Algebraic Methods in Philosophical Logic

by Dunn, J. Michael; Hardegree, Gary

• ISBN13: 9780198531920
• ISBN10: 0198531923
• Format: Hardcover
• Copyright: 2001-08-30
• Publisher: Oxford University Press

Summary

This comprehensive text demonstrates how various notions of logic can be viewed as notions of universal algebra. It is aimed primarily for logisticians in mathematics, philosophy, computer science and linguistics with an interest in algebraic logic, but is also accessible to those from anon-logistics background. It is suitable for researchers, graduates and advanced undergraduates who have an introductory knowledge of algebraic logic providing more advanced concepts, as well as more theoretical aspects. The main theme is that standard algebraic results (representations) translateinto standard logical results (completeness). Other themes involve identification of a class of algebras appropriate for classical and non-classical logic studies, including: gaggles, distributoids, partial- gaggles, and tonoids. An imporatant sub title is that logic is fundamentally informationbased, with its main elements being propositions, that can be understood as sets of information states. Logics are considered in various senses e.g. systems of theorems, consequence relations and, symmetric consequence relations.

Table of Contents

Introduction	1	(9)
Universal Algebra	10	(45)
Introduction	10	(1)
Relational and Operational Structures (Algebras)	10	(1)
Subrelational Structures and Subalgebras	11	(2)
Intersection, Generators, and Induction from Generators	13	(2)
Homomorphisms and Isomorphisms	15	(4)
Congruence Relations and Quotient Algebras	19	(6)
Direct Products	25	(3)
Subdirect products and the Fundamental Theorem of Universal Algebra	28	(5)
Word Algebras and Interpretations	33	(3)
Varieties and Equational Definability	36	(1)
Equational Theories	37	(2)
Examples of Free Algebras	39	(2)
Freedom and Typicality	41	(3)
The Existence of Free Algebras; Freedom in Varieties and Subdirect classes	44	(3)
Birkhoff's Varieties Theorem	47	(2)
Quasi-varieties	49	(2)
Logic and Algebra: Algebraic Statements of Soundness and Completeness	51	(4)
Order, Lattices, and Boolean Algebras	55	(70)
Introduction	55	(1)
Partially Ordered Sets	55	(3)
Strict Orderings	58	(2)
Covering and Hasse Diagrams	60	(3)
Infima and Suprema	63	(4)
Lattices	67	(3)
The Lattice of Congruences	70	(1)
Lattices as Algebras	71	(3)
Ordered Algebras	74	(3)
Tonoids	77	(5)
Tonoid Varieties	82	(3)
Classical Complementation	85	(3)
Non-Classical Complementation	88	(4)
Classical Distribution	92	(6)
Non-Classical Distribution	98	(7)
Classical Implication	105	(4)
Non-Classical Implication	109	(6)
Filters and Ideals	115	(10)
Syntax	125	(16)
Introduction	125	(1)
The Algebra of Strings	125	(5)
The Algebra of Sentences	130	(3)
Languages as Abstract Structures: Categorial Grammar	133	(3)
Substitution Viewed Algebraically (Endomorphisms)	136	(1)
Effectivity	137	(1)
Enumerating Strings and Sentences	138	(3)
Semantics	141	(43)
Introduction	141	(1)
Categorial Semantics	142	(2)
Algebraic Semantics for Sentential Languages	144	(2)
Truth-Value Semantics	146	(2)
Possible Worlds Semantics	148	(4)
Logical Matrices and Logical Atlases	152	(3)
Interpretations and Valuations	155	(3)
Interpreted and Evaluationally Constrained Languages	158	(4)
Substitutions, Interpretations, and Valuations	162	(4)
Valuation Spaces	166	(3)
Valuations and Logic	169	(3)
Equivalence	172	(4)
Compactness	176	(5)
The Three-Fold Way	181	(3)
Logic	184	(42)
Motivational Background	184	(1)
The Varieties of Logical Experience	185	(2)
What Is (a) Logic?	187	(2)
Logics and Valuations	189	(2)
Binary Consequence in the Context of Pre-ordered Sets	191	(3)
Asymmetric Consequence and Valuations (Completeness)	194	(2)
Asymmetric Consequence in the Context of Pre-ordered Groupoids	196	(3)
Symmetric Consequence and Valuations (Completeness and Absoluteness)	199	(3)
Symmetric Consequence in the Context of Hemi-distributoids	202	(6)
Structural (Formal) Consequence	208	(1)
Lindenbaum Matrices and Compositional Semantics for Assertional Formal Logics	209	(2)
Lindenbaum Atlas and Compositional Semantics for Formal Asymmetric Consequence Logics	211	(2)
Scott Atlas and Compositional Semantics for Formal Symmetric Consequence Logics	213	(1)
Co-consequence as a Congruence	214	(2)
Formal Presentations of Logics (Axiomatizations)	216	(8)
Effectiveness and Logic	224	(2)
