Interpolation and Definability Modal and Intuitionistic Logic,9780198511748

Interpolation and Definability Modal and Intuitionistic Logic

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ISBN13: 9780198511748
ISBN10: 0198511744
Format: Hardcover
Pub. Date: 2005-07-28
Publisher(s): Clarendon Press
List Price: $268.19

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Summary

This monograph is on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigious world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of Intuitionism (Second Edition) , J.M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic , H. Rott's Change Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , P.T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 , and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof, Theory, Semantics and Control.

Table of Contents

1 Introduction and discussion
1(34)
1.1 General discussion
1(13)
1.1.1 View 1: Common logical content
2(1)
1.1.2 View 2: Expressive power
2(2)
1.1.3 View 3: Quantifier elimination
4(2)
1.1.4 View 4: Artificial intelligence
6(1)
1.1.5 View 5: Proof theory
6(1)
1.1.6 View 6: Consistency
7(1)
1.1.7 View 7: Semantical view
8(1)
1.1.8 View 8: Algebraic view
9(3)
1.1.9 View 9: Definability
12(1)
1.1.10 View 10: Interpolation by translation
13(1)
1.1.11 View 11: Traditional studies
14(1)
1.1.12 Concluding remarks
14(1)
1.2 Interpolation in general logics
14(11)
1.2.1 Historical background
14(1)
1.2.2 General logics and interpolation
15(10)
1.3 Overview of the book
25(10)
2 Modal and superintuitionistic logics: basic concepts
35(26)
2.1 Introduction overview
35(1)
2.2 The Kripke semantics for quantified modal and intermediate logics
35(15)
2.2.1 Propositional modal logics
36(5)
2.2.2 Propositional intermediate logics
41(2)
2.2.3 Quantified modal logics
43(5)
2.2.4 Quantified superintuitionistic logics
48(2)
2.3 Algebraic interpretation of propositional logics
50(6)
2.3.1 Pseudoboolean algebras
50(3)
2.3.2 Modal algebras
53(3)
2.4 Inter-relation of relational and algebraic semantics
56(5)
2.4.1 From the Kripke semantics to the algebraic one
57(1)
2.4.2 Representation theorems
58(3)
3 Superintuitionistic logics and normal extensions of the modal logics S4
61(42)
3.1 Translation
61(9)
3.1.1 Pseudoboolean and topoboolean algebras
61(5)
3.1.2 Lattice of superintuitionistic logics and NE(S4)
66(4)
3.2 A classification of normal extensions of S4 according to their superintuitionistic fragments
70(9)
3.2.1 Characteristic formulas of pre-ordered frames
70(3)
3.2.2 Some properties of the classification
73(6)
3.3 Well-representable logics
79(10)
3.3.1 Algebras and pre-ordered frames
79(1)
3.3.2 Representing frames
80(4)
3.3.3 Well-representable logics and varieties
84(5)
3.4 Classification by slices
89(5)
3.5 Finite pseudoboolean and topoboolean algebras
94(9)
3.5.1 Finite algebras and finite frames
94(3)
3.5.2 Characteristic formulas of Gödelian pseudoboolean and topoboolean algebras
97(3)
3.5.3 Logics LC and KC
100(3)
4 The interpolation theorem in intuitionistic predicate calculus
103(26)
4.1 Interpolation in classical predicate logic
103(7)
4.1.1 Craig's interpolation and Robinson's joint consistency
103(3)
4.1.2 Lyndon's interpolation
106(4)
4.2 Interpolation theorem in the intuitionistic logic
110(11)
4.2.1 Definitions
110(3)
4.2.2 Robinson's theorem
113(6)
4.2.3 Equivalence of CIP and RCP
119(2)
4.3 Propositional intermediate logics
121(3)
4.4 Notes
124(1)
4.5 Implicit and explicit definability
125(4)
5 Interpolation and definability in quantified logics
129(42)
5.1 Inter-relations between interpolation, definability, and joint consistency
129(3)
5.2 Lyndon's interpolation in some modal systems
132(8)
5.2.1 LIP in quantified logics
132(7)
5.2.2 Lyndon's interpolation in propositional modal logics
139(1)
5.3 The Craig interpolation in modal logics
140(9)
5.3.1 Modal logics without LIP
140(2)
5.3.2 Craig's interpolation in some modal logics
142(7)
5.4 Failure of interpolation
149(3)
5.5 Preserving interpolation and definability
152(8)
5.5.1 Axioms preserving interpolation
153(4)
5.5.2 Interpolation and intersection of logics
157(3)
5.6 First-order logics with equality
160(11)
5.6.1 Preliminaries
161(1)
5.6.2 Formulas preserving interpolation
162(3)
5.6.3 Modal logics
165(1)
5.6.4 A counter-example
166(2)
5.6.5 Functional symbols
168(3)
6 Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudoboolean algebras
171(34)
6.1 Craig's theorem and amalgamation property
172(5)
6.2 Amalgamable varieties of PBA
177(6)
6.3 Characterization of the varieties H1-H8
183(2)
6.4 Necessary conditions for varieties of PBA to be amalgamable
185(14)
6.5 Logics with Craig's interpolation property
199(4)
6.6 Positive logics
203(2)
7 Interpolation, definability, amalgamation
205(20)
7.1 Inter-relation of Beth's and Craig's properties in propositional logics
205(6)
7.2 Varieties of modal algebras
211(14)
7.2.1 Interpolation, implicit and explicit definability
211(2)
7.2.2 The Beth property, interpolation, amalgamation in varieties
213(9)
