ROBERT RAY

ARE TRUTH

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OBJECTS?

(Received 30 August, 1977) In at least two articles, Gottlob Frege seems to claim that there are two objects which are signified (bedeuten) by certain sentences. Frege calls these objects 'the true' and 'the false'. Since it is contrary to ordinary intuitions that there are such objects, it would seem that there is a need to justify such a claim. However, those passages in which Frege seems to attempt to justify such a claim, e.g., '[Jber Sinn und Bedeutung', pp. 3 2 - 3 5 , 1 seem not to yield sufficient justification for this claim. One philosopher, E. Tugendhat, 2 interprets some o f these statements o f Frege's in such a way that they would not commit Frege to the existence of these objects. Michael Dummett 3 gives good reasons for not accepting Tugendhat's interpretation as a good interpretation o f Frege, but he concludes: Whether a given thought is true or false is, of course, an objective matter: but that the truth-value of the thought is itself an entity which is one of the components of reality is a conception which is held merely in order to complete the analogy accordifig to which the referents of all expressions are extratinguistic correlates of those expressions, belonging to the real world, and does not appear to be a thesis which has any significance on its own. (p. 413) Both Dummett and Tugendhat seem to conclude that Frege's thesis that truth values are objects which are signified by certain sentences is an assumption which was unjustified even for Frege. In this paper I wish to show that Frege's thesis was one o f several assumptions which led Frege to a complex semantic theory for the first order predicate calculus which is surpassed only by Tarski's truth and satisfaction definitions. 4 As such, this thesis receives its justification by being an essential part of a theory which as far as Frege knew was an adequate semantic theory for his version o f the first order predicate calculus. This justification, now, o f course, would not support the view that there are these objects. The reason for this is that Tarski's truth definition gives us a way of providing a semantics for a class of sentences which properly includes the class of sentences for which Frege's semantics is adequate without the need to assume the existence o f truth values as objects. Philosophical Studies 35 (1979) 199-211. 0031-8116/79/0352-0199501.30 Copyright 9 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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It seems to me that Dummett and, perhaps, Tugendhat arrive at their conclusions by trying to justify Frege's theses about language on the basis of our current knowledge of the properties of language. It also seems to me that this can lead to misrepresenting Frege's position and to overlooking justification that Frege may have felt he had for his theses. It seems, in this instance, that Frege thought that the way that language was related to the world so that we could gain knowledge of the world from language was that simple linguistic expressions signified extralinguistic entities and that complex linguistic expressions signified combinations of the things signified by the simple parts of the complex linguistic expressions. Such a theory seems false, or, at least, in great need of justification and elucidation, to many philosophers today. However, if one approaches Frege's work by assuming that he was working under the constraints of such a view, then Frege's problems appear in a different light. In particular, one may assume that the major task facing Frege was finding entities which various linguistic expressions could signify and which would "combine" to form entities for complex expressions to signify. These entities also had to be such that they would allow Frege to provide an adequate semantics for the formal language which he developed. While Frege was not completely successful in finding entities which would enable him to carry out the above task, his assumption that sentences signify truth values and that concepts are functions which have truth values as values enabled Frege to provide a semantics for a certain class of sentences which far surpassed any previous semantics and, in conception, at least, came quite close to the current paradigm of a semantic theory, viz., Tarski's truth definition. In order to indicate how these assumptions are essential to Frege's conception of this semantics, I will (1) sketch Frege's semantic theory from the Begriffsschrift, s (2) indicate the important shifts in theory which led to Frege's theory in 'Function und Begriff', 6 (3) develop a "definition of signification and designation" based on the semantic theory of 'Funktion und Begriff' and (4) examine the adequacy of the definitions of these notions. In the final section, I will compare Frege's assumptions to certain assumptions Tarski makes in his "definitions of truth and satisfaction" which are analogous to Frege's assumption of the existence of truth values as objects.

I

The basic part of Frege's semantic theory from the Begriffsschr~ft is contained in the following quote.

