Verfügbarkeit: A geometry of approximation

A geometry of approximation: Rough Set Theory: logic, algebra and topology of conceptual patterns

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Bibliographische Detailangaben
Hauptverfasser:	Pagliani, Piero (VerfasserIn), Chakraborty, Mihir (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin Springer 2008
Schriftenreihe:	Trends in Logic 27
Schlagworte:	Grobmenge Heyting-Arithmetik Frame > Mathematik Epistemologischer Kontextualismus Modelltheoretische Semantik Approximation Semitopologische Halbgruppe Grothendieck-Topologie Informationssystem
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	LXXXII, 704 S. graph. Darst.
ISBN:	9781402086212

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Datensatz im Suchindex

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adam_text	CONTENTS PREFACE LIST OF FIGURES NOTATION ABBREVIATIONS INTRODUCTION 1 PERCEPTION AND CONCEPTS: A PHENOMENOLOGICAL APPROACH . 1.1 MONOLOGICAL APPROACH AND DIALOGICAL APPROACH . 2 MONOLOGICAL APPROACH TO PERCEPTION AND CONCEPTS . 3 PHENOMENOLOGY AND LOGIC . 3.1 SEMANTICS VS SYNTAX . 3.2 INFORMATION AND INTERPRETATION: CORRESPONDENCE THEORY OF TRUTH VS PRAGMATISM . 3.2.1 MEANING-CONDITIONS VS TRUTH-CONDITIONS . 3.2.2 LOGIC, MEANING AND ROUGH SET THEORY . 4 THE LOGIC(}- ALGEBRAIC INTERPRETATION OF ROUGH SET SYSTEMS . 5 EQUIVALENCE CLASSES, ABSTRACTION AND MEANING . 5.1 TYPES, TOKENS AND ABSTRACT POINTS . 5.2 ABSTRACT POINTS AND MEANING . V XXIV XXIX XXXI XXXIII XXXIII XXXVI XXXVIII I I LVI LVII LXII LXIV !XVIII LXVIII !XXVI XIII XIV 6 7 5.3 ABSTRACT POINTS AND ROUGH SETS . ROUGH SETS AND LOGIE . CONCLUDING REMARKS . CONTENTS !XXVI !XXVII !XXXI I A MATHEMATICS OF PERCEPTION 1 1 OBSERVATIONS, NOUMENA AND PHENOMENA 3 1.1 FOREWORD . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 FORMAL RELATIONSHIPS BETWEEN NOUMENA AND PHENOMENA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 PROPERTY SYSTEMS - P-SYSTEMS . . . . . . .. . . 8 1.2.1.1 FUNCTIONAL PROPERTY SYSTEMS . . . 11 1.2.1.2 RELATIONAL PROPERTY SYSTEMS ... 12 1.2.1.3 DICHOTOMIE PROPERTY SYSTEMS .. 13 1.2.2 ATTRIBUTE SYSTEMS - A-SYSTEMS. .. . . . . . . . 13 1.3 FUNCTIONAL P-SYSTEMS AND CONCEPTUALISATION . . . . . . 14 1.3.1 CATEGORIZING THROUGH FUNCTIONAL P-SYSTEMS 15 1.4 CATEGORIZING THROUGH RELATIONAL P-SYSTEMS 20 1.4.1 TYPES AND APPROXIMATION . . . . . . . . . . . . . . 23 1.4.2 DIVISORS AND RESIDUALS 24 1.4.3 GALOIS ADJUNCTIONS AND GALOIS CONNECTIONS 29 1.4.4 AIGEBRAIC PROPERTIES OF ADJOINT MAPS ... . 33 2 CONCRETE AND FORMAL INFORMATION CONSTRUCTIONS 43 2.1 CONCRETE AND FORMAL OBSERVATION SPACES. .. . . . . . . 43 2.1.1 OBSERVATIONS AND PARTIAL OBSERVATIONS. . . 45 2.1.2 RELATIONS ANDGALOIS ADJUNCTIONS. . . . . . . . 47 2.2 THE BASIC PHENOMENOLOGICAL CONSTRUCTORS . . . . . . . . 50 2.2.1 A MODAL READING OF THE BASIC CONSTRUCTORS 53 2.3 FORMAL OPERATORS ON POINTS AND ON OBSERVABLES . . . 58 2.3.1 AIGEBRAIC PROPERTIES OF FORMAL PERCEPTION SYSTEMS . . . . . . . . . . . . . . . . . . . . 63 2.3.2 MULTI-AGENT PRE-TOPOLOGICAL APPROXIMATION SYSTEMS .... . . . . . . . . . . . . 70 3 PRE-TOPOLOGICALAND TOPOLOGICALAPPROXIMATION OPERATORS 73 3.1 INFORMATION, CONCEPTS AND FORMAL OPERATORS. . . . . 73 CONTENTS XV 3.1.1 CHOOSING THE INITIAL PERCEPTION ACT . . . . . . 75 3.1.2 INFORMATION QUANTUM RELATIONAL SYSTEMS 81 3.2 COMPARING PERCEPTION SYSTEMS. . . . . . . . . . . . . . . . . . 85 3.3 HIGHER LEVEL OPERATORS 90 3.4 TRANSFORMING PERCEPTION SYSTEMS 97 3.5 TOPOLOGICAL APPROXIMATION OPERATORS. . . . . . . . . . . . 100 3.6 TOPOLOGICAL APPROXIMATION SYSTEMS 103 4 FRAMES(PART I) 107 4.1 FRAME - APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 FRAME - CLASSIFIEATION. . . . . . . . . . . . . . . . . . . . .. . . . . 108 4.3 FRAME - CATEGORIZING THROUGH POINTLESS TOPOLOGY . 110 4.4 FRAME - OBSERVABLE PROPERTIES . . . . . . . . . . . . . . . . . . 113 4.4.1 DEEIDABLE AND SEMI-DECIDABLE PROPERTIES . 113 4.4.2 DECIDABLE PROPERTIES, TOPOLOGY, DOMAINS AND GEOMETRIE LOGIE . . . . . .. . . . . . . . . . . . . 115 4.5 FRAME - FINITE OBSERVATIONS: THE BINARY MACHINE EXAMPLE 116 4.6 FRAME - QUANTA OF INFORMATION 120 4.6.1 QUANTA AT A LOEATION AND ORTHOLATTIEES . . 120 4.6.2 A TOPO-ALGEBRAIC READING OF IQRSS . .. . . 123 4.6.3 DUALITY BETWEEN P-SYSTEMS AND PREORDERS . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.7 FRAME - INFORMATION SYSTEMS . . . . . . . . . . . . . . . . . . . 126 4.7.1 GENERALISING INFORMATION RELATIONS 126 4.7.2 GENERALIZING INDISEERNIBILITY RELATION 127 4.7.3 GENERALISING FROM SETS TO RELATIONS. . . . . . 128 4.8 FRAME - DICHOTOMIE, COMPLEMENTARY AND FUNETIONAL SYSTEMS. . . . . . . . . . . . . . . . . .. . . . . . 129 4.8.1 DIEHOTOMIE SYSTEMS. . . . . . . . . . . . . . . . . . . 129 4.8.2 COMPLEMENTARY SYSTEMS I . . . . . . . . . . . . . . 130 4.8.3 COMPLEMENTARY SYSTEMS II 130 4.8.4 COMPLEMENTARY SYSTEMS III. ............ 131 4.8.5 DICHOTOMIE SYSTEMS AND FUNETIONAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.8.6 FUNETIONAL SYSTEMS AND APPROXIMATIONS. . 132 4.8.7 DICHOTOMIE SYSTEMS AND APPROXIMATIONS . 132 XVI CONTENTS 4.9 FRAME - CONCEPT LATTICES 133 4.9.1 FORMAL CONTEXTS AND FORMAL CONCEPT ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.9.2 FORMAL CONCEPTS AND APPROXIMATION OPERATORS 135 4.9.3 COMBINING CLASSICAL APPROXIMATION SYSTEMS AND CONCEPT LATTICES. . . . . . . . . . . 