Index

abstract argument, 181 abstract statements, 181-82, 314 A-, E-, I-, and O-statements, 235-38, 298

Square of Opposition, 237 affirming the consequent, 58-60 algebraic class logic, 281 All except ι, 337 alternation (or exclusive disjunction), 95-96 ambiguous statements, 237-38 amphiboly, 361-64 ‘and,’ 75, 77, 392

Anselm, Saint, Archbishop of Canterbury,

363

antecedent, 51-53, 67 antisymmetric relations, 348-49, 354 ‘any,’ 304

replaced by ‘some,’ 305-06 appropriate standards of evidence, 35-37 arbitrary individual, 262, 270, 369, 371,

374, 379 argument, 6, 18, 72

abstract, 182

basic unit of reasoning, 9

definition, 4-5

argument forms, 32-33, 56, 59, 182, 315 and formal validity, 30-33 validity, 205-06

arguments with more than 3 predicates, 281-85

Carroll diagrams for 4 or 5 categories, 281-85

sorites, 286-87

Aristotelian logic, 358

Aristotelian sorites, 286

Aristotle, 156, 158-59,234-35,401-02

Aristotle’s logic, 233-35

Aspect, Alain, 388nl

A-statements, 235-38, 240, 244, 254, 257

symbolizing, 258 asyllogistic arguments, 303-20 asyIlogistic proofs: QN (Quantifier

Negation), 307-11 asymmetry, 328

as property ofbinary relations, 331 symbolizing, 329

At least one, two..., 338, 341

At most one, two..., 338

Augustine, Saint, Bishop of Hippo, City of

God, 5, 403n2

axiom schemas, 176, 187-90

baby-crocodile syllogism, 286

Bacon, Roger, 211

Bell, John, 391

Bell’s Theorem, 388nl

Bentley, Richard, 208

Biconditional Equivalence (BE), 131-32,

166, 169-72, 176

biconditionals, 125-35

negation of, 170, 172

symbolizing, 129-30 binary relations, 323, 326

nonconnexivity, 348 properties of, 327-31 simple connexivity, 348 total connexivity, 348 binary statement operators, 44, 62, 77 binary truth-functional operators, 45 biology, 234 bivalence, 390

law of, 388, 390

Principle OfBivalence, 65-66

Bloor, David, Knowledge and Social

Imagery, 151-52, 161

Boole, George, 239, 270, 281, 292, 303 Boolean algebra, 330

Boolean approach, 292 bound variables

definition, 319 branching rules, 220, 224, 367 BriefTruth Table Method (BTT), 207-10,

215

Brouwer, Luitzen Egbertus Jan, 66n3, 390 Brown, J.R., 151, 161

Buddhists, 55, 65

‘but,’ 75, 77

“but not both,” 94

capital letters (as logical symbols), 49, 56,

59, 109, 179, 234,314 Cardan, Jerome, 403 Carroll, Lewis, 4, 240nl, 281-82, 286,

291,294, 405

Symbolic Logic, 4, 257, 259nl, 314 Through the Looking Glass, 292n9, 361 Carroll Diagrams, 239-44, 249, 257, 260,

267-68, 297, 408-11

Carroll Diagrams for 4 or 5 categories, 281-85,287

“Cartesian Circle,” 30 categorical statement, 238 categorical syllogism, 234, 238

category logic

A-, E-, I-, and O-statements, 235-37

ambiguous statements, 237-38

Aristotle’s logic, 233-35 causal conditional, 397

Chain Rule, 86, 115.

Syllogism (HS)

Chomsky, Noam, 48, 323

Chrysippean analysis, 399

Chrysippean criterion, 398

Chrysippean criterion for validity, 27

Chrysippean definition, 26

Chrysippus, 3, 26, 42nl, 56, 64, 85-86, 98,

234

Church, Alonzo, 354nl

Cicero, 69, 86

On Fate, 85 circular arguments, 28, 30 City of God (Augustine), 5 Clarke, Samuel, 12, 161, 330 claustrophobic house example, 120-22,

149

closed paths, 224

complete closed tree, 368 complete open path, 218-19, 221, 224,

372-74

completeness, 183

of a system of inference, 183

truth tree method of determining, 227-

29

complex conditionals, 52-53 complex constructive dilemma, 136-37,

139, 141

complex destructive dilemma, 138-39 component statement, 44 components, 48

definition, 43 compound statements, 43-44, 48 compounds, 42-43

“Computing Machinery and Intelligence”

