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. 106
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proof, 46, 49, 103
advantages over truth tables, universal, 230
127 quanti¬ers, 3, 32, 227, 230“231, 239
by cases, 131“134 alternations of, 312
by contradiction, 136“138 alternative notation for, 255“
by induction 256
basis, 449 DeMorgan laws for, 279
induction step, 449 ¬rst-order, 257
conditional, 199 game rules for, 237“238
discovery of the notion of, 287 generalized



General Index
General Index / 581



adding to fol, 383 resolvent, 489, 490
and vagueness, 387 rules
logic of, 389 default use of, 143, 145, 153,
semantics of, 499“500 161, 208, 345, 349
mixed, 293, 296, 329 elimination, 142
monotone, 390 introduction, 142
multiple, 289 Russell
second-order, 257 paradox, 406, 432, 433
semantics for, 234“238 set for a set, 430, 431
absolute version, 432
range of a function, 428
Russell, Bertrand, 430
rational inquiry, 1“2, 4
Russell™s Theorem, 362
rational number, 131
Russellian analysis of de¬nite de-
reasoning
scriptions, 379, 382
invalid, 1
Strawson™s objection, 381
valid, 1
Russellian analysis of both and nei-
reductio ad absurdum, 136“138
ther, 382
reference columns, 96
referent of a name, 497, 498
re¬‚exivity, 52, 422 satisfaction, 234, 500, 502, 506
of identity, 50 algorithm for Horn sentences,
reiteration, 56, 151 479, 481, 484
relations satis¬ability, 373, 469
antisymmetric, 422
satis¬able
asymmetric, 422
truth table (tt-), 469
binary, 422
Saturday Night Live, 304
equivalence, 424“425
scope, 80
functional, 427
second-order logic, 257
inverse, 423
semantics
irre¬‚exive, 422
for connectives, 68, 72, 75, 178,
modeling in set theory, 422“
182
425
for quanti¬ers, 234“238
re¬‚exive, 422
semi-w¬, 449
transitive, 422
sentence, 232, 231“233
relative complement, 440
atomic, 19, 23, 23, 25, 32, 51
replacement method, 270“271
complex, 67
resolution clause, 488
existential, 400
resolution method, 54, 489“491, 519,
Horn, 479, 481
524
normal forms, 124
completeness of, 493
satis¬able, 373
for fol, 516, 521
soundness of, 493 universal, 520


General Index
582 / General Index


set structure
¬rst-order, 498, 495“498, 500,
conception of
506
naive, 405“411, 416, 432“434,
truth in, 504
436, 439
Submit, 5“10
von Neumann™s, 435, 436, 438
subproof, 149“151, 163, 206, 343
Zermelo™s, 438
end of, 164
empty, 407, 412
proper use of, 163“166
membership, 37
subset relation, 407, 412“414
singleton, 412, 420, 440
substitution, 49
set theory, 435
of logical equivalents, 118, 276
¬rst-order language of, 37
principle, 277
sets
su¬cient condition, 179
cummulative, 438“439
summary of game rules, 237
in¬nite, 437“438
symbolic sciences, 2“4
size of, 437
symmetry, 52, 422
simpli¬cation, 129
of identity, 50
singleton set, 412, 420, 440
Skolem
Tarski, Alfred, 400, 506, 556
function, 515
Tarski™s World, 5“10, 14, 24
normal form, 515
Taut Con, 114“116, 158, 171, 221,
paradox, 547
272
Skolem, Thoralf, 515
and truth-functional form al-
Skolem,Thoralf, 546
gorithm, 263
Skolemization, 514, 515, 521
tautological consequence, 113, 266,
Smullyan, Raymond, 449, 555
469
sound
tautology, 94, 97, 100, 101, 103,
argument, 43, 44, 140
137, 218, 219, 266, 469
deductive system, 214, 214, 361
and FO validity, 271
Soundness Theorem
and quanti¬cation, 257“264
for ¬rst-order logic, 509
of fol, 262
for propositional logic, 215
tense, 398
use of, 220
terms, 32, 229
spatial predicates, 401
complex, 32, 34
step-by-step translation, 298, 299
of ¬rst-order arithmetic, 39
Strawson, P. F., 380, 381
ternary connective, 195
Strawsonian analysis of de¬nite de-
theorem, 47, 193
scriptions, 382
theory
strengthening the antecedent, 203,
formally complete, 472
212
formally consistent, 471
strengthening the consequent, 203,
transitivity, 422
212

General Index
General Index / 583



of <, 52 something, 228
of ”, 203, 212 the, 379
of identity, 51 un-, 68
translation, 13, 28, 84 unless, 180
and meaning, 84 of complex noun phrases, 243“
and mixed quanti¬ers, 289“291, 244
308 of conditionals, 179
and paraphrase, 300 step-by-step method, 298
extra exercises, 315“318 using function symbols, 308
of truth, 500
a, 231 assignment, 468
all, 230 conditions, 84, 84, 188
an, 231 in a structure, 504, 506
and, 71 in all worlds, 363
any, 230, 243 logical, 93, 94, 103, 181, 182,
at least n, 366 267
at least one, 231 non-logical, 363
at most n, 366 unde¬nability of, 556
both, 379 value, 24, 67
but, 71 truth table, 67
each, 230, 243 disadvantages of, 127
every, 230, 239, 243 for §, 72
everything, 228 for ”, 182
exactly n, 366 for ¬, 68
few, 386 for ’, 178
if, 179 for ∨, 75
if and only if, 182 joint, 106, 110
i¬, 182 method, 481
just in case, 182 modeling in set theory, 468“
many, 386 469
moreover, 71 number of rows, 95
most, 386 reference columns, 96
neither, 379 truth-functional
neither. . . nor, 75 completeness, 190, 193
no, 239, 243 connective, 67
non-, 68 binary, 190
not, 68 semantics for, 68, 72, 75, 178,
only if, 180 182
or, 74 ternary, 195
provided, 179 form, 261
some, 231, 239 algorithm, 261“264


General Index
584 / General Index


empty, 501
tt-satis¬able, 469
bound, 232, 282
twin prime conjecture, 333
free, 231, 232
unary function symbol, 308 limited number of, 373
unde¬nability of truth, 556 reusing, 374
uni¬cation, 517 unbound, 231
algorithm, 518 Vaught, Robert, 435
uninterpreted language, 15 von Neumann, John, 434“436, 438
union, 415, 416
weakening the consequent, 203, 212
axiom, 435, 440
uniqueness Web address, 15
well-formed formula (w¬), 231, 231“
claim, 375
233
quanti¬er, ∃!, 375
atomic, 229, 231
universal
existential, 450
elimination, 342
generalization, 325, 326, 332, universal, 450
342 wellfounded sets, 450
winning strategy for playing the game,
instantiation, 321, 342
78
introduction, 325, 342, 343
witnessing constant, 528
noun phrase, 243, 248
quanti¬er, 230 adding to a language, 529“530
sentence, 520 for a w¬, 529
set, 432, 440
You try it exercises, 7, 12
w¬, 450
unordered pair, 419
Zermelo, Ernst, 438
axiom, 435
Zermelo-Frankel set theory, 435
uses of fol, 3“4
zfc, 435, 547
vacuously true generalization, 245
vagueness, 21
valid
argument, 140
¬rst-order, 505
proof, 46

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. 106
( 107 .)



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