137 4.9.4 COMBINING NON-CLASSICAL APPROXIMATION SYSTEMS AND CONCEPT LATTICES .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.9.5 NOMINAL SYSTEMS AND CONCEPTUAL SCALING. 140 4.10 FRAME - NEIGHBORHOOD SYSTEMS.. . . . . . . . . . . . . . . . 142 4.11 FRAME - BASIC PAIRS AND POINT-FREE TOPOLOGY . . . . . 143 4.12 FRAME - CHU SPACES. . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.13 FRAME - INTUITIONISM, MODALITIES AND RELATIONAL SEMANTIES. . .. . . . . .. . . . .. . . .. . . . . . . . . . . . . . . . . . 145 4.13.1 NECESSITY AND POSSIBILITY 146 4.13.2 BASIC OPERATORS AS MODAL OPERATORS. .. . . 147 4.13.3 RAMIFIED TENSE LOGIC 148 4.13.4 NECESSITY AND SUFFICIENEY OPERATORS. . . . . . 148 4.13.5 MODAL OPERATORS AND INFORMATION SYSTEMS 149 4.14 FRAME - GAJOISADJUNCTIONS 151 4.14.1 GALOIS ADJUNETIONS IN COMPUTER SCIENCE. . 152 4.14.2 GALOIS ADJUNCTIONS AND DEDEKIND CUTS.. . 154 4.14.3 GALOIS ADJUNETIONS AT LARGE 155 4.14.4 GALOIS CONNECTIONS AND REPRESENTATION THEOREMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.14.5 GALOIS ADJUNCTIONS, ISOMORPHISMS AND APPROXIMATION: A NOTE 157 4.15 FRAME - CATEGORIES AND ADJOINT FTMCTORS. . . . . . . . . 158 4.16 SOLUTIONS..................................... 161 11 THE LOGICO-ALGEBRAICTHEORY OF ROUGH SETS 167 5 LOGIC AND ROUGH SETS: AN OVERVIEW 169 5.1 FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2 ROUGH SET SYSTEMS AND THREE- VALUED LOGIES. . . . . . 172 CONTENTS XVII 5.3 EXACT AND INEXACT LOCAL BEHAVIOURS IN ROUGH SET SYSTEMS 174 5.4 REPRESENTING ROUGH SETS . . . . . . . . . . . . . . . . . . . . . . . 177 5.5 ROUGH SET SYSTEMS, LOCAL VALIDITY, AND LOGICO-ALGEBRAIC STRUCTURES. . . . . . . . . . . . . .. . . 181 6 BASIC LOGICO-ALGEBRAIC STRUCTURES 193 6.1 HEYTING AIGEBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.2 NELSON AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.3 N-VALUED LUKASIEWICZ AIGEBRAS 203 6.4 CHAIN- BASED LATTICES 204 6.5 RELATIONSHIPS, ANALOGIES AND DIFFERENCES BETWEEN STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7 LOCAL VALIDITY, GROTHENDIECK TOPOLOGIES AND ROUGH SETS 211 7.1 REPRESENTING ROUGH SETS. . . . . . . . . . . . . . . . . . . . . . . 211 7.1.1 LOCAL LOGICAL BEHAVIOURS IN ROUGH SET SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.2 SOME DUALITY OF DISTRIBUTIVE LATTICES . . . . . . . . . . . . 216 7.2.1 DUALITY FOR HEYTING AIGEBRAS. . . . . . . . . . . . 218 7.3 GROTHENDIECK TOPOLOGIES 219 7.3.1 A FUNDAMENTAL EXAMPLE: THE DENSE TOPOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.4 LAWVERE-TIERNEY OPERATORS AND ROUGH SET SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8 APPROXIMATION AND AIGEBRAIC LOGIC 237 8.1 APPROXIMATION OPERATORS. . . . . . . . . . . . . . . . . . . . . . 237 8.2 ADJOINTNESS, APPROXIMATIONS AND THE CENTER OF A ROUGH SET SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.3 MULTI- VALUED LOGICS: A KNOWLEDGE-ORIENTED INTERPRETATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.3.1 A TAXONOMY OF LOGICAL SYSTEMS. . . . . . . . . 242 8.3.2 EXACT AND INEXACT INFORMATION IN LOGICO-ALGEBRAIC SYSTEMS 247 XVIII CONTENTS 9 A LOGICO-PHILOSOPHICEXCURSUS 255 9.1 TRUTH-ORIENTED AND KNOWLEDGE-ORIENTED APPROACHES IN LOGIC . .. . . . . . . . . . . . . . . . . . . . . . . . . 255 9.2 UNDERSTANDING THE KNOWLEDGE-ORIENTED POINT OF VIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.3 SOME PROBLEMS ARISING FROM THE KNOWLEDGE- ORIENTED POINT OF VIEW . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.3.1 POSSIBLE SOLUTIONS 1: MAKING CLASSICAL AND CONSTRUCTIVE ATTITUDES COEXIST . . . . . . 260 9.3.2 POSSIBLE SOLUTIONS 2: STRENGTHENING INT WITH CLASSICAL PRINCIPLES . . . . . . . . . . . . . . . 261 9.4 A MIXED-RADIX ATTITUDE IN LOGIC. . . . . . . . . . . . . . 262 9.5 A MAXIMAL INTERMEDIATE CONSTRUCTIVE LOGIC 266 9.6 MIXED-RADIX INFORMATION SYSTEMS. . . . . . . .. . . . . . . 269 9.6.1 LOCAL VALIDITY IN NELSON LATTICES FROM HEYTING AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . 269 9.6.2 LOCAL VALIDITY AND MIXED LOGICAL BEHAVIOUR 276 9.7 CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10 FRAMES (PART 11) 281 10.1 FRAME ~ ROUGH SET SYSTEMS AND CHAIN-BASED LATTICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.2 FRAME - ROUGH SET SYSTEMS AS REGULAR DOUBLE STONE AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.3 FRAME - INFORMATION-ORIENTED DUALITY THEOREMS. . 284 10.3.1 INFORMATION-ORIENTED INTERPRETATION OF DUALITY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.3.2 DUALITY OF LOGIC(}-ALGEBRAIC STRUCTURES ... 286 10.3.3 COLLAPSE OF MAXIMAL ELEMENTS AND ATOMIC DECIDABILITY . . . . . . . . . . . . . . . . 290 10.3.4 ROUGH SETS, DUALITY AND DECIDABILITY .... 291 10.3.5 ROUGH SET SYSTEMS, POST AIGEBRAS AND TOTAL ATOMIC UNDECIDABILITY .... . . . . 294 10.4 FRAME - REPRESENTATION OF THREE- VALUED LUKASIEWICZ AIGEBRAS AS ROUGH SET SYSTEM. . . . . . . 296 10.5 FRAME - PROOF OF THE FACTS STATED IN WINDOW 7.1. . 299 10.6 FRAME - PROOF OF PROPOSITION 8.3.1 300 CONTENTS XIX 10.7 FRAME - GROTHENDIECK TOPOLOGIES AND LAWVERE-TIERNEY OPERATORS. . . . . . . . . . . . . . . . . 302 10.8 FRAME ~ REPRESENTATION OF ROUGH SETS . . . . . . . . . . . 303 10.9 FRAME - ROUGH SETS AND NON CLASSIEAL LOGIEO-ALGEBRAIE SYSTEMS. . . . . . . . . . . . . . . . . . . . . . . 