Page numbers in italics refer to illustrations.
AAA syllogism mood, 74–75, 76–77
Barbara as mood-name of, 76, 79–80, 81, 93, 158
abecedarium, 162, 229n
Achilles and the tortoise, paradox of, 32
ad hominem argument, fallacy of, 178–80
adjectives, autological vs. heterological, 189
Adversus Mathematicos (Sextus Empiricus), 224n
AII syllogism mood, 77–80
Darii as mood-name of, 77, 79–80, 93
Al-Farabi, 232n
algebra, 149, 151–52
see also Boolean algebra
“all” statements, 36–37, 40–52, 53–54, 64, 67, 137, 204, 212
class inclusion and, 44, 45, 46–47, 51
conversion of, 42–44, 49–50, 51, 60–61, 181
definition of, 39
diagrams of, 44–48, 47, 51, 150, 150
reasoning mistakes and, 41–42, 43–44, 43, 44, 48–52, 60–61
as transformed into conditionals, 116–17
as universal affirmations, 41–42, 51, 54, 59–61, 64, 65
see also A propositions
amphiboly, 122, 158, 230n
“and” statements, see conjunctive propositions
Anhalt-Dessau, Princess of, 45
antecedent, 98, 99 116, 125, 213
in Boolean algebra, 154
consequent as necessitated by, 111, 114–15, 215, 216
fallacy of denying, 130–31, 132, 135–36, 184–86, 199
negative, 110, 124
seen as unique cause of consequent, 112–15
see also modus ponens
antinomies, 189, 190
appeal to authority, fallacy of, 180
appeal to force, fallacy of, 180
A propositions, 59–61, 70–72, 180
as Boolean equation, 153
E propositions as contraries of, 65–66, 66
mnemonic for, 65
O propositions as contradictories of, 66, 66
in syllogisms, 74–85, 88–89
as transformed into conditional propositions, 116–17, 135
Arabic scholars, 206, 232n
Argent, Susan, 112
argument by refutation, 32–33
argument from ignorance (ignoratio elenchi) fallacy of, 180
Aristotle, 20, 22, 23, 30, 32, 45, 92, 138, 158, 195, 196, 202, 209, 223n
“all” defined by, 39
categorical reasoning of, 119
conditional propositions used by, 118
contradictory pairs of, 35
on contraposition, 115
contraries defined by, 36
fallacies identified by, 117–78, 180–81, 230n
figures recognized by, 75, 222n
“if” as undefined by, 95
language errors classified by, 230n
on law of noncontradiction, 31
modal theory of, 168–69
name of, as mnemonic device, 65
on negation, 52, 53, 54, 58–59
nonuniversal propositions and, 64
probabilities and, 175, 176
on propositions, 40
propositions classified by, 59, 60, 63, 70–72, 74, 118, 119, 153
on simple propositions, 53
on “some,” 65
reduced statements demonstrated by 78–79
Sophists and, 73, 177–78
syllogisms of, 73–74, 76, 94–95, 117, 118, 124, 125, 207
treatises of, 40
valid syllogisms identified by, 76–77
arithmetic, binary, 110, 148, 149, 152–56, 159, 165
Arnauld, Antoine, 194, 197–98, 208–9, 212
artificial intelligence, 126, 171
Art of Reason, rightly termed Witchcraft, The (Lever), 92–93
atmosphere effect, 88–89
autological (self-descriptive) adjectives, 189
Averroës, 232n
Avicenna, 232n
Bacon, Sir Francis, 193
bald man (falakros) paradox, 188
Barbara, as AAA mood-name, 76, 79–80, 81, 93, 158
Barnes, William, 224n
Baron, Jonathan, 64, 67–68, 116
begging the question (petitio principii), fallacy of, 178
Begriffschrift (Frege), 157
Bell, E. T., 22, 149
Bernoulli, Johann, 46
Berry’s paradox, 190
Bezdek, James, 176
biconditionals, 110–12, 116, 131–132, 158, 181, 199, 213, 214, 215
binary arithmetic, 110, 148, 149, 152–56, 159, 165
bivalued propositions, 168
“blue diamond” experiment, 15–16, 15, 98–99
Boole, George, 22–23, 24, 149, 151–59, 160, 161, 163, 208
De Morgan and, 151
exclusive “or” preferred by, 120
successors of, 156–57
Boolean algebra, 23, 151–59, 161, 165, 175
binary arithmetic in, 152–56, 159
classes in, 152–53, 156
computers and, 152, 154, 159
conditionals in, 154, 157, 158
conjunctions in, 152–53, 154–56, 157, 158, 159, 163, 164
disjunctions in, 153, 154–56, 157, 158, 159, 163, 164, 228n
duality in, 154–56, 159
equations in, 153
symbolic notation of, 157–59, 163
truth values in, 154, 173
Braine, Martin, 128, 200
brain hemispheres, 206
Buddhist logic, 128–29, 206–7
Burkhart, William, 165
calculating machines, 148, 149, 228n
calculus, 22
predicate, 157
propositional, 157
Callimachus, 119
Carroll, Lewis (Rev. Charles Lutwidge Dodgson), 23–24, 73, 96
diagrams of, 47, 47, 87, 87
Carry, Susan, 57–58
