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Index

actions, 129, 146 advance, 299 algebraic strong de Morgan law,

53, 57, 63, 76, 80 Aristotelian notion of truth, 308 atom,

sub-boolean, 14

bold algebra of fuzzy sets, 8, 23 bold field of fuzzy sets,

a-complete, 8

C*-algebra, 159, 161 categories of fuzzy sets, 235, 236 category

concrete, 220 local cartesian closed, 229 of L-fuzzy sets, 219 symmetric, monoidal closed,

137 characteristic map, 224, 225 characteristic morphism, 222,

224,225 checklist

paradigm, 320, 329, 330 abstract, 321

classifiable, 128, 150 classification of MV -algebras

with square roots, 65 classification problem, 154 cl-monoid, 79, 88

385

coherent theory, 304 colouring technique, 33, 36 commutative ring with unity, 67 compatible family, 202 completeness of

La, 277 LL,276 the infinite-valued sentential

calculus of Lukasiewicz, 107

conjunction, 154, 236 fuzzy (cf. Lukasiewicz arith­

metic conjunction), 238, 240

standard, 236, 238, 239, 257 consequence operator, 309 context

evaluation, 279, 282, 287 model, 278

continuity, 54, 239 copulas, 324 correspondence theory, 304 covers, 255 crispness of a fuzzy subset, 316

degree of overlap, 132 designated

values, 301 element, 307, 310

disjunction, 155 standard, 236, 238, 257

divisibility, 3, 53

386

downsegments, 219, 221, 225 Duns Scotus law, 3, 67

epistemology, 295 equality

L-valued (cf. fuzzy equality mapping), 134

canonical, 141 M-valued, 132 of a quantal set, 167

exportation law, 3, 67 extent of a quantal set (cf. local

existence), 167,177 extremal subobject classifier, 151

factorization structures, 255, 256 Farey

partition, 107 sequence, 108

filter (cf. ideal), 72, 73, 363, 365 prime, 72, 74 maximal, 72, 74, 365

fitting condition, 343, 368 Friedman's conjecture, 33 fuzziness of a fuzzy subset, 316 fuzzy

degree of validity, 310 equality mapping, 242 function, 236, 246, 249 powerset, 262,314,315,319 predicate calculus, 351 relation (cf. strict and exten-

sional L-valued maps), 245

binary, 245 injective, 246 single-valued, 246, 247 surjective, 246 total, 246, 247

set, 3, 127,236,292

INDEX

H-valued, 242 H-valued, totally, 242 of logical axioms, 350, 356,

358, 359 theory, 3, 4, 53

theory, 351, 353, 356, 359, 362

of Henkin type, 353 of types, 315

generalized characteristic functions, 4,

127 topological space, 308

Girard-monoids, 4, 60, 63, 69 Girard quantale, 79, 85, 90 GL-monoid, 79,87, 132 G8del implication (cf. standard

implication), 272, 273 Goguen-Gaines operator, 317

height of a fuzzy subset, 316 Heyting algebra, 4,33,53,69,77,

86, 131 complete, 235, 237, 242 finitely generated, 35, 36 monadic, 35, 49 projective, 35, 38, 41

Horn clauses, 271, 272 H-sets,243

discrete, 243 void, 250

hyperdoctrinal diagram, 223

ideal, 7 a-closed, 9, 11 maximal, 8, 11, 24 prime, 8, 11 Stone, 10

INDEX

implication, 236 fuzzy (cf. Lukasiewicz impli­

cation), 238, 240 standard, 239, 257

implicational fragment of logic of Hilbert, 309

implied attribute, 320 importation law, 3, 67 inheritable operation, 231 insertion, 252 interpretations of

statements, 266 terms, 265

Kleene algebra, 23 Kleene-Dienes operator, 317, 324

lattice-ordered Abelian group, 23, 24

Archimedean, 23, 25 projectable, 29

strongly, 29 semiprojectable, 28, 29

lattice-ordered monoid, 3 law of

double negation, 3, 53, 69 divisibility, 69 the excluded middle, 3

L-category, 313 Lindenbaum algebra, 54, 67, 68 l-monoids

residuated, commutative, 54 integral, 53, 55 integral, divisible, 58, 80 integral, locally finite, 74 integral, semi-simple, 76

local existence, 4, 128 logical matrix, 307 logic

equational, 111, 112

387

fuzzy, 4, 235, 237, 239, 240, 242, 263, 292, 313, 341, 342, 344

first-order, 341, 344, 355, 360, 369

intermediate, 42, 43 internal, 236, 251 intuitionistic, 3, 235, 236

modal, 34, 50 linear, 3, 69 Lukasiewicz, 3, 69, 105, 219,

235 many-valued, 4,291,292,

294, 331, 341 monoidal,67 non-classical, 3, 4, 67, 291 philosophical, 302, 303 probabilistic, 278 of fuzzy subsets, 235 [0,1]-valued,271

