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Concepts and Fuzzy Logic Index Abstraction level of categories, 96 Averaging operation on fuzzy sets, 55, Acquisition of concepts, 16, 23, 26 59–60, 139 Additive conjoint measurement (ACM), 157, 160 Basic-level categories, 31, 100, 103 Adjectival modifi cation, 212 Basic levels of concepts, 31, 202–203 Adjectives, 212–214, 247 Basic system of fuzzy logic, 70 private, 213 Bayesian reasoning, 32–33 Adjointness of conjunction and impli- Bivalence, 2, 46–47 cation, 68 Boundary conditions of intersections Aggregation operation, 60, 85, 139, 261 and unions of fuzzy sets, 58 Analogy, 15–17 Brain, 16 Analytic hierarchy process (AHP), 161–162 Cardinality of a fuzzy set, 54, 171 Antisymmetric fuzzy relation, 64 Categories Applications of fuzzy logic, 80–82, 122 abstraction levels of, 96 Approximate reasoning, 3, 77–78 artifi cial, 23 Arithmetic average, 139, 227 basic levels of, 31, 100, 103 Arithmetic operations on fuzzy inter- graded structure of, 94–96, 103–107 vals, 55, 60–61, 77 learning of, 21, 24, 27, 97 Arity of fuzzy relations, 62 mental representation of, 97 Association, 96 Categorization, 16, 19, 21–23, 27, Associativity of intersections and 29, 32, 34–35, 38n3, 216, 218, 238, unions of fuzzy sets, 55, 57 252–255 Attitudes toward fuzzy logic in the psy- Category learning, 21, 24, 27, 97 chology of concepts, 5–7, 122, 259 experiments, 17–18, 24, 27, 29–30, Attribute, 20–21, 226–228 37, 94 in FCA, 178, 181, 187 Category membership, 29–30, 94–95, in FCA, fuzzy, 180, 189 220, 222–223, 237 Attribute implication, 178, 187–188, 194 Causal reasoning, 32 nonredundant base of, 187–188 Central tendency, 101, 123, 246 270 Index Characteristic function, 49–50 lexical, 234, 237–238, 245–246 Classical logic, 1–2, 7, 46–48, 65–66, logically empty, 124–125, 136 68, 90 logically universal, 124–125, 136 Classical set, 2, 5, 49–51, 53, 139 prototype view of, 14, 19–25, 38n2, Classical set theory, 2, 61, 131, 139 91, 123–124, 126–127, 132, 146n1, Classical view (or theory) of concepts, 218, 246 5, 9, 14, 16–19, 23, 38n2, 91–94, 182, psychology of, 1, 4–5, 7–9, 15, 115, 200–201 122, 130, 218, 259–264 rejection of, 19, 114 theory view of, 14, 32–36 Cognitive process, 14, 22, 29, 33 Conceptual (or concept) category, 5, Cognitive psychology, 96, 151 91 Color categories, 94–95 Conceptual (or concept) combination, Combination of concepts. See Concep- 10, 21, 31, 94, 109, 131, 209–211, tual (or concept) combination 216, 223, 228, 260 Commonsense human reasoning, 3, adjective-noun, 109, 139, 212–213 78–79 conjunctive, 109–110, 215 Commutativity of intersections and heuristic strategies for, 212 unions of fuzzy sets, 58 intersective, 210–212, 215 Compensation, 217–218 nonintersective, 210, 211 Complement of a fuzzy set, 55–57 noun-noun, 109 Compositionality, 129, 214 Conceptual distance, 98 Compositional rule of inference, 78 Conceptual complexity, 18 Composition of fuzzy relations, 63 Conditional fuzzy proposition, 76–77 Concept-forming operators (in FCA), Conjunction, 2, 6, 65–68, 109, 215–216, 182, 191–192, 194 219, 222, 224, 226, 249 Concept lattice, 178, 183, 185 Conjunctive concept, 18, 110–113, Concepts, 1, 4–10, 14–38, 89–117, 179, 124–125, 131, 140–141 233–236 Constructing fuzzy sets, 9, 51, 149 acquisition of, 16, 23, 26 Context, 105, 107, 213, 252–253 and language, 233–235 formal (in FCA), 181, 190 basic