Abstract In this paper, a new approach to the issue of extra-logical information within analytic (i.e. obeying the sub-formula property) sequent systems is introduced. We prove that incorporating extra-logical axioms into a purely logical system can preserve analyticity, provided these axioms belong to a suitable class of formulas that can be decomposed into a set of equivalent initial sequents and are permutable over the cut rule. Our approach is applicable not only to first-order classical and intuitionistic logics, but also to substructural logics. Furthermore, we establish a limit for the augmented systems under analysis: exceeding the boundaries of their respective classes of extra-logical axioms leads to either a loss of analyticity or a loss of structural properties.

Abstract Enhancing cut-free expressiveness through minimal structural additions to sequent calculus is a natural step. We focus on Åqvist’s system $$\textbf{F}$$ F with cautious monotonicity (CM), a deontic logic extension of $$\textbf{S5}$$ S 5 , for which we define a sequent calculus employing (semi) analytic cuts.The transition to hypersequents is key to develop modular and cut-free calculi for $$\mathbf{F + (CM)}$$ F + ( CM ) and $$\textbf{G}$$ G , also supporting countermodel construction.

Abstract Sequent-style refutation calculi with non-invertible rules are challenging to design because multiple proof-search strategies need to be simultaneously verified. In this paper, we present a refutation calculus for the multiplicative–additive fragment of linear logic ($\textsf{MALL}$) whose binary rule for the multiplicative conjunction $(\otimes )$ and the unary rule for the additive disjunction $(\oplus )$ fail invertibility. Specifically, we design a cut-free hypersequent calculus $\textsf{HMALL}$, which is equivalent to $\textsf{MALL}$, and obtained by transforming the usual tree-like shape of derivations into a parallel and linear structure. Next, we develop a refutation calculus $\overline{\textsf{HMALL}}$ based on the calculus $\textsf{HMALL}$. As far as we know, this is also the first refutation calculus for a substructural logic. Finally, we offer a fractional semantics for $\textsf{MALL}$—whereby its formulas are interpreted by a rational number in the closed interval [0, 1] —thus extending to the substructural landscape the project of fractional semantics already pursued for classical and modal logics.

The Gödel-McKinsey-Tarski embedding allows to view intuitionistic logic through the lenses of modal logic. In this work, an extension of the modal embedding to infinitary intuitionistic logic is introduced. First, a neighborhood semantics for a family of axiomatically presented infinitary modal logics is given and soundness and completeness are proved via the method of canonical models. The semantics is then exploited to obtain a labelled sequent calculus with good structural properties. Next, soundness and faithfulness of the embedding are established by transfinite induction on the height of derivations: the proof is obtained directly without resorting to non-constructive principles. Finally, the modal embedding is employed in order to relate classical, intuitionistic and modal derivability in infinitary logic extended with axioms.

Abstract Neighbourhood semantics for intuitionistic logic extended with countable conjunctions and disjunctions is introduced and shown equivalent to topological semantics, with an indirect completeness proof as payoff. The new semantics is used to obtain a labelled sequent calculus with good structural properties. In particular, admissibility of weakening and contraction, invertibility with preservation of height for each rule and cut elimination are shown. Finally, a direct Tait–Schütte–Takeuti style form of completeness via the extraction of a countermodel from a failed proof search is proved.