Truth-relevant Logic, Propositional Calculus

TL;DR

This paper advocates for truth-relevant logic as a superior foundation for mathematics, introducing a three-valued system based on Kleene tables, and proves key tautological properties with a tableaux proof system.

Contribution

It introduces a truth-relevant logic framework with a three-valued system, providing a new proof system and philosophical justification for its use in mathematics.

Findings

Existence of tautologies with redundant variables that can be unknown.

Proof of a key tautology involving t-relevant variables.

Philosophical justification for truth-relevant logic.

Abstract

The thesis of this paper is that truth-relevant logic is a better foundation for mathematics than classical logic. It is a system proposed by Richard Diaz in 1981. In a certain sense t-relevant logic is based on Kleene strong tables. These define a system with three values: true, false, unknown. It turns out that there exist tautologies with the following property: there exists a proper subset of propositional variables (t-relevant variables) such that for all combinations true/false the tautology will be true, that is, the rest of the variables (redundant variables) occurring in the tautology can be unknown. We consider such compound sentences as neither true nor false. Philosophical justification is provided. Proof system based on tableaux is proposed. The following theorem is proved: tautology L = (R1 v ~R1) & (R2 v ~R2) & ... & (Rm v ~Rm), where {R1, R2, ... Rm} is a subset of t-relevant propositional variables, and = is equivalence according to Kleene strong tables.