328 The subjectivity of utterance

world. Before taking up these points, I, must say something about modality. ^:

The only kind of modality recognized in traditional modal logic is that which has to do with the notions of necessity and pos­sibility in so far as they relate to the truth (and falsity) of proposi­tions: aletheutic, or alethic, modality. (Both 'aletheutic' and 'alethic' come, indirectly, from the Greek word for truth: 'aletheutic' is etymologically preferable, but 'alethic' is now widely used in the literature.) We have already looked at the question of the necessary truth and falsity of propositions on sev­eral occasions, and with particular reference to entailment and analyticity in Chapter 4. In section 6.5, we noted that the modal operators N and M (or } and ), like the operator of negation in the prepositional calculus, are truth-functional.

It may now be added that aletheutic necessity and possibility are interdefinable under negation: they are inverse opposites or (to use the more technical terminology of mathematical logic) duals. To adapt one of the examples used in section 6.5:

13. "Necessarily, the sky is blue" is logically equivalent to

14. "It is not possible that the sky is not blue"

(i.e., (Np = ~M~p), or (}p =~~p)); and

15. "Possibly, the sky is blue" is logically equivalent to

16. "It is not necessarily the case that the sky is not blue"

(i.e., (Mp =~N~p) or (p = ~}~p)).

The question whether other kinds of necessity and possibility have the same logical properties with respect to negation as aletheutic necessity and possibility is somewhat more controver- sial; and we shall come back to it presently. The fact that aletheutic necessity and possibility are duals means that in this respect they are like the universal and existen- sional quantifiers ((x)or, alternatively, (x): "all"; and (Ex) or

10.5 Modality, modal expressions and mood 329

(x): "some") as these are standarly defined by logicians: (x) fx B ~ ((Ex)~fx), i.e. "For all x, x has the property f" is equivalent to "It is not the case that there is some x such that (i.e., there is no x such that) x does not have the property f". This parallelism between quantification and modality is by no means fortuitous. In traditional logic (based on a bipartite analysis of propositions into subject and predicate), modality was commonly described as quantification of the predicate. And, as we have seen, in some systems of modern intensional logic (including the one which underpins Montague semantics) necessity is defined (fol­lowing Leibniz) in terms of truth in all possible worlds, possibi­lity in terms of truth in some (i.e., at least one) possible world. Given that necessity and possibility are interdefinable, the ques­tion arises which, if either, should be regarded as being more basic than the other. Generally speaking, logicians take aletheu­tic modality to be necessity-based, rather than possibility-based. But from a purely formal point of view this is a matter of arbitrary decision.

Aletheutic modality, then, like prepositional negation, is by definition truth-functional. But what about modality in the everyday use of natural languages? Let us take another of the examples used in section 6.5: the sentence

(17) 'He may not come'.

Now, there is no doubt that this sentence can be used to assert a modalized negative proposition (with either external or internal negation: either ~Np or M~p). In this case both the negative particle not and the modal verb 'may' are construed as contribu­ting to the propositional content of the sentence.

But with this particular sentence (when it is uttered in most everyday contexts), the modality is more likely to be either epistemic or deontic than aletheutic. (The terms 'epistemic' and 'deontic' were explained in section 8.4. As we shall see, they are being used here in essentially the same sense.) Both kinds of mod­ality may be either objective or subjective. If our sample sen­tence is given an objective epistemic interpretation, its propositional content will be