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(1) A judgment is always expressed with the help of the sign F which stands to the left of the sign or combination of signs which indicate the content of the judgment. If the small vertical stroke on the left end of the horizontal stroke is omitted then this will transform the judgment into a mere combination o f ideas of which the author does not express whether he acknowledges its [viz., the combination of ideas'] truth or not. Let, e.g., FA signify (bedeute) the judgment, 'opposite magnetic poles attract each other', then -A will not express this judgment, but solely is supposed to call forth in the reader the idea (Vorstellung) of the mutual attraction of unlike magnetic poles, perhaps, to draw conclusions from it and to test the correctness of the thought on these. We paraphrase in this case by means of the words, 'der Umstand, dass', or 'der Satz, dass'. Not every content can become a judgment by placing Fbefore its sign, e.g., not the idea 'house'. We distinguish, therefore, judgment and non-judgment contents. The horizontal stroke from which the sign F is formed combines the signs following it into a whole, and the affirmation, which is expressed by means of the vertical stroke on the left end of the horizontal [stroke], relates to the whole. The horizontal stroke may be called the content stroke; the vertical stroke may be called the judgment stroke. The content stroke besides also serves to place whatever signs at all in relation to the whole of the signs following it. What follows the content stroke must always have a judgment content. 7 (pp. 1 2) The semantic t h e o r y that is suggested by this q u o t e is the following. Certain linguistic expressions signify or express extralinguistic objects. Sentences signify a ' c o m b i n a t i o n ' o f entities. I have argued elsewhere ~ that in general the 'ideas' signified by the signs usually are things such as tables, chairs, and numbers rather than subjective mental entities. In that same paper, I also argued that 'to signify' is the best translation of ' b e d e u t e n ' and that the objects signified by the signs are the c o n t e n t s o f the signs. Since talk a b o u t one sign c o m b i n i n g other signs is at best metaphorical and since sentences p r e c e d e d by the horizontal stroke signify a c o m b i n a t i o n o f entities and for other reasons discussed later, it is reasonable to conclude that the horizontal stroke signifies a c o m b i n i n g e l e m e n t and that in the last paragraph o f q u o t e (1), Frege is sliding b e t w e e n a discussion o f the signs and a discussion o f the c o n t e n t o f the signs. If this is the case then the following are suggested by q u o t e (1). Placing a horizontal stroke in front o f a string o f symbols yields a symbol that signifies the c o m b i n a t i o n o f the ideas that are signified by each o f the symbols in the string c o m b i n e d by the e l e m e n t signified by the horizontal stroke. Placing the vertical stroke on the left end o f the horizontal stroke which is in front o f the string o f symbols yields a s y m b o l which

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signifies the combination of the combination of ideas with affirmation, i.e., the judgment, It is clear also from quote (1) that the mere combination of ideas which are signified by sentences prefixed with a horizontal stroke without the vertical stroke are the primary bearers of truth and falsity. The notions of truth and falsity can be extended to sentences as follows. A sentence is true if and only if it signifies a true combination of ideas, and it is false if and only if it signifies a false combination of ideas. See the paper cited in footnote 8 for further details and arguments for this interpretation of Frege's semantic theory in the Begriffsschr(ft.

II

The most important notion that Frege considers in 'Funktion and Begriff' is the notion of 'function'. In the Begriffsschrift, Frege had already extended the notion of mathematical function to cover predicates of his symbolic language. However, he held that the linguistic expression was the function. (2) If a simple or compound sign occurs in one or more places in an expression, whose content need not be judicable, and we consider it replaceable in all or one of these positions by other [signs] - everywhere, however, by the same sign - then we call the part of the expression which appears unchanged in this 'a function', [and we call] the replaced [part] its argument. (p. 16)

Frege's first major shift in 'Funktion und Begriff' was to maintain that the function was what the expression designated (bezeichnen). 9 (3) Thus, if one wants to know what was first meant by the word 'function' in mathematics, one goes back to the time of the discovery of higher analysis. One perhaps would receive as an answer to this question: 'by a function of x, an expression of calculation, a formula, which included the letter x , was meant ...'. This answer cannot be satisfactory, because the form and the content, the sign and the designated, are not distinguished. [This isl a mistake which one certainly encounters now in mathematical writing by even noted writers ('Funktion und Begriff', p. 2.)