categorical proposition, 118
as Boolean equations, 153
categorical reasoning, 119, 136, 180, 206
categorical syllogisms, 117, 118, 144, 180
Categories (Aristotle), 40
causality, 96, 112–15
cause-and-effect reasoning, 97, 112, 216
Celarent, as EAE mood-name, 77, 79–80, 93, 158
Chinese, 146, 147
Yi Ching, or Book of Changes, of, 148
Chrysippus, 123–25, 162
circuit design, 159, 164, 165, 171
classes, Boolean, 152–53, 156
class inclusion, 44, 45, 46–47, 51, 195–96
as hyponymy, 195
cognitive development, 210, 215, 216
cognitive psychologists, 14–15, 55, 78, 98, 99
combinational logic, 162
common-element fallacy, 219n
commonsense knowledge, 200, 204, 213, 214, 215
complement, 149
composite number, 146–47
compound propositions, 119–36, 166
in fuzzy logic, 175
possible number of, 123–24
reasoning mistakes in, 120–21, 122
see also conditional syllogisms, Stoic; conjunctive propositions; disjunctive propositions
computers, 23, 69, 148, 162, 164, 171, 196, 200, 201, 209
Boolean algebra and, 152, 154, 159
FORTRAN language for, 133, 227n
Fortran 77 language for, 133–34
The Logic Theorist program for, 126, 226n
search systems of, 159
Concerning Demonstration (Posterior Analytics) (Aristotle), 40
Concerning Syllogisms (Prior Analytics) (Aristotle), 40, 73–74, 76
conclusion, 72, 74, 75, 78, 80, 81, 83, 85, 88–89, 90–91, 92, 125, 181–82, 198–99, 201, 217
fallacies of, 180, 184–86, 230n
Jevons’s Logic Machine and, 161
major and minor terms of, 222n
in mental-model theory, 210–11
in proof, 31, 34
conditional probabilities, 109–10
conditional (“if...then”) propositions, 37–38, 95, 96–117, 118, 119, 125, 163, 165, 202, 211, 212, 213
additional vs. alternative, 199
A propositions as transformed into, 116–17, 135
biconditionals, 110–12, 116, 131–132, 158, 181, 199, 213, 214, 215
“blue diamond” experiment and, 98–99
in Boolean algebra, 154, 157, 158
in bridge bidding system, 99
causality conveyed by, 96, 112–15
conjunctive propositions equivalent to, 124
contrapositive, 115–16, 204
conversion of, 108–12, 116, 181
cube task and, 101, 102
disjunctive propositions equivalent to, 124
entailment expressed by, 96, 113, 114–15
in Euathlus and Protagoras story, 96–97, 224n
evidence for a consequence presented by, 96
historical origins of, 114–15
logically equivalent forms of, 101–2
in natural language, 108–9, 115–16, 133, 200
obligation conveyed by, 215
permission conveyed by, 211, 213, 215
promises conveyed by, 96, 133
propositions linked in, 98
reasoning mistakes and, 97, 99–116, 181, 199, 213, 214–15, 216
temporal relations in, 96, 112, 113–14
“then” as implied in, 96, 133
threats conveyed by, 96, 133
training in use of, 214–15
truth values and, 169–70, 169, 188
Wason Selection Task and, 99–101, 100, 103–5, 104, 214
see also antecedent; consequent
conditional (hypothetical) syllogisms, Stoic, 114–15, 118–19, 124–36
Buddhist logic and, 128–29
conjunctive propositions in, 129–30
consequent as necessitated by antecedent in, 114–15
diagrams of, 134–36, 134, 135
disjunctive propositions in, 129–30
fallacies of, 130–34, 135–36, 216
in Fortran 77, 133–34
reasoning mistakes and, 128, 131–32
valid inference schema of, 124–30, 155
see also modus ponens; modus tollens
Confucianism, 207
conjunctive (“and”) propositions, 119, 121–24, 165, 171, 202
in Boolean algebra, 152–53, 154–56, 157, 158, 159, 163, 164
conditional propositions equivalent to, 124
in conditional syllogisms, 129–30
words equivalent to, 194–95
consequent, 98, 99, 116, 199
antecedent seen as unique cause of, 112–15
in Boolean algebra, 154
conversion of, 181
fallacies of affirming, 130–31, 132, 181, 184–85, 208, 230n
modus ponens and, 125–27
necessity of, 111, 114–15, 215, 216
consistency, 29–33, 69–70
contradiction, proof by, 33–36, 62
contradiction training, 112
contradictories, 35–36
Aristotle’s pairs of, 35
contrapositive conditional statements, 115–16, 204
contraries vs., 36
negations as, 54, 58–59
Square of Opposition diagram of, 65–66, 66
contraries, 36, 149, 155
negations as, 58–60
Square of Opposition diagram of, 65–66, 66
contrast classes, 57
controllers, fuzzy, 177
conversion:
of “all” statements, 42–44, 49–50, 51, 60–61, 181
