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 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.

The work reported in this paper refers to Massey's proof of the surface classification theorem based on the standard word-rewriting treatment of surfaces. We arrange this approach into a formal rewriting system and provide a new version of Massey's argument. Moreover, we study the computational properties of two subsystems of : orfor dealing with words denoting orientable surfaces and norfor dealing with words denoting non-orientable surfaces. We show how such properties induce an alternative proof for the surface classification in which the basic homeomorphism between the connected sum of three projective planes and the connected sum of a torus with a projective plane is not required.