Crítica, Revista Hispanoamericana de Filosofía, Volume 1, number 1, enero 1967
Dos problemas en la doctrina de Frege
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Thomas M. Simpson
Consejo Investigación Científica
Buenos Aires
Abstract: The problems discussed pertain to the two channels of the work of Frege: that of mathematical logic and that of semantic analysis. Part I points out difficulties standing in the way of a univocal interpretation of the concept of “range of values” (Wertverlauf) of a function, a basic concept in Frege’s definition of natural number. The examination of some interpretations and in particular the analysis of the function ∩ permits displaying the peculiar character of Frege’s notions. Part II is concerned with the theory of declarative sentences as proper names of truth values and shows that infinitely many entities satisfy the formal requirements adduced by Frege for the denotata of sentences.
I
Notwithstanding its importance, the concept “range of values” is not clear in the texts of Frege. The expression “range of values” adopted here is unfortunate, for it suggests the identification of Wertverlauf with the domain (codominio) of the function, constituted by the values that this latter puts into correspondence with its arguments. In volume I of Grundgesetze der Arithmetik, (§3), Frege establishes the criterion of identity that the Wertverläufe must satisfy. Symbolizing, in Frege’s notation, the valueranges of Φ(ξ) and ψ(ξ), respectively, by ἐΦ(ɛ) and ἀψ(α), this condition of identity stands expressed by the formula:
(1) (x) (Φ(x) = ψ (x)) ≡ (ἐΦ (ɛ) = ἀψ(α))
which expresses the basic law V of the system of the Grundgesetze. That the basic law V contains the sign of identity and not that of propositional equivalence is owing to that, according to Frege, declarative sentences are names of the True and the False. With identity in place of equivalence, the basic law V establishes that “(x) (Φ(x) = Ψ(x))” denotes the same truth value as “ἐΦ(ɛ) = ἀψ(ά)”, whether this be the True or the False. But this is what (1) establishes, namely: the valueranges of F(x) and Y(x) are identical if and only if these functions have the same values for the same arguments. As long as the intelligibility of our subject is not affected, the basic law V will be identified with (1). The view of Birjukov, according to which a range of values in Frege is what “usually is called the graph of the function ...”, is examined, and the opposing view of Angelelli, who observes that in Funktion und Begriff the graphic representation is considered to be a mere intuitive image, is commented.
But it is only fair to point out, in support of Birjukov, that other interpretations are more objectionable notwithstanding that they are suggested by texts of Frege. For example: in speaking of concepts, Frege suggests the identification of valueranges with extension in the ordinary sense. (See Grundlagen der Arithmetik —the Definition of Natural Number—, Funktion und Begriff, (p. 30); and Grundgesetze (I, §3). Still, it would be a mistake to base oneself on these texts to interpret the range of values as, in general, the set of arguments that satisfies the function. Among other reasons, his interpretation ought to be rejected as being inapplicable to nonpropositional functions such as x2, capital x, sen x, etc.
In Frege, the terms “concept”, “monadic propositional function” and “property” are interchangeable, as are likewise the expressions “x satisfies the monadic propositional function F(x)” and “x comes under the concept F(x)” and “x possesses the property F(x)”. In the case of this type of functions (concepts), one may speak of the set of objects that satisfy them, but it is meaningless to say that an object satisfies the function x2, for the result of applying this function is not a truth value. And since the concept of Wertverlauf is general and applies to all functions (propositional or not), this alternative is invalid. On the other hand, it does make sense to speak of the set of couples whose first member is a possible argument and whose second member is the result of the function x2 for this value of x. This interpretation has the advantage of greater generality and coincides with the ordinary mathematical concept of a function in extension.
Other alternatives can be eliminated. Thus the Wertverlauf cannot be the set of all the arguments, for in this case all the functions of level 1 (that is, those applicable only to “objects” which in Frege are distinguished from functions themselves), would have the same Wertverlauf, since the domain of the arguments of any function (monadic) of level 1 is universal. Therefore, this interpretation does not even satisfy the equivalence (1). Nor can the Wertverlauf be the domain of the function because all concepts possess the same domain.
Although Birjukov’s interpretation satisfies the criterion of identity (1) and can be applied to all functions, there remains the problem of establishing a relation between the range of values and the extension in the traditional sense. Frege does not clarify the point and one can only conjecture.