(Turing), 67

Conan Doyle, Sir Arthur, 211 conclusion, 5, 9, 14-15, 18 conclusion indicators, 9-10

concrete argument, 181, 314 conditional

negation of a, 170, 172

conditional proof, 109

symbolizing, 112

Conditional Proof(CP), 113, 117, 166, 169-70, 175, 392

justification of the formal validity, 115 conditional statements, 51-53, 72, 170, 171 conditionals

counter-factual, 397

conjunction

negation of, 169-70, 181

Conjunction (Conj), 79, 81, 169-170, 176, 392

conjunction, rules of inference for, 79-81 conjunctions, symbolizing, 75-77 Conjunctive Syllogism (CS), 80-81, 167, 171, 172, 176, 189

conjuncts, 75, 77

connexivity, 348-49

consequent, 51,53, 60, 64

Consequentia mirabilis (Marvellous

Consequence), 188,402-04

Consequentiae, 401

‘consequently,’ 88

consistency

of systems of rules of inference, 175, 183

of truth tree rules for SL, 226-29 truth tree method of determining, 228- 29

Constructivism, 390

contingent statement, 198, 200

contradiction, 197, 226-27, 229 symbol for, 150

contradictories, 62-63

contraries, 290

conversational implicature, 130-31 in evaluating natural arguments, 131 converse, 130, 135

conversion

in Carroll’s Diagrams, 243

conversion by limitation, 290, 298

Cotes, Roger, 9, 36-37

counter-factual conditionals, 397

“cult of the expert,” 4

Curd, Patricia, A Presocratics Reader, 110n2

Darapti form, 294-95

Darwin, Charles, 21

Darwin, Erasmus, 322, 327

Darwin, Robert, 322, 327

Davies, P.C.W.,212

De Morgan, Augustus, 103

De Morgan’s Laws (DM), 102-05, 166, 170-71, 176, 181,218, 330

decision procedure, 193

declarative sentences, 41

decomposition rules for truth tree method, 219-23

branching, 220, 223-24, 228 non-branching, 220, 223, 228

definite article ‘the,’ 340

definite description, 339-41

denial of the consequent, 64

denying the antecedent, 67

Deontic Logic, 387-88

derived rules, 174-76

Derrida, Jacques, 155

Descartes, Rene, 3, 30, 156, 403n2

Principles of Philosophy, 22

The Development of Logic (Kneale and Kneale), 149, 392-95, 397, 401, 403

dialectical rule, 57

Dilemma (DL), 137, 139, 166

dilemmas, 135-44

Disjunction (Disj), 101, 104, 165,170, 176, 392

disjunctions

definition, 95 negation of, 170-71 rules of inference for, 98-105 symbolizing, 93-96, 109

disjunctions in conditionals, 109 Disjunctive Syllogism (DS), 3, 98-100, 165, 171-72, 176, 392

disjuncts, 95

“The Doctrine of Fascism” (Mussolini), 21 Dodgson, Charles.

Duns Scotus, John, 188

dyadic predicates, 323

dyadic relations. See binary relations

Eco, Umberto, The Name of the Rose, 43-44

effective completeness, 372-74

EG. See Existential Generalization (EG)

Einstein’s formula, 26

Einstein’s General relativity, 390 elliptic geometry, 403-04

‘else’

symbolizing, 110

Emile (Rousseau), 11

The End ofScience (Horgan), 48 entailment, 186-87 enthymemes, 13-15

Epicurus, 85-86 equivalence class, 330-31 equivalence relations, 329, 331 Equivalence Rules, 166-67

Biconditional Equivalence (BE), 166 De Morgan’s Laws (DM), 166 DoubleNegation (DN), 166

