Introduction

Algebraic logic hi the modern sense can be said to have begun with Tarski's

1935 paper [43] on the foundations of the calculus of systems. Here axe found,

clearly discernible for the first time, the characteristic features of the subject

wre recognize today. In the paper Tarski introduced the algebra of propositional

formulas. He defined a relation = on the set of formulas by the condition

/= x/; & \- (/) - ip and h xj) - £, (1)

and asserted that = forms what we now call a congruence relation on the

algebra of formulas. He then went on to say in effect that the corresponding

quotient algebra

wras

a Boolean algebra, and that the theorems of logic (i.e.,

the tautologies) coincided exactly with the formulas equivalent to T (or some

fixed but arbitrarily chosen tautology). Conversely, Tarski indicated how a

deductive system for classical propositional logic can be constructed from any

axiomatization of Boolean algebras. Subsequently a number of different non-

classical propositional logics were algebraized in this way, the most important

being the intuitionistic logic of Heyting, the multiple-valued logics of Post and

Lukasiewicz, and the modal logics S4 and S5 of Lewis. Tarski himself directed

an extensive research program on the algebraization of the classical first-order

predicate logic.

The quotient algebra obtained by factoring the algebra of formulas by the

congruence (1) has come to be known as the Tarski-Lindenbaum algebra of the

logic, and the study of an algebraizable logic can to a large extent be reduced

to the study of this algebra. There is for example a correspondence between

theorems and algebraic identities that allows the deductive apparatus of each

algebraizable logic to be interpreted in the equational theory of its Tarski-

Lindenbaum algebra. Many higher order metalogical notions also have natural

algebraic interpretations; such an interpretation of the deduction theorem is

given in [7]. Consequently, when a logic is algebraizable, the powerful methods

of modern algebra can be used in its investigation, and this has had a profound

influence on the development of these logics.1

1 This is certainly true of the non-classical propositional logics. Algebraic methods have

not had as yet a strong impact on metalogical investigations of predicate logic, except in

some specialized areas.

I