Treating functions as something which an expression designates is part of what allows Frege to clarify the notion of 'combination' in the claim that complex expressions signify a combination of the entities signified by the simple parts of the expression. It seems fairly clear that in the Begriffsschrift general expressions and relation expressions signified concepts and relations. The basic change made from the Begrfffsschrift to 'Function und Begriff' is that in the latter work, Frege maintains that concepts and relations are functions which have as values only the two truth values. The reason that truth values had to be components o f

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reality for Frege was that they were to be values and arguments of functions. However, this is not evidence for the claim that there are truth values which are components of reality. To see the evidence for this claim, we need to see the importance of there being functions which have truth values as values. Part of the importance of this will be seen next. It is not explicit in the Begriffsschrift what the logical connectives are supposed to signify, or even that they are to signify anything at all. Of course, if two sentences differ in truth value, it is clear that according to the theory in the Begriffsschrift, one of the sentences would have to signify a different combination of ideas from the other. Thus, if two sentences differed in truth value, either their parts would have to signify different ideas or there would have to be something in one of the sentences which would indicate that it signifies a combination of the same ideas which is different from the combination of these ideas signified by the other sentence. In the case of negation, for instance, one would expect the negative particle to signify something. In fact, one would expect that the spatial or temporal order of the semantically relevant linguistic expressions in the sentences determines the kind of combination and that each unit signifies or designates something. It becomes plausible that Frege held such a theory as one examines the theory of 'Funktion und Begriff'. As with most of the logical connectives, it is not clear from the Begriffssehrift whether the horizontal stroke, ' - ' , is supposed to signify something or whether it just combines with a string of symbols to form a symbol which signifies a certain combination of ideas. The following things suggest that Frege thought that the horizontal stroke signifies a combining element. One might speculate that the last paragraph in quote (1) involves some use-mention confusions and that Frege did not (only) intend that the horizontal stroke was to combine the signs following it into a whole but that he (also) intended that the horizontal stroke was to signify something which would combine the things signified by the other signs into a whole. In 'Funktion und Begriff', due to Frege's new conception of 'function' and due t o Frege regarding truth values as ob/eets which could be values and arguments of functions, Frege was able to find an entity for the horizontal stroke to designate. (4) l i n t r o d u c e -x as such in t h a t I s t i p u l a t e t h a t t h e value of the f u n c t i o n is to be the true w h e n the true is t a k e n as a r g u m e n t ; on the o t h e r h a n d , in all o t h e r cases, the value o f this f u n c t i o n is the

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false; hence, it is this when the argument is the false, as well as when it is not a truth value. ('Funktion und Begriff', p. 21.) It should be noted that a combining element does seem to appear in ' F u n k t i o n und Begriff' in that functions are incomplete or unsaturated ( ' F u n k t i o n und Begriff', p. 6). 1~ Frege's conception o f 'function' and his regarding truth values as objects also enabled Frege to make a great advance in his treatment of negation. In the Begriffsschrift, Frege explained negation as follows. (5) If a small vertical stroke is affixed to the underside of the content stroke then the circumstance that the content does not obtain is to be expressed therewith. Thus, e.g., ~-r-A signifies '.4 does not obtain'. I call this small vertical stroke the negation stroke. schrift, p. 10)

(Begriffs-

Nowhere in the Begriffsschrift are we told what the negation stroke itself signifies if anything. In ' F u n k t i o n und Begriff', we are told (6) The next simplest function may be the one whose value is the false for just the arguments for which the value of - x is the true and, conversely, whose value is the true for the arguments for which the value of - x is the false. I designate it, thusly, -Vx I call the small vertical stroke, the negation stroke. ('Funktion und Begriff', p. 22.) The case is similar for the conditional sign. It is not clear in the Begriffsschrift what the conditional sign signifies, if anything; Whereas in ' F u n k t i o n und Begriff' this sign designates a function. The conditional sign

