Logic, Unity in Three Persons

Abstract

This paper is mainly a review of some well-known facts concerning interconnections between such basic syntactic notions of logic as relation of logical consequence, consequence operator, and the lattice of theories under a logic. In doing so, we seek to provide evidence for the fact that, to define the logic syntactically, it is necessary and sufficient to define one of these three notions: namely, if one of them is defined, it unambiguously determines the other two. We consider in detail conditions that are both necessary and sufficient to prove the following statement: a closure operator generated by a class of sets of formulas can be interpreted as a consequence operator. To that end, we introduce the notion of a system of sets of formulas forming a lattice of theories. We prove that such a system defines a logic and consider some possible approaches to constructing such systems.
The paper draws attention to the fact that the most popular syntactic definitions of logics (such as sequent calculi, Frege-type calculi, closures of sets with respect to inference rules) can be equally well understood as defining relations of logical consequence, consequence operators and compact elements of the lattice of theories under a logic. DOI: 10.21146/2074-1472-2018-24-1-9-25