A cut-free modal theory of consequence

Abstract

The cut-free validity theory STV proposed by Barrio, Rosenblatt, and Tajer suffers from incompleteness with respect to its object language validity predicate. The validity predicate of STV fails in validating some valid inferences of its underlying logic, the Strict Tolerant logic ST. In this paper, we will present the non-normal modal logic ST[]<> whose modalities [] and <> capture the tautologies/valid inferences and the consistent formulas of the logic , respectively. We show that does not trivialize when extended with self-referential devices. We also show that such a solution poses a dilemma. If we extend ST[]<> in such a way that it allows iterated modal formulas among its theorems, then the resulting interpretation of [] as validity implies that metametainferences of ST behave like classical logic. On the other hand, if we allow these modalities to receive intermediate truth values, we obtain formulas incompatible with the proposed reading of [].