IyX designates a two-place function whose value is the false when the y-argument is the true and the x-argument is not the true and whose value is the true in all other cases ( ' F u n k t i o n und Begriff', p. 28). Frege's treatment of the universal quantifier signs is somewhat unclear. In 'Funktion und Begriff', he first introduced the symbol '-~r-F(a)' as follows. "I mean then by the sign ' @ - F(a)' the true, if the function F(x) always has the true as value whatever its argument may be; in all other cases it is to signify the false" ( ' F u n k t i o n und Begriff', p. 23). Later he said of ' q - ~ F(a)', "Now, just as in x 2 we have a function whose argument is indicated by 'x', I also conceive of ' - r ~ F(a)' as an expression of a function whose argument is indicated by ' F ' " ( ' F u n k t i o n und Begriff', p. 26). This suggests that ' @ - F ( a ) ' designates a second order or level function which takes the value

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the true for those unary first order functions which have the true as value for every argument and which takes the value the false for all other unary first order functions. Since Frege had only considered unary functions up to and including this part of 'Funktion und Begriff' - his discussion of the conditional came later - Frege did not say what the value of these second order functions is for arguments which are binary functions. As we shall see in Section IV, this is one reason his treatment is inadequate. 11 It is clear from this that one very good reason Frege had for treating truth values as objects was that it enabled him to specify clearly what the toNcal connectives designated or stood for in the world. The advantage that he gained, however, was not simply to clarify what the logical connectives signified. The 'combining' properties of functions are such that Frege's selection of functions to be the designations of the logical connectives yielded a great insight into the semantics of sentences.

iii Functions have two 'combining properties' which arc important for Frege's new semantics. They take objects and functions as arguments and yield the same or other objects and functions as values, e.g., the function l + x yields the value 3 for the argument 2. Also, some functions compose with other functions to yield the same or different functions, e.g., 1 + x composes with x - y to yield the function 1 + ( x - y ) . In order to show the importance of these two properties. I wilI characterize a certain class of sentences in Frege's artificial language and then "defines the notions of signification and designation" for the complex signs. The importance of these combining properties of functions will be seen in the use that is made of them in these 'definitions'. The importance of treating truth values as objects will also be clear. While f Frege may not have had a clear conception of the need to r these terms, it seems clear to me that he was aware o f the importance revealed in these definitions of the combining properties of functions. It is useful to divide the signs in a Fregian language into the categories: primitive proper name; primitive predicate symbols of degree n; free variables, x ,. s x . , x. . ...... bound a. , a. , a ..... lo gical symbols, - , -r- ,-E , . . . variables, . . . a a . a.r- , -vr- , ..., primitive predicate symbols with free variables; atomic formulas and logically complex formulas. Except for the last two categories, these are pair wise disjoint categories. I will assume that there are objects in these 9

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categories. I will assume also that we have characterizations for the primitive proper names, the primitive predicate symbols, the free variables, the bound variables and the logical symbols. The primitive predicate symbols with free variables may be characterized as follows. x is a primitive predicate symbol with free variables if and only if x is a primitive predicate symbol of degree n followed by n occurrences of free variables. The atomic formulas may be characterized as follows. x is an atomic formula if and only if x is a primitive predicate symbol with variables, or x is the result of replacing all occurrences of some one free variable in an atomic formula by (1) a primitive proper name, or (2) an atomic formula, or (3) a logically complex formula. The logically complex formulas may be characterized as follows. x is a logically complex formula if and only if (1) there is a y such that y is an atomic formula or y is a logically complex formula and x = the concatenation of '--' or '-1' with y, or (2) there is a y and a z and a w such that y is an atomic formula or y is a logically complex formula, z is a free variable which occurs in y , w is a bound variable which does not occur in y, and x = the concatenation of a quantifier in which w occurs with the result of replacing some and all occurences of z in y by w, or (3) there is a y and a z such that y and z are either atomic or logically complex formulas and x = the result of replacing ' P ' and 'Q' in '-12~2' by y and z respectively. In the following 'definition of signification and designation', I will assume that the significations and designations of the primitive proper names, the primitive predicate symbols, the logical connectives and the primitive predicate symbols with free variables have been given. The semantics of the atomic formulas exactly parallels the semantics for the logically complex formulas and it has the same difficulties. These difficulties will be pointed out in the next section. I will assume that the semantics for the atomic formulas has been given and I will classify the atomic formulas along with the other kinds of expressions just mentioned. They wilt all be called 'primitive expres-