304 10.9.1 ROUGH SETS AND BROUWER-ZADEH LATTICES . . 305 10.9.2 LATTICES AND NON-CLASSIEAL LOGIES................................ 305 10.9.3 LATTIEES WITH STRONG NEGATION. . . . . . . . . . . 306 10.10 FRAME - REPRESENTATION THEOREMS AND DECOMPOSITION OF DISTRIBUTIVE LATTIEES . . . . . . . 307 10.11 FRAME - REPRESENTATION OF LOGICAL VALUES BY ORDERED PAIRS , . . . . . . . . . . . . . . 311 10.12 FRAME - NEGATION 312 10.12.1 CLASSIFYING FORMAL NEGATIONS. . . . . . . . . . . . 314 10.12.2 A GEOMETRIE INTERPRETATION OF NEGATION. . 316 10.12.3 STRONG NEGATIONS AND KNOWLEDGE STATES IN ARTIFICIAL INTELLIGENCE 317 10.12.4 NEGATIONS, BODIES AND BOUNDARIES. . . . . . . . 318 10.12.5 NEGATIONS, MODALITIES AND STALK SPACES. . . 320 10.13 FRAME - INTUITIONISTIC LOGIC: NATURAL DEDUCTION SYSTEM INT ... .............................. 323 10.14 FRAME - CLASSIEAL LOGIE: NATURAL DEDUCTION SYSTEM CF:, 324 10.14,1 DERIVING THE PRINCIPLE OF EXCLUDED MIDDLE FROM CF:, . . . . . . . . . . . . . . . . . . . . . . . 324 10.15 FRAME - NELSON LOGIC: NATURAL DEDUCTION SYSTEM CLSN 324 10.15.1 CONSTRUETIVE LOGICS WITH STRONG NEGATION 324 10.15.2 NATURAL DEDUETION SYSTEM FOR CF:,SN . . . . . 325 10.16 FRAME - THE SYSTEM &0 . . . . .. . . . . . . . . . . . . . . . . . . 326 10.16.1 THE NATURAL DEDUETION SYSTEM &0 . . . . . . . 327 10.16.2 RELATIONAL MODELS FOR NELSON LOGIE 327 10.16.2.1 RELATIONAL MODELS FUER &0 . * . . . . . 327 10.16.3 LOGIE AND ALGEBRA IN PARTNERSHIP . . . . . . . . 327 10.16.4 ALGEBRAIE ANALYSIS OF THE CHARACTERISTIE PROOFS OF T:O . . . . . . . . . . . . . . 328 10.17 FRAME - THE LOGIE FA ........................ 332 XX CONTENTS 10.17.1 THE NATURAL DEDUCTION SYSTEM E* - ALIAS FA. ............................ 333 10.17.2 EVALUATION FORM SEMANTIES. . . . . . . . . . . . . 333 10.18 FRAME - MEDVEDEV S LOGIC OF FINITE PROBLEMS. . . . . 337 10.19 FRAME - ATOMICDECIDABILITY AND NON-STANDARD SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 10.19.1 ATOMIC DECIDABILITY AND THE FAILURE OF UNIFORM SUBSTITUTION . . . . . . . . . . . . . . . . 340 10.20 FRAME - AN APPLICATIONS OF THE ALGEBRAIC APPROACH TO PARTIAL INFORMATION SYSTEMS , . . . . . . . . 341 10.20.1 P-SLLSTEMS WITH PARTIAL INFORMATION. . . . . . 345 10.20.2 ADEQUACY OFTHE KLEENE FRAGMENT W.R.T. DEFINITE ANSWERS 351 10.21 FRAME - LOGICAL OPERATIONS IN A PURE ALGEBRAIC SETTING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35~ 10.22 SOLUTIONS..................................... 355 III THE MODAL LOGIC OF ROUGH SETS 361 11 MODALITY AND KNOWLEDGE 363 11.1 FOREWORD..................................... 363 11.2 MODALITIES AND ASSERTIONS 365 11.3 INTERNAL MODALITIES VS EXTERNAL MODALITIES . . . . . . . . 367 11.4 KNOWLEDGE AND INFORMATION. . . . . . . . . . . . . . . . . . . . . 371 11.5 KNOWLEDGE AND MODAL SYSTEMS. . . . . . . . . . . . . . . . . . 375 11.5.1 INTERNAL MODALITIES AND NON-DISTRIBUTIVITY . . . . . . . . . . . . . . . . . 381 11.5.2 EXTERNAL MODALITIES, DISTRIBUTIVITY AND POSSIBLE WORLDS SEMANTICS . . . . . . . . . . . . . . 382 11.5.3 REPRESENTING MODAL SYSTEMS. . . . . . . . . .. . 383 12 MODALITIES AND RELATIONS 389 12.1 MODAL SYSTEMS AND BINARY RELATIONS. . . . . . . . . .. . 389 12.2 FROM LOOSELY STRUCTURED SPACES TO STRUCTURED SPACES: A VARIETY OF MODAL PROPERTIES . . . .. . . . . . . 402 12.3 RELATIONS, PRE- TOPOLOGIES AND TOPOLOGIES . . . . . . . . . 406 12.4 PRE- TOPOLOGIEAL SPACES. . . . . . . . . . . . . . . . . . . . . . . . . 407 CONTENTS XXI 12.4.1 EXCURSUS. DYNAMICS 1: THE FAILURE OF THE ISOTONICITY LAW . . . . . . . . . . . . . . . . . . 423 12.5 TOWARDS TOPOLOGY 1 426 12.5.1 EXCURSUS: REFLEXIVITY, DISTRIBUTION AND PERCEPTION . . . . . . . . . . . . . . . . . . . . . . . . 430 12.6 TOWARDS TOPOLOGY 2 431 12.6.1 BASES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 12.6.2 EXCURSUS. DYNAMICS 2: THE FAILURE OF THE DISTRIBUTIVITY LAWS . . . . . . . . . . . . . . . . . 437 12.7 PRE-TOPOLOGICAL SPACES AND BINARY RELATIONS ..... 441 12.7.1 EXCURSUS: PRE-TOPOLOGICAL SPACES AND MODAL AIGEBRAS 449 12.7.2 EXCURSUS. PRE-TOPOLOGICAL SPACES AND APPROXIMATION SPACES 453 12.8 TOPOLOGICAL SPACES AND BINARY RELATIONS . . . . . . . . . 458 13 MODALITIES, TOPOLOGIES AND AIGEBRAS 471 13.1 TOPOLOGICAL BOOLEAN AIGEBRAS . . . . . . . . . . . . . . . . . . . 471 13.2 MONADIE TOPOLOGICAL BOOLEAN AIGEBRAS . . . . . . . . . . . 472 14 THE PROPOSITIONAL MODAL LOGIC OF ROUGH SETS 479 14.1 INTRODUCTION.................................. 479 14.2 FROM SYNTAX TO SEMANTIES. . . . . . . . . . . . . . . . . . . . . . 480 14.3 ROUGH ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 14.4 THE SYSTEMS LI, L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 14.4.1 THE SYSTEM L1 . . . . . . . . . . . . . . . . . . . . . . . . 492 14.4.2 THE SYSTEM L2 .. . . . . . . . . . . . . . . . . . . . . . . 497 14.4.3 ROUGH SET SEMANTICS FOR L2 . . . . . . . . . . . . . 499 14.5 AIGEBRAIC INTERPRETATION AND MODAL INTERPRETATION OF ROUGH SET SYSTEMS . . . . . . . . . . . . . 502 15 FRAMES (PART III) 505 15.1 FRAME - PROOF OF THE DUALITY BETWEEN LR(X) = U{Z : R(Z) 5; X} AND MR(X) = N{-Z: R(Z) 5; -X}............. 505 15.2 FRAME - RELATIONAL PROPERTIES AND LOGICAL CHARACTERISTICS . . . . . . . . .. . . . . . . . . . . . 506 15.2.1 PROOF OF KT5 = KTB4 . . . . . . . . . . . . . . . . . . 