Minimal and Subminimal Logic of Negation
Almudena Colacito
Abstract:
Starting from the original formulation of minimal propositional logic proposed by Johansson, this thesis aims to investigate some of its relevant subsystems. The main focus is on negation, defined as a primitive unary operator in the language. Each of the subsystems considered is defined by means of some ‘axioms of nega- tion’: different axioms enrich the negation operator with different properties. The basic logic is the one in which the negation operator has no properties at all, except the property of being functional. A Kripke semantics is developed for these subsystems, and the clause for negation is completely determined by a function between upward closed sets. Soundness and completeness with respect to this se- mantics are proved, both for Hilbert-style proof systems and for defined sequent calculus systems. The latter are cut-free complete proof systems and are used to prove some standard results for the logics considered (e.g., disjunction property, Craig’s interpolation theorem). An algebraic semantics for the considered sys- tems is presented, starting from the notion of Heyting algebras without a bottom element. An algebraic completeness result is proved. By defining a notion of descriptive frame and developing a duality theory, the algebraic completeness result is transferred into a frame-based completeness result which has a more generalized form than the one with respect to Kripke semantics.