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sions'. I will call anythirtg that is either a primitive expression or a logically complex expression a 'significant expression'. The 'definition of signification and designation' is the following. I f x is a significant expression then (A) (1) (2)

(3)

(B) (1) (2)

x signifiesy if and only if x is a primitive expression and x signifies y ; or there is a z and a u such that z and u are significant expressions which signify objects and (a) if x = the concatenation of ' - ' (or ' T ' ) with z then y = the value of the function designated by ' - ' (or ' T ' ) for the argument signified by z, and (b) if x = the result of replacing ' P ' and 'Q' in ' T ~ ' by z and u respectively then y = the value of the function designated by T y for the y-argument the signification of u and the xargument the signification of z; or if there is a z such that z is a significant expression and x -- the concatenation of a quantifier with the result of replacing some and all occurrences of some one free variable in z by a bound variable which occurs in the quantifier but not in z and the value of the function designated by the quantifier for the argument designated by z is an object t h e n y = this object. and x designates y if and only if x is a primitive expression a n d y is the designation of x; or there is a z and a u such that z and u are significant expressions and z signifies an object and u designates a function and i f x = the result of replacing ' P ' and 'Q' in 'q5~2' by z and u respectively then y = the composition 12 of the function whose course of values is {: <, v} is a member of the course values of the function designated by ' qSyx'} with the function designated by u; or

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(3)

there is a significant expression z and a significant expression u such that z designates a function and u signifies an object and (a) if x = the concatenation of ' - ' (or ' T ' ) and z then y = the composition of the function designated by ' - ' (or ' T ' ) with the function designated by z, and (b) if x = the result of replacing 'P' and '(2' in '-E ~2' by z and u respectively then y = the composition of the function whose course of values is {(w, v}: ((the signification of u, w}, v} is a member of the course of values of the function designated by ' T f ' } with the function designated by z, and (c) if x = the concatenation of a quantifier with the result of replacing some and all occurrences of some one free variable in z by a bound variable which occurs in the quantifier but not in z and the value of the function designated by the quantifier for the argument the designation o f z is a function t h e n y = that function;

(4)

there is a z and a u such that z and u are significant expressions and z and u designate functions and if x = the result of replacing 'P' and 'Q' in '-E~2' by z and u respectively then y = the 'composition' of the function designated by -Ey ,with the functions designated by z and u.

or

IV

While Frege's idea for the semantics of his artificial language was very ingenious and successful for certain sentences of his language, there are some basic difficulties with his idea. These difficulties arise with clauses (B) (3) (c) and (B) (4) and with what the quantifiers are supposed to'designate. It is these difficulties that I wish to consider now. The difficulty with (B) (3) (c) is the following. Suppose that there are only natural numbers, functions and Frege's language. Then, presumably, 'x is less than or equal to y ' and 'y is less than or equal to x ' would designate the same function ('Funktion und Begfiff', p. l 1). However, '@-x is less than or equal to a' and '-t~-a is less than or equal to x' would designate different functions,

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OBJECTS'?