7.2.3 Independence of amalgamation property and the Beth property in equational theories of modal algebras
222(3)
8 Interpolation in normal extensions of the modal logic S4
225(40)
8.1 Interpolation and amalgamability
226(1)
8.2 Necessary conditions for amalgamability
227(12)
8.3 Classification of varieties of topoboolean algebras
239(2)
8.4 Interpolation theorems in modal logics
241(2)
8.5 Sufficient conditions for amalgamation
243(12)
8.5.1 Well-representable logics and varieties
244(3)
8.5.2 Sufficient conditions for amalgamability and superamalgamability
247(1)
8.5.3 Lemmas on (α1, α2)-products
248(4)
8.5.4 Stable and superstable classes of frames
252(3)
8.6 Logics with interpolation in NE(S4)
255(2)
8.7 Decidability of interpolation over S4
257(1)
8.8 NE(S4) versus E(Int)
258(7)
8.8.1 More on Gödel's translation
259(2)
8.8.2 IPN in NE(S4)
261(4)
9 Complexity of some problems in modal and intuitionistic calculi
265(20)
9.1 Main results
265(2)
9.2 Reducibilities
267(2)
9.3 Complexity
269(3)
9.4 Tabularity and related properties
272(5)
9.5 Interpolation and amalgamation
277(8)
10 Interpolation in modal infinite slice logics containing the logic K4 285(26)
10.1 K4 and S4
285(2)
10.2 Logics and varieties of infinite slice
287(6)
10.3 Necessary condition of interpolation
293(18)
11 An analogue of Beth's theorem in normal extensions of the modal logic K4 311(22)
11.1 Preliminaries
311(2)
11.2 The Replacement theorem and its corollaries
313(2)
11.3 The main theorem
315(12)
11.3.1 Case 1
318(3)
11.3.2 Case 2
321(6)
11.4 A counter-example to the Beth property
327(1)
11.5 Explicit definitions
328(5)
11.5.1 Logics of finite slices
329(2)
11.5.2 Constructing explicit definitions
331(2)
12 Extensions of the provability logic 333(38)
12.1 Two extensions of G
333(5)
12.2 Interpolation in infinite-slice extensions of provability logic
338(15)
12.2.1 Definitions and notations
338(2)
12.2.2 Description of Gγ and Gδ
340(7)
12.2.3 Interpolation theorem
347(6)
12.3 Continuum of extensions of the provability logic with interpolation
353(10)
12.3.1 Interpolation and amalgamation properties
354(1)
12.3.2 The Logic Gλ
355(3)
12.3.3 Continuum of extensions of the logic Gλ that have the Craig interpolation property
358(3)
12.3.4 Amalgamation and superamalgamation properties
361(2)
12.4 Boxed formulas
363(8)
12.4.1 Preliminaries
363(1)
12.4.2 The main lemmas
364(2)
12.4.3 CIP and IPB
366(3)
12.4.4 The property B2
369(2)
13 Syntactic proof of interpolation for the intuitionistic predicate logic 371(10)
13.1 Formal system S
371(2)
13.2 Proof of interpolation
373(6)
13.3 Fragments of IntQ
379(2)
14 Interpolation by translation 381(22)
14.1 Introduction
381(3)
14.2 Interpolation by quantifier elimination
384(4)
14.2.1 SCAN: Second-order quantifier elimination
384(1)
14.2.2 SCAN can interpolate
385(3)
14.3 Case study: Quantified S5
388(9)
14.3.1 Preliminary discussion
388(2)
14.3.2 Interpolation for QS5
390(7)
14.4 Case study: Propositional modal logic S4.3
397(1)
14.5 Interpolation by translation: General theory
398(5)
15 Interpolation in (intuitionistic) logic programming 403(22)
15.1 Introduction
403(1)
15.2 N-prolog
404(5)
15.3 Interpolation for propositional Horn programs
409(3)
15.4 Alternative proof
412(2)
15.5 Controlled interpolation for propositional Horn clauses
414(2)
15.6 Failure of interpolation for fragment of predicate intuitionistic logic
416(2)
15.7 Weak interpolation for intuitionistic logic programs
418(7)
16 Interpolation in goal-directed proof systems 425(46)
16.1 Introduction
425(7)
16.1.1 General background
425(2)
16.1.2 Specific background
427(5)
16.2 Interpolation for linear and for intuitionistic implication
432(6)
16.2.1 Interpolation for linear implication
432(3)
16.2.2 Interpolation for intuitionistic logic
435(3)
16.2.3 Interpolation for classical logic
438(1)
16.3 Interpolation for the Lambek calculus
438(7)
16.4 Interpolation for strict implication
445(19)
16.5 Concluding discussion, chain interpolation
464(7)
16.5.1 Structural interpolation
464(2)
16.5.2 Chain interpolation
466(2)
16.5.3 Beth definability
468(1)
16.5.4 Standard interpolation in classical logic
469(1)
16.5.5 Concluding remarks
470(1)
17 Further results and discussion 471(12)
17.1 Introduction
471(1)
17.2 Further results
471(7)
17.2.1 Temporal logics
472(1)
17.2.2 Beth properties and epimorphisms surjectivity
472(1)
17.2.3 Projective Beth property over Int
473(1)
17.2.4 Positive and paraconsistent logics
473(1)
17.2.5 Modal logics and projective Beth property
474(1)
17.2.6 Restricted interpolation and restricted amalgamation
475(1)
17.2.7 Variable separation
476(1)
17.2.8 Decidable properties of logics and of varieties
477(1)
17.3 Further discussion
478(5)
17.3.1 Interpolation and artificial intelligence
478(1)
17.3.2 Interpolation for classical theories
478(1)
17.3.3 A semantic/categorial engine for interpolation
478(1)
17.3.4 Interpolation in computer science
479(1)
17.3.5 Case study: Implementation of constant domains modal K4 in classical logic
480(3)
Appendix 483(1)
References 484(19)
Index 503

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