scholarly journals Generalized quantification as substructural logic

1996 ◽  
Vol 61 (3) ◽  
pp. 1006-1044 ◽  
Author(s):  
Natasha Alechina ◽  
Michiel Van Lambalgen
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Substructural Logic ◽  
Generalized Quantifiers ◽  
Accessibility Relation ◽  
Dependence Relation ◽  
Structural Rules ◽  
First Order ◽  
Proof Systems ◽  
Structured Objects

AbstractWe show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

Introduction and Overview

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Scientific Method ◽  
Proof Theory ◽  
Valid Argument ◽  
First Order ◽  
Proof Systems ◽  
Formal Systems ◽  
Deductive Logic ◽  
Order Language ◽  
Foundational Work ◽  
The Right

This is a foundational work, written not just for philosophers of logic, but for logicians and foundationalists generally. Like Frege we seek to deal with the formal first-order language of mathematics. We revisit Gentzen’s proof theory in order to build relevance into proofs, while leaving intact all the logical power one is entitled to expect of a deductive logic for mathematics and for scientific method generally. Proof systems are constituted by particular choices of rules of inference. We raise the issue of the reflexive stability of any argument for a particular choice of logic as the ‘right’ logic. We examine the question of pluralism v. absolutism in choice of logic, and suggest that the informal notion of valid argument is stable and robust enough for us to be able to ‘get it right’ with our formal systems of proof for both constructive and non-constructive reasoning.

Complexity of translations from resolution to sequent calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1061-1091
Author(s):  
GISELLE REIS ◽  
BRUNO WOLTZENLOGEL PALEO
Keyword(s):  
Automated Reasoning ◽  
Theoretical Basis ◽  
Automated Theorem Proving ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Formal Proof ◽  
Human Beings ◽  
Proof Systems ◽  
Translation Methods ◽  
Resolution Calculus

Resolution and sequent calculus are two well-known formal proof systems. Their differences make them suitable for distinct tasks. Resolution and its variants are very efficient for automated reasoning and are in fact the theoretical basis of many theorem provers. However, being intentionally machine oriented, the resolution calculus is not as natural for human beings and the input problem needs to be pre-processed to clause normal form. Sequent calculus, on the other hand, is a modular formalism that is useful for analysing meta-properties of various logics and is, therefore, popular among proof theorists. The input problem does not need to be pre-processed, and proofs are more detailed. However, proofs also tend to be larger and more verbose. When the worlds of proof theory and automated theorem proving meet, translations between resolution and sequent calculus are often necessary. In this paper, we compare three translation methods and analyse their complexity.

scholarly journals SOME OBSERVATIONS ABOUT GENERALIZED QUANTIFIERS IN LOGICS OF IMPERFECT INFORMATION

2019 ◽  
Vol 12 (3) ◽  
pp. 456-486 ◽  
Author(s):  
FAUSTO BARBERO
Keyword(s):  
Imperfect Information ◽  
Proof Theory ◽  
Higher Order ◽  
Generalized Quantifiers ◽  
First Order ◽  
Generalized Quantifier ◽  
Team Semantics ◽  
Special Cases ◽  
Qualitative Component ◽  
Definition Of

AbstractWe analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engström, comparing them with a more general, higher order definition of team quantifier. We show that Engström’s definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engström’s quantifiers only range over the latter. We further argue that Engström’s definitions are just embeddings of the first-order generalized quantifiers into team semantics, and fail to capture an adequate notion of team-theoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engström’s quantifiers.

scholarly journals Pretopologies and completeness proofs

1995 ◽  
Vol 60 (3) ◽  
pp. 861-878 ◽  
Author(s):  
Giovanni Sambin
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Double Negation ◽  
Open Problems ◽  
Completeness Proof ◽  
Structural Rules ◽  
First Order ◽  
Definition Of

Pretopologies were introduced in [S], and there shown to give a complete semantics for a propositional sequent calculus BL, here called basic linear logic, as well as for its extensions by structural rules, ex falso quodlibet or double negation. Immediately after Logic Colloquium '88, a conversation with Per Martin-Löf helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for first order BL and virtually all its extensions, including the usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as the usual two-valued models.To make the paper self-contained, I briefly review in §1 the definition of pretopologies; §2 deals with syntax and §3 with semantics. The completeness proof in §4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In §5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in §6 connections with proofs with respect to more traditional semantics are briefly investigated, and some open problems are put forward.

LINEAR TIME IN HYPERSEQUENT FRAMEWORK

2016 ◽  
Vol 22 (1) ◽  
pp. 121-144 ◽  
Author(s):  
ANDRZEJ INDRZEJCZAK
Keyword(s):  
Proof Theory ◽  
Linear Time ◽  
Expressive Power ◽  
Sequent Calculi ◽  
Temporal Logics ◽  
Structural Rules ◽  
Proof Systems ◽  
Nonclassical Logics ◽  
Time Flow ◽  
Basic Calculus

AbstractHypersequent calculus (HC), developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi (SC). In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and provide a cut-free HC formalization ofKt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite (multi)sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules (three for future-necessity and three dual rules for past-necessity). Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents.

PRESERVATION OF STRUCTURAL PROPERTIES IN INTUITIONISTIC EXTENSIONS OF AN INFERENCE RELATION

2018 ◽  
Vol 24 (3) ◽  
pp. 291-305 ◽  
Author(s):  
TOR SANDQVIST
Keyword(s):  
Structural Properties ◽  
Sequent Calculus ◽  
General Nature ◽  
Cut Elimination ◽  
Formal Proofs ◽  
Structural Rules ◽  
Logical Operators ◽  
First Order ◽  
Order Language ◽  
The Given

AbstractThe article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are.Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus.In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind.

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

scholarly journals The Method of Socratic Proofs Meets Correspondence Analysis

2019 ◽  
Vol 48 (2) ◽  
pp. 99-116
Author(s):  
Dorota Leszczyńska-Jasion ◽  
Yaroslav Petrukhin ◽  
Vasilyi Shangin
Keyword(s):  
Correspondence Analysis ◽  
Boolean Functions ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Sequent Calculi ◽  
Classical Propositional Logic ◽  
Logic Of Questions ◽  
Proof Systems

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for designing proof systems. In this paper it is used to consider sequent calculi with non-branching (the only exception being the rule of cut), invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions.

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