A Note on Always Decidable Propositional Forms

TL;DR

This paper investigates whether propositional formulas that are decidable under all instantiations or have a computable decision procedure are necessarily tautologies or contradictions, affirming this under certain conditions.

Contribution

It proves that if all instantiations of a propositional formula are decidable in a strong recursive theory, then the formula is tautological or contradictory, establishing a new criterion for propositional form decidability.

Findings

Decidability of all instantiations implies tautology or contradiction.

Existence of a Turing program deciding truth values leads to the same conclusion.

Supports the idea that decidability across all instances indicates trivial propositional forms.

Abstract

We ask the following question: If all instantiations of a propositional formula $A(x_1,...,x_n)$ in $n$ propositional variables are decidable in some sufficiently strong recursive theory, does it follow that $A$ is tautological or contradictory? and answer it in the affirmative. We also consider the following related question: Suppose that for some propositional formula $A(x_1,...,x_n)$, there is a Turing program $P$ such that $P([\phi_{1}],...,[\phi_{n}])\downarrow=1$ iff $\mathbb{N}\models A(\phi_{1},...,\phi_{n})$ and otherwise $P([\phi_{1}],...,[\phi_{n}])\downarrow=0$ (where $[\phi]$ denotes the G\"odel number of $\phi$), does it follow that the truth value of $A(\phi_{1},...,\phi_{n})$ is independent of $\phi_1,...,\phi_{n}$ and hence that $A$ is tautological or contradictory?