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E-raamat: Connectives

(Monash University)
  • Formaat: 1512 pages
  • Sari: The Connectives
  • Ilmumisaeg: 29-Jul-2011
  • Kirjastus: MIT Press
  • Keel: eng
  • ISBN-13: 9780262298834
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  • Formaat: 1512 pages
  • Sari: The Connectives
  • Ilmumisaeg: 29-Jul-2011
  • Kirjastus: MIT Press
  • Keel: eng
  • ISBN-13: 9780262298834

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In The Connectives, Lloyd Humberstone examines the semantics and pragmatics of natural language sentence connectives (and, or, if, not), giving special attention to their formal behavior according to proposed logical systems and the degree to which such treatments capture their intuitive meanings. It will be an essential resource for philosophers, mathematicians, computer scientists, linguists, or any scholar who finds connectives, and the conceptual issues surrounding them, to be a source of interest.This landmark work offers both general material on sentence connectives in formal logic, such as truth-functionality and unique characterization by rules, and information on specific connectives (including conjunction and disjunction), considering their pragmatic and semantic properties in natural language as well as various attempts to simulate the latter in the formal languages of different systems of propositional logic. Chapters are divided into sections, and each section ends with notes and references for material covered in that section. If a section covers numerous topics separately, the notes and references are divided into parts, each with its own topic-indicating heading. When topics are not covered in detail but are relevant to matters under discussion, the notes and references provide pointers to the literature. Readers may find it useful to browse through a topic of interest and then follow the references within it forward and backward on the topic in question, or those to the extensive literature outside it.

Preface and Navigation Guide xiii
0 Preliminaries
1(46)
0.1 Connections And Combinations
1(16)
0.1.1 Relational Connections and Posets
1(2)
0.1.2 Galois Connections
3(4)
0.1.3 Lattices and Closure Operations
7(4)
0.1.4 Modes of Object Combination
11(5)
Notes and References for §0.1
16(1)
0.2 Some Algebraic Concepts
17(30)
0.2.1 Algebras
17(6)
0.2.2 Derived Operations
23(3)
0.2.3 Homomorphisms, Subalgebras, Direct Products
26(3)
0.2.4 Equational Classes of Algebras
29(3)
0.2.5 Horn Formulas
32(5)
0.2.6 Fundamental and Derived Objects and Relations: Tuple Systems
37(6)
Notes and References for §0.2
43(4)
1 Elements of Sentential Logic
47(148)
1.1 Truth And Consequence
47(56)
1.1.1 Languages
47(7)
1.1.2 Consequence Relations and Valuations
54(7)
1.1.3 #-Classical Consequence Relations and #-Boolean Valuations
61(6)
1.1.4 Forcing Consistent Valuations to Be #-Boolean
67(5)
1.1.5 Two Equations and the `Converse Proposition' Fallacy
72(1)
1.1.6 Generalized Consequence Relations
73(6)
1.1.7 Generalized Consequence Relations: Supplementary Discussion
79(3)
1.1.8 Truth-Tables, and Two More Connectives: Implication and Equivalence
82(5)
1.1.9 A More Hygienic Terminology
87(13)
Notes and References for §1.1
100(3)
1.2 Rules And Proof
103(92)
1.2.1 Sequents and Frameworks
103(9)
1.2.2 Sequents and (Generalized) Consequence Relations
112(2)
1.2.3 A Sample Natural Deduction System in SET-FMLA
114(8)
1.2.4 A Closer Look at Rules
122(5)
1.2.5 Semantic Apparatus for Sequents and for Rules
127(5)
1.2.6 Two Relational Connections
132(8)
1.2.7 Sample Natural Deduction and Sequent Calculus Systems in SET-SET
140(10)
1.2.8 Some Other Approaches to Logic in SET-SET
150(6)
1.2.9 Axiom Systems and the Deduction Theorem
156(24)
Appendix to §1.2: What Is a Logic?