loss of the information variety, 323

Lukasiewicz arithmetic conjunction, 54,

65, 72, 131 conjunction (cf. arith-

metic conjunction), 239, 345

connectives, 128 implication, 239, 273, 279,

287,317,324,342 multiplication, 342

MacNeille completion, 53, 81 map

extensional, 236, 247 of quantal sets, 168 strict, extensional, L-valued,

138 McNaughton's theorem, 107 mean

388

arithmetic, 54 geometric, 54

mediant, 108 Medvedev's logic, 33, 42, 46, 48 membership relation, 261 meta-language, 307, 313 meta-metalanguage, 307 Minkowski multiplication, 84 Mitchell-Benabou language, 237,

263 modal propositional language, 34 modification, 256 monoidal closed structures, 228 MV-algebra, 4, 7, 13, 23, 53, 63,

69, 112, 116, 130, 343 atomless, 15, 100

strongly-, 16, 17, 78, 101 complete, 15, 23

with square roots, 99 a-complete, 8 IT-complete, 23, 90 hypernormal, 28 locally finite, 8, 24, 87 semi-simple, 8,24,25,54,77,

91 Stonian, 10, 11, 19,23

strongly, 19 strict, 65, 94, 99, 101 subb-atomic, 14, 101 sub-divisible, 17 weakly divisible, 17

MV-term, 111, 116

necessity, 34 measure, 283, 284, 285

negation, 155, 343 fuzzy, 239

nilpotent minimum, 72 node of a piecewise linear func­

tion, 109 non-classical model theory, 3

nonexpansive maps, 128

object language, 307, 313 ontology, 295

INDEX

perfect Kripke model, 33, 36, 41 Pinkava algebras, 296 Poincare's paradox, 53 Popper's axioms, 328 possibility, 34 powerset object, 261, 262

weak, 261, 262 pragmatic theory, 304 predicate calculus w.r.t.

(infinite-valued) Luka-siewicz logic, 104

presheaf freely generated, 140 over a GL-monoid, 128

separated, 145 over L, 128, 139

separated, 128 over M, 141

freely generated, 145 separated, 145

on a right Gelfand quantale, 185

on a quantale, 184 probability distribution func­

tions, 90 projectale, 220,221,225,226,227

structure, 219 projectoriallattice, 225, 227 Prolog, 271, 273

fuzzy, 272

quantale, 159,161,219, 230, 232 Gelfand, 159 idempotent, 164

INDEX

left-sided, 162 right Gelfand, 159, 166 right-sided, 162 structure, 219, 233 two-sided, 162

quantifier existential, 236, 258 universal, 236, 258

quantum logic, 230 quasitopos, 128, 151,219223,

243, 251 topological, 251

radical (of an MV-algebra), 24 recursively enumerable, 355, 359

weakly, 358, 359 redundancy theory, 304 representation of semi-simple

MV-algebras, 100 residuation (cf. commutative,

residuated l-monoids), 329, 343

right-symmetricity, 165

satisfaction, 307 graphs, 320

Schauder hat, 108, 110 terms, 116, 121

semantic

set

consequences, 347 theory, 304, 306, 307

L-valued, 134 M-valued, 128, 132

separated, 135 of truth values, 4 quantal, 160, 167

complete, 176 sheaf

389

on a local, 159 on a right Gelfand quantale,

203 singleton of a

quantal set, 175 totally fuzzy set, 248

slice categories, 219, 221, 228, 232, 252

soundness of LG ,276 LL 275

square roots, 54, 61, 80, 86 statement, 263, 264

atomic, 264 strongly isotone, 33 sub algebra of an MV-algebra, 26

dense, 26 large, 26

submodel, 361 elementary, 361

subobject classifiable, 128 unbalanced, 219, 221

subset of a quantal set, 174 subsheaves, 127 syntactic

consequences, 350 truth value, 348

Tarski's lemma, 54, 101 taut lift theorem, 143 t-norms, 53, 71, 239, 296, 298,

301 continuous, 72, 131,273 left-continuous, 54, 71

topological category over sets, 243, 244

topos, 215, 251 theory, 237

truth arrows, 154

390

valuation, 345 values, 4, 342, 369

typed terms, 263

ultrafilter, 363, 364, 367, 368 ultra-infinite order, 9 ultraproduct, 367

unit theorem, 341, 342, 363, 367

strong, 24 weak, 24

weak representation of unba­lanced subobjects, 230

weak representer, 222

INDEX

THEORY AND DECISION LIBRARY

SERIES B: MATHEMATICAL AND STATISTICAL METHODS Editor: H. J. Skala, University of Paderbom, Germany

