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
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.
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