Natural Deduction, Hybrid Systems and Modal Logics

516 Pages·2010·2.12 MB·English

by  Andrzej Indrzejczak

Natural Deduction, Hybrid Systems
and Modal Logics
TRENDSINLOGIC
StudiaLogicaLibrary
VOLUME30
ManagingEditor
RyszardWo´jcicki,InstituteofPhilosophyandSociology,
PolishAcademyofSciences,Warsaw,Poland
Editors
VincentF.Hendricks,DepartmentofPhilosophyandScienceStudies,
RoskildeUniversity,Denmark
DanieleMundici,DepartmentofMathematics“UlisseDini”,
UniversityofFlorence,Italy
EwaOrłowska,NationalInstituteofTelecommunications,
Warsaw,Poland
KristerSegerberg,DepartmentofPhilosophy,UppsalaUniversity,
Sweden
HeinrichWansing,InstituteofPhilosophy,DresdenUniversityofTechnology,
Germany
SCOPEOFTHESERIES
Trends in Logic is a bookseries covering essentially the same area as the jour-
nalStudiaLogica–thatis,contemporaryformallogicanditsapplicationsand
relations to other disciplines. These include artificial intelligence, informatics,
cognitivescience,philosophyofscience,andthephilosophyoflanguage.How-
ever,thislistisnotexhaustive,moreover,therangeofapplications,comparisons
andsourcesofinspirationisopenandevolvesovertime.
VolumeEditor
HeinrichWansing
Forfurthervolumes:
http://www.springer.com/series/6645
Andrzej Indrzejczak
Natural Deduction, Hybrid
Systems and Modal Logics
123
Dr.AndrzejIndrzejczak
UniversityofLo´dz
Dept.Logic
Kopcinskiego16/18
90-232Lo´dz
Poland
indrzej@filozof.uni.lodz.pl
ISBN978-90-481-8784-3 e-ISBN978-90-481-8785-0
DOI10.1007/978-90-481-8785-0
SpringerDordrechtHeidelbergLondonNewYork
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Contents
Introduction xi
1 Preliminaries 1
1.1 Classical and Free Logic . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Basic Propositional Language . . . . . . . . . . . . . . 1
1.1.2 The Language of First-Order Logic . . . . . . . . . . . 4
1.1.3 Some Reasons for Introducing FQL . . . . . . . . . . . 6
1.1.4 Formalization of CQLI and FQLI . . . . . . . . . . . . 8
1.1.5 Important Derived Notions . . . . . . . . . . . . . . . 14
1.2 Deductive Systems, Rules, Proofs . . . . . . . . . . . . . . . . 17
1.2.1 Deductive Systems . . . . . . . . . . . . . . . . . . . . 17
1.2.2 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3 Realization . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.4 Extensions and Simulations . . . . . . . . . . . . . . . 22
1.2.5 Semantical Side . . . . . . . . . . . . . . . . . . . . . . 24
1.2.6 Types of Deductive Systems . . . . . . . . . . . . . . . 25
2 Standard Natural Deduction 28
2.1 Origins of ND . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Preliminary Characterization . . . . . . . . . . . . . . . . . . 31
2.3 Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 F-Systems . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 S-Systems . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Trees or Sequences? . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Problems with Trees . . . . . . . . . . . . . . . . . . . 38
2.4.2 Problems with Linear Proofs . . . . . . . . . . . . . . 40
2.4.3 Suppes’ Format . . . . . . . . . . . . . . . . . . . . . . 43
2.5 System KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.1 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
v
vi CONTENTS
2.5.2 Realization . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.4 The Original Formulation of KM . . . . . . . . . . . . 53
2.6 Adequacy of KM . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 ND for First-Order Logic . . . . . . . . . . . . . . . . . . . . 57
2.7.1 Gentzen Systems . . . . . . . . . . . . . . . . . . . . . 57
2.7.2 Kalish/Montague Rules for CQL . . . . . . . . . . . . 59
2.7.3 Gentzen’s Variant of KM . . . . . . . . . . . . . . . . 62
2.7.4 KM for Free Logic . . . . . . . . . . . . . . . . . . . . 64
2.7.5 Introduction of Parameters . . . . . . . . . . . . . . . 66
2.7.6 Gentzen’s Variant of KMP . . . . . . . . . . . . . . . 68
2.7.7 KM with Parameters for Free Logic . . . . . . . . . . 72
2.7.8 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Other Deductive Systems 75
3.1 Sequent Systems and Tableaux . . . . . . . . . . . . . . . . . 76
3.1.1 Sequent Calculus . . . . . . . . . . . . . . . . . . . . . 76
3.1.2 Tableau Systems . . . . . . . . . . . . . . . . . . . . . 82