i.e., the first would have the true as value for the argument 0 and the second would have the false as value for the argument 0. According to (B) (3) (c) each of these designates the value of the function designated by ' - ~ - F ( a ) ' for the argument designated by 'x is less than or equal to y ' , i.e., the function designated by 'y is less than or equal to x'. Since functions only have one value for each argument, the two quantified expressions should designate the same thing, t3 The difficulty with (B) (4) is the following. The composition of a unary function, i.e., a function whose course of values is a set of ordered pairs and not a set of 3 tuples, with an n-ary function can be defined fairly easily. However, it is not clear how one is to make sense of the notion of composition when it is applied to a binary function and two other functions so that the following problem is taken care of. 'f(g(x), h(y))' supposedly designates a binary function and 'f(g(x), h (x))' would designate a unary function. According to Frege, 'h(x)' and ' h ( y ) ' would designate the same function. Thus, it would seem that 'f(g(x), h (y))' should designate the same function as 'f(g(x), h(x))', but it does not. These two difficulties can be brought out more fully by the following example. 'x = 1' and 'y = 1' designate the same function. Thus,'-ExX 11' and ' T ~ = I ' designate the same function. However, ' -~rCa a a = I' signifies the true, while ' - ~ = I' does not signify anything. It designates a function, t4 ~

CONCLUSION

Despite the difficulties which have been pointed out, Frege's ideas for the semantics of his formal language was a great advancement in semantics. His semantics, for instance, is adequate for the propositional calculus and for some sentences of the first order predicate calculus. Two important assumptions ,mere made in arriving at this semantics. One was that functions are designated by certain expressions. The second was that there are two objects, viz., truth values which are signified by certain sentences. The fact that these two assumptions are central to Frege's semantics and the fact that Frege's semantics is adequate for many sentences of his language is good evidence that there are these two objects which are signified by certain sentences. However, Tarski was able to give definitions of truth and satisfaction' for the first order predicate calculus which (1) are not committed to there being truth values that are signified by sentences, and (2) give a different account

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o f the s e m a n t i c s o f t h e s e n t e n c e s Frege's s e m a n t i c s is a d e q u a t e for, a n d (3) give a way o f p r o v i d i n g a s e m a n t i c s for s e n t e n c e s t h a t give rise to t h e difficulties in sec. IV w h i c h is a d e q u a t e to o v e r c o m e the difficulties. This o v e r c o m e s t h e e v i d e n c e t h a t Frege's w o r k gives us for t h e e x i s t e n c e o f these entities.

However,

Tarski's t h e o r y

is c o m m i t t e d

to t h e r e

being infinite

s e q u e n c e s o f objects w h i c h satisfy s e n t e n t i a l f u n c t i o n s . Tarski's success gives us e v i d e n c e t h a t t h e r e are these s e q u e n c e s j u s t as Frege's success gave h i m e v i d e n c e t h a t t h e r e were t r u t h values. T h u s , it seems to m e t h a t Frege's thesis is no more a thesis w h i c h has n o significance o n its o w n than Tarski's thesis t h a t t h e r e are i n f i n i t e s e q u e n c e s o f objects w h i c h satisfy s e n t e n t i a l f u n c t i o n s is a thesis w h i c h h a s n o significance o n its own.

General Applied Science Labs, Inc. (Westbury, N. Y.)