506 15.2.2 PROOF OF KDB4 = KTB4 . . . . . . . . . . . . . . . . 507 XXII CONTENTS 15.3 FRAME - PROOF OF PROPOSITION 12.7.10 507 15.4 FRAME -- PROOFS OF THE PROPOSITIONS ABOUT THE UNIQUENESS OF P(RT(P)) AND P(RB(P)) 508 15.5 FRAME - ALTERNATIVE PROOFS OF COROLLARY 12.8.2.(1) . 508 15.6 FRAME - DIRECTPROOF OF MR(X) = MR(X), FOR R A PREORDER RELATION 509 15.7 FRAME - TRANSFORMING APRE- TOPOLOGICAL SPACE OF TYPE M SINTO A TOPOLOGICAL SPACE. . . . . . . . . . . . . 509 15.8 FRAME - MODAL INTERPRETATIONS OF APPROXIMATION SPACES AND ROUGH SETS. . . . . . . . . 511 15.9 FRAME - KRIPKE-JOYAL MODELS . . . . . . . . . . . . . . . . . . . 512 15.10 FRAME - QUANTUM LOGIC AND INTERNAL MODALITIES .. 513 15.11 FRAME - PERSISTENCE OF MODALISED FORMULAS . . . . . . . 515 15.12 FRAME - COHERENCE BETWEEN INFORMATION AND KNOWLEDGE 518 15.13 FRAME - NEIGHBORHOOD SYSTEMS. . . . . . . . . . . . . . . . . 522 15.13.1 SOME HISTORY AND RECENT APPLICATIONS ... 522 15.13.2 NEIGHBORHOOD SYSTEMS AND APPROXIMATION OF INFORMATION. . . . . . . 525 15.13.3 NEIGHBORHOOD SYSTEMS AND MODAL SYSTEMS . . . . . . . . . . . . . . . . . . . . 526 15.13.4 THE OMNISCIENCE PROBLEM IN EPISTEMIC AND DOXASTIC LOGICS . . . . . . . . 529 15.14 FRAME - PRE-TOPOLOGIES AND INTUITIONISTIC FORMAL SPACES 530 15.14.1 FORMAL COVERING RELATIONS. . . . . . . . . . . . . . 531 15.14.2 SEMI-TOPOLOGICAL FORMAL SYSTEMS AND NEIGHBORHOOD SYSTEMS . . . . . . . . . . . . . . . . . 541 15.14.3 A NEW AGE: GETTING RID OF . AND 1.. ..... 549 15.14.4 THE LOGICAL INTERPRETATION OF THE PRE- TOPOLOGY FORMAL APPROACH 552 15.14.5 A SAMPIE FORMAL NEIGHBORHOOD SYSTEM. . 555 15.14.6 A PSEUD(}- TOPOLOGICAL NEIGHBORHOOD SYSTEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 15.14.7 A TOPOLOGICAL FORMAL NEIGHBORHOOD SYSTEM IN WHICH N3 DOES NOT HOLD 559 15.14.8 A TOPOLOGICAL FORMAL NEIGHBORHOOD SYSTEM IN WHICH NI DOES NOT HOLD. . . . . . . 560 CONTENTS XXIUE 15.14.9 A NON-TOPOLOGIEAL FORMAL NEIGHBORHOOD SYSTEM SUEH THAT OINT(U) IS A HEYTING ALGEBRA OF SETS. . . . . . . . . . . . . . . . . . . . . . . . 561 15.14.10 A LOGICAL INTERPRETATION OF CONERETE PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 15.15 FRAME - MODAL STRUETURES AND PRE- TOPOLOGIEAL SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 15.15.1 MODAL AIGEBRAS AND NEIGHBORHOOD SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 15.15.2 MODAL SYSTEMS, PRE-MONADIE BOOLEAN ALGEBRAS AND KRIPKE FRAMES . . . . . . . . . . . . 569 15.16 FRAME - DUALITY OF OPERATIONS AND AIGEBRAIE STRUETURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 15.17 FRAME - COMPUTING DEPENDENEY RELATIONS IN A FRAGMENT OF INTUITIONISTIE LOGIC . . . . . . . . . . . . . . . . . 571 15.18 FRAME - APPROXIMATION, FORMAL CONEEPTS, MODALITIES AND RELATION AIGEBRAS. . . . . . . . . . . . . . . . 574 15.18.1 RELATION AIGEBRAS 575 15.18.2 MODALIZING RELATIONS BY MEANS OF RELATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . 577 15.18.3 COMPARING FORMAL CONEEPTS AND ROUGH SETS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 15.18.4 APPROXIMATION OF RELATIONS. . . . . . . . . . . . . 588 15.18.5 MAKING FORMAL CONEEPTS AND ROUGH SETS INTERACT . . . . . . . . . .. . . . . . . . . . . . . . . . 589 15.18.6 WHAT THIS APPROACH MAY SUGGEST 590 15.18.7 COMPUTING DEPENDENCIES IN RELATION AIGEBRAS 591 15.19 FRAME - RELATIONAL PROOF THEORY. . . . . . . . . . . . . . . . 595 15.20 FRAME - SOME HISTORY OF THE AIGEBRAIE CONEEPTS USED IN THIS PART 600 15.21 SOLUTIONS..................................... 601 16 MATHEMATIEAL TOOLKITS 615 16.1 A MATHEMATICAL TOOLKIT: ORDERS. . . . . . . . . . . . . . . . . 615 16.2 A MATHEMATIEAL TOOLKIT: FUNCTIONS . . . . . . . . . . . . . . 616 16.3 A MATHEMATIEAL TOOLKIT: LATTIEES. . . . . . . . . . . . . . . . 618 16.4 A MATHEMATIEAL TOOLKIT: TOPOL~GY 621 16.5 A MATHEMATIEAL TOOLKIT: RELATIONS 623 XXIV CONTENTS 16.5.1 PULL-BACKS AND KERNELS. . . . . .. . . . . . . . . . . 625 16.5.1.1 CATEGORIZATION AND KERNEIS . . . . . 625 16.5.1.2 CATEGORIZATION AND PULL-BACKS . . 627 BIBLIOGRAPHY INDEX 631 681
adam_txt	CONTENTS PREFACE LIST OF FIGURES NOTATION ABBREVIATIONS INTRODUCTION 1 PERCEPTION AND CONCEPTS: A PHENOMENOLOGICAL APPROACH . 1.1 MONOLOGICAL APPROACH AND DIALOGICAL APPROACH . 2 MONOLOGICAL APPROACH TO PERCEPTION AND CONCEPTS . 3 PHENOMENOLOGY AND LOGIC . 3.1 SEMANTICS VS SYNTAX . 3.2 INFORMATION AND INTERPRETATION: CORRESPONDENCE THEORY OF TRUTH VS PRAGMATISM . 3.2.1 MEANING-CONDITIONS VS TRUTH-CONDITIONS . 3.2.2 LOGIC, MEANING AND ROUGH SET THEORY . 4 THE LOGIC(}- ALGEBRAIC INTERPRETATION OF ROUGH SET SYSTEMS . 5 EQUIVALENCE CLASSES, ABSTRACTION AND MEANING . 5.1 TYPES, TOKENS AND ABSTRACT POINTS . 5.2 ABSTRACT POINTS AND MEANING . V XXIV XXIX XXXI XXXIII XXXIII XXXVI XXXVIII I I LVI LVII LXII LXIV !XVIII LXVIII !XXVI XIII XIV 6 7 5.3 ABSTRACT POINTS AND ROUGH SETS . ROUGH SETS AND LOGIE . CONCLUDING REMARKS . CONTENTS !XXVI !XXVII !XXXI I A MATHEMATICS OF PERCEPTION 1 1 OBSERVATIONS, NOUMENA AND PHENOMENA 3 1.1 FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 FORMAL RELATIONSHIPS BETWEEN "NOUMENA" AND "PHENOMENA" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 PROPERTY SYSTEMS - P-SYSTEMS . . . . . . . . . 8 1.2.1.1 FUNCTIONAL PROPERTY SYSTEMS . . . 11 1.2.1.2 RELATIONAL PROPERTY SYSTEMS . 12 1.2.1.3 DICHOTOMIE PROPERTY SYSTEMS . 