3.2 Resolution and Davis/Putnam Procedure . . . . . . . . . . . 85
3.2.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Davis/Putnam System . . . . . . . . . . . . . . . . . . 88
3.3 Cut and Complexity of Proof . . . . . . . . . . . . . . . . . . 89
4 Extended Natural Deduction 95
4.1 Analytic and Universal Versions of ND . . . . . . . . . . . . . 96
4.1.1 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.2 KE and ND . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 System AND1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Proof Search Procedure for AND1 . . . . . . . . . . . 109
4.2.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 System AND2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.4 Resolution and ND Combined . . . . . . . . . . . . . . . . . . 124
4.4.1 Clauses Introduced . . . . . . . . . . . . . . . . . . . . 125
4.4.2 System RND . . . . . . . . . . . . . . . . . . . . . . . 128
4.4.3 Simulation of Resolution and DP in RND . . . . . . . 131
4.4.4 RND for First-Order Logic . . . . . . . . . . . . . . . 135
CONTENTS vii
5 Survey of Modal Logics 137
5.1 Basic Modal and Tense Language . . . . . . . . . . . . . . . . 137
5.2 Modal Logics in General . . . . . . . . . . . . . . . . . . . . . 140
5.3 Axiomatic Approach to Modal Logics . . . . . . . . . . . . . 143
5.3.1 Deducibility . . . . . . . . . . . . . . . . . . . . . . . . 149
5.4 Relational Semantics . . . . . . . . . . . . . . . . . . . . . . . 150
5.4.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . 152
5.4.2 Normal Logics . . . . . . . . . . . . . . . . . . . . . . 153
5.4.3 Expressive Strength of Ordinary Modal Language . . 155
5.4.4 Regular Logics . . . . . . . . . . . . . . . . . . . . . . 159
5.4.5 Weak Logics . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.6 Entailment . . . . . . . . . . . . . . . . . . . . . . . . 162
5.5 Completeness, Decidability and Complexity . . . . . . . . . . 163
5.6 First-Order Modal Logics . . . . . . . . . . . . . . . . . . . . 167
5.6.1 Introductory Remarks . . . . . . . . . . . . . . . . . . 167
5.6.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.6.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.6.4 Some Logics . . . . . . . . . . . . . . . . . . . . . . . . 178
6 Standard Approach to Basic Modal Logics 182
6.1 Standard Sequent Calculi and Tableau Systems . . . . . . . . 183
6.1.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . 183
6.1.2 Standard SC for Basic Modal Logics . . . . . . . . . . 184
6.1.3 SC for Weak Basic Logics . . . . . . . . . . . . . . . . 187
6.2 Some Standard ND for Modal Basic Logics . . . . . . . . . . 188
6.2.1 Modal Assumptions . . . . . . . . . . . . . . . . . . . 188
6.2.2 Modalization of Rules . . . . . . . . . . . . . . . . . . 192
6.3 Modalization of Reiteration Rule . . . . . . . . . . . . . . . . 195
6.4 Rules for Possibility . . . . . . . . . . . . . . . . . . . . . . . 203
6.4.1 Original Fitch’s System . . . . . . . . . . . . . . . . . 203
6.4.2 Fitch’s System Generalized . . . . . . . . . . . . . . . 205
6.4.3 Modal Assumptions . . . . . . . . . . . . . . . . . . . 209
6.5 Standard ND for Weak Logics . . . . . . . . . . . . . . . . . . 211
6.6 First-Order Modal Logics . . . . . . . . . . . . . . . . . . . . 217
7 Beyond Basic Logics and Standard Systems 221
7.1 Beyond Basic Normal Logics . . . . . . . . . . . . . . . . . . 222
7.1.1 Almost Basic Logics . . . . . . . . . . . . . . . . . . . 223
7.1.2 Provability Logics . . . . . . . . . . . . . . . . . . . . 224
viii CONTENTS
7.1.3 Logics with Branching TS Rules . . . . . . . . . . . . 224
7.1.4 Logics of Linear Frames . . . . . . . . . . . . . . . . . 226
7.1.5 Temporal Logics . . . . . . . . . . . . . . . . . . . . . 227
7.2 Limitations of Standard Approach . . . . . . . . . . . . . . . 230
7.3 Redundancy of Standard Systems . . . . . . . . . . . . . . . . 236
7.3.1 Admissibility of Proof Construction Rules . . . . . . . 236
7.3.2 Interdefinability Problem . . . . . . . . . . . . . . . . 241
7.4 RND for Modal Logics . . . . . . . . . . . . . . . . . . . . . . 244
7.4.1 RND Systems for M, R and K . . . . . . . . . . . . . 244
7.4.2 RND for Other Modal Logics . . . . . . . . . . . . . . 251
7.5 Nonstandard Deductive Systems . . . . . . . . . . . . . . . . 254
