Logics of Variable Inclusion (Trends in Logic, 59)

✍ Scribed by Stefano Bonzio, Francesco Paoli, Michele Pra Baldi

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 227
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This monograph shows that, through a recourse to the concepts and methods of abstract algebraic logic, the algebraic theory of regular varieties and the concept of analyticity in formal logic can profitably interact. By extending the technique of Plonka sums from algebras to logical matrices, the authors investigate the different classes of models for logics of variable inclusion and they shed new light into their formal properties.

The book opens with the historical origins of logics of variable inclusion and on their philosophical motivations. It includes the basics of the algebraic theory of regular varieties and the construction of Plonka sums over semilattice direct systems of algebra. The core of the book is devoted to an abstract definition of logics of left and right variable inclusion, respectively, and the authors study their semantics using the construction of Plonka sums of matrix models. The authors also cover Paraconsistent Weak Kleene logic and survey its abstract algebraic logical properties. This book is of interest to scholars of formal logic.

✦ Table of Contents

Contents
Acknowledgements
Chapter 1 Logic, analyticity, and significance
1.1 Logic and analyticity
1.1.1 Informational explications
1.1.2 Semantic explications
1.1.3 Syntactic explications
1.2 Logic and significance
1.2.1 The family of Kleene logics
1.2.2 Weak Kleene logics: B3 and PWK
1.2.3 The interpretation of the third value in B3
1.2.4 The interpretation of the third value in PWK
1.2.5 Other logics with infectious values
1.3 From significance to analyticity via variable inclusion
1.3.1 Syntactic characterisations of B3 and PWK
1.3.2 Pure variable inclusion companions of classical logic
1.3.3 Extensions of B3 and PWK
1.4 Logics of variable inclusion: A general frame work
Chapter 2 Płonka sums and regular varieties
2.1 Semilattice direct systems and Płonka sums
2.2 The Płonka decomposition theorem
2.3 Regular varieties
2.3.1 τ-semilattices
2.3.2 Subdirectly irreducible algebras
2.3.3 Subvarieties and equational bases
2.3.4 An example: Bisemilattices
2.4 Generalised involutive bisemilattices
2.4.1 Definition and elementary properties
2.4.2 The structure of the Boolean subalgebras
2.4.3 Characterising Boolean algebras and semilattices
2.4.4 The Płonka sum representation
Chapter 3 Dualities for regular varieties
3.1 Background
3.1.1 Basic notions
3.1.2 The Stone duality
3.1.3 The Priestley duality
3.2 Semilattice systems
3.3 Duality
3.3.1 Other dualities
3.4 Dual spaces
3.4.1 Left normal bands and GR spaces
3.4.2 GR spaces with involution
3.5 A topological counterpart of Płonka sums
Chapter 4 An interlude: Abstract Algebraic Logic
Chapter 5 Logics of left variable inclusion
5.1 Płonka sums of matrices and l-direct systems
5.1.1 General results
5.1.2 Left partition functions
5.2 Hilbert-style axiomatisations
5.3 Suszko reduced models of Ll
5.4 Some well-behaved cases
5.4.1 Equivalential logics
5.4.2 Logics with antitheorems
5.5 Classification in the Leibniz hierarchy
Chapter 6 Logics of right variable inclusion
6.1 Płonka sums of matrices and r-direct systems
6.1.1 General results
6.1.2 Right partition functions
6.2 Hilbert-style axiomatisations
6.3 The algebraic counterpart
6.3.1 Logics without antitheorems
6.3.2 Logics with antitheorems
6.4 Leibniz reduced models
6.5 Suszko reduced models
6.5.1 Truth-equational logics
6.5.2 Two well-behaved cases
Chapter 7 Paraconsistent Weak Kleene Logic
7.1 Abstract Algebraic Logic properties
7.1.1 Basic properties
7.1.2 Deductive filters and matrix models
7.1.3 Suszko reduced models
7.2 Hilbert-style calculi
7.3 Sequent calculi
7.3.1 Systems with linguistic restrictions
7.3.2 Systems without linguistic restrictions
7.4 Other proof-theoretic presentations
7.4.1 Natural deduction calculi
7.4.2 Tableaux
Chapter 8 Conclusions and open problems
8.1 Open problems
8.1.1 Universal Algebra
8.1.2 Abstract Algebraic Logic
8.1.3 Proof Theory
8.1.4 Duality Theory
Bibliography