Sequent Calculus System Research Articles

Non-classical Elegance for Sequent Calculus Enthusiasts

In this paper we develop what we can describe as a "dual two-sided" cut-free sequent calculus system for the non-classical logics of truth lp, k 3, stt and a non-reflexive logic ts which is, arguably, more elegant than the three-sided sequent calculus developed by Ripley (The Review of Symbolic Logic 5:354---378, 2012) for the same logics. Its elegance stems from how it employs more or less the standard sequent calculus rules for the various connectives and truth, and the fact that it offers a rather neat connection between derivable sequents and validity in comparison to the calculus developed by Ripley (The Review of Symbolic Logic 5:354---378, 2012).

Cut Elimination for GLS Using the Terminability of its Regress Process

The system GLS, which is a modal sequent calculus system for the provability logic GL, was introduced by G. Sambin and S. Valentini in Journal of Philosophical Logic, 11(3), 311–342, (1982), and in 12(4), 471–476, (1983), the second author presented a syntactic cut-elimination proof for GLS. In this paper, we will use regress trees (which are related to search trees) in order to present a simpler and more intuitive syntactic cut derivability proof for GLS1, which is a (more connectively and inferentially economical) variant of GLS without the cut rule.

HOW A SEMANTICS FOR TONK SHOULD BE

AbstractThis paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion.

A logical basis for constructive systems

The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article elaborates some basic complexity theory for the CoL framework. Then it proves soundness and completeness for the deductive system CL12 with respect to the semantics of CoL, including the version of the latter based on polynomial time computability instead of computability-in-principle. CL12 is a sequent calculus system, where the meaning of a sequent intuitively can be characterized as "the succedent is algorithmically reducible to the antecedent", and where formulas are built from predicate letters, function letters, variables, constants, identity, negation, parallel and choice connectives, and blind and choice quantifiers. A case is made that CL12 is an adequate logical basis for constructive applied theories, including complexity-oriented ones.

Maude as a Platform for Designing and Implementing Deep Inference Systems

Deep inference is a proof theoretical methodology that generalizes the traditional notion of inference in the sequent calculus: in contrast to the sequent calculus, the deductive systems with deep inference do not rely on the notion of main connective, and permit the application of the inference rules at any depth inside logical expressions, in a way which resembles the application of term rewriting rules. Deep inference provides a richer combinatoric analysis of proofs for different logics. In particular, construction of exponentially shorter proofs becomes possible. In this paper, aiming at the development of computation as proof search tools, we propose the Maude language as a means for designing and implementing different deep inference deductive systems and proof strategies that work on these systems. We give Maude implementations of deep inference systems together with an implementation that simulates sequent calculus proofs to serve as a benchmark. We demonstrate these ideas on classical logic, and argue that they can be analogously carried to other deep inference systems for other logics, as well as sequent calculus systems.

Admissibility of structural rules for extensions of contraction-free sequent calculi

The contraction-free sequent calculus G4 for intuitionistic logic is extended by rules following a general rule-scheme for nonlogical axioms. Admissibility of structural rules for these extensions is proved in a direct way by induction on derivations. This method permits the representation of various applied logics as complete, contraction- and cut-free sequent calculus systems with some restrictions on the nature of the derivations. As specific examples, intuitionistic theories of apartness and order are treated.

A Sequent Calculus for Subtyping Polymorphic Types

We present two complete systems for polymorphic types with sub- typing. One system is in the style of natural deduction, while another is a Gentzen-style sequent calculus system. We prove several meta- mathematical properties for these systems including cut elimination, subject reduction, coherence, and decidability of type reconstruction. Following the approach by J. Mitchell, the sequents are given a simple semantics using logical relations over applicative structures. The systems are complete with respect to this semantics. The logic which emerges from this paper can be seen as a successor to the original Hilbert style system proposed by J. Mitchell in 1988, and to the “half way” sequent calculus of G. Longo, K. Milsted, and S. Soloviev proposed in 1995.

A multimodal logic for reasoning about complementarity

ABSTRACT Two objects o1, o2 of an information system are said to be complementary with respect to attribute a if α(o1) = -α(o2), where α(o) is the set of values of attribute a assigned to o. They are said to be complementary with respect to a set of attributes A if they are complementary with respect to each attribute α ε A. A multi-modal logical language for reasoning about complementarity relations is presented, with modalities [A] and ⟨A⟩ parameterised by subsets of a given set ATTR of attributes. Two complete deduction system for the language are presented: a Rasiowa-Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories.