Abstract
Minimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the most important ones are contraposition logic and negative ex falso, a weak form of the ex falso principle. Completeness is proved, and the created semantics is further studied. The negative translation of classical logic into intuitionistic logic is made part of a chain of translations by introducing translations from minimal logic into contraposition logic and intuitionistic logic into minimal logic, the latter having been discovered in the correspondence between Johansson and Heyting. Finally, as a bridge to the work of Franco Montagna a start is made of a study of linear models of these logics.
Highlights
We study minimal logic in its two equivalent formulations: one with a basic symbol for the contradiction the other with a basic symbol for negation
Given a countable set of propositional variables, the formulation used nowadays is based on the propositional language of the positive fragment of intuitionistic logic, i.e., L+ = {∧, ∨, →}, with an additional propositional constant f, representing falsum
If IPC+ denotes the positive fragment of intuitionistic logic, minimal logic has the same axioms as IPC+, and f does not have the same properties as the intuitionistic ⊥
Summary
We study minimal logic in its two equivalent formulations: one with a basic symbol for the contradiction the other with a basic symbol for negation. Given a countable set of propositional variables, the formulation used nowadays is based on the propositional language of the positive fragment of intuitionistic logic, i.e., L+ = {∧, ∨, →}, with an additional propositional constant f , representing falsum In this setting, negation of φ is defined as φ → f and denoted by ¬φ. The main purpose of the paper is to study a weak form of negation, considering subsystems of minimal logic while keeping the IPC+ axioms fixed. We call such forms of negation subminimal negation. We present in this paper minimal logic, CoPC and its subsystems as paraconsistent variations of intuitionistic logic
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