Material Implication (MI), 166 Transposition (TR), 166

E-statements, 235-38, 241, 244, 254, 298 symbolizing, 258

Euclid, 149

Euclid, Elements, 330, 403

Euclidean geometry, 404

Euclides ab Omni Nosvo J ^rindicatus

(Saccheri), 403

evaluating extended arguments, 84-88 evaluating natural arguments, 34-37 conversational implicature in, 131

evaluating validity of syllogisms, 245-49 everyone except, 341

exactly one, two..., 338, 341

exhortations or commands, 5

existence and non-existence

in Carroll’s Diagrams, 242

existential conclusion from universal premises, 297

Existential Generalization (EG), 273,276,392 existential import, 258, 290-93

conversationally implied, 293 particular (I- and O-) statements, 298

Existential Instantiation (El), 269-73, 276 existential quantification, 267-69, 296 existential quantifiers, 267, 305-06 explanations, 10, 15 explicit condition

symbol for, 150

extended arguments, 84-88

natural arguments as, 23

F allacy of Affirming the Consequent (FAC), 58, 60

Fallacy OfDenying the Antecedent (FDA), 67 Feyerabend, Paul, 48

Fisher, Alec, The Logic of Real Argument, 4, 36n3, 84nl

“following from” or consequence, 52, 72,

186

Fonesca, Peter, 163

‘for,’ 85

formal invalidity, 31, 205, 206

formal proof, 287

formal validity, 26, 30-33, 36-37, 57-58, 60, 67, 204

definition, 55

formula, 179-80,314-15,319

free variable, 319

Frege, Gottlob, 187-88, 253, 257

full truth table method, 201-04

Galileo Galilei, 147, 156, 158-59, 162

Gauss, Carl Friedrich, 404

Goclenian sorites, 286

Godel, Kurt, 393

“Gorbachev-Brezhnev” argument, 368-69 Gould, Stephen Jay, 75

governing operators, 93, 181

Grice, H.P., 130, 293

Groucho Marx arguments, 30, 359-60 groupers, 179, 316

convention regarding outermost grou­pers, 47, 316

Halley, Edmond, 59

Heller, Joseph, 154-56

Hendrix, Jimi, 61

Heraclitus, 8

Heyting, Arend, 390, 392-94

Hitler, Adolf, 109

Hofstadter, Douglas, Gδdel, Escher, Bach, 115, 154

Holbach, Paul Henri Thiry, Baron d’, 12

Horgan, John, The End of Science, 48 ‘however,’ 77

Hume, David, Dialogues Concerning Natural Religion, 88

hyperbolic geometry, 403-04

Hypothetical Syllogism (HS), 115-17, 166, 175, 188,401-02

proof of, 186

identifying arguments

enthymemes, 13-15 explanations and, 10-11 implicit arguments, 11-13

inference indicators, 9-10

identity

reflexivity of, 345

identity and quantity

Russell’s theory of definite description, 339-40

symbolizing identities and quantities, 335-39

identity of indiscernibles, 344, 385-86, 391 identity relations, 345-46

transitivity of, 345-46

“if and only if,” 129-30

‘if’ and ‘since’ confusion, 72

‘if...then...,’ 392, 398

“iff,” 130

immortal philosophers and the dead

Socrates example, 196, 396, 398

implication, 70, 72

symbolizing, 84 implicit arguments, 11-13 Implicitconclusion, 13, 15 Implicitpremises, 14-15, 35, 131 incompatibility, 26 indirect proof, 390 Indiscernibility of identicals, 385 individual names, 258

symbolizing, 256 inductive logic, 36 inference, 15, 18-19 forms of, 314 symbolizing, 84 inference and implication, 71-72 inference indicators, 9-10, 15

conclusion indicators, 10 premise indicators, 9-10 inference to the best explanation, 11 inferences involving identity properties of identity, 345-46 rule of inference SI, 343-45 inferring, 72 infinitesimals, 391 “Informal Logic” movement, 19 instance of a quantification, 260, 264 intermediate conclusion, 20 intransitive, 327 Intuitionistic Logic (IL), 390, 392 Intuitionists, 65nl, 66n3, 390 intuitive assessment, 84 invalid argument forms, 204-06 Inwood and Gerson, Hellenistic

Philosophy, 42, 85 Irreflexibility

symbolizing, 329 “Is Logic Empirical?” (Putnam), 390 “is of identity,” 335,341 “is of predication,” 335 !-statements, 241, 244, 298

symbolizing, 267, 269

“Jack Layton” argument, 25, 30 judgement, 35, 37

Kleene, Stephen Cole, 389

Kneale, William and Martha, 397, 403n2 The Development of Logic, 149, 392, 394, 395,397,401,403

Knowledge and Social Imagery (Bloor), 151-52

Kranish, Michael, 13

Kuhn, Thomas, 48

Law of Double Negation.