13 1.2.2 ATTRIBUTE SYSTEMS - A-SYSTEMS. . . . . . . . . 13 1.3 FUNCTIONAL P-SYSTEMS AND CONCEPTUALISATION . . . . . . 14 1.3.1 CATEGORIZING THROUGH FUNCTIONAL P-SYSTEMS 15 1.4 CATEGORIZING THROUGH RELATIONAL P-SYSTEMS 20 1.4.1 TYPES AND APPROXIMATION . . . . . . . . . . . . . . 23 1.4.2 DIVISORS AND RESIDUALS 24 1.4.3 GALOIS ADJUNCTIONS AND GALOIS CONNECTIONS 29 1.4.4 AIGEBRAIC PROPERTIES OF ADJOINT MAPS . . 33 2 CONCRETE AND FORMAL INFORMATION CONSTRUCTIONS 43 2.1 CONCRETE AND FORMAL OBSERVATION SPACES. . . . . . . . 43 2.1.1 OBSERVATIONS AND PARTIAL OBSERVATIONS. . . 45 2.1.2 RELATIONS ANDGALOIS ADJUNCTIONS. . . . . . . . 47 2.2 THE BASIC PHENOMENOLOGICAL CONSTRUCTORS . . . . . . . . 50 2.2.1 A MODAL READING OF THE BASIC CONSTRUCTORS 53 2.3 FORMAL OPERATORS ON POINTS AND ON OBSERVABLES . . . 58 2.3.1 AIGEBRAIC PROPERTIES OF FORMAL PERCEPTION SYSTEMS . . . . . . . . . . . . . . . . . . . . 63 2.3.2 MULTI-AGENT PRE-TOPOLOGICAL APPROXIMATION SYSTEMS . . . . . . . . . . . . . 70 3 PRE-TOPOLOGICALAND TOPOLOGICALAPPROXIMATION OPERATORS 73 3.1 INFORMATION, CONCEPTS AND FORMAL OPERATORS. . . . . 73 CONTENTS XV 3.1.1 CHOOSING THE INITIAL PERCEPTION ACT . . . . . . 75 3.1.2 INFORMATION QUANTUM RELATIONAL SYSTEMS 81 3.2 COMPARING PERCEPTION SYSTEMS. . . . . . . . . . . . . . . . . . 85 3.3 HIGHER LEVEL OPERATORS 90 3.4 TRANSFORMING PERCEPTION SYSTEMS 97 3.5 TOPOLOGICAL APPROXIMATION OPERATORS. . . . . . . . . . . . 100 3.6 TOPOLOGICAL APPROXIMATION SYSTEMS 103 4 FRAMES(PART I) 107 4.1 FRAME - APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 FRAME - CLASSIFIEATION. . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 FRAME - CATEGORIZING THROUGH POINTLESS TOPOLOGY . 110 4.4 FRAME - OBSERVABLE PROPERTIES . . . . . . . . . . . . . . . . . . 113 4.4.1 DEEIDABLE AND SEMI-DECIDABLE PROPERTIES . 113 4.4.2 DECIDABLE PROPERTIES, TOPOLOGY, DOMAINS AND GEOMETRIE LOGIE . . . . . . . . . . . . . . . . . . 115 4.5 FRAME - FINITE OBSERVATIONS: THE BINARY MACHINE EXAMPLE 116 4.6 FRAME - QUANTA OF INFORMATION 120 4.6.1 QUANTA AT A LOEATION AND ORTHOLATTIEES . . 120 4.6.2 A TOPO-ALGEBRAIC READING OF IQRSS . . . . 123 4.6.3 DUALITY BETWEEN P-SYSTEMS AND PREORDERS . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.7 FRAME - INFORMATION SYSTEMS . . . . . . . . . . . . . . . . . . . 126 4.7.1 GENERALISING INFORMATION RELATIONS 126 4.7.2 GENERALIZING INDISEERNIBILITY RELATION 127 4.7.3 GENERALISING FROM SETS TO RELATIONS. . . . . . 128 4.8 FRAME - DICHOTOMIE, COMPLEMENTARY AND FUNETIONAL SYSTEMS. . . . . . . . . . . . . . . . . . . . . . . 129 4.8.1 DIEHOTOMIE SYSTEMS. . . . . . . . . . . . . . . . . . . 129 4.8.2 COMPLEMENTARY SYSTEMS I . . . . . . . . . . . . . . 130 4.8.3 COMPLEMENTARY SYSTEMS II 130 4.8.4 COMPLEMENTARY SYSTEMS III. . 131 4.8.5 DICHOTOMIE SYSTEMS AND FUNETIONAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.8.6 FUNETIONAL SYSTEMS AND APPROXIMATIONS. . 132 4.8.7 DICHOTOMIE SYSTEMS AND APPROXIMATIONS . 132 XVI CONTENTS 4.9 FRAME - CONCEPT LATTICES 133 4.9.1 FORMAL CONTEXTS AND FORMAL CONCEPT ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.9.2 FORMAL CONCEPTS AND APPROXIMATION OPERATORS 135 4.9.3 COMBINING CLASSICAL APPROXIMATION SYSTEMS AND CONCEPT LATTICES. . . . . . . . . . . 137 4.9.4 COMBINING NON-CLASSICAL APPROXIMATION SYSTEMS AND CONCEPT LATTICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.9.5 NOMINAL SYSTEMS AND CONCEPTUAL SCALING. 140 4.10 FRAME - NEIGHBORHOOD SYSTEMS. . . . . . . . . . . . . . . . 142 4.11 FRAME - BASIC PAIRS AND POINT-FREE TOPOLOGY . . . . . 143 4.12 FRAME - CHU SPACES. . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.13 FRAME - INTUITIONISM, MODALITIES AND RELATIONAL SEMANTIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.13.1 NECESSITY AND POSSIBILITY 146 4.13.2 BASIC OPERATORS AS MODAL OPERATORS. . . . 147 4.13.3 RAMIFIED TENSE LOGIC 148 4.13.4 NECESSITY AND SUFFICIENEY OPERATORS. . . . . . 148 4.13.5 MODAL OPERATORS AND INFORMATION SYSTEMS 149 4.14 FRAME - GAJOISADJUNCTIONS 151 4.14.1 GALOIS ADJUNETIONS IN COMPUTER SCIENCE. . 152 4.14.2 GALOIS ADJUNCTIONS AND DEDEKIND CUTS. . 154 4.14.3 GALOIS ADJUNETIONS AT LARGE 155 4.14.4 GALOIS CONNECTIONS AND REPRESENTATION THEOREMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.14.5 GALOIS ADJUNCTIONS, ISOMORPHISMS AND APPROXIMATION: A NOTE 157 4.15 FRAME - CATEGORIES AND ADJOINT FTMCTORS. . . . . . . . . 158 4.16 SOLUTIONS. 161 11 THE LOGICO-ALGEBRAICTHEORY OF ROUGH SETS 167 5 LOGIC AND ROUGH SETS: AN OVERVIEW 169 5.1 FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2 ROUGH SET SYSTEMS AND THREE- VALUED LOGIES. . . . . . 172 CONTENTS XVII 5.3 EXACT AND INEXACT LOCAL BEHAVIOURS IN ROUGH SET SYSTEMS 174 5.4 REPRESENTING ROUGH SETS . . . . . . . . . . . . . . . . . . . . . . . 177 5.5 ROUGH SET SYSTEMS, LOCAL VALIDITY, AND LOGICO-ALGEBRAIC STRUCTURES. . . . . . . . . . . . . . . . 181 6 BASIC LOGICO-ALGEBRAIC STRUCTURES 193 6.1 HEYTING AIGEBRAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.2 NELSON AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.3 N-VALUED LUKASIEWICZ AIGEBRAS 203 6.4 CHAIN- BASED LATTICES 204 6.5 RELATIONSHIPS, ANALOGIES AND DIFFERENCES BETWEEN STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7 LOCAL VALIDITY, GROTHENDIECK TOPOLOGIES AND ROUGH SETS 211 7.1 REPRESENTING ROUGH SETS. . . . . . . . . . . . . . . . . . . . . . . 