Semantics, possible worlds

Possible worlds semantics (PWS) is a family of ideas and methods that have been used to analyse concepts of philosophical interest. PWS was originally focused on the important concepts of necessity and possibility. Consider:

Intuitively, (a) is true but (b) is false. There is simply no way that 2 and 2 can add up to anything but 4, so (a) is true. But although Socrates did in fact have a snub nose, it was not necessary that he did; he might have had a nose of some other shape. So (b) is false.

Sentences (a) and (b) exhibit a characteristic known as intensionality: sentences with the same truth-value are constituent parts of otherwise similar sentences, which nevertheless have different truth-values. Extensional semantics assumed that sentences stand for their truth-values, and that what a sentence stands for is a function of what its constituent parts stand for and how they are arranged. Given these assumptions, it is not easy to explain the difference in truth-value between (a) and (b), and hence not easy to give an account of necessity.

PWS takes a sentence to stand for a function from worlds to truth-values. For each world, the function yields the truth-value the sentence would have if that world were actual. ‘2 + 2 = 4’ stands for a function that yields the truth-value ‘true’ for every world, while ‘Socrates had a snub nose’ stands for a different function that yields ‘true’ for some worlds and ‘false’ for others, depending on what Socrates’ nose is like in the world. Since these two sentences stand for different things, sentences that have them as constituents, such as (a) and (b), can also stand for different things.

This basic idea, borrowed from Leibniz and brought into modern logic by Carnap, Kripke and others, has proven extremely fertile. It has been applied to a number of intensional phenomena in addition to necessity and possibility, including conditionals, tense and temporal adverbs, obligation and reports of informational and cognitive content. PWS spurred the development of philosophical logic and led to new applications of logic in computer science and artificial intelligence. It revolutionized the study of the semantics of natural languages. PWS has inspired analyses of many concepts of philosophical importance, and the concept of a possible world has been at the heart of important philosophical systems.