The First Incompleteness Theorem

Quine makes this same point again in his contribution to the Carnap-Schilpp volume (Quine, 1963,§IV). The majority of this paper directly concerns Carnap’s philosophy of mathematics, or what Quine calls the “linguistic doctrine of logical truth”. In§VII he specifically discusses the technical situation in Logical Syntax, noting first that “[w]hatever our difficulties over the relevant distinctions, it must be conceded that logic and mathematics do seem qualitatively different from the rest of science.” (p. 397) However, Quine concludes that this seeming distinction amounts to no more than a difference of degree, rather than kind.14 _{The problem, according to Quine, is}

that both mathematical and empirical theories submit to formalization according to the same general scheme: Supplement a basic logical framework (say instantiating the first-order predicate calculus) with the choice of a further set of L- or P-Rules. The resulting frameworks can in both cases be thought of as just formal axiomatic theories, and so it is unclear why we should consider the one (with only L-Rules) analytic, or conventional, while the other (including P-Rules) is partially empirical. Another way to see Quine’s point is to recall from the Logico-Mathematical In- terlude that Carnap expresses the complete division between the formal and factual components of his languages via the fundamental theorems which show that the logical sentences of their respective languages are L-Determinate. This shows that the logical sentences can act as formal auxiliaries, because they are completely de- termined by the syntactical rules of their language. In the case of LII this proof relies upon the prior definition of ‘Analytic’ in LII, which amounts to what Quine calls a “truth-definition” for the language. Quine observes that for a language as powerful as Carnap’s LII, the proof that every canonical logico-mathematical sen- tence is L-Determinate (in the case of LII, Theorem 34e.11) must be carried out in a meta-language more powerful than the object-language. As we saw, this is because of G¨odel’s first incompleteness theorem, which tells us that any definite language of sufficient strength will include sentences that are irresoluble within the language.

13_{Cf. Carroll ([1895]1979).}

14_{The most famous expression of Quine’s arguments in this direction are of course in his “Two}

Dogmas of Empiricism” ([1951]1980). As noted in chapter 1, the argument with which we are presently concerned is distinct from these arguments against the analytic/synthetic distinction and for his brand of holism.

But if this is the thesis forwarded by the linguistic doctrine of logical truth, then [. . . ]the thesis that logico-mathematical truth is syntactically specifiable becomes uninteresting. For, what it says is that logico-mathematical truth is specifiable in a notation consisting solely of [names of signs], [an operator expressing concatenation of expressions], and the whole logico- mathematical vocabulary itself. (1963, p. 400. Original emphasis.)

So along the same lines as Quine’s original 1935 objection to conventionalism, and G¨odel’s objection to the first thesis of the syntactic viewpoint, the need for Carnap to appeal to mathematics as tacitly understood at the meta-level is here taken to undermine the purpose of the linguistic doctrine of logical truth, which according to Quine was to show that logico-mathematical truth is “grounded in language”.

Although Potter (2000, chp. 11) devotes a good deal of space to discussing G¨odel’s objection, he argues that Carnap’s logico-mathematical program in Logical Syntax

founders foremost upon a more sophisticated form of this Quinean criticism.15 _Note

in the above that Quine, like G¨odel, assumes Carnap must limit himself to finitary

notions in order that his conventionalism have a chance to succeed. This is because an appeal to infinitary notions like a indefinite consequence relation strains the idea that the methods of syntax are merely the formally-directed combination and manip- ulation of signs. Potter sees this technical situation as placing Carnap in a dilemma: Either he goes ahead and specifies the definition of ‘Analytic’ for a language by invok- ing an indefinite consequence relation anyway,16 or he must acquiesce to accepting that some of the vocabulary customarily taken to be logico-mathematical will end up descriptive in that language.

This latter result follows just because, again owing to G¨odel’s first incomplete- ness theorem, the language will be incomplete and so certain canonically logico- mathematical sentences will not be decided by the syntactical rules of the language. This is just to say that those sentences, while containing only customarily logical expressions, are indeterminate. Therefore Carnap will end up having to count the ostensibly logical expressions which figure in such sentences as descriptive.17 Now,

15_{As far as I am aware, Potter does not attribute the criticism to Quine. Michael Friedman}

(1999) also independently discusses this same objection.

16_{As we saw in the Logico-Mathematical Interlude, one can accomplish this in at least two ways.}

From within a language, appeal to an infinitary rule of inference like Carnap’sω-rule (Cf.Logical Syntax,§14). From without, define a notion of ‘Analytic’ using a meta-language (Cf.Logical Syntax,

§34a–f). We saw Carnap pursue both strategies, in LI and LII respectively.

17_{Carnap in fact addresses this issue inLogical Syntax (seeExample on pp. 231–232). However,}

since Carnap is arguing that logic and mathematics are non-factual, Potter observes that Carnap must pursue the former strategy of defining ‘Analytic’ for LII in an in- definite way—as he in fact does. This strategy requires Carnap to give up the ability to “explain how a finite intelligence can grasp arithmetical truths which appear to refer to an infinite domain of objects” (Potter, 2000, p. 286). But if his goal is to recover a plausible story as to how mathematical truths and their objects are know- able, presumably Carnap should neither appeal to blatantly infinitary reasoning, as in LI, nor presuppose the very notions for which he is attempting to account, as in LII. And thus Carnap’s program seems to fail.