Law OfExcludedMiddle, 151, 390-91

Law OfMaterial Implication (MI). See Material Implication (MI)

Law ofTautology, 151

Laws and Symmetry (Bas van Fraassen), 129nl

Leibniz, Gottfried, 12, 156-57, 161, 286, 330, 344, 364

Labyrinth of the Continuum, 157n9 Leibniz’s Law, 344

Monadology, 247

New Essays, 363

Principle of Sufficient Reason, 295 Principles of the Identity of

Indiscernibles, 344, 385, 391

Lewis, C.I., 397

Liar Paradox, 154, 209

Lincoln, Abraham, 137 literals in truth tree method, 218, 224 Livio, Mario, 163

Lobachevsky, Nikolai, 404

Locke, John, 363-64 logic diagrams, 405-11 logic is not Boolean, 391 logic of moral obligation. See Deontic Logic The Logic of Real Argument (Fisher), 4, 36n3,84nl

Logica Demonstrativa (Saccheri), 403 logical contradictions, 147, 149 logical equivalence, 195-99

truth tree method of determining, 227 logical falsehood. See contradiction logical force, 76

logical inconsistency, 26, 36-37 logical negation symbol, 61 logical symbols, 179 logically equivalent, 200 logically true, 187

Lucanus, Ocellus, 162

Lucretius, Titus Carus, 162

Lukasiewicz, Jan, 188, 404

three axioms for statement logic, 404

Maimonides, 163 main conclusion, 23

Margulis, Lynn, 48

Marx, Chico, 322, 328

Marx, Groucho, 30, 322-23, 328, 359-60

Marx, Gummo, 328

Marx, Harpo, 322, 328

Marx, Minnie Schoenberg, 328

Marx, Zeppo, 328

Material Conditional, 52-53, 399

Material Implication (MI), 166, 172, 176, 196,395-99

mathematics, 147-48, 339, 390

Megarians, 234

Meno (Plato), 149

‘mercenary’ argument, 256-57

a Merman I should turn to be, 61

Modal Inference (Mod), 387

Modal Logic, 386-87

Modus Ponens (MP), 55-57, 59, 65, 165, 171, 176, 188, 392

Modus Tollens (MT), 64-67, 111, 165, 171, 176

monadic or unary predicates, 327

Monadic Predicate Logic, 376

Monadology (Leibniz), 247

Monty Python and the Holy Grail (1975), 27,58

Monty Python's Flying Circus, “Argument Sketch,” 4, 5nl, 14, 19

Monty Python's Flying Circus: Just the Words, 202-03

“moral God” argument, 369

‘moreover,’ 77

Mussolini, Benito, “The Doctrine of

Fascism,” 2

The Name of the Rose (Eco), 43-44 natural arguments, 18-23, 37 evaluating, 34-37 supposition in, 119-22 techniques of diagramming, 19-23 natural deduction, 57 natural dilemmas, 141-44 natural reasoning, 14 natural reductio arguments, 156-60 technique for diagramming, 157 necessary and sufficient conditions, 125-

29

necessary conditions, 127

‘only if,’ 125-26

Sufiicientconditions, 127

necessity, 386 negated quantifier decomposition rules,

370-72 negation, 61-72

Ofabiconditional, 170, 172

Ofaconditional, 170, 172 Ofaconjunction, 170-71, 181 Ofadisjunction, 170-71 inference and implication, 70-72 symbolizing negations, 61-64 nested quantifiers, 324 New Essays (Leibniz), 363 Newton, Isaac, Sir, 36, 208

The Principia, 9, 59

Nicod, J.G.P., 190 non-branching rules, 218, 223, 367 non-classical statements, 303 non-emptiness of the UD, 295-353 nonreflexivity

symbolizing, 329 nonsymmetry

symbolizing, 329 ‘not,’ 62, 392 ‘nothing but,’ 255

On Fate (Cicero), 85 Onlyι, 126, 255,336 “Only bis S,” 341 “only if’ statements, 125-26, 131 “Only M are S,” 255

The only ι..., None but ι..., 336

‘or,’94-96, 111,392

ordering relations, 347-55

O-statements, 241, 267, 298 symbolizing, 269

‘otherwise’

symbolizing, 110

p if q, 126-27

p only if q, 126-27

ⁱp otherwise q' symbolizing, 111 pairs of individuals, 321 paradoxes of material implication, 188, 196, 220, 395-99

