Logic, philosophy of

1. The impact of modal logic

In the broad area of mathematical logic, the biggest philosophical punch is packed by modal logic, including tense logic (see Modal logic; Modal logic, philosophical issues in; Tense and temporal logic). Modal logic has been important since Aristotle (see Logic, ancient; Logic in the 17th and 18th centuries; Logic in the 19th century; Logic in the early 20th century) but has only been put on a rigorous footing in the second half of the twentieth century, by such figures as Hintikka, Kanger, Prior, and most especially Kripke (see Semantics, possible worlds). The most important philosophical outgrowth of this mathematical work is contained in Kripke’s three lectures from January 1970 published as ‘Naming and Necessity’, in which Kripke draws out some ways in which possible worlds semantics is in tension with then-prevailing orthodoxies in the philosophy of language and mind. Some of Kripke’s views have become new orthodoxies since (see Essentialism; Proper names; Reference §§2–4; for related work by David Lewis, Robert Stalnaker, David Kaplan and others that uses the possible worlds framework, see Counterfactual conditionals; Demonstratives and indexicals; Descriptions).

To give some flavour of developments here, consider the familiar Fregean view that the relation of reference which holds between a name and its bearer is sustained by the relation of presentation which holds between the sense of the name and the bearer of the name: the name refers to such-and-such an object precisely because it expresses a sense which presents that object (see Frege, G. §3; Sense and reference). When pressed for an explanation of what the senses of names are like, the natural Fregean response is to specify them, as Frege himself did in some cases, using definite descriptions (see Descriptions). So, for instance, the sense of the name ‘Aristotle’ might be ‘the pupil of Plato who taught Alexander’. However, though it may well in fact have been Aristotle who taught Alexander, there are many ways things might have gone (many ‘possible worlds’) in which someone other than Aristotle is taught by Plato and teaches Alexander: suppose Aristotle had got the appointment but been killed in an accident before he could take it up, and had been replaced at Philip’s insistence by another pupil of Plato. The description ‘the pupil of Plato who taught Alexander’ is therefore ‘non-rigid’, in Kripke’s terminology. That is, it can pick out different individuals in different possible worlds, and in some worlds may pick out no one (Philip for some reason comes to distrust Platonic pedagogy and fails to conduct an equal opportunity search). But it is clear from the formal semantics for modal logic that there is conceptual ‘room’ for a category of expression which is ‘rigid’, in the sense that it picks out the same object in every possible world, or at least in every possible world where it picks out any object at all. So the formal semantics prompts the question whether names in natural language behave as if their reference is determined by a sense which presents different individuals at different worlds, or whether they behave as if they are rigid designators. With a series of brilliant examples Kripke demonstrates that names are rigid designators and therefore do not express reference-determining senses which are non-rigid (see Proper names).

The idea that a formal semantics for a kind of logic provides an account of a possible semantics for a category of natural-language expression, opening the door to debate on whether it is the right account or not, also captures some of the philosophical bearing of kinds of logic other than modal logic. Thus free logic shows how name-like expressions can function without standard existential commitment (see Free logics, philosophical issues in); intuitionistic logic and many-valued logic show how a language can have a compositional semantics even if its sentences are not used to make statements with verification-transcendent truth-conditions which always either obtain or fail to obtain (see Compositionality; Intuitionistic logic and antirealism; Many-valued logics, philosophical issues in; Presupposition). And second-order logic offers a particular way of understanding the semantic import of a range of puzzling locutions, such as plural quantifiers (see Second-order logic, philosophical issues in). In all these cases the formal semantics for the logical system prompts debates about how well the semantics carries over to natural language.