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Formal Semantics
cal vocabulary of the object language will have counterparts in the metalanguage, for instance, the material implication symbol -» at object-language level reflects the only if or ^>at metalevel, the material implication symbol = at object level the ij^or o at metalevel, and so on.
The question of which concepts belong to the object language and which to the metalanguage is a key problem of logic. Alfred Tarski has argued that truth is essentially a metalanguage concept. His argument is based on a well-known semantic paradox dating from Antiquity, the Paradox of the Liar. Consider sentence (1):
This sentence is false. (1)
Sentence (1) is paradoxical. This can be seen as follows. First assume that (1) is true. Then because of what (1) says, it must be false. Contradiction. Now assume that (1) is false. Then what the sentence says is not true, that is, it is not true that (1) is false. In other words, (1) is true. Contradiction again.
Given the presence of a truth predicate T in the object language, the Liar Paradox can be formalized in first-order logic. Let f ~| be a function mapping the formula <pto a term f <p~| . Then the fact that T is the truth predicate makes the following principle (2) true, for arbitrary sentences (p.
<p=r|>~|. (2)
Modulo some assumptions about the possibility of encoding syntactic operations as functions on codes of syntactic objects, it can be shown that there exists a sentence ^ for which the equivalence (3) is true (in some suitable model).
(3)
Equivalence (3) is a formalized version of Liar sen- tence (1), and the combination of (2) and (3) gives rise to a paradox in first-order logic. (Further details can be found in Formal Semantics.)
In Tarski's view, the Formalized Liar Paradox arises because the concept of truth is used at the wrong level. It cannot be denied that natural languages do contain a truth predicate, so Tarski's argument is rel- evant for natural language semantics. Tarski simply dismissed natural languages as unable to withstand logical scrutiny, but there are several ways to get around his conclusion. First, one can take care always to restrict semantic accounts for natural language to 'fragments,' in such a way that no fragment contains a truth predicate for the sentences of the fragment itself, although a fragment might contain a truth predicate for embedded natural-language fragments. This approach might lead to a hierarchy of an object fragment, a metafragment, a meta-metafragment, and so on. A second solution is the observation that one half of (2) is harmless: weakening (2) to <p-» T [<p~\ blocks the paradox. Saul Kripke has shown that it is indeed possible to define a partial truth predicate in the object language satisfying <p-» T f<ji>l - His defi- nition is outside the scope of the present article.
See also: Formal Semantics; Paradoxes, Semantic; Truth.
Bibliography
Kripke S A 1975 Outline of a theory of truth. The Journal of Philosophy 72: 690-716
Tarski A 1956 The concept of truth in formalized languages. In: Tarski A (ed.) Logic, Semantics, Metamathematics. Clarendon Press, Oxford
Modal logic is the logic of necessity and possibility— intuitively, of the ways things must be, and the ways things might have been. For example, it is not a mere contingency about the world that all grandmothers are mothers of parents: that is something that must be the case. On the other hand, if Jane is not in fact a grandmother, that is merely contingent: Jane might have been a grandmother if things had gone differ- ently. Modal logic is a means of formalizing the claim that it is necessary that all grandmothers are mothers of parents, and the claim that it is possible that Jane should have been a grandmother.
1. The Origins of Modal Logic
In ancient philosophy, modal notions loom large in discussions of fatalism, determinism, and divine fore- knowledge, although it is arguable that before the work of Duns Scotus in the late thirteenth century these discussions did not clearly distinguish between genuinely modal ideas (e.g., what is not actually so, but might have been so) and temporal ideas (e.g., what is not now so, but sometime will be so). In any case, the modern origins of modal logic lie in the work of Lewis on the notion of strict implication (see Lewis 1912; Lewis and Langford 1932).
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Modal Logic
M. Davies