211 7.1.1 LOCAL LOGICAL BEHAVIOURS IN ROUGH SET SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.2 SOME DUALITY OF DISTRIBUTIVE LATTICES . . . . . . . . . . . . 216 7.2.1 DUALITY FOR HEYTING AIGEBRAS. . . . . . . . . . . . 218 7.3 GROTHENDIECK TOPOLOGIES 219 7.3.1 A FUNDAMENTAL EXAMPLE: THE DENSE TOPOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.4 LAWVERE-TIERNEY OPERATORS AND ROUGH SET SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8 APPROXIMATION AND AIGEBRAIC LOGIC 237 8.1 APPROXIMATION OPERATORS. . . . . . . . . . . . . . . . . . . . . . 237 8.2 ADJOINTNESS, APPROXIMATIONS AND THE CENTER OF A ROUGH SET SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.3 MULTI- VALUED LOGICS: A KNOWLEDGE-ORIENTED INTERPRETATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.3.1 A TAXONOMY OF LOGICAL SYSTEMS. . . . . . . . . 242 8.3.2 EXACT AND INEXACT INFORMATION IN LOGICO-ALGEBRAIC SYSTEMS 247 XVIII CONTENTS 9 A LOGICO-PHILOSOPHICEXCURSUS 255 9.1 TRUTH-ORIENTED AND KNOWLEDGE-ORIENTED APPROACHES IN LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.2 UNDERSTANDING THE KNOWLEDGE-ORIENTED POINT OF VIEW " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.3 SOME PROBLEMS ARISING FROM THE KNOWLEDGE- ORIENTED POINT OF VIEW . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.3.1 POSSIBLE SOLUTIONS 1: MAKING CLASSICAL AND CONSTRUCTIVE ATTITUDES COEXIST . . . . . . 260 9.3.2 POSSIBLE SOLUTIONS 2: STRENGTHENING INT WITH CLASSICAL PRINCIPLES . . . . . . . . . . . . . . . 261 9.4 A "MIXED-RADIX" ATTITUDE IN LOGIC. . . . . . . . . . . . . . 262 9.5 A MAXIMAL INTERMEDIATE CONSTRUCTIVE LOGIC 266 9.6 MIXED-RADIX INFORMATION SYSTEMS. . . . . . . . . . . . . . 269 9.6.1 LOCAL VALIDITY IN NELSON LATTICES FROM HEYTING AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . 269 9.6.2 LOCAL VALIDITY AND MIXED LOGICAL BEHAVIOUR 276 9.7 CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10 FRAMES (PART 11) 281 10.1 FRAME ~ ROUGH SET SYSTEMS AND CHAIN-BASED LATTICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.2 FRAME - ROUGH SET SYSTEMS AS REGULAR DOUBLE STONE AIGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.3 FRAME - INFORMATION-ORIENTED DUALITY THEOREMS. . 284 10.3.1 INFORMATION-ORIENTED INTERPRETATION OF DUALITY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.3.2 DUALITY OF LOGIC(}-ALGEBRAIC STRUCTURES . 286 10.3.3 COLLAPSE OF MAXIMAL ELEMENTS AND ATOMIC DECIDABILITY . . . . . . . . . . . . . . . . 290 10.3.4 ROUGH SETS, DUALITY AND DECIDABILITY . 291 10.3.5 ROUGH SET SYSTEMS, POST AIGEBRAS AND TOTAL ATOMIC UNDECIDABILITY . . . . . 294 10.4 FRAME - REPRESENTATION OF THREE- VALUED LUKASIEWICZ AIGEBRAS AS ROUGH SET SYSTEM. . . . . . . 296 10.5 FRAME - PROOF OF THE FACTS STATED IN WINDOW 7.1. . 299 10.6 FRAME - PROOF OF PROPOSITION 8.3.1 300 CONTENTS XIX 10.7 FRAME - GROTHENDIECK TOPOLOGIES AND LAWVERE-TIERNEY OPERATORS. . . . . . . . . . . . . . . . . 302 10.8 FRAME ~ REPRESENTATION OF ROUGH SETS . . . . . . . . . . . 303 10.9 FRAME - ROUGH SETS AND NON CLASSIEAL LOGIEO-ALGEBRAIE SYSTEMS. . . . . . . . . . . . . . . . . . . . . . . 304 10.9.1 ROUGH SETS AND BROUWER-ZADEH LATTICES . . 305 10.9.2 LATTICES AND NON-CLASSIEAL LOGIES. 305 10.9.3 LATTIEES WITH STRONG NEGATION. . . . . . . . . . . 306 10.10 FRAME - REPRESENTATION THEOREMS AND DECOMPOSITION OF DISTRIBUTIVE LATTIEES . . . . . . . 307 10.11 FRAME - REPRESENTATION OF LOGICAL VALUES BY ORDERED PAIRS , . . . . . . . . . . . . . . 311 10.12 FRAME - NEGATION 312 10.12.1 CLASSIFYING FORMAL NEGATIONS. . . . . . . . . . . . 314 10.12.2 A GEOMETRIE INTERPRETATION OF NEGATION. . 316 10.12.3 STRONG NEGATIONS AND KNOWLEDGE STATES IN ARTIFICIAL INTELLIGENCE 317 10.12.4 NEGATIONS, BODIES AND BOUNDARIES. . . . . . . . 318 10.12.5 NEGATIONS, MODALITIES AND STALK SPACES. . . 320 10.13 FRAME - INTUITIONISTIC LOGIC: NATURAL DEDUCTION SYSTEM INT . . 323 10.14 FRAME - CLASSIEAL LOGIE: NATURAL DEDUCTION SYSTEM CF:, 324 10.14,1 DERIVING THE PRINCIPLE OF EXCLUDED MIDDLE FROM CF:, . . . . . . . . . . . . . . . . . . . . . . . 324 10.15 FRAME - NELSON LOGIC: NATURAL DEDUCTION SYSTEM CLSN 324 10.15.1 CONSTRUETIVE LOGICS WITH STRONG NEGATION 324 10.15.2 NATURAL DEDUETION SYSTEM FOR CF:,SN . . . . . 325 10.16 FRAME - THE SYSTEM &0 . . . . . . . . . . . . . . . . . . . . . . . 326 10.16.1 THE NATURAL DEDUETION SYSTEM &0 . . . . . . . 327 10.16.2 RELATIONAL MODELS FOR NELSON LOGIE 327 10.16.2.1 RELATIONAL MODELS FUER &0 . * . . . . . 327 10.16.3 LOGIE AND ALGEBRA IN PARTNERSHIP . . . . . . . . 327 10.16.4 ALGEBRAIE ANALYSIS OF THE CHARACTERISTIE PROOFS OF T:O . . . . . . . . . . . . . . 328 10.17 FRAME - THE LOGIE FA . 332 XX CONTENTS 10.17.1 THE NATURAL DEDUCTION SYSTEM E* - ALIAS FA. . 333 10.17.2 EVALUATION FORM SEMANTIES. . . . . . . . . . . . . 333 10.18 FRAME - MEDVEDEV'S LOGIC OF FINITE PROBLEMS. . . . . 337 10.19 FRAME - ATOMICDECIDABILITY AND NON-STANDARD SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 10.19.1 ATOMIC DECIDABILITY AND THE FAILURE OF UNIFORM SUBSTITUTION . . . . . . . . . . . . . . . . 340 10.20 FRAME - AN APPLICATIONS OF THE ALGEBRAIC APPROACH TO PARTIAL INFORMATION SYSTEMS ',' . . . . . . . . 341 10.20.1 P-SLLSTEMS WITH PARTIAL INFORMATION. . . . . . 345 10.20.2 ADEQUACY OFTHE KLEENE FRAGMENT W.R.T. DEFINITE ANSWERS 351 10.21 FRAME - LOGICAL OPERATIONS IN A PURE ALGEBRAIC SETTING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35~ 10.22 SOLUTIONS. 355 III THE MODAL LOGIC OF ROUGH SETS 361 11 MODALITY AND KNOWLEDGE 363 11.1 FOREWORD. 363 11.2 MODALITIES AND ASSERTIONS 365 11.3 INTERNAL MODALITIES VS EXTERNAL MODALITIES . . . . . . . . 367 11.4 KNOWLEDGE AND INFORMATION. . . . . . . . . . . . . . . . . . . . . 371 11.5 KNOWLEDGE AND MODAL SYSTEMS. . . . . . . . . . . . . . . . . . 375 11.5.1 INTERNAL MODALITIES AND NON-DISTRIBUTIVITY . . . . . . . . . . . . . . . . . 381 11.5.2 EXTERNAL MODALITIES, DISTRIBUTIVITY AND POSSIBLE WORLDS SEMANTICS . . . . . . . . . . . . . . 382 11.5.3 REPRESENTING MODAL SYSTEMS. . . . . . . . . . . 383 12 MODALITIES AND RELATIONS '389 12.1 MODAL SYSTEMS AND BINARY RELATIONS. . . . . . . . . . . 389 12.2 FROM LOOSELY STRUCTURED SPACES TO STRUCTURED SPACES: A VARIETY OF MODAL PROPERTIES . . . . . . . . . . 402 12.3 RELATIONS, PRE- TOPOLOGIES AND TOPOLOGIES . . . . . . . . . 406 12.4 PRE- TOPOLOGIEAL SPACES. . . . . . . . . . . . . . . . . . . . . . . . . 407 CONTENTS XXI 12.4.1 EXCURSUS. DYNAMICS 1: THE FAILURE OF THE ISOTONICITY LAW . . . . . . . . . . . . . . . . . . 423 12.5 TOWARDS TOPOLOGY 1 426 12.5.1 EXCURSUS: REFLEXIVITY, DISTRIBUTION AND PERCEPTION . . . . . . . . . . . . . . . . . . . . . . . . 430 12.6 TOWARDS TOPOLOGY 2 431 12.6.1 BASES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 12.6.2 EXCURSUS. DYNAMICS 2: THE FAILURE OF THE DISTRIBUTIVITY LAWS . . . . . . . . . . . . . . . . . 437 12.7 PRE-TOPOLOGICAL SPACES AND BINARY RELATIONS . 441 12.7.1 EXCURSUS: PRE-TOPOLOGICAL SPACES AND MODAL AIGEBRAS 449 12.7.2 EXCURSUS. PRE-TOPOLOGICAL SPACES AND APPROXIMATION SPACES 453 12.8 TOPOLOGICAL SPACES AND BINARY RELATIONS . . . . . . . . . 458 13 MODALITIES, TOPOLOGIES AND AIGEBRAS 471 13.1 TOPOLOGICAL BOOLEAN AIGEBRAS . . . . . . . . . . . . . . . . . . . 471 13.2 MONADIE TOPOLOGICAL BOOLEAN AIGEBRAS . . . . . . . . . . . 472 14 THE PROPOSITIONAL MODAL LOGIC OF ROUGH SETS 479 14.1 INTRODUCTION. 479 14.2 FROM SYNTAX TO SEMANTIES. . . . . . . . . . . . . . . . . . . . . . 480 14.3 ROUGH ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 14.4 THE SYSTEMS LI, L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 14.4.1 THE SYSTEM L1 . . . . . . . . . . . . . . . . . . . . . . . . 492 14.4.2 THE SYSTEM L2 . . . . . . . . . . . . . . . . . . . . . . . 497 14.4.3 ROUGH SET SEMANTICS FOR L2 . . . . . . . . . . . . . 499 14.5 AIGEBRAIC INTERPRETATION AND MODAL INTERPRETATION OF ROUGH SET SYSTEMS . . . . . . . . . . . . . 502 15 FRAMES (PART III) 505 15.1 FRAME - PROOF OF THE DUALITY BETWEEN LR(X) = U{Z : R(Z) 5; X} AND MR(X) = N{-Z: R(Z) 5; -X}. 505 15.2 FRAME - RELATIONAL PROPERTIES AND LOGICAL CHARACTERISTICS . . . . . . . . . . . . . . . . . . . . 506 15.2.1 PROOF OF KT5 = KTB4 . . . . . . . . . . . . . . . . . . 506 15.2.2 PROOF OF KDB4 = KTB4 . . . . . . . . . . . . . . . . 507 XXII CONTENTS 15.3 FRAME - PROOF OF PROPOSITION 12.7.10 507 15.4 FRAME -- PROOFS OF THE PROPOSITIONS ABOUT THE UNIQUENESS OF P(RT(P)) AND P(RB(P)) 508 15.5 FRAME - ALTERNATIVE PROOFS OF COROLLARY 12.8.2.(1) . 508 15.6 FRAME - DIRECTPROOF OF MR(X) = MR(X), FOR R A PREORDER RELATION 509 15.7 FRAME - TRANSFORMING APRE- TOPOLOGICAL SPACE OF TYPE M SINTO A TOPOLOGICAL SPACE. . . . . . . . . . . . . 509 15.8 FRAME - MODAL INTERPRETATIONS OF APPROXIMATION SPACES AND ROUGH SETS. . . . . . . . . 511 15.9 FRAME - KRIPKE-JOYAL MODELS . . . . . . . . . . . . . . . . . . . 512 15.10 FRAME - QUANTUM LOGIC AND INTERNAL MODALITIES . 513 15.11 FRAME - PERSISTENCE OF MODALISED FORMULAS . . . . . . . 515 15.12 FRAME - COHERENCE BETWEEN INFORMATION AND KNOWLEDGE 518 15.13 FRAME - NEIGHBORHOOD SYSTEMS. . . . . . . . . . . . . . . . . 522 15.13.1 SOME HISTORY AND RECENT APPLICATIONS . 522 15.13.2 NEIGHBORHOOD SYSTEMS AND APPROXIMATION OF INFORMATION. . . . . . . 525 15.13.3 NEIGHBORHOOD SYSTEMS AND MODAL SYSTEMS . . . . . . . . . . . . . . . . . . . . 526 15.13.4 THE "OMNISCIENCE PROBLEM" IN EPISTEMIC AND DOXASTIC LOGICS . . . . . . . . 529 15.14 FRAME - PRE-TOPOLOGIES AND INTUITIONISTIC FORMAL SPACES 530 15.14.1 FORMAL COVERING RELATIONS. . . . . . . . . . . . . . 531 15.14.2 SEMI-TOPOLOGICAL FORMAL SYSTEMS AND NEIGHBORHOOD SYSTEMS . . . . . . . . . . . . . . . . . 541 15.14.3 A NEW AGE: GETTING RID OF . AND 1. . 549 15.14.4 THE LOGICAL INTERPRETATION OF THE PRE- TOPOLOGY FORMAL APPROACH 552 15.14.5 A SAMPIE FORMAL NEIGHBORHOOD SYSTEM. . 555 15.14.6 A PSEUD(}- TOPOLOGICAL NEIGHBORHOOD SYSTEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 15.14.7 A TOPOLOGICAL FORMAL NEIGHBORHOOD SYSTEM IN WHICH N3 DOES NOT HOLD 559 15.14.8 A TOPOLOGICAL FORMAL NEIGHBORHOOD SYSTEM IN WHICH NI DOES NOT HOLD. . . . . . . 560 CONTENTS XXIUE 15.14.9 A NON-TOPOLOGIEAL FORMAL NEIGHBORHOOD SYSTEM SUEH THAT OINT(U) IS A HEYTING ALGEBRA OF SETS. . . . . . . . . . . . . . . . . . . . . . . . 561 15.14.10 A LOGICAL INTERPRETATION OF CONERETE PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 15.15 FRAME - MODAL STRUETURES AND PRE- TOPOLOGIEAL SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 15.15.1 MODAL AIGEBRAS AND NEIGHBORHOOD SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 15.15.2 MODAL SYSTEMS, PRE-MONADIE BOOLEAN ALGEBRAS AND KRIPKE FRAMES . . . . . . . . . . . . 569 15.16 FRAME - DUALITY OF OPERATIONS AND AIGEBRAIE STRUETURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 15.17 FRAME - COMPUTING DEPENDENEY RELATIONS IN A FRAGMENT OF INTUITIONISTIE LOGIC . . . . . . . . . . . . . . . . . 571 15.18 FRAME - APPROXIMATION, FORMAL CONEEPTS, MODALITIES AND RELATION AIGEBRAS. . . . . . . . . . . . . . . . 574 15.18.1 RELATION AIGEBRAS 575 15.18.2 MODALIZING RELATIONS BY MEANS OF RELATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . 577 15.18.3 COMPARING FORMAL CONEEPTS AND ROUGH SETS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 15.18.4 APPROXIMATION OF RELATIONS. . . . . . . . . . . . . 588 15.18.5 MAKING FORMAL CONEEPTS AND ROUGH SETS INTERACT . . . . . . . . . . . . . . . . . . . . . . . . . 589 15.18.6 WHAT THIS APPROACH MAY SUGGEST 590 15.18.7 COMPUTING DEPENDENCIES IN RELATION AIGEBRAS 591 15.19 FRAME - RELATIONAL PROOF THEORY. . . . . . . . . . . . . . . . 595 15.20 FRAME - SOME HISTORY OF THE AIGEBRAIE CONEEPTS USED IN THIS PART 600 15.21 SOLUTIONS. 601 16 MATHEMATIEAL TOOLKITS 615 16.1 A MATHEMATICAL TOOLKIT: ORDERS. . . . . . . . . . . . . . . . . 615 16.2 A MATHEMATIEAL TOOLKIT: FUNCTIONS . . . . . . . . . . . . . . 616 16.3 A MATHEMATIEAL TOOLKIT: LATTIEES. . . . . . . . . . . . . . . . 618 16.4 A MATHEMATIEAL TOOLKIT: TOPOL~GY 621 16.5 A MATHEMATIEAL TOOLKIT: RELATIONS 623 XXIV CONTENTS 16.5.1 PULL-BACKS AND KERNELS. . . . . . . . . . . . . . . . 625 16.5.1.1 CATEGORIZATION AND KERNEIS . . . . . 625 16.5.1.2 CATEGORIZATION AND PULL-BACKS . . 627 BIBLIOGRAPHY INDEX 631 681
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spelling	Pagliani, Piero Verfasser aut A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns Piero Pagliani ; Mihir Chakraborty Berlin Springer 2008 LXXXII, 704 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Trends in Logic 27 Grobmenge (DE-588)4362502-2 gnd rswk-swf Heyting-Arithmetik (DE-588)4576522-4 gnd rswk-swf Frame Mathematik (DE-588)4528312-6 gnd rswk-swf Epistemologischer Kontextualismus (DE-588)4998367-2 gnd rswk-swf Modelltheoretische Semantik (DE-588)4221369-1 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Semitopologische Halbgruppe (DE-588)4180976-2 gnd rswk-swf Grothendieck-Topologie (DE-588)4158325-5 gnd rswk-swf Informationssystem (DE-588)4072806-7 gnd rswk-swf Heyting-Arithmetik (DE-588)4576522-4 s Informationssystem (DE-588)4072806-7 s Modelltheoretische Semantik (DE-588)4221369-1 s Grothendieck-Topologie (DE-588)4158325-5 s Semitopologische Halbgruppe (DE-588)4180976-2 s Epistemologischer Kontextualismus (DE-588)4998367-2 s Approximation (DE-588)4002498-2 s Frame Mathematik (DE-588)4528312-6 s DE-604 Grobmenge (DE-588)4362502-2 s Chakraborty, Mihir Verfasser aut Trends in Logic 27 (DE-604)BV011512969 27 Digitalisierung UB Erlangen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016552902&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Pagliani, Piero Chakraborty, Mihir A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns Trends in Logic Grobmenge (DE-588)4362502-2 gnd Heyting-Arithmetik (DE-588)4576522-4 gnd Frame Mathematik (DE-588)4528312-6 gnd Epistemologischer Kontextualismus (DE-588)4998367-2 gnd Modelltheoretische Semantik (DE-588)4221369-1 gnd Approximation (DE-588)4002498-2 gnd Semitopologische Halbgruppe (DE-588)4180976-2 gnd Grothendieck-Topologie (DE-588)4158325-5 gnd Informationssystem (DE-588)4072806-7 gnd
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title	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns
title_auth	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns
title_exact_search	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns
title_exact_search_txtP	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns
title_full	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns Piero Pagliani ; Mihir Chakraborty
title_fullStr	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns Piero Pagliani ; Mihir Chakraborty
title_full_unstemmed	A geometry of approximation Rough Set Theory: logic, algebra and topology of conceptual patterns Piero Pagliani ; Mihir Chakraborty
title_short	A geometry of approximation
title_sort	a geometry of approximation rough set theory logic algebra and topology of conceptual patterns
title_sub	Rough Set Theory: logic, algebra and topology of conceptual patterns
topic	Grobmenge (DE-588)4362502-2 gnd Heyting-Arithmetik (DE-588)4576522-4 gnd Frame Mathematik (DE-588)4528312-6 gnd Epistemologischer Kontextualismus (DE-588)4998367-2 gnd Modelltheoretische Semantik (DE-588)4221369-1 gnd Approximation (DE-588)4002498-2 gnd Semitopologische Halbgruppe (DE-588)4180976-2 gnd Grothendieck-Topologie (DE-588)4158325-5 gnd Informationssystem (DE-588)4072806-7 gnd
topic_facet	Grobmenge Heyting-Arithmetik Frame Mathematik Epistemologischer Kontextualismus Modelltheoretische Semantik Approximation Semitopologische Halbgruppe Grothendieck-Topologie Informationssystem
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016552902&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV011512969
work_keys_str_mv	AT paglianipiero ageometryofapproximationroughsettheorylogicalgebraandtopologyofconceptualpatterns AT chakrabortymihir ageometryofapproximationroughsettheorylogicalgebraandtopologyofconceptualpatterns

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