NOTES Frege, Gottlob: 1892, '~lber Sinn und Bedeutung', Zeitschrift fiir Philosophie und Philosophische Kritik, C. pp. 2 5 - 5 0 . 2 Tugendhat, Ernst: 1970, 'The meaning of 'Bedeutung' in Frege', Analysis XXX, pp. 177-184. Dummett, Michael: t973, F~ege: Philosophy of Language (Harper and Row, New York). 4 Tarski, Alfred: 1956, 'The concept of truth in formalized Languages', Logic, Semantics, Metamathematics, ed. and trans, by J. H. Woodger (Oxford University Press, London), pp. 152-278, Section 3. s Frege, Gottlob: 1879, Begriffsschrift, eine der arithmetischen nachgebildet Formalsprache des reinen Denkens, Halle a/S. 6 Frege, Gottlob: 1891, Funktion und Begriff: Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft ffir Medicine und Naturwissenschaft (Herman Pohle, Jena). 7 I compared reprints of Frege's work to the translations in Geach and Black, Translations from the Philosophical Writings of Gottlob Frege (Blackwell, Oxford, 1966), and retranslated the quoted passages if I thought the translations needed improvement. 8 'Frege's difficulties with identity', Philosophical Studies 31 (1977), pp. 219-234. 9 Frege uses the word 'bezeichnen' to signify the relation between a function expression and a function in 'Funktion und Begriff' and in Grundgesetze derArithmetik. This also is the word he uses in quote (3) where he says that taking a function to be an expression confuses form and content. As far as I know in the works that he had published during his lifetime, he did not use the word 'bedeuten' to express this relation. In 'Ausffihrung fiber Sinn und Bedeutung' from the Nachlass, however, he does reverse the use of these terms; he uses 'bezeichnen' to signify the relation between a proper name and an object and 'bedeuten' to signify the relation between a function expression and the function. Frege may have a reason for using two distinct terms. In the writings he had published during his lifetime, 'bedeuten' may designate a two place function which takes objects only as arguments. 'Bezeichnen' would stand .for a two place

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function which takes objects for its first a r g u m e n t and first level functions for its second argument. If Frege had this in mind, he was not too firm on it, for there are places where he seems to suggest that certain expressions designate objects, e.g., ' F u n k t i o n u n d Begriff', F o o t n o t e 7. As is indicated in the sentence before last by 'stand for', new words are needed as we go up levels. This feature of Frege's semantics will n o t be relevant b e y o n d the first two levels for m y purposes. 10 Frege, in ' F u n k t i o n und Begriff', p. 17, even speaks of function symbols as being u n s a t u r a t e d and as n o t having a complete sense. The former corresponds to the talk in the 'Begriffsschrift' o f the horizontal stroke combining the signs into a whole. ~1 In the Grundgesetze der Arithmetik, Section 21, Frege suggests that these f u n c t i o n s do not take binary functions as arguments. The suggestion seems to be that "@-F(a, x)' signifies a different function from "~..arF(a)'. However, this move would complicate Frege's semantics considerably. For this reason, I will ignore it. 12 ' C o m p o s i t i o n ' is defined in Halmos, Paul R.: 1960, Naive Set Theory (Princeton), m 40, for functions in the set theoretical sense, i.e., what I am referring to here as 'the course of values o f the function'. See Wells, R u l o n S.: 1951, 'Frege's ontology', The Review of Metaphysics IV, Section 11, pp. 5 4 0 - 5 4 6 , for the identification of course o f values with sets of ordered n-tuples. I am using the usual reduction of an n + 1-tuple to an ordered pair whose first element is an n-tuple. It also should be noted that I am not identifying functions with their courses of values. Courses o f values are objects for Frege; they are n o t functions. However, in so far as it is legitimate to talk about Frege's functions being identical, they are identical if and only if their course of values are identical (see 'Ausffihrung tiber Sinn u n d Bedeutung', Schriften zur Logik und Sprachphilosophie ed. by Gottfried Gabriel (Hamburg, 1971), p. 25. 13 It is to be noted that this difficulty does not result from the simplification made in Note 4. If the simplification were n o t made, we would be concerned with one function which takes binary functions only as arguments. 14 The discussion in this section is of an historical position. The difficulties I point o u t here can be overcome in several ways if one is willing to give up certain of Frege's claims. One m a y follow Richard Montague, e.g., in 1974, 'The proper t r e a t m e n t o f quantification in ordinary language', Formal Philosophy, Selected Papers of Richard Montague, ed. by T h o m a s o n , R i c h m o n d (Yale University Press, New Haven), and give up the claim that the logical connectives signify something. Or, starting with functions from variables to objects, one m a y define concepts as functions from these functions to t r u t h values, and by giving up the claim that negation and the conditional signify t r u t h functions, i.e., functions from t r u t h values to truth values, one can go on to maintain that negation and the conditional signify functions from concepts to concepts.