Logic, Modal

the branch of logic devoted to the study of modalities, the construction of calculi in which modalities are applied to propositions, in addition to logical operations, and the comparative study of such calculi. Modal operators, such as “possible” and “necessary,” may refer to propositions or predicates and to words that express certain actions or acts. Interest in modal logic is chiefly due to the natural relation between modalities such as “necessary” and the concept of logical law (that is, an identically valid proposition of some logical system), on the one hand, and between modalities such as “possible” and such epistemological and general scientific concepts as “(effectively) realizable” and “calculable,” on the other.

In classical systems of modal logic (for which the law of the excluded middle A V ┐ A or the law of double negation ┐ ┐ A ⊃ A is valid), duality relations—analogous to De Morgan’s laws ┐(A V B)↔(┐A & ┐B) and ┐(A & B)↔(┐A V ┐B) of the algebra of logic and to the corresponding equivalencies for quantifiers — relate the possibility operator ✧ and the necessity operator ☐ to negation ┐ obtain for modalities:

☐A ↔ ┐ ✧ ┐A and ✧ A ↔ ┐ ☐ ┐A

Therefore, one modal operation is usually introduced as the initial operation in axiomatic systems of modal logic (by using one of these equivalencies to define the other operation). Other modal operations— which are not logical operations and cannot be expressed in terms of them —also are introduced in a similar manner.

The systems of modal logic may be interpreted in terms of many-valued logic. The simplest systems may be interpreted as three-valued systems: “true,” “false,” “possible.” This fact, as well as the possibility of applying modal logic to the construction of a theory of “probable” conclusions, points to its strong kinship with probability logic.

In addition to the “absolute” modalities considered above, modal logic also deals with relative modalities— that is, modalities linked to certain conditions, such as “if B, then A is possible.” The formalization of the rules for dealing with such modalities does not create additional difficulties and is carried out by means of restricted quantifiers (using predicates expressing restrictions and the logical operations of material implication).