Abstract Constrained Input/Output (I/O) logics address scenarios involving conflicting conditional obligations. By allowing the withdrawal of norms to preserve consistency, these logics exhibit a close relationship with default logics. In this paper, we provide a formal account of this relationship to develop a uniform Gentzen-style proof theory for the entire family of constrained I/O logics under the credulous approach. Specifically, we introduce hypersequent calculi that integrate extra-logical rules to directly capture conditional obligations. The parallel composition of sequents and antisequents formalizes the dynamic updating of conclusions under consistency constraints. Crucially, such an approach avoids any ad hoc extension of the underlying language. Moreover, we establish the admissibility of structural rules and the invertibility of logical rules, showing that cut-free proofs maintain a weakened form of analyticity. Finally, we leverage straightforward translations between hypersequent calculi for constrained I/O logics and those for default logics, as developed in Piazza and Sabatini (2025, ACM Trans. Comput. Log., 26, 1–36), to provide a modular treatment of disjunctive default logics and disjunctive normative inference.

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.

In this article, we investigate default reasoning from a structural proof-theoretic perspective. We introduce hybrid hypersequent calculi for propositional default logics, where extra-logical rules directly capture default rules, while parallel composition of sequents and antisequents formalizes contrary updating on the conclusions of extra-logical rules. We establish the admissibility of structural rules and the invertibility of logical rules, showing that cut-free proofs exhibit a weakened form of analyticity. Next, we prove that specific hybrid hypersequent calculi are sound and weakly complete with respect to credulous consequence based on Łukaszewicz extensions. Lastly, we propose a hypersequent-based decision method for skeptical consequence which circumvents the need for early computation of all extensions.

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.

Abstract In this paper, we propose and explore a new approach to abstract machines and optimal reduction via streams, infinite sequences of elements. We first define a sequential abstract machine capable of performing directed virtual reduction (DVR) and then we extend it to its parallel version, whose equivalence is explained through the properties of DVR itself. The result is a formal definition of the λ-calculus interpreter called Parallel Environment for Lambda Calculus Reduction (PELCR), a software for λ-calculus reduction based on the Geometry of Interaction. In particular, we describe PELCR as a stream-processing abstract machine, which in principle can also be applied to infinite streams.

The goal of this article is to design a uniform proof-theoretical framework encompassing classical, non-monotonic and paraconsistent logic. This framework is obtained by the control sets logical device, a syntactical apparatus for controlling derivations. A basic feature of control sets is that of leaving the underlying syntax of a proof system unchanged, while affecting the very combinatorial structure of sequents and proofs. We prove the cut-elimination theorem for a version of controlled propositional classical logic, i.e. the sequent calculus for classical propositional logic to which a suitable system of control sets is applied. Finally, we outline the skeleton of a new (positive) account of non-monotonicity and paraconsistency in terms of concurrent processes.

Abstract In this paper, we show by a proof-theoretical argument that in a logic without structural rules, that is in noncommutative linear logic with exponentials, every formula A for which exchange rules (and weakening and contraction as well) are admissible is provably equivalent to? A. This property shows that the expressive power of “noncommutative exponentials” is much more important than that of “commutative exponentials”.

In this note, we show by a proof-theoretical argument that in full linear logic the set of formulas for which contraction and weakening are admissible (the set of saturated formulas) does not coincide (up to equivalences) with the set of exponentiated formulas. This solves an open problem of Schellinx.

In this paper we introduce and investigate new examples of concrete quantales and we show their potential use, in a framework of algebraic semantics, by giving a characterization of the formulas in multiplicative fragment of linear logic for which the dismissed weakening and contraction are admissible.