A fundamental non-classical logic

TL;DR

This paper introduces a fundamental non-classical logic with a proof system based solely on introduction and elimination rules, providing algebraic and relational semantics that unify intuitionistic, orthologic, and classical logics.

Contribution

It presents a new foundational logic with a Fitch-style natural deduction system and algebraic semantics based on bounded lattices with weak pseudocomplementation, unifying various logical systems.

Findings

The base logic is characterized by introduction and elimination rules only.

Adding Reiteration yields intuitionistic logic; adding Reductio yields orthologic; adding both yields classical logic.

The algebraic semantics are based on bounded lattices with weak pseudocomplementation, representable via relational structures.

Abstract

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.