180(8)
Notes and References for §1.2
188(7)
2 A Survey of Sentential Logic
195(180)
2.1 Many-Valued Logic And Algebraic Semantics
195(80)
2.1.1 Many-Valued Logic: Matrices
195(16)
2.1.2 Many-Valued Logic: Classes of Matrices
211(8)
2.1.3 Algebraic Semantics: Matrices with a Single Designated Value
219(27)
2.1.4 ≤-Based Algebraic Semantics
246(4)
2.1.5 A Third Version of Algebraic Semantics: Indiscriminate Validity
250(7)
2.1.6 From Algebraic Semantics to Equivalent Algebraic Semantics
257(11)
Notes and References for §2.1
268(7)
2.2 Modal Logic
275(23)
2.2.1 Modal Logic in FMLA: Introduction
275(7)
2.2.2 Modal Logic in FMLA: Kripke Frames
282(6)
2.2.3 Other Logical Frameworks
288(8)
Notes and References for §2.2
296(2)
2.3 Three Rivals To Classical Logic
298(77)
2.3.1 Quantum Logic
298(4)
2.3.2 Intuitionistic Logic
302(24)
2.3.3 Relevant and Linear Logic
326(43)
Notes and References for §2.3
369(6)
3 Connectives: Truth-Functional, Extensional, Congruential
375(136)
3.1 Truth-Functionality
375(69)
3.1.1 Truth-Functional Connectives
375(7)
3.1.2 Pathologies of Overdetermination
382(3)
3.1.3 Determinant-induced Conditions on a Consequence Relation
385(18)
3.1.4 Functional Completeness, Duality, and Dependence
403(15)
3.1.5 Definability of Connectives
418(5)
3.1.6 Defining a Connective
423(5)
3.1.7 Truth-Functions, Rules, and Matrices
428(8)
3.1.8 Intuitionistically Dangerous Determinants
436(6)
Notes and References for §3.1
442(2)
3.2 Extensionality
444(40)
3.2.1 A Biconditional-based Introduction to Extensionality
444(4)
3.2.2 A Purified Notion of Extensionality (for GCRs)
448(5)
3.2.3 Some Extensionality Notions for Consequence Relations
453(8)
3.2.4 Hybrids and the Subconnective Relation
461(22)
Notes and References for §3.2
483(1)
3.3 Congruentiality
484(27)
3.3.1 Congruential Connectives
484(6)
3.3.2 Some Related Properties
490(5)
3.3.3 The Three Properties Compared
495(2)
3.3.4 Operations vs. Relations
497(11)
Notes and References for §3.3
508(3)
4 Existence and Uniqueness of Connectives
511(120)
4.1 Philosophical Proof Theory
511(25)
4.1.1 Introduction
511(2)
4.1.2 Normalization of Proofs in Positive Logic
513(3)
4.1.3 A Proof-Theoretic Notion of Validity
516(3)
4.1.4 Further Considerations on Introduction and Elimination Rules
519(16)
Notes and References for §4.1
535(1)
4.2 Existence Of Connectives
536(42)
4.2.1 The Conservative Extension Proposal
536(12)
4.2.2 Conditions not Corresponding to Rules: An Example
548(18)
4.2.3 Other Grounds for Claiming Non-existence
566(5)
4.2.4 Non-partisan Existence Questions
571(5)
Notes and References for §4.2
576(2)
4.3 Uniqueness Of Connectives
578(53)
4.3.1 Unique Characterization: The Basic Idea
578(6)
4.3.2 Stronger-than-Needed Collections of Rules
584(4)
4.3.3 Intermission: Rules Revisited
588(1)
4.3.4 Stronger-than-Needed Collections of Rules (Continued)
589(8)
4.3.5 Unique Characterization by Zero-Premiss Rules: A Negative Example
597(4)
4.3.6 Unique Characterization in Modal Logic
601(4)
4.3.7 Uniqueness and `New Intuitionistic Connectives'
605(9)
4.3.8 Postscript on `New Intuitionistic Connectives'
614(12)
Notes and References for §4.3
626(5)
5 "And"
631(136)
5.1 Conjunction In Natural Language And In Formal Logic
631(46)
5.1.1 Syntax, Semantics, Pragmatics
631(8)
5.1.2 Temporal and Dynamic Conjunction
639(6)
5.1.3 Why Have Conjunction?
645(5)
5.1.4 Probability and ˆ-Introduction
650(8)
5.1.5 `Intensional Conjunction'and ˆ-Elimination
658(3)
5.1.6 Conjunction and Fusion in Relevant Logic
661(12)
Notes and References for §5.1
673(4)
5.2 Logical Subtraction
677(31)
5.2.1 The Idea of Logical Subtraction
677(3)
5.2.2 Four Choices to Make
680(3)
5.2.3 Cancellation, Independence, and a `Stipulated Equivalence' Treatment
683(6)
5.2.4 Content Subtraction and Converse Implication
689(3)
5.2.5 Requirement Semantics: First Pass
692(6)
5.2.6 Requirement Semantics: Second Pass
698(2)
5.2.7 Requirement Semantics: Final Considerations
700(7)
Notes and References for §5.2
707(1)
5.3 ˆ-Like Connectives And The Like
708(59)
5.3.1 #-Like and #-Representable Connectives
708(12)
5.3.2 Non-creative Definitions
720(9)
5.3.3 #-Representable Binary Relations
729(9)
5.3.4 Ordered Pairs: Theme and Variations
738(12)