levels of, 31, 202–203 Cooperation between psychology classical view of, 5, 9, 14, 16–19, 23, of concepts and fuzzy logic, 8–10, 38n2, 91–94, 182, 200–201 89–90, 263–264 combination of, 10, 21, 31, 94, 109, Core of a fuzzy set, 51 131, 209–211, 216, 223, 228, 260 Crisp set, 50–53, 112 conjunctive, 18, 110–113, 124–125, Cylindric closure, 63 131, 140–141 Cylindric extension of fuzzy relation, disjunctive, 18, 124–126 62–63 ecological view of, 91, 107 exemplar view of, 14, 24–31 Decision rule, 21–22, 29 in FCA, 178–179 Deduction, 107–112, 188 intensional view of, 225–227 Defuzzifi cation, 84 Index 271 Degree of membership, 6, 47, 49–50, Fuzzy attribute implication (in FCA), 194 73, 90, 95, 106, 109–110, 116, 124, Fuzzy compatibility relation, 65 140, 155, 173, 218, 220, 237–238, 260 Fuzzy concept lattice (in FCA), 193 Degree of subsethood. See Inclusion of Fuzzy conjunction, 65–68 fuzzy sets Fuzzy equivalence relation, 65 Degree of truth, 6, 46–50, 65–66, 73, Fuzzy implication, 66, 68, 77. See also 190. See also Truth-value Residuum Diagnosticity, 20 Fuzzy interval, 54 Direct assignment, 168 Fuzzy logic, 1–10, 46–49, 73, 90–91, Direct scaling, 159 99, 104–105, 108–115, 122, 214–216, Disjunction, 2, 6, 220–221, 224 218–219, 222, 228, 234, 237, 245, Disjunctive concept, 18, 124–126 247, 250, 259, 261–264 Distance, 17–28, 98–99, 154 Fuzzy logic connectives, 65–69, 191, 203 Drastic intersection of fuzzy sets, 58–60 Fuzzy logic in the broad sense, 3, 49, Drastic union of fuzzy sets, 58–60 73–80 Fuzzy logic in the narrow sense, 3, 48, Ecological view of concepts, 91, 107 65–73, 143–145, 189 Emergent attribute, 111 erroneous claims about, 143–145 Exemplar, 26–28, 37, 38n4, 116 Fuzzy logic of concepts, 109–112 Exemplar view of concepts, 14, 24–31 Fuzzy negation, 66–69 Expressive power of fuzzy logic, Fuzzy number, 54 138–143 Fuzzy partial ordering, 65 Extension of concept, 205n1, 223, 224, Fuzzy power set, 53 226, 239, 242, 245 Fuzzy predicate, 74 Extension of predicate, 2 Fuzzy predicate logic, 71 Extent of formal concept (in FCA), 178, Fuzzy probability, 3, 74 182, 192, 198, 205n1 Fuzzy proposition, 74–78 Externalist theory of meaning, 236–237 Fuzzy quantifi er, 3, 71–72, 74, 135–136 Fuzzy relation, 1, 62–65, 77 Factor analysis (in FCA), 195–200 Fuzzy set, 1–6, 49–63, 77, 94–96, 99, Family resemblance, 97, 101, 247 112, 116, 152, 245 Formal concept (in FCA), 178, 182 Fuzzy set theory, 5–6, 61, 73, 77, 90, Formal concept analysis (FCA), 9–10, 107, 114, 124, 127, 130–132, 139, 178–179 146n1, 152, 165, 167, 170–172 Formal context (in FCA), 181, 190 Fuzzy system, 78–80 Formal deductive system, 1 Fuzzy truth-values, 74 Formal fuzzy concept (in FCA), 192 as a factor, 197 Galois connection, 182 Formal fuzzy context (in FCA), 190, 196 fuzzy, 192 Formulas in fuzzy logic, 65–67 Generalized context model, 27, 29 Fundamental measurement, 156 Geometric average (or mean), 59, 216 Fuzzifi cation, 84 Goal-derived categorization, 102 272 Index Gödel system of fuzzy logic, 69–70 Law of contradiction, 61, 125, 137–138 Goguen system of fuzzy logic, 69–70 Law of excluded middle, 61, 125, Goodness of example, 95–99, 108, 110, 137–138 112, 116, 260 Learning, 18, 96 Graded membership, 219, 246 of categories, 21, 24, 27, 97 Graded structure of categories, 94–95 of concepts, 18, 23 Grade of membership. See Degree of Level-cut of a fuzzy set, 51–53, 154 membership Level-cut