of conditional propositions, 108–12, 116, 181
fallacies of, 180–81
inversion vs., 115–16, 130
of negations, 61
of premises, 89
in reasoning mistakes, 43, 49–50, 51, 60–61, 89, 108–12, 115, 130, 181, 196
cooperation principle of discourse, 197–99
Corax, 224n
counterexamples, 36–38, 49, 50, 105, 112, 217
lack of, in reasoning mistakes, 183–84, 214
in reasoning theories, 210–11
Couturat, Louis, 147–48, 227n
Cox, James, 104–5
crisp concepts, 174
crocodile and the baby, paradox of, 187
crytologic bridge bidding system, 99
cube task, 101, 102
Darii, as AII mood-name, 77, 79–80, 93
De Formae Logicae Comprobatione per Linearum Ductus (Leibniz), 45–46
degree of membership, in fuzzy set theory, 174–76
demonstrative mathematics, 31–32
Demonstrator, Stanhope, 93–94, 160–61
De Morgan, Augustus, 87, 139, 149–51, 160, 161, 192, 228n
algebra and, 149, 152
Boole and, 151
diagrams of, 150, 150, 151
Hamilton’s feud with, 71–72, 149, 151
negative terms concept of, 149–150, 152, 163
riddle of, 139
symbolic logic of, 149–51, 154–57
universe of discourse concept of, 87, 149, 150, 152, 174–75
De Morgan’s Rules, 154–56
de omni, 41
Descartes, René, 147
De Vita et Moribus Philosophorum Libri (Laertius), 224n
Devlin, Keith, 200
Dharmakīrti, 128
diagrams, 143, 162, 163, 202, 223n
of “all” statements, 44–48, 47, 51, 150, 150
of conditional syllogisms, 134–36, 134, 135, 136
of disjunction, 150, 151
of negations, 60, 60, 61
of particular propositions, 65–67, 66, 68, 68, 82–84, 83, 84
of sorites, 85–87, 86, 87
Square of Opposition diagram, 65–66, 66
of syllogisms, 80–87, 81, 82, 83, 84, 86, 87, 94
dialectic, 79, 192
Dignāga, 128, 129
disjunctive (“or”) propositions, 119–21, 122, 124, 165, 171, 202
in Boolean algebra, 153, 154–56, 157, 158, 159, 163, 164, 228n
conditional propositions equivalent to, 124
in conditional syllogisms, 129–30
diagram of, 150, 151
exclusive vs. inclusive, 119–21, 130, 195, 228n
in natural language, 194, 195, 200
in reasoning mistakes, 120–21, 122
in THOG problem, 120–21, 121
disproof, 36–39
see also counterexamples
double negatives, 33, 61–63, 124
duality between disjunction and conjunction, 154–56, 159
Dunham, William, 31
EAE syllogism mood, 77–78
Celarent as mood-name of, 77, 79–80, 93, 158
Eddy, David, 110
Educational Testing Service (ETS), 17–18, 69–70, 140, 227n
Einstein, Albert, 29, 37
EIO syllogism mood, 77–78
Ferio as mood-name of, 77, 79–80, 93
electric logical machine, Marquand’s, 164, 165
Elements (Euclid), 33
Elements of Philosophy Concerning Body (Hobbes), 138, 227n
emotional factor, 54–55, 108, 182
end-anchoring, 144
English Cyclopaedia, 155–56
entailment, 96, 113, 114–15
Epimenides the Cretan, 188–89
Epp, Susanna, 50, 108
E propositions, 59–61, 70–72, 180
A propositions as contraries of, 65–66, 66
as Boolean equation, 153
I proposition as contradictories of, 66, 66
mnemonic for, 65
in syllogisms, 74–85, 88–89
equations, Boolean, 153
equivocation, 178
Esperanto, 147
Euathlus, 97, 224n
Eubulides of Miletus, 188
Euclid, 33–34, 39
Euler, Leonhard, 19
on Aristotle’s syllogism, 94
rules for valid syllogisms of, 88
on syllogistic forms, 74
as teacher, 46, 47–48
Euler, Leonhard, diagrams of, 150
“all” statements, 44–48, 51
negations, 60, 60, 61
particular propositions, 66–67, 66, 68, 68
valid syllogisms, 80
Euler’s circles, 45–47, 80, 87
Euthydemus (Plato), 30–31
existential assumptions, 51
violation of, 170
existential quantifiers, 65–68
expert systems, fuzzy, 176–77
falakros (bald man) paradox, 188
fallacies, 23, 50, 83, 88, 112, 138, 177–86, 194, 199, 200, 213
ad hominem argument, 178–80
of denying the antecedent, 130–31, 132, 135–36, 184–86, 199
appeal to authority, 180
appeal to force, 180
argument from ignorance (ignoratio elenchi), 180
Aristotle’s identification of, 177–78, 180–81, 230n
begging the question (petitio principii), 178
common-element, 219n
of conclusion, 180, 184–86, 230n
of conditional syllogisms, 130–34, 135–36, 216
of affirming the consequent, 130–31, 132, 181, 184–85, 208, 230n
of conversion, 180–81
equivocation in, 178
of language, 178–80
non sequitur argument, 180
red herring argument, 180
of series syllogisms, 138–39, 178
see also reasoning mistakes
falsification, 183–84
