Rudolf Carnap: Modal Logic

In two works, a paper in The Journal of Symbolic Logic in 1946 and the book Meaning and Necessity in 1947, Rudolf Carnap developed a modal predicate logic containing a necessity operator N, whose semantics depends on the claim that, where α is a formula of the language, Nα represents the proposition that α is logically necessary. Carnap’s view was that Nα should be true if and only if α itself is logically valid, or, as he put it, is L-true. In the light of the criticisms of modal logic developed by W.V. Quine from 1943 on, the challenge for Carnap was how to produce a theory of validity for modal predicate logic in a way which enables an answer to be given to these criticisms. This article discusses Carnap’s motivation for developing a modal logic in the first place; and it then looks at how the modal predicate logic developed in his 1946 paper might be adapted to answer Quine’s objections. The adaptation is then compared with the way in which Carnap himself tried to answer Quine’s complaints in the 1947 book. Particular attention is paid to the problem of how to treat the meaning of formulas which contain a free individual variable in the scope of a modal operator, that is, to the problem of how to handle what Quine called the third grade of ‘modal involvement’.

In an important article (Carnap 1946) and in a book a year later, (Carnap 1947), Rudolf Carnap articulated a system of modal logic. Carnap took himself to be doing two things; the first was to develop an account of the meaning of modal expressions; the second was to extend it to apply to what he called “modal functional logic” — that is, what we would call modal predicate logic or modal first-order logic. Carnap distinguishes between a logic or a ‘semantical system’, and a ‘calculus’, which is an axiomatic system, and states on p. 33 of 1946 that  “So far, no forms of MFC [modal functional calculus] have been constructed, and the construction of such a system is our chief aim.” In fact, in the preceding issue of The Journal of Symbolic Logic, the first presentation of Ruth Barcan’s axiomatic systems of modal predicate logic had already appeared, although they contained only an axiomatic presentation. (Barcan 1946.) The principal importance of Carnap’s work is thus his attempt to produce a semantics for modal predicate logic, and it is that concern that this article will focus on.

Nevertheless, first-order logic is founded on propositional logic, and Carnap first looks at non-modal propositional logic and modal propositional logic. I shall follow Carnap in using ~ and ∨ for negation and disjunction, though I shall use ∧ in place of Carnap’s ‘.’ for conjunction. Carnap takes these as primitive together with ‘t’ which stands for an arbitrary tautologous sentence. He recognises that ∧ and t can be defined in terms of ~ and ∨, but prefers to take them as primitive because of the importance to his presentation of conjunctive normal form. Carnap adopts the standard definitions of ⊃ and ≡. I will, however, deviate from Carnap’s notation by using Greek in place of German letters for metalinguistic symbols. In place of ‘valid’ Carnap speaks of L-true, and in place of ‘unsatisfiable’, L-false. α L-implies β iff (if and only if) α ⊃ β is valid. α and β are L-equivalent iff α ≡ β is valid.

One might at this stage ask what led Carnap to develop a modal logic at all. The clue here seems to be the influence of Wittgenstein. In his philosophical autobiography Carnap writes:

For me personally, Wittgenstein was perhaps the philosopher who, besides Russell and Frege, had the greatest influence on my thinking. The most important insight I gained from his work was the conception that the truth of logical statements is based only on their logical structure and on the meaning of the terms. Logical statements are true under all conceivable circumstances; thus their truth is independent of the contingent facts of the world. On the other hand, it follows that these statements do not say anything about the world and thus have no factual content. (Carnap 1963, p. 25)

Wittgenstein’s account of logical truth depended on the view that every (cognitively meaningful) sentence has truth conditions. (Wittgenstein 1921, 4.024.) Carnap certainly appears to have taken Wittgenstein’s remark as endorsing the truth-conditional theory of meaning. (See for instance Carnap 1947 p. 9.) If all logical truths are tautologies, and all tautologies are contentless, then you don’t need metaphysics to explain (logical) necessity.

One of the features of Wittgenstein’s view was that any way the world could be is determined by a collection of particular facts, where each such fact occupies a definite position in logical space, and where the way that position is occupied is independent of the way any other position of logical space is occupied. Such a world may be described in a logically perfect language, in which each atomic formula describes how a position of logical space is occupied. So suppose that we begin with this language, and instead of asking whether it reflects the structure of the world, we ask whether it is a useful language for describing the world. From Carnap’s perspective, (Carnap 1950) one might describe it in such a way as this. Given a language £ we may ask whether £ is adequate, or perhaps merely useful, for describing the world as we experience it. It is incoherent to speak about what the world in itself is like without presupposing that one is describing it. What makes £ a Carnapian equivalent of a logically perfect language would be that each of its atomic sentences is logically independent of any other atomic sentence, and that every possible world can be described by a state-description.