Logic, philosophy of

2. Logic and language

There is also a collection of long-established topics discussion of which can be much improved, in rigour at least, in the light of the development of modern logic. For example, a distinction between propositions (or statements, or sentential contexts) which are de dicto and propositions (and so on) which are de re originates in medieval philosophy. But only contemporary modal logic affords the tools for a precise characterization of this distinction, although it must be granted that the distinction remains a puzzle in epistemic contexts (see De re/de dicto; Descriptions §2; Propositional attitude statements). Other topics which can be classified in this way include Essentialism, Existence, Identity, Indicative conditionals, Modal operators, Quantifiers and Vagueness. Again, to give some of the flavour of this kind of work, consider the de re/de dicto contrast. There is an evident syntactic difference between ‘It is necessary that parents have children’ and ‘Parents are such that it is necessary that they have children’, but just because there is a syntactic difference, it does not follow that there is any interesting difference in meaning. But the difference can be brought out quite precisely in possible worlds semantics. To say that it is necessary that parents have children is to say that in every possible world, the people who are parents in that world have children in that world; and this is an obvious truth. On the other hand, to say that parents are such that it is necessary that they have children is to say that the people who are parents in the actual world are such that they have children in every possible world. This is clearly false, even putting aside contingency of existence of actual parents. For given anyone who is actually a parent, there is a way things could have gone – a possible world – in which that person is childless, hence not a parent (see Quantifiers, substitutional and objectual; Modal operators).

When a formal semantics for a system of logic is applied to a fragment of natural language, a very precise account of the literal content of sentences in that fragment is given. But there may be aspects of the meanings of those sentences which are omitted. Philosophical views may then divide over whether the formal semantics has been shown to be wanting as an account of the semantics of the fragment, or whether instead the aspects of meaning not captured have been shown not to belong to literal content (see Presupposition). In the case of indicative conditionals, for instance, the formal semantics that is relevant is the simplest possible kind, namely, the truth-functional account of ‘if…then…’. According to this account, ‘If p then q’ is true if p is false or if q is true, regardless of the actual meanings of p and q. So in particular, any indicative conditional with a true consequent is true; examples would include ‘If lead floats in water then lead sinks in water’ and ‘If the solar system has nine planets then the Conservative Party lost the British elections in 1997’. Barring an astrological justification of the latter, both these conditionals look decidedly odd. But oddness is one thing, falsity another. The idea that such conditionals are false is based on the thought that if a conditional is true, then in establishing it in the most direct manner, non-redundant use has to be made of the antecedent. Spelling this out leads to relevance logic (see Relevance logic and entailment; Indicative conditionals). On the other hand, if we say the conditionals are merely odd, we are led to some theory of communication to explain the oddness (see Grice, H.P.; Implicature).

But we should not take away the impression that the traffic is all one way, from logic to language or from pure mathematics to pure philosophy. There is a two-way street here, with the above comments on conditionals representing a common phenomenon; that of a concern in the philosophy of language giving rise to a formal development which in turn feeds back into philosophy. For example, the idea that for a conditional to be true, the most direct way of establishing it must make non-redundant use of its antecedent seems clear enough on the face of it, but familiarity with logic of conditionals literature may well lead one to reconsider. This kind of dialectical interplay should continue to be a fruitful source of philosophical research for the foreseeable future.