see also counterexamples
Ferio, as EIO mood-name, 77, 79–80, 93
figures, of a syllogism, 75–85, 222n
FORTRAN computer language, 133, 227n
Fortran 77, computer language, 133–34
France, Abraham, 193, 231n
Frederick the Great, King of Prussia, 45
Frege, Gottlob, 157
fuzzy controllers, 177
fuzzy expert systems, 176–77
fuzzy logic, 172, 173–77
applications of, 176–77
compound propositions in, 175
probabilities in, 175–76
truth degrees in, 173–75, 176, 177, 188
vague concepts in, 174, 177, 188
fuzzy set theory, 174–76, 177
Galen, 75, 118–19, 124, 226n
series syllogisms of, 138
Gardner, Martin, 164–65, 173, 174, 200, 220n, 229n
Geis, M., 115
Gellius, Aulus, 96–97
“General Theory of Elementary Propositions, A” (Post), 171
geometry, 20, 21, 76, 118 194
axioms (postulates), 32, 33
Euclidean, 33–34, 39
Geulincx, Arnold, 228n
Gigerenzer, Gerd, 208
Gödel, Escher and Bach (Hofstadter), 19
Graduate Management Admissions Test (GMAT), 16–17
Graduate Records Examination (GRE), 16–17, 18
General Test of, analytical reasoning measures in, 139–42, 143
Practice General Test of, 122–23
Granger, Thomas, 193
Greeks, ancient, 19–21, 73, 76, 118–19, 210
debates of, 35
forum of, 207
negation as viewed by, 54
proofs and, 30–36, 39
Greene, Judith, 58
Grelling, Kurt, 189
Grice, H. Paul, 197
Gricean maxims, 197
Griggs, Richard, 104–5
Hamilton, Sir William, 71–72, 149, 151, 222n
heap (sorites) paradox, 188
Henle, Mary, 181–82, 184, 186, 201
heterological adjectives, 189
Hilbert, David, 157, 158
Hipparchus of Nicea and Rhodes, 123–24, 162
Hobbes, Thomas, 138–39, 227n
Hofstadter, Douglas, 19
hooded man paradox, 187
horned man paradox, 187
hyponymy, 195
hypothetical syllogisms, see conditional syllogisms, Stoic
Idiom Neutral, 147
Ido, 147
“if” statements, see conditional propositions
ignoratio elenchi (argument from ignorance), fallacy of, 180
III syllogism forms, 81, 84
illicit major, fallacy of, 180
illicit minor, fallacy of, 180
indices, 157n
indirect proof, 33
individual (singular) propositions, 42
Inhelder, Bärbel, 43–44, 48
Interlingua, 147, 227n
Introductiones in Logicam (William of Shyreswood), 79–80
Introduction to Dialectic (Galen), 118–19, 138
inversion, 115–16, 130
Investigation of the Laws of Thought, An (Boole), 22–23, 152, 161, 208
invited inferences, 132, 197, 199
I propositions, 65–68, 70–72, 180
as Boolean equation, 153
E propositions as contradictories of, 66, 66
O propositions as contraries of, 65–66, 66
in syllogisms, 74–85, 88–89
“is” meaning of, 195–96
Jevons, William Stanley, 157, 229n
Logical Abacus of, 162
Logical Abecedarium of, 162
Logical Slate of, 162
Logic Machine of 160–63, 164
jigsaw puzzle, Venn’s, 162
Johnson-Laird, Philip, 15, 55, 57, 68, 91, 98–99, 113–14, 144, 202, 216–17, 219n, 232n
Jones, Sheila, 56
Kalin, Theodore A., 165
Kneale, William and Martha, 54, 76, 79, 114–15
knowledge, commonsense, 200, 204, 213, 214, 215
cultural, 192, 198
interference of, 49–50, 89–90, 181, 184, 198
laboratory test results, evaluation of, 109–110
Laertius, Diogenes, 224n
Lambert, Johann, 228n
language, natural, 175, 177, 191, 192–201, 202, 209, 211–12, 216, 217, 230n
conditional propositions in, 108–9, 115–16, 133, 200
connectives in, 194–95, 200
context of, 192, 196, 197, 213, 216
in conversation, 133, 192, 196–99
and cooperative principle of discourse, 197–99
cultural knowledge and, 192, 198
fallacies of, 178–80
international, 145–48, 149
pragmatic reasoning (natural logic) and, 196–97, 200, 211
in reasoning mistakes, 90, 108–9, 115–16, 186, 195–96
semantics of, 194–96
“some” statements in, 67–68, 69, 200
and suppression of premises, 198
Lawiers Logike, The (Fraunce), 193
law of noncontradiction, 29–30, 32–33, 39, 40, 53, 79, 168, 170
Aristotle on, 31
fuzzy logic and, 177
law of the excluded middle, 34, 39, 40, 51, 53, 168, 170
as Boolean equations, 153
in horned man paradox, 187
modus ponens and, 126–27
in negative terms concept, 150
Law School Admissions Test (LSAT), 17, 18, 202–5
Leibniz, Gottfried Wilhelm, 22, 145–49, 150, 212, 230n
algebra and, 149, 152
on Aristotle’s syllogism, 94
binary system of, 148, 149
on probabilities, 175–76
stepped drum calculating machine of, 148, 149, 228n
