Non-standard Interpretations

The final circularity objection we will consider is once again found in Carnap’s Schilpp volume, this time in the contribution by E.W. Beth (1963).18 Beth’s is unique amongst our collection of circularity objections in that it is cast partially in model- theoretic terms, and in the fact that it concerns not thestrength of the mathematics which must be presupposed by Carnap to carry through his program, but rather that aparticular interpretationof the meta-language must be assumed in the investigation of a rational reconstruction.

The technical details of this objection are quite involved, but the basic idea is that the languages Carnap constructs in Logical Syntax are such that they admit of non-standard interpretations. In the simplest case we need only recognize that formal languages admit of arithmetization, and so take a non-standard interpretation of _N.19 Keeping this in mind, Beth introduces a fictional logician, Carnap*, whose intuitive understanding of the language LII is non-standard. Specifically, his intuitive understanding (which Beth identifies with some model M*) is guided by the extension of LII, called LII*, which includes as an axiom the negation of the arithmetization in LII of a sentence expressing the consistency of LII. What is essential here is that Carnap* takes his intuitive interpretation also as a guide to the meta-language Mathematica (Whitehead & Russell, [1910–1913]1997)) because they include only finitary rules of deduction without specifying any indefinite rules of consequence needed to provide a language with a complete criterion of validity.

18_{Ricketts (2004) and Friedman (2009) also discuss this objection.}

19_{This informative example derives from the discussion in Friedman (2009). The situation is in}

fact a bit more complex than this, since Carnap’s LII is a higher-order theory of types. Given the standard semantics for such a language, Peano Arithmetic is categorical and so does not admit non-standard models. Assuming a Henkin semantics however, which Beth does, higher-order Peano Arithmetic does indeed admit non-standard models as in the first-order case.

within which LII is investigated.20 _{The consequences of all this are that Carnap* will}

systematically misinterpret the main inductive definitions and lines of argument in

Logical Syntax, leading up to Carnap’s proof that LII is non-contradictory (Theorem 34i.23), and any subsequent theorems which rely upon this result.

Supposing that we take LII to in fact be consistent, Beth argues that in the situation as described, Carnap and Carnap* will disagree with regard to certain properties of LII:

Now Carnap* could settle the dispute at once in his favor by exhibiting the inconsistency which, according to him, exists in Language II, that is, by actually deriving a contradiction. But this he is unable to perform [because LII is in fact consistent]. So he is compelled to proceed entirely by indirect argument, guided by his intuitive model M*. But in doing so, he will again and again resort to assertions which Carnap cannot accept and for which no basis can be found inLogical Syntax. Therefore, he will not be able to convince Carnap of an error.

On the other hand, Carnap will no more be able to convince Carnap* for the statements to which Carnap* appeals are consistent with every state- ment made inLogical Syntax; if Carnap is to refute Carnap*’s assertions, he must resort to his “sufficiently rich syntax language” for Language II, and some statements provable in this syntax language are, for Carnap*, either false or devoid of meaning. (Beth, 1963, p. 481)

The point here is just that the Carnaps are approaching LII from different intuitive perspectives, owing to their conflicting informal understandings. Carnap* supposes the language to be inconsistent, while Carnap obviously does not—the assumptions made in their respective informal meta-languages reflect these divergent perspectives. The situation is analogous to two logicians arguing over certain properties of the natural numbers while one of the participants holds a non-standard interpretation. When the numerals are listed: ‘0,00,000,and so on’, the first logician will understand the expression ‘and so on’ in the customary way, while the second logician will include an additional, non-finite numeral. In this case however, the disputants can compare their divergent models in a suitable semantic theory. In the dispute between Carnap and Carnap* there is no such recourse, since Logical Syntax explicitly disavows the

20_{In his reply to Beth, Carnap notes (p. 929) that this assumption is essential for Beth’s argument,}

semantic treatment of formal languages.21 _{Instead, the Carnaps can only resort}

to formalizing their meta-languages. Carnap can here display the consistency of LII—and Carnap* its inconsistency—but at this stage each will of course reject the other’s formalization as an adequate characterization of the informal meta-language used throughout Logical Syntax.

Beth’s conclusion is that Logical Syntax relies upon a tacitly understood inter- pretation (the so-called “standard interpretation”) of the concepts and words of the informal meta-language used to convey the ideas in the book. But the assumption of such a tacit understanding is not innocent, since it amounts to a prior understanding of the mathematics in question. So Beth, as with G¨odel and Quine above, concludes that “[. . . ]Carnap has not been able to avoid every appeal to logical or mathemat- ical intuitions[. . . ]” (p. 502), or what amounts to the same thing for Beth, to the

ontological assumption that mathematical objects exist in some robust sense.