226 The formalization of sentence-meaning

The next step is to invoke the notion of possible worlds. As we saw earlier, necessarily true (or false) propositions are proposi­tions that are true (or false) in all possible worlds. The notion has also been applied, in an intuitively plausible way, in the defi­nition of descriptive synonymy, as follows: expressions are descriptively synonymous if, and only if, they have the same extension in all possible worlds. Since expressions are descrip­tively synonymous if, and only if, they have the same sense (which we have identified with their intension), it follows that the intension of an expression is either its extension in all possible worlds or some function which determines its extension in all possible worlds. The second of these alternatives is the one that is adopted in Montague grammar. The intension of an expres­sion is defined to be a function from possible worlds to extensions. But what does this mean?

The answer that I will give to this question is somewhat differ­ent from the answer that is given in standard accounts of formal semantics, but it is an answer that is faithful to the spirit of Mon­tague semantics and philosophically defensible. My deliberately non-technical explanation of the basic notions of Montague's version of model-theoretic possible-worlds semantics is couched as far as possible in terms of the notions and distinctions that have been explained and adopted in earlier chapters.

7.6 Possible worlds

Leibniz introduced the notion of possible worlds for primarily theological purposes, arguing that God, being omniscient (and beneficent), would necessarily actualize the best of all possible worlds and, though omnipotent, was none the less subject in his creativity to the constraints of logic: he could create, or actua­lize, only logically possible worlds. As exploited by modern logi­cians, the notion of possible worlds has, of course, been stripped of its theological associations, and it has been converted into a highly technical, purely secular and in itself non-metaphysical, concept. But some knowledge of its philosophical and theologi- cal origins may be helpful (especially when it comes to the use

7.6 Possible worlds 227

that is made of the notion of possible worlds in epistemic and deontic logic). Hence this brief philosophical interlude.

Every natural language, let us assume, provides those who are competent in it with (a) the means of identifying the world that is actual at the time of speaking - the extensional world -and distinguishing it from past and future worlds, and (b) the means of referring to individual entities and sets of entities, on the one hand, and to substances, on the other, in whatever world has been identified. We may refer to whatever means is used to identify temporally distinct worlds (tense, adverbs of time, etc.) as an index - more precisely, a temporal index - to the world in question. I shall have more to say about this in Chapter 10: here I will simply draw readers' attention to the connexion between the term 'index', as I have just used it, and 'indexicality'. An alternative to 'index', in this sense, is 'point of reference': possible worlds are identified from a particular point ofreference.

Granted that one can identify the world that is explicitly or implicitly identified, how does one know what is being referred to by the expression that is used, when a sentence is uttered? For example, how does one know what 'those cows' refers to in the utterance of

(10) 'Those cows are pedigree Guernseys'?

The traditional answer, as we have seen, is that one knows the concept "cow" and that this, being the intension (or sense) of 'cow', determines its extension. (One also needs to be able to interpret the demonstrative pronoun 'that' and the grammatical category of plurality. But let us here assume - and it is a not inconsiderable assumption - that the meaning of 'that' and plurality, not to mention the grammatical category of tense, can be satisfactorily handled in model-theoretic terms.) Con­cepts are often explained in terms of pictures or images, as in cer­tain versions of the ideational theory of meaning (see 1.7). But we can now think of them more generally, as functions (in the mathematical sense): that is, as rules, or operations, which assign a unique value to the members of their domain. It is as if we had a book of rules for all the expressions in the language (the