We introduce two different calculi for a first-order extension of inquisitive pair semantics (Groenendijk 2008): Hilbert-style calculus and Tree-sequent calculus. These are first-order generalizations of (Mascarenhas 2009) and (Sano 2009), respectively. First, we show the strong completeness of our Hilbert-style calculus via canonical models. Second, we establish the completeness and soundness ...

The paper Proofs Without Syntax [Annals of Mathematics, to appear] introduced the notion of a combinatorial proof for classical propositional logic. The present paper uses combinatorial proofs to define a semantics for classical propositional sequent calculus, an inductive translation from sequent proofs to combinatorial proofs. The semantics is abstract and efficient: abstract in the sense tha...

We present a uniform algorithm for transforming machine-found matrix proofs in classical, constructive, and modal logics into sequent proofs. It is based on unified representations of matrix characterizations, of sequent calculi, and of prefixed sequent systems for various logics. The peculiarities of an individual logic are described by certain parameters of these representations, which are su...

We give a calculus of proof-nets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relationship between sequent proofs and proof-nets as an annotated sequent calculus, deriving formulae de...

A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn uniform interpolation, which is stronger than interpolation. We develop a constructive method proving via sequents apply it reprove the property $\mathsf{K...

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged rule. D. Prawitz calls such improper, and others proper. Improper more complicated often harder to understand than proper ones. present paper, we distinguish between improper derivations using sequent systems. Specifically, introduce a \(\vdash_{\bf S...

The logic of bunched implications (BI) of O’Hearn and Pym [5] is a substructural logic which freely combines additive connectives ⊃ , ∧, ∨ from propositional logic and multiplicative connectives −?, ? from linear logic. Because of its concise yet rich representation of states of resources, BI is regarded as a logic suitable for reasoning about resources. For example, by building a model for BI ...

. We discuss different options for two-sided sequent systems of noncommutative linear logic and prove a restricted form of cut elimination. By "classical Lambek logic" we denote a sequent system with sequences of propositional formulas on the right and left side of the sequent sign, which has no structural rule except cut. We credit this logic to J. Lambek since he was the first to investigate ...