universally characteristic language of, 145–47, 149
Leibniz, Gottfried Wilhelm, diagrams of, 149, 150
“all” statements, 45–47, 51
negations, 60, 60, 61
particular propositions, 66–67, 66, 68, 68
valid syllogisms, 80–81, 81
Letter to a German Princess (Euler), 44–45, 48, 74
Lever, Ralph, 92–93
liar’s paradox, 188–89
Liber de Divisione Scientiarum (Al-Farabi), 232n
linear syllogisms, see series syllogisms
logic:
Buddhist, 128–29, 206–7
closed fist as symbol of, 193
combinational, 162
derivation of term, 92
in English university curricula, 79
fuzzy, see fuzzy logic
in history, 19–24
human illogic vs., 24
as ideal, 209
necessity of, 18–19, 217
rhetoric vs., 192–93
symbolic, see symbolic logic
teaching of, 214–15
terminology of, 20, 91–95
Logical Abacus, Jevons’s, 162
Logical Abecedarium, Jevons’s, 162
logical-diagram machine, Marquand’s, 163–64, 168
logical-diagram machine, Venn’s, 162–63
logical product, 152
Logical Slate, Jevons’s, 162
logical spectrum table, Macfarlane’s, 223n
logical sum, 153
logical truth calculator, Kalin-Burkhart, 165
Logic Machine, Jevons’s, 160–63, 164
Logic Machines and Diagrams (Gardner), 164–65
logic of everyday conversations, 197
Logic Theorist, The, 126, 226n
Logique, La ou L’Art de Penser (Logic, or the Art of Thinking) (Arnauld and Nicole), 194, 208–9, 212
Logique de Leibniz, La (Leibniz), 148
Lukasiewicz, Jan, 168–69, 169
Macfarlane, Alexander, 223n
major premise, 222n
major term, 222n
Marquand, Allan, 165–68
diagrams of, 87, 163
electric logical machine, 164, 165
logical-diagram machine of, 163–64, 168
matching bias, 101
Mathematical Analysis of Logic (Boole), 151
mathematics, 22, 64, 149, 150–51, 157, 171, 173, 190
demonstrative, 31–32
proof in, 20, 25, 31–32, 126, 128–29
mechanical devices, 159, 160–65
electric, 164–65
Jevons’s Logic Machine, 160–63, 164
Kalin-Burkhart logical truth calculator, 165
Leibniz’s calculating machine, 148
limitations of, 171–72, 209
Marquand Logic Machine, 163–64, 168
Stanhope Demonstrator, 93–94, 160–61
Venn’s logical-diagram machine, 162–63
medical research, evaluation of, 109–10
mental-model theories, 210–11, 232n
metaphysical argument, 20, 32, 118–19
Method of Agreement, Buddhist, 128–29, 207
Method of Difference, Buddhist, 128–29, 207
middle term, 75, 222n
Mill, John Stewart, 36, 181
minor premise, 222n
minor term, 222n
mnemonic devices, 65
mood-names as, 76, 77, 79–80, 81, 93, 223n
mnemonic verses, 79–80, 81, 93
modal propositions, 168–69, 171
modus ponens, 125–27, 130–31, 132, 183, 199, 213
as Buddhist Method of Agreement, 128–29, 207
Venn diagrams of, 134–35, 134, 135
modus tollens, 127–29, 130–31, 132, 208, 213, 226n, 227n
as Buddhist Method of Difference, 128–29, 207
Chernobyl disaster and, 206
in reasoning mistakes, 128, 131
mood-names, 76, 77, 79–80, 81, 93, 223n
moods, 74–85, 88, 223n
“most” statements, 64, 85
Naiyāyiks, 128–29
NAND operations, 159
natural logic (pragmatic reasoning), 113–14, 196–97, 200, 210, 211
negations, 53–63, 69–70, 88, 92, 155, 158, 163, 168, 202, 204, 212
affirmations vs., 52, 53–54, 55–57, 61–63
in antecedent, 110, 124
Aristotle on, 52, 53, 54, 58–59
changed vs. preserved meaning in, 58
as contradictories, 54, 58–59
as contraries, 58–60
contrast classes and, 57
conversion of, 61
definition of, 53
diagrams of, 60, 60, 61
double negatives in, 33, 61–63, 124
emotional factor in, 54–55
exceptional vs. unexceptional cases in, 57–58
explicit vs. implicit, 55–57, 56
more than one quantifier with, 71
natural vs. unnatural, 58
negative pregnant in, 61–63
of “possible,” 169–70
preconceived notions corrected by, 57, 58
reasoning mistakes and, 54–58, 56, 63, 110, 181, 186
scope of, 58–59
as simple propositions, 53–54
“some” in, 54, 63
truth values and, 169–70, 170, 171
visualization and, 57
see also E propositions; modus tollens; O propositions
negative terms, concept of, 149–50, 152, 163
New Analytic of Logical Forms (Hamilton), 71
Newell, Allen, 226n
New England Journal of Medicine, 109
New Testament, 188–89
Newton, Issac, 22, 228n
Nicole, Pierre, 194, 197–98, 208–9, 212
Nisbett, Richard, 207, 215
nonsensical propositions, 51
non sequitur argument, fallacy of, 180
NOR operations, 159
Notations de Logique (Peano), 157