Parmenides, 149

particular affirmative, 235 particular negative, 235 particular statements

existential quantification, 267-69 particular (I- and O-) statements

existential import, 298

path

in truth tree method, 22, 218

Peirce, Charles Sanders, 189

Peirce’s Arrow operator, 189 penevalid arguments

existential import, 293, 298

Penrose, Roger, 48

Philo, 395

Philonian conditional, 395-96, 396-99 Philosophical Dictionary (Voltaire), 143 Plato, Meno, 149

polyadic predicates, 326, 376

Pospesel, Propositional Logic, 100n4, 313nl

possibility, 386 predicate logic, 32, 253, 260, 391, 401 defined rules in, 309

extending also to relations, 322 predicate logic as a formal system

propositional functions and quantifier scope, 318

symbols, formulas and wffs, 314-17 predicate logic proof, 276

predicate logic truth trees, 367

additional rules for quantifications, 368-70

decomposition rules for universal and existential quantifications, 374 effective completeness, 372-74 negated quantifier decomposition rules, 370-72

premise indicators, 9-10, 15 premise-directed strategies

Biconditional Equivalence (BE), 170 Conjunctive syllogism (CS), 171-72 De Morgan’s Laws (DM), 170, 176 Disjunctive Syllogism (DS), 171 Material Implication (MI), 172

Modus Ponens (MP), 171

ModusTollens (MT), 171

Simplification (Simp), 170-71, 176 premises, 5, 9, 14, 18

truth of, 28-29

prenex forms, 354-57

Prenex Normal Form, 354nl

A Presocratics Reader (Curd), 46 Prigogine, Ilya, 48

A Primer of Infinitesimal Analysis (Bell),

391

The Principia (Newton), 9, 59

Principle of Bivalence, 64-66

Principle of Charity, 13, 15, 35, 37

Principle of Sufficient Reason, 295

Principle of the Identity of Indiscernibles, 385

Principles of Mathematics (Russell), 339 Principles of Philosophy (Descartes), 22 proof, definition of, 183 proof of transitivity, 346

proof strategies, 276

goal-directed strategies, 170 premise-directed strategies, 171-72 properties ofbinary relations, 327-31 reflexivity, 327-28, 331 symmetry, 327-28, 331 transitivity, 327-28, 331

properties of identity, 345-46

proposition, 5-6, 42

Propositional Calculus, 187-90 propositional challenge, 188 propositional function in x, 260, 264, 318 “Propositional Logic,” 42

propositions, 41-42 Protagoras, 142-43 Putnam, Hilary, “Is Logic Empirical?,” 390 Pythagoras, 147-48

Pythagoras’ Theorem, 253

quantification, 317 quantificational logic for analysis of prob­lems in philosophy, 340

QuantifierNegation (QN), 309, 311, 391-92 quantifier scope, 318

quantifier scope fallacy, 361-64

Quantifier Theory, 253

quantifiers, 253

Quantum Logic, 388-90

Quantum Mechanics, 386

Quantum Theory, 388

reductio ad absurdum (RA), 147-53, 166, 169-70, 176, 390, 392

Descartes’ use of, 156

Galileo’s use of, 147, 156, 158-59 Leibniz’s use of, 156

proof of validity, 151

Pythagoras Theorem and, 147-48, 156 rhetorical force, 147

Zeno of Elea’s use of, 149, 156, 160 reductio ad impossibile, 149, 151 reflexivity, 327

of identity, 345-46

as property ofbinary relations, 331 symbolizing, 329

Reichenbach, Hans, 388 relation of identity, 336 relational arguments, 351-64

quantifier scope fallacy and, 361

Relational Logic, 205, 253, 321-31 truth tree rules in, 378 used to reveal certain fallacies, 364 relational or polyadic predicates, 326, 376 relational proofs, 324 relations, 321-22, 326

Relevance Logic, 397

rhetoric, 142

rhetorical force, 76

rhetorical questions, 12

Riemann, Bernhard, 404

Rousseau, Jean-Jacques, Emile, 11

Rule of Substitution, 188

Rules OfInference, 174-75

Biconditional Equivalence (BE), 131- 35,166

Conditional Proof (CP), 117, 166

Conjunction (Conj), 79, 165

Conjunctive Syllogism (CS), 80-81,

165

Dilemma (DL), 166 disjunction (See Rules of Inference for disjunctions)