representation of a fuzzy set, Granulation, 79 53, 61 Level of measurement, 13, 153, 155 Hasse diagram, 185. See also Line Lexical concept, 234, 238, 245 diagram Life game, 107–108, 117 Heterogeneity hypothesis, 36–37 Line diagram (in FCA), 178, 185–186. Hierarchy of formal concepts, 184–185, See also Hasse diagram 193 Linguistic hedge, 55, 98, 158 Higher cognitive competencies, 14–16, Linguistics, 23, 91, 93, 211 26, 31, 36 Linguistic terms, 3, 47–48, 79 Human reasoning, 3, 47 Linguistic variable, 79 Logic Idempotence of averaging operations, classical, 1, 2, 7, 46–48, 65–66, 68, 90 59–60 fuzzy, 1–10, 46–49, 73, 90–91, 99, Inclusion of fuzzy sets, 72–73, 133–135 104–105, 108–115, 122, 214–216, Indirect scaling, 159, 169 218–219, 222, 228, 234, 237, 245, Induction, 15–17, 24–25, 31, 35–36 247, 250, 259, 261–264 Inference, 24–25, 32, 70, 78, 92, 98, 105 intensional, 228 Institute of Electrical and Electronic many-valued (or multi-valued), 2–3, Engineers (IEEE), 83 65 Intension of concept, 205n1, 223–224, Port-Royal, 178–179, 182 226, 239, 242, 245, 247 Logically empty concepts, 124–125, Intent of concept (in FCA), 178, 182, 136 192, 198, 205n1 Logically universal concepts, 124–125, Intermediary degree of truth (or truth- 136 value), 1–2, 46–48, 65 Long-term memory, 14–15, 29, 37 International Fuzzy Systems Association Łukasiewicz system of fuzzy logic, (IFSA), 83 69–70, 143 Intersection of fuzzy sets, 55, 57–60, 131, 216, 217, 260. See also t-norm Many-valued logic, 2, 65. See also Multi- Intersubjectivity, 237 valued logic Intransitivity of categorization, 224 Mathematical fuzzy logic, 65. See also Inverse of a fuzzy relation, 64 Fuzzy logic in the narrow sense Involutive complement of a fuzzy set, 57 Mathematical psychology, 152 Mathematical theory, 10, 130–131, 260, Join of fuzzy relations, 64 262 Index 273 Meaning of a word, 93, 212, 236, 239, Object (in FCA), 178, 181, 190 245 Osherson and Smith’s 1981 paper Measurement (OSP), 122–127, 223 additive conjoint (ACM), 157, 160 fallacies in, 129–145 by fi at, 157, 161 infl uence of, 127–129 fundamental, 156 Overextension, 217, 227 level of, 13, 153, 155, 158 psychological, 156 Pavelka-style system of fuzzy logic, Measurement scale 70–71 absolute, 156 Physiology of perception, 101 interval, 155–156, 161 Polysemy of adjectives, 213 log-interval, 156 Port-Royal logic, 178, 179, 182 ordinal, 155–156 Possibility theory, 85 ratio, 156, 161 Predicate, 2, 74 Measurement theory, 9, 155, 157, 203 Predicate fuzzy logic, 71 Membership assignment, 152–153, 166 Probability, 33, 97, 106, 154, 160, 221, methods for, 158–162 223, 254 Membership degree (or grade), 6, 47, Probability logic, 66 49–50, 73, 90, 95, 106, 110, 116, 124, Probability qualifi er, 76 140, 155, 173, 218, 220, 237–238, 260 Projection of a fuzzy relation, 62–63 Membership function, 50–53, 168 Property ranking, 152, 163–165, 167 Membership interpretation, 153 Propositional fuzzy logic, 65 Membership measurement, 153 Prototype, 19–25, 31, 37, 38n4, 99–106, Minimal description length, 18 112, 116, 155, 218–219, 222, 226, Modifi er, 210 240, 246 of a fuzzy set, 55–56 Prototype view (or theory) of concepts, of a noun, 210–214 14, 19–25, 38n2, 91, 123–124, 126– Monotonicity of intersections and 127, 132, 146n1, 218, 246 unions of fuzzy sets, 57 Psychological distance, 27–28 Multi-valued logic, 3.
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