Sequent Calculus Rules Research Articles

Semantic Information and the Complexity of Deduction

AbstractIn the chapter “Information and Content” of their Impossible Worlds, Berto and Jago provide us with a semantic account of information in deductive reasoning such that we have an explanation for why some, but not all, logical deductions are informative. The framework Berto and Jago choose to make sense of the above-mentioned idea is a semantic interpretation of Sequent Calculus rules of inference for classical logic. I shall argue that although Berto and Jago’s idea and framework are hopeful, their definitions do not capture what is intended. This is so because the definitions are solely based on the logical complexity of an argument and they fail to capture the richness of the non-logical content of that argument. Then I will suggest some amendments to address this problem. Finally, I will extend the application of the definitions to first-order logic. It shall be observed that in some classical deductions, applying contraction may lead to hiding some epistemic impossibilities. This happens when formulae which contribute different propositions to the proof get contracted at the quantified level.

CUT ELIMINATION AND NORMALIZATION FOR GENERALIZED SINGLE AND MULTI-CONCLUSION SEQUENT AND NATURAL DEDUCTION CALCULI

AbstractAny set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for natural deduction proofs as in the Curry–Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases.

Functional Completeness in CPL via Correspondence Analysis

Kooi and Tamminga's correspondence analysis is a technique for designing proof systems, mostly, natural deduction and sequent systems. In this paper it is used to generate sequent calculi with invertible rules, whose only branching rule is the rule of cut. The calculi pertain to classical propositional logic and any of its fragments that may be obtained from adding a set (sets) of rules characterizing a two-argument Boolean function(s) to the negation fragment of classical propositional logic. The properties of soundness and completeness of the calculi are demonstrated. The proof of completeness is conducted by Kalmár's method.
 Most of the presented sequent-calculus rules have been obtained automatically, by a rule-generating algorithm implemented in Python. Correctness of the algorithm is demonstrated. This automated approach allowed us to analyse thousands of possible rules' schemes, hundreds of rules corresponding to Boolean functions, and to nd dozens of those invertible. Interestingly, the analysis revealed that the presented proof-theoretic framework provides a syntactic characteristics of such an important semantic property as functional completeness.

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).

Symmetry vs. Duality in Logic

The author discusses the problem of symmetry, namely of the orientation of the logical consequence. The author shows that the problem is surprisingly entangled with the problem of “being infinite”. The author presents a model based on quantum states and shows that it features satisfy the requirements of the symmetric mode of Bi-logic, a logic introduced in the '70s by the psychoanalyst I. Matte Blanco to describe the logic of the unconscious. The author discusess symmetry, in the model, to include correlations, in order to obtain a possible approach to displacement. In this setting, the author finds a possible reading of the structural rules of sequent calculus, whose role in computation, on one side, and in the representation of human reasoning, on the other, has been debated for a long time.

Algebraic proof theory for substructural logics: Cut-elimination and completions

We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices. We introduce the substructural hierarchy — a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of cut-elimination for extensions of FL and the MacNeille completion for subvarieties of pointed residuated lattices coincide up to the level N 2 in the hierarchy. Negative results, which indicate limitations of cut-elimination and the MacNeille completion, as well as of the expressive power of structural sequent calculus rules, are also provided. Our arguments interweave proof theory and algebra, leading to an integrated discipline which we call algebraic proof theory.

Deep Inference and Probabilistic Coherence Spaces

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BV-category is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of probabilistic coherence spaces and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.

Varieties of Linear Calculi

A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and sequent calculus that are obtained as special instances from the uniform calculus. Other instances give all the invertibilities and partial invertibilities for the sequent calculus rules of linear logic. The calculus is normalizing and satisfies the subformula property for normal derivations.