DoubleNegation (DN), 166

Existential Generalization (EG), 272- 73, 276

Existential Instantiation (El), 269-73, 276,368

Hypothetical Syllogism (HS), 166 of Intuitionist Logic (IL), 392 Material Implication (MI), 166, 172, 176, 196, 395-99

for Modal Logic, 387

Modus Ponens (MP), 165 ModusTollens (MT), 165

Quantifier Negation (QN), 311 for questions of identity, 336

Reductio ad Absurdum (RA), 150, 152,

166

Simplification (Simp), 79, 165

Substitution of Identicals (SI), 343-46, 385

Transposition (TR), 166

Universal Generalization (UG), 262-64

Universal Instantiation (UI), 260-62, 368

Rules of Inference for disjunctions, 98-105

De Morgan’s Laws (DM), 102-05, 166

Disjunction (Disj), 101, 104, 165 Disjunctive Syllogism (DS), 98-100, 104, 165

rules of statement formation, 47

Russell, Bertrand, 161, 390

definite description, 339

“logicist” approach to the foundations of mathematics, 339

Principles of Mathematics, 339 Russell’s theory of definite description, 339-40

Saccheri, Gerolamo

Euclides ab Omni Noevo I ^rindicatus, 403

Logica Demonstrativa, 403 Sankhyas, 55, 65

Sankhyas’ syllogism, 66

Scientific Revolution, 235 second-order logic, 385 self-contradictory form, 198, 200 sentence fragments, 5

Sequentform, 187 sequent schema, 402 sequents, 186-88

Sextus’ dilemma, 135-36

Sextus Empiricus, 135, 395

Shakespeare sorites, 286 Shcherbatskoi, Fedor, 55, 60, 65n2

Sheffer Stroke, 190

Sheffer’s Stroke operator, 189 simple constructive dilemma, 136, 139

as a rule of inference, 137 simple destructive dilemma, 138-39 simple statement, 49

definition, 43 Simplification (Simp), 79, 81, 165, 171, 176, 392

‘since,’ 71

singular statement

symbolizing, 256 singular terms, 258

symbolizing, 256

“Sir Bedevere” argument, 27, 31-33, 58-59

SL as a formal system, 179-92

Smolin, Lee, 141, 163

Snell, Bruno, The Discovery of Mind, 110n2

Socrates, 135, 149, 196, 396, 398

Socratic ‘elenchus,’ 149

Sophists, 142

sorites, 286-87

soundness, 35

of arguments, 27-29

Spinoza, Baruch, 22

Square of Opposition, 237, 290 standards of evidence, 35-37

Star Trek, 3, 209

statement connectives, 44

statement form, 182

Statement Logic, 32, 44, 228, 253, 392

as a formal system, 179-92

statement operators, 44-49, 179

statement variables, 56, 59, 182 statements, 6, 41-42

definition, 5, 42

Stephenson, Neal, 8

Stipulative definition, 130

Stoic Logic, 149

Stoics, 57, 80, 234, 402

“strict implication,” 397 strict partial ordering, 349 strict total ordering, 349 subalternation, 290 subcontraries, 290

substitution instance, 56, 59, 182

Substitution OfIdenticals (SI), 344, 346, 385 sufficient conditions, 127 sufficient reason, 295

superlatives, 337 supposition, 5, 113, 150

in natural argument, 119-22

for the sake of argument, 119

for the sake of example, 119 supposition indicators, 120 supposition rule, constraints on the use of, 114, 116

suppositional argument, 262

syllogisms, 234

evaluating validity of, 245-49 syllogistic logic

Carroll’s Diagrams, 239-44 category logic, 233-37

Symbolic Logic (Carroll), 4, 257, 314

Symbolic Logic (Venn), 292 symbolization in predicate logic, 315 symbolizing conjunctions, 75-77 symbolizing conventions, 47, 49 symbolizing disjunctions, 93-96 symbolizing identities and quantities, 335-39 symbolizing negations, 61-64

contradictories, 62-63

symbolizing non-classical statements, 303 symbolizing relational statements, 351-53 prenex forms, 354-57

symbolizing relations, 323

dyadic predicates, 323

symmetry, 327

as property ofbinary relations, 331 symbolizing, 329

symmetry of identity relations, 345-46

tautologous form, 197, 200

tautology, 187, 197, 200, 229

Law ofTautology, 151

truth tree method of determining, 226 techniques of diagramming, 19-23 telescoped rules of inference, 325-26 ‘the,’ 340