numerically definite quantifiers, 84–85, 160, 228n
obligation-based training, 215
O’Brien, David, 41–42, 107
obversion, 226n
Ockham’s Razor, 155, 158
O’Daffer, Phares, 186
On Interpretation (Aristotle), 40, 220n
On Language (Safire), 61
On Sophistical Refutations (Sophistical Elenchi) (Aristotle), 40, 177–78, 202
“On the Diagrammatic and Mechanical Representation of Propositions and Reasoning” (Venn), 46
On the Logic of Probability (Leibniz), 230n
O propositions, 65–67, 68–72, 180
A propositions as contradictories of, 66, 66
as Boolean equations, 153
diagrams of, 68, 68
I propositions as contraries of, 65–66, 66
as not reduceable, 78
in syllogisms, 74–85, 88–89
Organon (Aristotle), 40, 92
“or” propositions, see disjunctive propositions
Outline of Redecraft, An (Barnes), 224n
paradoxes, 171, 187–90
antinomies, 189, 190
bald man (falakros), 188
Berry’s, 190
crocodile and the baby, 187
heap (sorites), 188
hooded man, 187
horned man, 187
liar’s 188–89
Russell’s, 190
village barber, 189–90
Zeno’s, 32–33
Parmenides, 54
particular propositions, 35–36, 65–68, 88, 92
diagrams of, 65–67, 66, 68, 68, 82–84, 83, 84
see also I propositions; O propositions; “some” statements
Pascal, Blaise, 147, 194
Peano, Giuseppe, 157, 158, 227n
Peirce, Charles Sanders, 181, 191, 216
diagrams of, 47
Marquand and, 164, 165
on reasoning machines, 160, 163–64, 171–72
series syllogisms of, 139
“some” definition preferred by, 72
symbolic logic of, 157
symbolic notation of, 157n, 159, 163, 170–71
three-valued system of, 170–71
Peirce arrow, 159
petitio principii (begging the question), fallacy of, 178
Philosophical Magazine, 46
Piaget, Jean, 43–44, 48, 210, 215, 216
Plato, 20, 32
Sophists and, 30–31, 73
Platonic Academy, 32
Politzer, Guy, 37–38, 48, 112, 220n
Port Royal Logic, The (Arnauld and Nicole), 194, 208–9, 212
“possible,” as truth value, 168–70
Post, Emil Leon, 165, 171
Posterior Analytics (Concerning Demonstration) (Aristotle), 40
pragmatic reasoning (natural logic), 113–14, 196–97, 200, 210, 211
Praxis I test, 16–17
predicate, quantification of, 71
predicate calculus, 157
premises, 20, 74, 75–76, 77, 78, 80, 81, 82–85, 88–89, 91, 92, 125, 199, 201
conversion of, 89
end-anchoring of, 144
as inferred, 90, 137–38
in The Logic Theorist computer program, 126
major vs. minor, 222n
in mental-model theory, 210, 211
in pragmatic reasoning, 197
in proof, 31, 32, 33, 34
in reasoning mistakes, 180, 181–82, 186, 212
of series syllogisms, 137–38, 143–44
suppression of, in conversation, 198
prime numbers, 33–34, 147
Principia Mathematica (Russell and Whitehead), 157
Prior Analytics (Concerning Syllogisms) (Aristotle), 40, 73–74, 76
probabilities, 175–76
conditional, 109–10
Proceedings of the American Academy of Sciences, 163
proof, 29–39
ancient Greeks and, 30–36, 39
and argument by refutation, 32–33
conclusion in, 31, 34
consistency in, 29–33
by contradiction, 33–36, 62
contradictories in, 35–36
disproof vs., 36–39
Euclidean, 33–34, 39
law of noncontradiction in, 29–30, 31, 32–33, 39
law of the excluded middle in, 30–31, 34, 39
mathematical, 20, 25, 31–32, 126, 128–29
premises in, 31, 32, 33, 34
Pythagorean, 32, 33, 39
propositional calculus, 157
propositional reasoning, 119, 134, 136, 144, 180
propositions, 20, 40–52, 92, 165
Aristotle’s classification of, 59, 60, 63, 70–72, 74, 118, 119, 153
bivalued, 168
conversion of, 42–44
definition of, 40
diagrams of, see diagrams
existential assumptions in, 51
more than one quantifier in, 70–71
nonsensical, 51
quantifiers in, 41–42
simple, 53–54, 96, 119, 123
singular (individual), 42
subjects and predicates in, 42
as true or false, 40–41, 51, 134, 168
see also specific types of propositions
Protagoras, 96–97, 224n
Pythagoras, Pythagoreans, 32, 33, 39, 220n, 224n
quantifiers in, 41–42, 64, 204
existential, 65–68
numerically definite, 84–85, 160, 228n
of the predicate, 71
see also particular propositions; universal propositions
Quine, Willard Van Orman, 80, 165, 189, 190
RAND Corporation, 226n
reasoning, 186, 192–217
analytical, 139–42, 143, 202–5
brain hemispheres in, 206
categorical, 119, 136, 180, 206
cause-and-effect, 97, 112, 216
cognitive development and, 210, 215, 216