5.3.5 Hybridizing the Projections
750(15)
Notes and References for §5.3
765(2)
6 "Or"
767(158)
6.1 Distinctions Among Disjunctions
767(53)
6.1.1 Introduction
767(13)
6.1.2 Inclusive/Exclusive
780(9)
6.1.3 Intensional/Extensional
789(10)
6.1.4 Conjunctive-Seeming Occurrences of Or
799(13)
Appendix to §6.1: Conjunctive-Seeming Or with Apparent Wide Scope
812(4)
Notes and References for §6.1
816(4)
6.2 Argument By Cases In Theory And In Practice
820(23)
6.2.1 Problematic Applications of V-Elimination
820(3)
6.2.2 Commentary on the Examples
823(7)
6.2.3 Supervaluations
830(3)
6.2.4 Distributive Lattices of Theories
833(9)
Notes and References for §6.2
842(1)
6.3 Commas On The Right
843(18)
6.3.1 GCRs Agreeing with a Given Consequence Relation
843(7)
6.3.2 The Model-Consequence Relation in Modal Logic
850(4)
6.3.3 Generalizing the Model-Consequence Relation
854(6)
Notes and References for §6.3
860(1)
6.4 Disjunction In Various Logics
861(64)
6.4.1 The Disjunction Property and Hallden-Complcteness
861(11)
6.4.2 The `Rule of Disjunction' in Modal Logic -- and More on the Disjunction Property
872(21)
6.4.3 Disjunction in the Beth Semantics for Intuitionistic Logic
893(6)
6.4.4 Disjunction in the `Possibilities' Semantics for Modal Logic
899(6)
6.4.5 Disjunction in Urquhart-style Semantics for Relevant Logic
905(5)
6.4.6 `Plus' Semantics and Valuational Semantics for Disjunction
910(8)
6.4.7 Quantum Disjunction
918(4)
Notes and References for §6.4
922(3)
7 "If"
925(238)
7.1 Conditionals
925(132)
7.1.1 Issues and Distinctions
925(23)
7.1.2 Even If, Only If, and Unless
948(27)
7.1.3 Material Implication and the Indicative Conditional
975(8)
7.1.4 Positive Implication and Clear Formulas
983(4)
7.1.5 From Strict to Variably Strict Conditionals
987(11)
7.1.6 Interlude: Suppositional Proof Systems
998(9)
7.1.7 Possible Worlds Semantics for Subjunctive Conditionals
1007(27)
7.1.8 `Counterfactual Fallacies' and Subjunctive/Indicative, Revisited
1034(10)
7.1.9 `Ordinary Logic': An Experiment
1044(9)
W. S. Cooper
Notes and References for §7.1
1053(4)
7.2 Intuitionistic, Relevant, Contractionless
1057(70)
7.2.1 Intuitionistic and Classical Implication
1057(11)
7.2.2 Ternary and Binary Connectives Involving Intuitionistic Implication
1068(20)
7.2.3 Relevant Implication in SET-FMLA: Proofs of Results in 2.33
1088(3)
7.2.4 Relevant Implication in FMLA: Various Topics
1091(7)
7.2.5 Contractionless Logics
1098(23)
Notes and References for §7.2
1121(6)
7.3 Biconditionals And Equivalence
1127(36)
7.3.1 Equivalence in CL and IL
1127(23)
7.3.2 A Connective for Propositional Identity
1150(8)
7.3.3 Reducibility without Tabularity: An Austere Propositional Identity Logic
1158(3)
Notes and References for §7.3
1161(2)
8 "Not"
1163(124)
8.1 Introduction To Negation
1163(51)
8.1.1 Contraries and Subcontraries
1163(11)
8.1.2 Negative Miscellany
1174(12)
8.1.3 Negation in Quantum Logic and Relevant Logic
1186(25)
Notes and References for §8.1
1211(3)
8.2 Negation In Intuitionistic Logic
1214(38)
8.2.1 Glivenko's Theorem and Its Corollaries
1214(8)
8.2.2 Dual Intuitionistic Negation
1222(6)
8.2.3 Strong Negation
1228(13)
8.2.4 Variations on a Theme of Sheffer
1241(9)
Notes and References for §8.2
1250(2)
8.3 The Falsum
1252(35)
8.3.1 Negation and the Falsum Connective
1252(5)
8.3.2 Minimal Logic
1257(6)
8.3.3 Variations on a Theme of Johansson
1263(11)
8.3.4 Extensions of Minimal Logic
1274(8)
8.3.5 The Falsum: Final Remarks
1282(2)
Notes and References for §8.3
1284(3)
9 Universally Representative and Special Connectives
1287(50)
9.1 Universally Representative Classes Of Formulas
1287(14)
9.1.1 Introduction
1287(2)
9.1.2 Universally Representative Connectives
1289(6)
9.1.3 Are There Conjunctive (Disjunctive, etc.) Propositions?
1295(6)
Notes and References for §9.1
1301(1)
9.2 Special Classes Of Formulas
1301(36)
9.2.1 Introduction
1301(2)
9.2.2 Special Connectives
1303(4)
9.2.3 The Biconditional in R
1307(5)
9.2.4 Connectives Neither Universally Representative Nor Special
1312(6)
9.2.5 Special Relations Between Formulas
1318(9)
9.2.6 Special `in a Given Respect'
1327(2)
9.2.7 Pahi's Notion of Restricted Generalization
1329(6)
Notes and References for §9.2
1335(2)
References 1337(103)
Index 1440