Many Faces of Logic
Dirk Hofmann, Manuel A. Martins · 2009
In this paper we present logic from various perspectives, starting from the standard way typically taught in an undergraduate course. We expose the relation- ship with other mathematical structures, namely closure relations, closure operators, coalgebras and bialgebras. more intuitive and therefore more often used in class room, while the latter considers only the formal properties of deduction ignoring the structure of formulae. The algebraic approach to sentential logic is very powerful, since it allows the use of tools and results from universal algebra (e.g. ultraproducts) to study logical systems. One important methodology is the classical Lindenbaum-Tarski process which associates to a sentential logic a class of algebras. Paradigmatic examples are the Boolean algebras in clas- sical propositional logic and Heyting algebras for the intuitionistic propositional logic. These classes of algebras can be viewed as the algebraic counterpart of its corresponding logic in the sense that there is a close relationship between the deductive theory of the logic and the equational theory of the algebras. Abstract algebraic logic goes further, the focus is no longer on the algebraic form of specic logical systems, but on the process of algebraisation itself (cf. (1)). In the second part of this paper we present logic (to be more precise: a consequence relation) on an abstract set rather then a set of formulae. We exhibit various ways to encode this structure which have roots in dierent elds of mathematics, namely topology and (co)algebra. We emphasise that properties of these structures and of maps between them can be expressed by simply (in)equalities and with the help of suitable dened compositions, which is useful when transporting notions or ideas from one structure to the other since the transition maps preserve both composition and inequalities. Our presentation here rests partially on general results of (8).