Frege: Una estipulación viable
Authors: Valdivia Lourdes 
Published: March 2020

These two objects [The True and the False] are recognized, if only implicitly, by everybody who judges something to be true —and so even by a sceptic. (Frege, G., “On Sense and Reference”, Trans. Geach & Black, p. 63.)
One of the most striking Fregean theses is the claim that declarative sentences (if true or false) are proper names of truth-values (I call this thesis (TP) for short). I here analyse Frege’s reasons for (TP) following Simpson’s reconstruction. Simpson has shown that Frege’s argument (if he has one) in favour of (TP) is not conclusive. But if (TP) cannot be established then other important Fregean claims, general semantic theses, fail as well. Thus, the aim of my paper is to introduce the aforementioned (TP) via a stipulation. The stipulation is not a merely ad hoc solution but is based on three main considerations: (i) Frege’s conjecture as it stands is either in-conclusive or lacks supporting argument. (ii) There is a well established practice in Mathematics that allows stipulations whenever they are needed in the theory. And finally, (iii) Frege’s notion of naming (namingF) is a theoretical concept which admits of an explication in Carnap’s sense. According to Frege the main function of language is that of naming. Thus semantically significant parts of language are names. Language is partitioned into types: names are either complete or incomplete. Incomplete names are function names which include function expressions (in a Mathematical sense), predicates, connectives and quantifiers. Complete names are called proper names and include: (1) grammatically ordinary proper names (genuine names in Frege’s terminology); (2) what we would nowadays call definite descriptions; (3) mathematical terms which describe or name numbers, like ‘2’, ‘2+5’; and (4) declarative sentences. The Proper names (1), (2) and (3) satisfy the two following semantic principles reconstructed by Church from Frege’s article “On Sense and Reference”:
(I) When a constituent name of a compound name lacks denotation the compound name does not denote either.

(II) When a constituent name of a compound name is replaced by another name having the same denotation, the denotation of the compound name does not change. (But sense may change.)
According to II, the compound name ‘The wife of Ulises’ lacks denotation since ‘Ulises’ does not denote. According to III, the denotation of the compound name ‘The capital of England’ does not change when substituting name ‘Churchill’s country’ for ‘England’. Simpson argues that Frege introduces (TP) by showing that declarative sentences and their truth-values satisfy principles II and III. Simpson reformulates these principles for declarative sentences replacing ‘compound name’ by ‘sentences’ and ‘denotation’ by ‘truth-value’ in the relevant places as follows:
(II’) When a constituent name of a sentence lacks denotation the sentence has no truth- value.
(III’) When a constituent name of a sentence is replaced by another name having the same denotation, the truth-value of the sentence (if there is such a truth-value) does not change.
According to II’ the sentence ‘Odysseus was set ashore at Ithaca while sound asleep’ lacks truth-value since ‘Odysseus’ does not denote. The sentence ‘The morning star is far from earth’ does not change its truth-value when ‘evening star’ replaces ‘morning star’ in accordance with principle III’. Simpson holds that for Frege (TP) follows from the premise that declarative sentences and their truth-values satisfy principles II’ and III’. However my reading of Frege’s texts does not agree with Simpson’s interpretation on this point. Frege considers the above mentioned principles only as necessary conditions for proper names. Hence I cannot see any argument here but and experimental development of (TP). Setting aside this point, the importance of Simpson’s reconstruction remains since it allows us to analyse Frege’s reason for (TP). Simpson finds Frege’s claim to be twofold: (a) declarative sentences are proper names and (b) their truth-values are their denotation; and he further argues that to conclude (TP) even were we to assume (a) we should also need to establish the truth of (b). In order to establish (b) Simpson argues that Frege must prove: (b’) that the naming relation holding between sentences and their truth-values satisfies principles II’ and III’, and (b’’) that there are no other entities satisfying principles II’ and III’ holding the naming relation to sentences. Simpson rejects (b’’) arguing two issues. First, that there are other kinds of entities satisfying principles II’ and III’, and which hold the naming relation to sentences; and second, that there is an infinity of such entities. His argument is as follows. Let us suppose that sentence A is a compound proper name. Let us also suppose that the denotation of A is the equivalence class of A mod. truth-value, i.e., the class of all the sentences which have the same truth-values as A. If the equivalence class of A mod. truth-value is the denotation of A, principles II’ and III’ are satisfied. Principle III’ is satisfied by sentence A and its supposed denotation since if the denotation of A is the equivalence class of all the sentences that have the same truth-value that A has, then if we replace a constituent name of A by other name having the same denotation