Some embedding theorems for modal logic.
David Makinson · Notre Dame Journal of Formal Logic · 1971
We* shall prove some embedding theorems for modal logic, that is, theorems to the effect that every consistent modal logic satisfying certain general conditions is a sub logic of certain particular logics.Our results are related to those of McKinsey [1] and Tarski [2], We begin with terminology.Formulae are understood to be built from a denumerable list of elementary letters by means of the operators i, Λ, D, with other operators introduced as usual.By a modal logic we mean any set S of formulae that contains all the tautologies in i and Λ and is closed under the operations of substitution (of arbitrary formulae for elementary letters) and detachment (a, a^β/β).We say that a set S of formulae is closed under congruence if whenever (a= β) e S then (Πa = Oβ) e S; closed under monotony if whenever (a^> β) e S then (Dα^Π/3)eS; closed under antitony if whenever (a^> β) e S then (D β3 D a) e S.By a modal algebra we mean a structure 8 = (A, -, Π, *) where (A, -, (Ί) is a Boolean algebra and * is a unary operation over A. A modal algebra is said to be monotonic if for all x, ye A, x^y implies *ΛΓ^ *y, and is said to be antitonic if for all x, y e A, x^ y implies *;y<*#.Among the modal algebras there are clearly just four that can be obtained by adding a unary operation to the two-element Boolean algebra: we shall call these the unit algebra (*1 = 1, *0 = 1), the identity algebra (*1 = 1, *0 = 0), the complement algebra (*1 = 0, *0 = 1), and the zero algebra (*1 = 0, *0 = 0).Each of these four algebras determines a corresponding set of formulae, consisting of just those formulae that are valid in the algebra, that is, just those formulae a such that for every homomorphism h from formulae into that algebra, h(a) = 1.It is easy to verify that each of these four sets of formulae is a modal logic in the sense defined, is closed either under monotony or under antitony, and can be axiomatized in a trivial way: we refer to these four sets of formulae as the unit, identity, complement, and zero modal logics respectively.