commonsense knowledge in, 200, 204, 213, 214, 215
cultural universality of, 207
heuristic, 200
inferential rules of, 207–10, 211, 215
optimal judgments in, 208
pragmatic (natural logic), 113–14, 196–97, 200, 210, 211
propositional, 119, 134, 136, 144, 180
see also language, natural
reasoning, theories of, 210–15
counterexamples in, 210–11
mental-model, 210–11, 232n
pragmatic, 211
rule-based, 210, 211
reasoning machines, 160–65, 171–72, 209
see also mechanical devices
reasoning mistakes, 13–18, 181–86, 212–17
abstract material in, 48, 49, 50, 103, 104, 106, 107, 110, 114, 132, 181, 214
“all” statements and, 41–42, 43–44, 43, 44, 48–52, 60–61
anomalous material and, 106–7
atmosphere effect in, 88–89
biconditionals in, 110–12, 181, 199, 213, 215
brevity vs. clarity in, 50–51
causal interpretation of conditionals in, 112–15
in compound propositions, 120–21, 122
conditional probabilities in, 109–10
conditional propositions and, 97, 99–116, 181, 199, 213, 214–15, 216
conditional syllogisms and, 128, 131–32, 181
conversion in, 43, 49–50, 51, 60–61, 89, 108–12, 115, 130, 181, 196
in different domains, 107–8
disjunctions in, 120–21, 122
double negatives in, 63
emotional factors in, 54–55, 108, 182
exclusive disjunctive in, 120–21
expert witness testimony and, 110
failure to accept logical task in, 181–82, 184, 186
familiar material and, 48–50, 89, 103–8, 182–83, 213
focus of attention in, 214
form of problem in, 183
implicit assumptions in, 181
incentive for high performance vs., 182
inversion in, 115–16, 130
knowledge interference in, 49–50, 89–90, 181, 184, 198
lack of counterexamples in, 183–84, 214
matching bias in, 101
meaningful content in, 49, 103–7, 114
in medical research evaluation, 109–10
modus tollens in, 128–131
natural language in, 90, 108–9, 115–16, 186, 195–96
and necessity of consequent, 111
negations and, 54–58, 56, 63, 110, 181, 186
number of required reasoning steps in, 183, 211
premises in, 180, 181–82, 186, 212
probabilistic, 109
relevant information selection in, 214
rule evaluation and, 104–6
short-term memory in, 183, 211, 214
“some” statements and, 68–70
syllogisms and, 88–91, 180, 183–85, 186
truth confused with validity in, 184, 204
truth interference in, 90–91
universal propositions and, 41–42
on visual material, 112
see also fallacies
red herring argument, fallacy of, 180
reduced statements, 78–79, 80
reductio ad absurdum, 32, 33
reductio ad impossibile, 32, 33, 220n
reference boxes experiment, 111, 111
referendum questions, 62, 62
refutation argument by, 32–33
Reichenbach, Hans, 230n
relational syllogisms, see series syllogisms
relevant information, selection of, 214
Renaissance, 192, 193
Revlis, Russell, 91
rhetoric, 21, 72, 73, 192–93, 194, 224n
open hand as symbol of, 193
Royal Society of London, 148, 161
Rule of Reason, containing the Art of Logic set forth in English, The (Wilson), 21, 92, 224n
Russell, Bertrand, 22, 157, 158, 227n
paradoxes of, 189–90
Russell’s paradox, 190
Safire, William, 61
self-descriptive (autological) adjectives, 189
self-reference, in paradoxes, 189
Sells, S. B., 88
semantics, 194–96
series (linear) syllogisms, 136, 137–44
in De Morgan’s riddle, 139
diagrams of, 143
fallacies of, 138–39, 178
of Galen, 138
in GRE analytical reasoning measures, 139–42, 143
of Hobbes, 138–39
of Peirce, 139
premises of, 137–38, 143–44
reasoning mistakes and, 144
spatial inclusion in, 143
temporal relations in, 143
visualization and, 142–43
set theory:
fuzzy, 174–76, 177
self-swallowing sets in, 190
Sextus Empiricus, 118, 119, 224n, 226n
Shapiro, Diana, 103–4
Shaw, Clifford, 226n
Sheapheardes Logike, The (Fraunce), 231n
Sheffer, Henry M., 158
Sheffer stroke operation, 158–59
Simon, Herbert, 226n
simple prepositions, 53–54, 96, 119, 123
singular (individual) propositions, 42
Socrates, 32
“some” statements, 35–36, 41, 54, 63, 64–72, 88, 134, 204, 228n
context of, 68–69
definitions of, 64, 67–68, 70, 71, 72, 195, 197
diagrams of, 65–67, 66, 68, 68, 82–84, 83, 84
as existential quantifier, 65–68
in I propositions, 65–68, 70–72
in natural language, 67–68, 69, 200
in O propositions, 65–67, 68–72
reasoning mistakes and, 68–70
in syllogisms, 82–85
