Preface and Navigation Guide |
|
xiii | |
|
|
1 | (46) |
|
0.1 Connections And Combinations |
|
|
1 | (16) |
|
0.1.1 Relational Connections and Posets |
|
|
1 | (2) |
|
|
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) |
|
|
17 | (6) |
|
|
23 | (3) |
|
0.2.3 Homomorphisms, Subalgebras, Direct Products |
|
|
26 | (3) |
|
0.2.4 Equational Classes of Algebras |
|
|
29 | (3) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
767 | (158) |
|
6.1 Distinctions Among Disjunctions |
|
|
767 | (53) |
|
|
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) |
|
|
830 | (3) |
|
6.2.4 Distributive Lattices of Theories |
|
|
833 | (9) |
|
Notes and References for §6.2 |
|
|
842 | (1) |
|
|
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) |
|
|
925 | (238) |
|
|
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) |
|
|
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) |
|
|
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) |
|
|
1228 | (13) |
|
8.2.4 Variations on a Theme of Sheffer |
|
|
1241 | (9) |
|
Notes and References for §8.2 |
|
|
1250 | (2) |
|
|
1252 | (35) |
|
8.3.1 Negation and the Falsum Connective |
|
|
1252 | (5) |
|
|
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) |
|
|
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) |
|
|
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 | |