Index

–, 59

algebraic semantics, 61

fuzzy, 242

!, 219

algebraic sum, 196

strongly, 26

¬, 12

ambiguous, 6

weakly, 26

, 12

antecedent, 13

conditional, 12

, 12

antisymmetric, 68

conjunction, 12

→, 12

argument, 17

bold, 94–95

↔, 12

arity, 39

weak, 94

, 39

associative, 60

Conjunction Introduction

, 39

axiom schema, 22

(CI), 150, 156

, 60

derived, 28

connective

∩, 60

instance of, 22

binary, 12

, 94

defined, 15

, 204

Baaz, Matthias, 204

main, 13

, 204

Baaz delta operation, 204

normal, 73

α, 219

BE3, 82

primitive, 15

=def, 15

BE3 , 139

truth-functional, 13

&, 94

BEA, 166

unary, 12

[0. .1], 178

BI3, 80

uniform, 73

|=, 75

BI3 , 138

consequent, 13

, 60

BIA, 166

Consequent Value (CV), 238

, 59

biconditional, 12

Conservative Extension

, 59

binding priority, 12

Theorem for FLPA, 244

1-projection, 204

BLA, 245

constant, 39

Black, Max, 2, 4, 5n6, 309

continuous function, 193

absorption, 60

BLG A, 294–295

contradiction, 15, 47, 75, 180,

adequate derivation system,

BLGA, 249

266

26

BLLA, 249

n-, 184

adjointness condition, 197

BLPA, 252

quasi-, 87, 141

adjunct residuum, 198

Bochvar, Dmitri, 80, 82

Contraposition (CON), 104,

algebra

Bold Conjunction

123

BL-, 216

Introduction (BCI), 233

countable, 315

Boolean, 60

borderline, 1, 5

counterextension, 130

DeMorgan, 164

CL A, 49–51

crisp set, 177

Godel,¨ 218

Heyting, 218

CLA, 22

decidable, 35–36

Kleene, 165

clause, 20

decision procedure, 35

Lindenbaum, 68

commutative, 60

Deduction Theorem

MV-, 167, 212

compact, 228

for BLGA, 251

product, 218

fuzzy, 243

for CLA, 24

unit projection, 219

complementation, 60

Modified, for BLPA, 253

algebraic interpretation, 61

complete, 26

Modified, for FLA, 228

algebraic product, 196

functionally, 32–35

Modified, for L3A, 114

328

Index

definitely operator, 306

formula of propositional

L3, 76

degree

logic, 13

L3 , 131

of membership, 177

atomic, 13

L3 A, 146

of truth, 178

compound, 13

L3 PA, 153–154

degree-entailment, 88, 142,

Frege, Gottlob, 22

L3A, 100

186, 267

fringe, 1, 5, 130

L3MV, 167

degree-validity, 88, 142, 186,

higher-order, 175

L3PA, 115–117

267

fuzzy consequence, 190

lattice, 65

DeMorgan’s Law, 19

fuzzy logic in the broad sense,

bounded, 65

derivable, 23

9–10

complemented, 65

to degree n, 241

fuzzy logic in the narrow

distributed, 65

to degree v, 121

sense, 8

residuated, 214

derivation, 23

fuzzy set, 177

semi-, 166

derivation system

FuzzyBE, 193

lattice ordering relation, 68

axiomatic, 21

FuzzyBI, 192

Law of Excluded Middle, 3

natural deduction, 21n5

FuzzyG, 200

Law of Noncontradiction, 7,

derived rule, 28

FuzzyG , 278

16

designated truth-value, 85

FuzzyGL, 216

least upper bound, 184

disjunction, 12

FuzzyK, 192

Left Conjunct Simplification

bold, 94–95

FuzzyL, 179

(LSIMP), 104

Disjunctive Consequence

FuzzyL , 263

linguistic truth-value, 303

(DC), 112, 234

FuzzyP, 202

literal, 18

Disjunctive Syllogism (DS),

FuzzyP , 280

complementary, 20

111

FuzzyPL, 216

LKL,

164

distribution, 60

LKL , 165

domain, 41, 60

general, 6

lower bound, 183

Double Negation (DN), 108

Generalized Contraposition

greatest, 183

downward distance, 187

(GCON), 108

lub, 184

maximum, 188

Generalized Hypothetical

Lukasiewicz,

Jan, 8, 21, 76, 78,

dual, 66

Syllogism (GHS), 109

179n6, 223

dual t-norm, t-conorm pair,

Generalized Modus Ponens

196

(GMP), 109

McNaughton, Robert, 193

Godel,¨ Kurt, 200n20

McNaughton’s Theorem, 193

entailment, 17, 48, 75, 180,

Goguen, Joseph, 3, 8n5,

meet, 60

266

202n24, 309

bold, 169

degree-, 88, 142, 186, 267

graded formula, 115

Modified Constructive

n-degree-, 188, 267

greatest lower bound, 183

Dilemma (MCD), 111,

quasi-, 87, 141

125

equivalence, 16

Hajek,´ Petr, 9, 216n4, 245

Modus Ponens (MP), 22,

express a truth-function, 33

hedge, 300

116–117, 231

extension, 130

Hypothetical Syllogism (HS),

Modus Tollens, 31

extension of a predicate, 43

28, 123, 233

external assertion, 82

n-contradiction, 184

fuzzy, 204

idempotent, 60

n-degree-entailment, 188,

External Assertion (EA), 297

implication, 197

267

interpretation, 43, 130,

n-degree-validity, 188, 267

first-order logic, 39

262

negation, 12

FL PA, 288–289

classical, 140

external, 82

FLA, 223

involutive, 203

FLPA, 230–231

join, 60

New Problem of the Fringe,

formula of first-order logic,

bold, 169

174

40

normal, 140

atomic, 40

Kleene, Stephen, 71, 74

normal form, 18–21

closed, 41

KS3, 71

conjunctive, 20

compound, 40

KS3 , 138

disjunctive, 18