law V is the sole logical principle governing concept-extensions that Frege need appeal to in his formal derivation of arithmetic from logic. So it would seem that we have, to a first approximation at least, the clarrifications that we sought: We now know what concepts are—they are functions from objects to truth-values; we now know what concept-extensions are—they are courses of values, saturated entities that represent the pairing of objects with truth-values; and we have some idea of the logical principle governing them—Basic Law V. I say that we have these clarrifications to a first approximation since the crucial decision to take truth-values as the values of concepts needs further defense—a task to which Frege’s essay “Sense and Reference” is dedicated.
Unfortunately, things have taken a tragic turn. Basic Law V is inconsistent—it is subject to Russell’s paradox. Let R be the extension consisting of all extensions that do not contain themselves as members:
Basic Law V does not rule out the existence of such an extension. Now consider: Is R a member of itself? If R is a member of R, then, according to the definition of R, R is not a member of R. But if R is not a member of R, then it is a member of R, again by its very definition. However, the statements “R is a member of R” and “R is a not a member of R” cannot both be true, and so we have a contradiction.
Russell describes the discovery of the paradox in his autobiography:
At the end of the Lent Term, Alys and I went back to Femhurst, where I set to work to write out the logical deduction of mathematics which afterwards became Principia Mathematica. I thought the work was nearly finished, but in the month of May I had an intellectual set-back almost as severe as the emotional set-back which I had had in February. Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world ought to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavoured to apply it to the class of all the things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies its contradictory. At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case. Burali-Forti had already discovered a similar contradiction, and it turned out on logical analysis that there was an affinity with the ancient Greek contradiction about Epimenides the Cretan, who said that all Cretans are liars. A contradiction essentially similar to that of Epimenides can be created by giving a person a piece of paper on which is written: ‘The statement on the other side of this paper is false.’ The person turns the paper over, and finds on the other side: ‘The statement on the other side of this paper is true.’ It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do? There was something wrong, since such contradictions were unavoidable on ordinary premises. Trivial or not, the matter was a challenge. Throughout the latter half of 1901 I supposed the solution would be easy, but by the end of that time I had concluded that it was a big job. I therefore decided to finish The Principles of Mathematics, leaving the solution in abeyance. In the autumn Alys and I went back to Cambridge, as I had been invited to give two terms’ lectures on mathematical logic. These lectures contained the outline of Principia Mathematica, but without any method of dealing with the contradictions.
Frege received Russell’s letter reporting this paradox when the Basic Laws of Arithmetic was in press. He added a hastily composed appendix concerning the paradox that contained no real solution. Indeed, Frege never discovered an adequate response to Russell’s paradox consistent with his development of logicism.
Summary
In order to complete his logicist program Frege needed to do three things:
Specify the values of the functions that concepts are identified with—truth-values
Specify what concept-exptensions are—courses of values
Specify the logical principle(s) governing concept-extensions—Basic Law V
It is Frege’s identification of truth-values as the values of concepts that leads him to draw the sense/reference distinction.