sophisms, 135
crocodilites, 187
hooded man paradox in, 187
Sophistical Elenchi (On Sophistical Refutations) (Aristotle), 40, 177–78, 202
Sophists, 20, 73, 177–78
law of the excluded middle in, 30–31
sorites, 85–87, 86, 87, 89, 188
definition of, 85
sorites (heap) paradox, 188
spatial inclusion, 143
spatial relations, 195, 206
split-brain research, 206
Square of Opposition diagram, 65–66, 66
Stanhope, Charles, Earl, 93–94, 160–61, 222n
Staudenmayer, Herman, 106–7
stepped drum calculator, 148, 149, 228n
Stoics, 20–21, 31, 33, 39, 45, 62, 94–95, 114–15, 118–36, 138, 155
exclusive “or” used by, 119–20, 130
founder of, 123
metaphysical argument of, 118–19
propositional reasoning of, 119, 134, 136, 144
see also compound propositions; conditional syllogisms, Stoic
Student’s Oxford Aristotle, The (Ross, trans.), 77, 221n
summation symbol, 157n
syllogism machines, 160–65
syllogisms, 72, 73–95, 117, 118–36, 160, 168–69, 194, 206, 207
Aristotle’s, 73–74, 76, 94–95, 117, 118, 124, 125, 207
Buddhist, 206–7
definition of, 74
diagrams of, 80–87, 81, 82, 83, 84, 86, 87, 94
figures of, 75–85, 222n
five-step, of Naiyāyiks, 128–29
invalid, 80, 81, 83–84, 89–90, 180
“is” in, 196
middle term of, 75, 222n
mnemonic devices for, 76, 77, 79–80, 81, 93
mnemonic verses for, 79–80, 81, 93
moods of, 74–85, 88, 223n
numerically definite quantifiers in, 84–85
possible number of, 78, 79
reasoning mistakes and, 88–91, 180, 183–85, 186
reduced statements and, 78–79, 80
rules for judging validity of, 80, 88
“some” statements in, 82–85
sorites as, 85–87, 86, 89, 188
subject and predicate of, 75, 77, 81–82
valid, 74, 76–87, 89, 186
variables as terms in, 76
see also conclusion; premises; specific types of syllogisms
Sylvie and Bruno (Carroll), 73
symbolic logic, 22–24, 144, 145–59, 165, 175
of De Morgan, 149–51, 154–57
of Leibniz, 145–49, 150
of Peirce, 157
see also Boolean algebra
Symbolic Logic (Carroll), 23
symbolic notation, 157–59, 163, 170–71
Syntagma Logicum, or The Divine Logike (Granger), 193
Tarski, Alfred, 168
temporal relations, 143, 195
in conditional propositions, 96, 112, 113–14
terminology, 59, 91–95
of Lever, 92–93
of Stanhope, 93–94
Tests at a Glance (Educational Testing Service), 69–70
Thales, 31–32, 39
Thinking and Deciding (Baron), 64
THOG problem, 120–21, 121, 214
Thornquist, Bruce, 186
threats, 96, 133
Through the Looking Glass (Carroll), 96
Titus, Letter of Paul to, 188–89
Topics (Aristotle), 30, 40, 176
truth:
interference of, 90–91
validity confused with, 184, 204
truth degrees, 173–75, 176, 177, 188
truth tables, 165–68, 229n
truth values, 67, 85, 102, 113, 166, 168–72, 220n
in Boolean algebra, 154, 173
conditional propositional and, 169–70, 169, 188
in fuzzy logic, 173–75
infinite, 188
of many valued systems, 168–71, 173
mechanical devices and, 171–72
negations and, 169–70, 170, 171
“possible” as, 168–70
and violation of existential presuppositions, 170
Uber das Unendliche (Hilbert), 157
undistributed middle, fallacy of, 180
union, Boolean, 153
universally characteristic language, 145–47, 149
universal propositions, 35–36, 41–42, 51, 59, 59–61, 64, 65, 88, 92, 116–17, 181, 194
see also “all” statements; A propositions; E propositions
universe class, Boolean, 152
universe of discourse, 87, 149, 150, 152, 174–75
vague concepts, 174, 177, 188
Venn, John, 46, 66, 82, 85–87, 162
logical-diagram machine of, 162–63
Venn diagrams, 46–47, 47, 51, 220n
of conditional syllogisms, 134–36, 134, 135, 136
jigsaw puzzle version of, 162
of modus ponens, 134–35, 134, 135
of negation, 60, 60, 61
of particular propositions, 66, 66, 67, 68, 68
of sorites, 85–87, 86
of valid syllogisms, 80, 82–84, 82, 83, 84
village barber paradox, 189–90
visualization, 57, 142–43, 209
Volapük, 147
Wason, Peter C., 15, 55, 57, 68, 91, 98–101, 113–14, 144, 219n
THOG problem and, 120–21
Wason Selection Task, 99–101, 100, 103–5, 104, 214
Whitehead, Alfred North, 157
Wilkins, M. C., 49
William of Ockham, 154–55
William of Shyreswood, 79–80
Wilson, Thomas, 21, 92, 224n
Winkler, Peter, 99, 224n
Woodworth, R. S., 88
Yi Ching, or Book of Changes, 148
yin and yang, 148
Zadeh, Lotfi, 174
Zeno of Elea, 32–33, 192, 220n
Zwicky, A. M., 115