In order to rectify the omission, Birjukov attempts to show (p. 23) that in the case of concepts it is possible to construct a onetoone correspondence between the valueranges (as a set of couples) and the extensions in the traditional sense (a set of objects that possess a common property). The construction of this correspondence is based on the supposition that all concepts have the same domain of arguments; but it is only valid for concepts of level 1. Nevertheless, the correspondence could be maintained if it were repeated on all the levels in a way analogous to that proved for level 1.
The peculiar character of valueranges is manifested in relation to the functional sign “∩”, whose meaning could be confused with the usual sign for membership in a class. Frege introduces the function by means of the identity,
(2) ψ (x) = x ∩ ἀψ (α),
that seems to be no more than a disguise for the formula
(3) ψ (x) . ≡ . xεẑ (Ψ) (x)),
with which Russell introduces the relation of membership ε; “ẑ(ψ)” designates the extension of ψ, just as as “ἀψ (α)” designates its range of value. But in (2) the identity cannot be substituted by the equivalence, for, unlike what happens in (3), ψ may be a nonpropositional function. Cases in which ψ is a concept and in which it is not are examined.
II
The most disconcerting feature of Frege’s semantics is found in his conception of sentences as proper names of truth values. Two aspects ought to be distinguished: (i) that the sentences are names; (ii) that the objects named by them are the True and the False. Frege assumes (i) and presents (ii) as the result of that hypothesis —which is in agreement with Birjukov’s interpretation (p. 94, n. 31).
In order to establish (ii), Frege ought to prove (a) that the relation of declarative sentences with truth values is governed by the same principles that govern the relation of proper names with their denotata and (b) that there are no other entities that have this type of relation with declarative sentences. In its turn, this proof ought to make (i) plausible.
The principles used by Frege are:
(1) When a name constitutive of a compound name has no denotation, neither has the compound name: “(predecessor of 0) + 1”. (2) When a name constitutive of a compound name is substituted by another that has the same denotation, the denotation of the compound name does not change (although the sense may change). Admitting (1) and (2), Frege asks himself what the object denoted by a declarative sentence can be. This object ought to satisfy principles (1) and (2). The crux of Frege’s argument consists in showing that:
(1′) When a name constitutive of a sentence has no denotation, the sentence has no truth value.
(2′) When a name constitutive of a sentence is substituted by another that has the same denotation, the truth value of the sentence does not change (although the sense may change). (1′) is supported by examples like “Odysseus was thrown onto the beach of Ithaca while he slept”. But this type of example is not sufficient to establish (1′), for the negative existentials of the form “The such and such does not exist” are true if and only if the grammatical subject lacks a denotation. As for (2′), it is a matter of the principle of interchangeability salva ueritate.
The parallelism between (1)–(2) and (1′)–(2′) is so complete that (1′) and (2′) seem like reformulations of the principles for the special case of declarative sentences. This parallelism proves —with the reservation noted with respect to (1′)— that truth values satisfy the requirements imposed for any type of objects that one may desire to postulate as the denotation of sentences. But Frege believes he has proved something more: that the truth value constitutes the reference of a sentence (pp. 63–64).
Birjukov believes that this is a necessary consequence. It might be said that the difficulty of finding another candidate proves nothing. But it can be shown that the relations between the denotata of the parts and of the whole that principles (1) and (2) put down, are also fulfilled if we consider as denotata of the sentences their respective equivalence classes.
We will begin with principle (2). Since the equivalence class of a sentence A is simply the class of all sentences that have the same truth value as A, Leibniz’ law assures that this class remains invariant under substitutions of terms having the same denotation. As for principle (1), the question can be formulated in this way: for the equivalence class of a sentence A to exist, it is a necessary and sufficient condition that A be true or false: therefore, if a name constitutive of A has no denotation, then (by virtue of 1′) its equivalence class does not exist. This turns out to be natural in the context of Frege’s theory, for if A is neither true nor false, then the description “The truth value of A” does not denote, and consequently (in accordance with principle 2) neither does the compound name, “The class of all sentences whose truth value is the same as the truth value of A”, have denotation. It follows that such a class does not exist, just as the number that results from adding 1 to the predecessor of 0 does not exist.
Starting from this, the possible denotata of A are now infinite: the unit class whose only element is the equivalence class of A also satisfies principles (1) and (2), and, in general, so does any member of the infinite succession {CA}, {{CA}}, {{{CA}}}, ..., where “CA” symbolizes the equivalence class of A.
[Traducción de J. Schönberg]
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