1. D. Rasch and M.L. Tiku (eds.): Robustness of Statistical Methods and Nonparametric Statistics. 1984 ISBN 90-277-2076-2

2. J.K. Sengupta: Stochastic Optimization and Economic Models. 1986 ISBN 90-277-2301-X

3. J. Aczel: A Short Course on Functional Equations. Based upon Recent Applications to the Social Behavioral Sciences. 1987

ISBN Hb 90-277-2376-1; Pb 90-277-2377-X

4. J. Kacprzyk and S.A Orlovski (eds.): Optimization Models Using Fuzzy Sets and Possibility Theory. 1987 ISBN 90-277-2492-X

5. A.K. Gupta (ed.): Advances in Multivariate Statistical Analysis. Pillai Memorial Volume. 1987 ISBN 90-277-2531-4

6. R. Kruse and K.D. Meyer: Statistics with Vague Data. 1987 ISBN 90-277-2562-4

7. J.K. Sengupta: Applied Mathematics for Economics. 1987 ISBN 90-277-2588-8

8. H. Bozdogan and AK. Gupta (eds.): Multivariate Statistical Modeling and Data Analysis. 1987 ISBN 90-277-2592-6

9. B.R. Munier (ed.): Risk, Decision and Rationality. 1988 ISBN 90-277-2624-8

10. F. Seo and M. Sakawa: Multiple Criteria Decision Analysis in Regional Planning. Concepts, Methods and Applications. 1988 ISBN 90-277-2641-8

11. I. Vajda: Theory of Statistical Inference and Information. 1989 ISBN 90-277-2781-3

12. J.K. Sengupta: Efficiency Analysis by Production Frontiers. The Non-parametric Approach. 1989 ISBN 0-7923-0028-9

13. A Chikan (ed.): Progress in Decision, Utility and Risk Theory. 1991 ISBN 0-7923-1211-2

14. S.E. Rodabaugh, E.P. Klement and U. Hahle (eds.): Applications of Category Theory to Fuzzy Subsets. 1992 ISBN 0-7923-1511-1

15. A Rapoport: Decision Theory and Decision Descriptive Approaches. 1989

16. A Chikan (ed.): Inventory Models. 1990

Behaviour. Normative and ISBN 0-7923-0297-4

ISBN 0-7923-0494-2

17. T. Bromek and E. Pleszczynska (eds.): Statistical Inference. Theory and Practice. 1991 ISBN 0-7923-0718-6

THEORY AND DECISION LffiRARY: SERIES B

18. J. Kacprzyk and M. Fedrizzi (eds.): Multiperson Decision Making Models Using Fuzzy Sets and Possibility Theory. 1990 ISBN 0-7923-0884-0

19. G.L. G6mez M.: Dynamic Probabilistic Models and Social Structure. Essays on Socioeconomic Continuity. 1992 ISBN 0-7923-1713-0

20. H. Bandemer and W. Nlither: Fuzzy Data Analysis. 1992 ISBN 0-7923-1772-6

21. A.G. Sukharev: Minimax Models in the Theory of Numerical Methods. 1992 ISBN 0-7923-1821-8

22. J. Geweke (ed.): Decision Making under Risk and Uncertainty. New Models and Empirical Findings. 1992 ISBN 0-7923-1904-4

23. T. Kariya: Quantitative Methods for Portfolio Analysis. MTV Model Approach. 1993 ISBN 0-7923-2254-1

24. M.J. Panik: Fundamentals of Convex Analysis. Duality, Separation, Represen-tation, and Resolution. 1993 ISBN 0-7923-2279-7

25. lK. Sengupta: Econometrics of Information and Efficiency. 1993 ISBN 0-7923-2353-X

26. B.R. Munier (ed.): Markets, Risk and Money. Essays in Honor of Maurice Allais.1994 ISBN 0-7923-2578-8

27. D. Denneberg: Non-Additive Measure and Integral. 1994 ISBN 0-7923-2840-X

28. V.L. Girko, Statistical Analysis of Observations of Increasing Dimension (forthcoming) ISBN 0-7923-2886-8

29. B.R. Munier and MJ. Machina (eds.): Models and Experiments in Risk and Rationality. 1994 ISBN 0-7923-3031-5

30. M. Grabisch, H.T. Nguyen and E.A. Walker: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. 1995 ISBN 0-7923-3175-3

31. D. Helbing: Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes. 1995 ISBN 0-7923-3192-3

32. U. Hohle and E.P. Klement (eds.): Non-Classical Logics and Their Applica-tions to Fuzzy Subsets. 1995 ISBN 0-7923-3194-X

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