  Thus, by imposing an adequate equivalence relation upon proofs, any deductive system can be turned into a category. It is therefore legitimate to think of a category as an algebraic encoding of a deductive system. This phenomenon is already well-known to logicians, but probably not to its fullest extent. An example of such an algebraic encoding is the Lindenbaum-Tarski algebra, a Boolean algebra corresponding to classical propositional logic. Since a Boolean algebra is a poset, it is also a category. (Notice also that Boolean algebras with appropriate homomorphisms between them form another useful category in logic.) Thus far we have merely a change of vocabulary. Things become more interesting when first-order and higher-order logics are considered. The Lindenbaum-Tarski algebra for these systems, when properly carried out, yields categories, sometimes called “conceptual categories” or “syntactic categories” (Mac Lane &amp; Moerdijk 1992, Makkai &amp; Reyes 1977, Pitts 2000).