Refutation Systems in Propositional Logic

Dual Erotetic Calculi and the Minimal $${\mathsf{LFI}}$$ LFI

“…Observe that, at any stage of this process, if the answer to a certain question is negative (that is, one cannot refute formula(s) concerned by the question), then the answer to the previous question must be 4 The difference between the notion of refutability considered here and the one known from the tradition of refutation systems (see e.g. [22]) is clear in this context, as one can have p and ¬p refuted in a refutation system (both are non-theses), but one cannot have them both falsified under the same valuation. In other words, the rules of refutation systems preserve non-validity, whereas the rules presented in this paper preserve-as we shall see-falsifiability under a valuation.…”

On Refutation Rules

“…However, normal forms based on clausal forms (see [4,6]) are more useful. They are both simple and general.…”

“…Of course, axiom systems are notoriously bad for proof search. And so are refutation systems containing the rules: R s and R L mp (although they have some interesting theoretical aspects (see [5,6])). However, the rule R d is more promising for proof search (or, rather, refutation search).…”

“…In this paper we study refutation rules of this kind. We are interested in refutation rules of the form F 1 · · · F k F where F is a certain normal form of rank n > 0 and each F i is a normal form of rank < n. Specific rules of this kind can be extracted from [4,6]. Here we generalize these techniques obtaining a simple method of proving that a refutation system is complete for a logic L (that is, it axiomatizes the complement −L of L).…”

Simple Decision Procedure for S5 in Standard Cut-Free Sequent Calculus

“…On the other hand, different approaches to the problem of decision procedure for S5 were proposed on the ground of tableau methods (Kripke [15], Zeman [36], Fitting [9], Goré [11]), refutation metods (Skura [30]), resolution (Farinas del Cerro and Herzig [8]) or application of normal forms. Tableau-based approaches are quite close to approaches based on (cut-free) SC and may be essentially treated as belonging to the same group.…”