Abbreviation conventions, in demonstrations, 83, 84, 86–88, 203, 204
fields, operators, 197
other, 204
parentheses, 198
(See also Notational conventions)
group operators, 185
inequalities, 225
inference rules, 50
multiple quantification, 158, 159
parentheses in quantified statements, 138
quantifiers, 158
restricted, 239
statement calculus, 38–40
symbolic form of miniature geometry, 245
Absolute value, 228
Abstract mathematical system, 102
Antecedent, 27
Arrangement (Ex. 1), 191
Bound variable, confusion of, 163, 164
Bound variable, confusion of, with IS, 174, 175
occurrence of, 139
Cancellation, in fields, 199
in groups, 187
Cases, proof by, 68–70
Conditional, interpretation of, 26–30
truth table for, 30
Conditional statement function, 133
Confusion of bound variables, 163, 164
Conjecture, 2
Conjunction, interpretation of, 22, 23
truth table for, 30
Conjunctive inference, 51, 55, 144
Conjunctive simplification, 51, 55
Connective, 30
Consequent, 27
Contrapositive inference, 52, 53, 56, 144
in Aristotelian logic, 73–76
per accidens, 75
Conversion, of quantifiers, 137, 138
Correlative definitions, 119, 120
Correspondence, one-to-one, 192
Deduction principle, 89
scope of assumption in, 179–181
De Morgan, A., 46
Detachment, law of, 50
Disjunction, exclusive, statement formulas for, 25
interpretation of, 23–25
truth table for, 30
Duality in miniature geometry, 108–111, 252
Duality principle, 110
Equality, inference rule IS, 174
inference rule ISC, 176
interpretation, 13
standard, 166
reflexivity axiom R, 174
Equivalence, definition and truth table, 33
Euler, L., 3
Existential quantifiers, 135
Fermat conjecture, 3
axioms, 197
cancellation in, 199
ordered, 220–233
absolute value defined in, 228
axioms of order, 221
solution of equations, linear, 205–207
simultaneous linear, 214–217
subfield, 220
Free variable, occurrence of, 139, 140
Frege, G., 29
Generalization principle (see PGU)
Goldbach conjecture, 2
Groups, 183–196
Abelian, 185
cancellation in, 187
isomorphic interpretations, 189–196
formal definition of, 193
subgroup (Ex. 1 f.), 191
Hilbert, D., 123
Hypothetical syllogism, 52, 56
IE (inference rule), 150–152, 165
definition, 152
basic form, 57–61
special cases, 61–63
summary of, 63
by elimination, 63–67
summary of, 67
summary of, 71
Individual, 128
Inference rules, predicate calculus (see IE; IS; ISC; IU; PGE; PGU)
statement calculus, derived, 51, 52
extended for predicate calculus, 143, 144
Interpretation of a system, 119–124
Inverse, of a conditional, 76, 77
of a field element, 198
of a group element, 188
operators (−, ÷), 213
IS (inference rule), 174
modified for terms, 184
ISC (inference rule), 176
modified for terms, 184
IU (inference rule), 143, 165, 166
modified for terms, 184
Lemma, definition of, 104
Limit of a sequence, 234
Major premise, 50
Mention, 9–15
Miniature geometry, axioms, 103
symbolic form of, 246
perspectivity in, 111
Minor premise, 50
(See also Interpretation)
extended to predicate calculus, 144
Modus tollens, 56
Necessary condition, 34
Negation, field operation, 213, 218
in statement calculus, interpretation, 20, 21
truth table for, 30
Notational conventions, 38–41
Numeral, 11
One-to-one correspondence, 192
Order axioms, 221
Ordered fields (see Fields)
Permutation (Ex. 1), 191
Perspectivity, 111
PGE (inference rule), 153, 166
with restricted quantifiers, 241
PGU (inference rule), final form of, 166
first statement of, 146
restricted, by IE, 152
for several variables, 165
with restricted quantifiers, 241
Placeholder, 31, 32, 41, 42, 128
Pole and polar, 103
Prime numbers, 2n.
relative, 160n.
Proposition, Aristotelian forms, 7–9, 73, 75
translations in predicate calculus, 135
Quantifiers, existential, 135
restricted, 238–242
transformation (conversion) of, 137, 138
universal, 132
R (reflexivity axiom for equality), 174
Restricted quantifiers, 238–242
Russell, B., 30
Scope, of assumption in a demonstration, 179
of a quantifier, 139, 158, 159
Sequence, 234
Sheffer, H., 30
Simplification, conjunctive, 51, 55
Socrates argument, 5, 6, 127, 134, 142
translation in predicate calculus, 134
Statement, simple, 18
Statement formulas, definition of, 31
truth value of, 32
validity of, 35–37
extended for statement functions, 145
Statement function, 128
conditional, 133
of several variables, 157–161
substitution instance of, 128, 129, 140
Subfield, 220
Subgroup (Ex. 1 f.), 191
Substitution of names, 12
Substitution instance of statement function, 128, 129, 140
Substitution principle, 43, 44
Sufficient condition, 34
Terms, 184
Translation, 14
Trichotomy, 221
Truth table, 30
Universal quantifiers, 132
Unknowns, 130
Use and mention, 9–15
Valid interpretation, 120
Valid statement formulas, 35–37
extended for statement functions, 145
Variable, bound (see Bound variable)
individual, 128
value of, 130
Veblen, O., 123
Well-formed formula, 31