Abstract

The cut-free validity theory $$\textsf{STV}$$ proposed by Barrio, Rosenblatt, and Tajer suffers from incompleteness with respect to its object language validity predicate. The validity predicate of $$\textsf{STV}$$ fails in validating some valid inferences of its underlying logic, the Strict Tolerant logic $$\textsf{ST}$$. In this paper, we will present the non-normal modal logic $$\textsf{ST}^{\Box \Diamond }$$ whose modalities $$\Box $$ and $$\Diamond $$ capture the tautologies/valid inferences and the consistent formulas of the logic $$\textsf{ST}$$, respectively. We show that $$\textsf{ST}^{\Box \Diamond }$$ does not trivialize when extended with self-referential devices. We also show that such a solution poses a dilemma. If we extend $$\textsf{ST}^{\Box \Diamond }$$ in such a way that it allows iterated modal formulas among its theorems, then the resulting interpretation of $$\Box $$ as validity implies that metametainferences of $$\textsf{ST}$$ 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 $$\Box $$.