Implicational propositional calculus

Abstract: In this paper we consider propositional calculi, which are finitely axiomatizableextensions of intuitionistic implicational propositional calculus together with the rulesof modus ponens and substitution. We give a proof of undecidability of the followingproblem for these calculi: whether a given finite set of propositional formulas consti-tutes an adequate axiom system for a fixed propositional calculus. Moreover, we provethe same for the following restriction of this problem: whether a given finite set oftheorems of a fixed propositional calculus derives all theorems of this calculus. Theproof of these results is based on a reduction of the undecidable halting problem forthe tag systems introduced by Post. Keywords: Classical and intuitionistic propositional calculi, implicational calculus, finiteaxiomatization, tag system. 1 Introduction In general, a propositional calculus is given by a finite set of propositional formulas oversome signature together with a finite set of rules of inferences. The problem of recognizingaxiomatizations for a propositional calculus is formulated as follows: whether a given finiteset of propositional formulas constitutes (axiomatizes) an adequate axiom system for thiscalculus, i.e., each formula of the calculus is derivable from a given set of formulas by therules of the calculus. The question of decidability of this problem was proposed by Tarski in1946 [13]. In this paper we consider only the propositional calculus with the rules of modusponens and substitution.The undecidability of recognizing axiomatizations for the classical propositional calculuswas obtained due to Linial and Post in 1949 [7]. They gave sketch of proofs for a numberof results, one of them expressible in the form that it is undecidable whether a given finiteset of propositional formulas axiomatizes all classical tautologies. Note that they considered1

Abstract: We study $Q$-tableaux and axiom systems that they engender, producing a new proof that the Implicational Propositional Calculus is complete.