the truth-value of A will not change nor will its equivalence class. Principle II’ is satisfied by A and its equivalence class given that in order to construct the denotation of A it is a necessary and sufficient condition that A have a truth-value. If a constituent name of the compound name A lacks denotation then A lacks denotation in the just described sense since we cannot construct the equivalence class of A mod. truth-value. Thus following Frege’s strategy Simpson concludes that the equivalence class mod. truth-value of a given sentence is the denotation of such a sentence. From this starting point he also argues that the number of possible denotata is infinite because principles II’ and III’ are also satisfied by the unitary class whose unique element is the equivalence class of A mod. truth-value, and in general they are satisfied by every member of the infinite series: {CA}…{{CA}}…{{{CA}}}… where “CA” stands for equivalence class of A mod. truth-value. Simpson’s reconstruction shows: (a) that Frege’s argument is not conclusive and (b) that there is an ad hoc flavour in Frege’s strategy. But if there is such and ad hoc flavour, and if (TP) is necessary for the theory, why do we not just stipulate (TP)? An affirmative answer is based on three main reasons: (i) Frege’s “argument” is not conclusive. (ii) In Mathematics there are cases where a stipulation is needed to validate a law. For instance using the intuitive notion of exponentiation the algebraic law: am = am-n an fails for a0 when m=n. Thus it is necessary to stipulate that a0=1 to validate the aforementioned law. This is not alien to Frege’s aim since he introduces “the special stipulation that divergent infinite series shall stand for number 0” (Cfr. “On Sense and Reference”, Trans. Geach & Black, p. 70). (iii) Frege’s notion of naming (namingF) is a theoretical concept that fulfills the following requirements for an explication in Carnap’s sense:
(1) NamingF is similar to our intuitive notion of naming (namingI) in such a way that in most cases in which namingI has so far been used, namingF can be used; however, close similarity is not required and considerable differences are permitted.
(2) The characterization of namingF is given in the form of a definition so as to introduce namingF into a well-connected system of theoretical concepts, as follows: a namesF x= df. x is an object and: if a is a genuine name x is the denotation of a, if a is a definite description x is the unique entity which satisfies the predicates occurring in a, if a is a declarative sentence x is the truth-value of sentence a.
(3) NamingF is a concept useful for the formulation of many universal statements, among others following: every predicate is a function name, every concept is a function whose value is always a truth-value, every equivalence statement is an identity statement. (4) NamingF is a simple as is possible, that means, as simple as the more important requirements (1), (2) and (3) permit.
(4) NamingF is a simple as is possible, that means, as simple as the more important requirements (1), (2) and (3) permit. Frege’s (TP) is used to subsume predicates to function names and their referents (concepts) to functions, as follows. According to Frege a first level monadic function name is an expression which contains a gap and its denotation is a function. Traditionally predicates have been regarded as monadic expressions containing a gap, and concepts are the denotations of predicates. When a function name is saturated by a proper name the resulting expression is a compound proper name. When a predicate is saturated by a singular term the result is a declarative sentence. If (TP) is the case then predicates and first level function names behave the same way. Thus since function names denote functions, predicates also denote functions. The subsumption of predicates to function names has two profitable consequences: there is an ontological reduction when concepts are considered as functions, and the traditional notion of a function is extended in two ways, by admitting a new kind of expression (predicates) as function expressions and by admitting a new kind of thing (truth-values) as arguments. Frege also uses (TP) to treat every equivalence statement as an identity statement, as follows. Statements of the form ‘P≡Q’ are true when P and Q have the same truth-value. If P and Q have the same truth-value and (TP) is the case, then P and Q have the same denotation. Then any assertion that results from replacing ‘=’ by ‘≡’ will also be true since ‘P≡Q’ is true if and only if P and Q have the same denotation. And finally, on the base of the definition of namingF, principles II’ and III’ result in general semantic laws. To conclude: the stipulation of (TP) has the advantage of avoiding the dubious Fregean conjecture. It is dubious in several ways: (i) our pre-theoretical conception of naming does not agree with the idea that sentences perform that role; (ii) even assuming that sentences could do that job, it is not clear what the reasons are for considering just truth-values to be their denotation instead of facts or other more intuitive candidates; (iii) even taking sentences as names and truth-values as their denotation, Frege’s reasons for (TP) seem to be inconclusive. Only if we consider theoretical needs and the naming notion as a theoretical one, can we dispel our intuitive rejection of (TP). [L.V.]