7.5.1 Semantic Tableaux of Kripke . . . . . . . . . . . . . . 255
7.5.2 Tableaux with Boxes . . . . . . . . . . . . . . . . . . . 256
7.5.3 Systems of Higher Level . . . . . . . . . . . . . . . . . 257
8 Labelled Systems in Modal Logics 259
8.1 Kinds of Labelling . . . . . . . . . . . . . . . . . . . . . . . . 260
8.2 Weak and Strong Labelling . . . . . . . . . . . . . . . . . . . 263
8.2.1 Some Weakly Labelled Systems . . . . . . . . . . . . . 263
8.2.2 Strong Labelling . . . . . . . . . . . . . . . . . . . . . 267
8.3 Medium Labelling – Fitting’s Approach . . . . . . . . . . . . 269
8.4 Labelled ND-K . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.4.1 LND System for K . . . . . . . . . . . . . . . . . . . . 273
8.5 Other Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.5.1 Basic Normal Logics . . . . . . . . . . . . . . . . . . . 278
8.5.2 Regular Basic Logics . . . . . . . . . . . . . . . . . . . 279
8.5.3 Temporal Logics . . . . . . . . . . . . . . . . . . . . . 280
8.5.4 Some Other Logics . . . . . . . . . . . . . . . . . . . . 282
8.6 LND for Weak Modal Logics . . . . . . . . . . . . . . . . . . 285
8.7 MRND Systems with Labels . . . . . . . . . . . . . . . . . . . 290
8.7.1 Local Labelling . . . . . . . . . . . . . . . . . . . . . . 290
8.7.2 Global Labelling . . . . . . . . . . . . . . . . . . . . . 293
9 Logics of Linear Frames 297
9.1 Deductive Systems for Logics of Linear Frames . . . . . . . . 298
9.1.1 Survey of Systems . . . . . . . . . . . . . . . . . . . . 298
9.1.2 A Comparison of System’s Properties and Strategies
of Linearization . . . . . . . . . . . . . . . . . . . . . . 305
9.2 LND-System for S4.3 . . . . . . . . . . . . . . . . . . . . . . 312
CONTENTS ix
9.2.1 Characteristic Rule and Its Correctness . . . . . . . . 312
9.2.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 315
9.3 LND for Linear Temporal Logics . . . . . . . . . . . . . . . . 317
9.3.1 Formalization of Kt4.3 . . . . . . . . . . . . . . . . . . 317
9.3.2 Other Linear Logics . . . . . . . . . . . . . . . . . . . 319
9.4 Analytic Version of LND for Linear Logics . . . . . . . . . . . 320
9.5 Extensions and Limitations . . . . . . . . . . . . . . . . . . . 326
10 Analytic Labelled ND and Proof Search 332
10.1 Analytic LND . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
10.1.1 Labelled Hintikka Sets . . . . . . . . . . . . . . . . . . 334
10.1.2 Basic Procedures . . . . . . . . . . . . . . . . . . . . . 339
10.2 Logics K, D, T . . . . . . . . . . . . . . . . . . . . . . . . . . 342
10.2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 345
10.3 Transitive Logics and Loop-Control . . . . . . . . . . . . . . . 348
10.4 Symmetric and Euclidean Logics . . . . . . . . . . . . . . . . 351
10.4.1 No Transitivity . . . . . . . . . . . . . . . . . . . . . . 351
10.4.2 Transitive Symmetric or Euclidean Logics . . . . . . . 354
10.5 Linear Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
10.5.1 Finite Chains . . . . . . . . . . . . . . . . . . . . . . . 357
10.5.2 Proof Search Algorithm . . . . . . . . . . . . . . . . . 359
10.5.3 Worst Case Analysis . . . . . . . . . . . . . . . . . . . 361
11 Modal Hybrid Logics 363
11.1 Hybrid Logic in Nutshell . . . . . . . . . . . . . . . . . . . . . 364
11.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 364
11.1.2 Historical Remarks . . . . . . . . . . . . . . . . . . . . 366
11.2 Basic Hybrid Logic . . . . . . . . . . . . . . . . . . . . . . . . 367
11.2.1 Basic Hybrid Language . . . . . . . . . . . . . . . . . 367
11.2.2 Hybrid Models . . . . . . . . . . . . . . . . . . . . . . 369
11.2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
11.3 Complete Hilbert Calculi for KH@ and KH . . . . . . . . . . 371
11.4 General Completeness Results . . . . . . . . . . . . . . . . . . 374
11.5 Hybrid Tense Logic . . . . . . . . . . . . . . . . . . . . . . . . 379
11.5.1 Impact of Past Operators . . . . . . . . . . . . . . . . 379
11.5.2 Tenses . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
11.6 Language Extensions . . . . . . . . . . . . . . . . . . . . . . . 381
11.6.1 Global Modalities . . . . . . . . . . . . . . . . . . . . 382
11.6.2 Difference Modality . . . . . . . . . . . . . . . . . . . 383