Modal Logic

In classical propositional LOGIC all the operators are truth-functional. That is to say, the truth or falsity of a complex formula depends only on the truth or falsity of its simpler propositional constituents. Modal logic is concerned to understand propositions about what must or might be the case. We might, for example, have two propositions alike in truth value, both true say, where one is true and could not possibly be false, while the other is true but might easily have been false. Thus it must be that 2 + 2 = 4, but while it is true that I am writing this entry, it might easily not have been. Modal logic extends the well-formed formulas (wff) of classical logic by the addition of a one-place sentential operator L (or ), interpreted as meaning "It is necessary that." Using this operator, a one-place operator M (or ) meaning "It is possible that" may be defined as ~L~, where ~ is a (classical) negation operator, and a two-place operator meaning "entails" may be defined as = , where is classical material implication. In fact, any one of L, M, or can be taken as primitive, and the others defined in terms of it.

Although Lp is usually read as "Necessarily p," it need not be restricted to a single narrowly conceived sense of "necessarily." Very often, for example, when we say that something must be so, we can be taken to be claiming that it is so; and if we take L to express "must be" in this sense, we shall want to have it as a principle that whenever Lp is true, so is p itself. A system of logic that expresses this idea will have Lpp as one of its valid formulas. On the other hand, there are uses of words such as "must" and "necessary" that express not what necessarily is so but rather what ought to be so, and if we interpret L in accordance with these uses, we shall want to allow the possibility that Lp may be true but p itself false because people do not always do what they ought to do. And fruitful systems of logic have been inspired by the idea of taking the necessity operator to mean, for example, "It will always be the case that," "It is known that," or "It is provable that." In fact, one of the important features of modal logic is that out of the same basic material can be constructed a variety of systems that reflect a variety of interpretations of L, within a range that can be indicated, somewhat loosely, by calling L a "necessity operator."

In the early days of modal logic, disputes centered round the question of whether a given principle of modal logic was correct. Typically, these disputes involved formulas in which one modal operator occurs within the scope of another -- formulas such as Lp LLp. Is a necessary proposition necessarily necessary? A number of different modal systems were produced that reflected different views about which principles were correct. Until the early sixties, however, modal logics were discussed almost exclusively as axiomatic systems without access to a notion of validity of the kind used, for example, in the truth table method for determining the validity of wff of the classical propositional calculus. The semantical breakthrough came by using the idea that a necessary proposition is one true in all possible worlds. But whether another world counts as possible may be held to be relative to the world of origin. Thus an interpretation or model for a modal system would consist of a set W of possible worlds and a relation R of accessibility between them. For any wff and world w, L will be true at w iff itself is true at every w ' such that wRw'. It can then happen that whether a principle of modal logic holds can depend on properties of the accessibility relation. Suppose that R is required to be transitive, that is, suppose that, for any worlds w1, w2, and w3, if w1Rw 2 and w2Rw3, then w 1Rw3. If so, then Lp LLp will be valid, but if nontransitive models are permitted, it need not be. If R is reflexive, that is, if wRw for every world w, then Lp p is valid. Thus different systems of modal logic can represent different ways of restricting necessity.

It is possible to extend modal logic by having logics that involve more than one modal operator. One particularly important class of multimodal systems is the class of tense logics. A tense logic has two operators, L 1 and L2, where L1 means "it always will be the case that" and L2 means "it always has been the case that". (In a tense logic L1 and L2 are often written G and H, with their possibility versions as P for ~H~, and F for ~G~.) More elaborate families of modal operators are suggested by possible interpretations of modal logic in computer science. In these interpretations the "worlds" are states in the running of a program. If is a computer program, then [] means that after program has been run, a will be true. If w is any "world," then wR w' means that state w' results from the running of program . This extension of modal logic is called "dynamic logic."

First-order predicate logic can also be extended by the addition of modal operators. The most interesting consequences of such extensions are those which affect "mixed" principles, principles that relate quantifiers and modal operators and that cannot be stated at the level of modal propositional logic or nonmodal predicate logic. Thus where is any wff, is valid, but for some wff need not be. (Even if a game must have a winner, there need be no one who must win.) In some cases the principles of the extended system will depend on the propositional logic on which it is based. An example is the schema (often known as the "Barcan formula"), which is provable in some modal systems but not in others. If both directions are assumed, so that we have , then this formula expresses the principle that the domain of individuals is held constant as we move from one world to another accessible world.

When identity is added, even more questions arise. The usual axioms for identity easily allow the derivation of (x = y) L(x = y), but should we really say that all identities are necessary? Questions like this bring us to the boundary between modal logic and metaphysics and remind us of the rich potential that the theory of possible worlds has for illuminating such issues. POSSIBLE WORLDS SEMANTICS can be generalized to deal with any operators whose meanings are operations on propositions as sets of possible worlds, and form a congenial tool for those who think that the meaning of a sentence is its truth conditions, and that these should be taken literally as a set of possible worlds -- the worlds in which the sentence is true. Such generalizations give rise to fruitful tools in providing a framework for semantical theories for natural languages.

See also

Additional links

-- Max Cresswell

Further Readings

Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge: Cambridge University Press.

Hughes, G. E., and M. J. Cresswell. (1996). A New Introduction to Modal Logic. London: Routledge.