Matrices and Atlases	226	(51)
Matrices	226	(11)
Background	226	(1)
Lukasiewicz matrices/submatrices, isomorphisms	227	(3)
Godel matrices/more submatrices	230	(1)
Sugihara matrices/homomorphisms	230	(2)
Direct products	232	(1)
Tautology preservation	232	(1)
Infinite matrices	233	(1)
Interpretation	234	(3)
Relations Among Matrices: Submatrices, Homomorphic Images, and Direct Products	237	(2)
Proto-preservation Theorems	239	(4)
Preservation Theorems	243	(3)
Varieties Theorem Analogs for Matrices	246	(3)
Unary assertional logics	246	(1)
Asymmetric consequence logics	247	(2)
Symmetric consequence logics	249	(1)
Congruences and Quotient Matrices	249	(5)
The Structure of Congruences	254	(3)
The Cancellation Property	257	(5)
Normal Matrices	262	(4)
Normal Atlases	266	(4)
Normal Characteristic Matrices for Consequence Logics	270	(1)
Matrices and Algebras	271	(2)
When is a Logic ``Algebraizable''?	273	(4)
Representation Theorems	277	(44)
Partially Ordered Sets with Implication(s)	277	(10)
Partially ordered sets	277	(1)
Implication structures	278	(9)
Semi-lattices	287	(1)
Lattices	288	(5)
Finite Distributive Lattices	293	(2)
The Problem of a General Representation for Distributive Lattices	295	(2)
Stone's Representation Theorem for Distributive Lattices	297	(3)
Boolean Algebras	300	(2)
Filters and Homomorphisms	302	(1)
Maximal Filters and Prime Filters	302	(1)
Stone's Representation Theorem for Boolean Algebras	303	(2)
Maximal Filters and Two-Valued Homomorphisms	305	(8)
Distributive Lattices with Operators	313	(4)
Lattices with Operators	317	(4)
Classical Propositional Logic	321	(35)
Preliminary Notions	321	(1)
The Equivalence of (Unital) Boolean Logic and Frege Logic	322	(2)
Symmetrical Entailment	324	(2)
Compactness Theorems for Classical Propositional Logic	326	(7)
A Third Logic	333	(1)
Axiomatic Calculi for Classical Propositional Logic	334	(1)
Primitive Vocabulary and Definitional Completeness	335	(2)
The Calculus BC	337	(4)
The Calculus D(BC)	341	(5)
Asymmetrical Sequent Calculus for Classical Propositional Logic	346	(2)
Fragments of Classical Propositional Logic	348	(1)
The Implicative Fragment of Classical Propositional Logic: Semi-Boolean Algebras	349	(1)
Axiomatizing the Implicative Fragment of Classical Propositional Logic	350	(2)
The Positive Fragment of Classical Propositional Logic	352	(4)
Modal Logic and Closure Algebras	356	(24)
Modal Logics	356	(2)
Boolean Algebras with a Normal Unitary Operator	358	(3)
Free Boolean Algebras with a Normal Unitary Operator and Modal Logic	361	(1)
The Kripke Semantics for Modal Logic	361	(2)
Completeness	363	(1)
Topological Representation of Closure Algebras	364	(3)
The Absolute Semantics for S5	367	(1)
Henle Matrices	367	(2)
Alternation Property for S4 and Compactness	369	(1)
Algebraic Decision Procedures for Modal Logic	370	(5)
S5 and Pretabularity	375	(5)
Intuitionistic Logic and Heyting Algebras	380	(14)
Intuitionistic Logic	380	(1)
Implicative Lattices	381	(2)
Heyting Algebras	383	(1)
Representation of Heyting Algebras using Quasi-ordered Sets	383	(1)
Topological Representation of Heyting Algebras	384	(2)
Embedding Heyting Algebras into Closure Algebras	386	(1)
Translation of H into S4	386	(1)
Alternation Property for H	387	(1)
Algebraic Decision Procedures for Intuitionistic Logic	388	(2)
LC and Pretabularity	390	(4)
Gaggles: General Galois Logics	394	(37)
Introduction	394	(1)
Residuation and Galois Connections	395	(3)
Definitions of Distributoid and Tonoid	398	(2)
Representation of Distributoids	400	(6)
Partially Ordered Residuated Groupoids	406	(2)
Definition of a Gaggle	408	(1)
Representation of Gaggles	409	(3)
Modifications for Distributoids and Gaggles with Identities and Constants	412	(2)
Applications	414	(1)
Monadic Modal Operators	415	(2)
Dyadic Modal Operators	417	(3)
Identity Elements	420	(1)
Representation of Positive Binary Gaggles	421	(1)
Implication	422	(3)
Implication in relevance logic	423	(1)
Implication in intuitionistic logic	424	(1)
Modal logic	424	(1)
Negation	425	(5)
The gaggle treatment of negation	425	(1)
Negation in intuitionistic logic	426	(1)
Negation in relevance logic	427	(2)
Negation in classical logic	429	(1)
Future Directions	430	(1)
Representations and Duality	431	(14)
Representations and Duality	431	(2)
Some Topology	433	(2)
Duality for Boolean Algebras	435	(3)
Duality for Distributive Lattices	438	(3)
Extensions of Stone's and Priestley's Results	441	(4)
References	445	(10)
Index	455