Undecidable iterative propositional calculus

TL;DR

This paper constructs an undecidable iterative propositional calculus with axioms in three variables, demonstrating that derivability in such calculi cannot be algorithmically decided.

Contribution

It introduces a specific undecidable iterative propositional calculus and proves the general derivability problem is undecidable.

Findings

Constructed an undecidable iterative propositional calculus

Proved the derivability problem is undecidable for these calculi

Uses axioms in three variables

Abstract

In this paper, we consider iterative propositional calculi, which are finite sets of propositional formulas together with the rules of modus ponens and weak substitution (when formula being substituted must be already inferred). We construct an undecidable iterative propositional calculus using axioms in 3 variables. Particulary, we show that the general problem of derivability for these calculi is undecidable.