Theophrastus, 401-02 theorems, 176, 187

‘therefore,’ 85, 88

Thomas Aquinas, Saint, 363-64 three-valued logic, 387-89. See also

Quantum Logic

Through the Looking Glass (Carroll), 361 Humpty Dumpty, 292n9

transitivity, 322, 327-28, 330-31

of identity relations, 345-46

Transposition (TR), 166 triple-dot sign, 179 truth tables, 193-94

BriefTruth Table Method (BTT), 207-10 for negation, 193-94 test for statement logic, 204 for a truth-functional compound state­ment, 195,200

and validity, 201-06 truth tree method, 215-18

decomposition rules for, 219-23 of determining completeness, 227-29 of determining logical equivalence, 227 determining soundness, 227-29 to prove a statement is a tautology, 226 truth tree rules for identity and diversity, 378 truth tree rules in relational logic, 376 truth trees for predicate logic, 367-81 truth value, 45, 193 truth-functional conditional, 52-53 truth-functional operators, 45, 49, 61 Turing, Alan, “Computing Machinery and

Intelligence,” 67 turnstile (sign), 179, 186

UD.

392

in Predicate Logic, 122

Universal Generalization rule, 307 restrictions, 311

Universal Instantiation (UI), 260-62, 286-

87, 392 universal negative, 235 universal quantification, 253-54, 258, 260,

264 universal quantifier

symbolizing, 254, 258 universal statements, 254

existential import, 290 Universal (À-and E-) statements, 298 Universe of Discourse (UD), 244, 254,

258, 268

non-emptiness of the, 295-96

‘unless’

symbolizing, 110

‘unless p, q^f symbolizing, 110

valid enthymemes, 293, 295

validity, 60, 67, 79 argument forms and, 30-33 of categorical syllogisms, 245-49 defining, 25-27 evaluating natural arguments, 34-37 soundness, 27-29 truth tables and, 201-06

van Fraassen, Bas C., 130

Laws and Symmetry, 129nl variables, symbolizing, 258 Venn, John, 281-82, 293, 303

Symbolic Logic, 292

Venn diagrams, 405-08

Voltaire, Philosophical Dictionary, 143 Vorobej, Mark, A Theory of Argument, 13,37

weak partial ordering, 349 weak total ordering, 349 well-formed formula (wff), 179-82, 318 recursive definition, 316 symbolization, 315

Whitehead, Alfred North, 339

Wilson, E.O., 155

Wilson, James Q., 19, 35

Witten, Edward, 48

Yates, Edmund, 286

‘yet,’ 77

Yourgrau, Palle, A World Without Time, 52

Zeno of Citium, 362-63

Zeno ofElea, 149, 156, 160, 362 Zenonian elenchus, 149

² Although it does not add to the persuasive force of an argument, there may still be good reason to reiterate a statement made earlier. In chapter 11 we will introduce a rule, Reiteration, that allows us to do just that in the course of a proof of validity.

EXERCISES 2.1

1. Based on your understanding of this section, say whether each of the following state­ments is true or false:

(a) An argument is valid if the conclusion follows from the premises.

(b) An argument is valid if all its premises are true.

(c) If an inference has only one premise but this is incompatible with its conclusion, then it is valid.

(d) An argument is valid if it is sound.

(e) An argument is valid if and only if the denial of the conclusion is incompatible with the acceptance of all its premises.

(f) All circular arguments are valid.

2. Make a true statement by filling in the gaps below with words taken from among the following set: {conclusion, deduce, infer, premises, follow from, valid, sound}:

In a good argument the________ should______ the_____. If you are given the premises,

you should be able to________ the conclusion, provided the argument is________.

EXERCISES 22.2

12. Symbolize and prove the validity of the ⁱⁱFrank- Judo” argument from the text:

Every animal in the WILD kingdom is an EXPERT₂ at something. Frank is an EXPERT₂ at Judo. Therefore either a wild animal is expert at Judo or Frank is not a wild animal. [Wx := x is an animal in the wild kingdom]

¹ This result, when made suitably precise, is the content of Bell’s Theorem, subsequently veri­fied by the experiments of Alain Aspect and others.