A cut-free modal theory of consequence

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