Index A Constructive dilemma, 32 Absorption, 32 Contingent, 28 Addition, 33 Contradiction, 28, 46 Adequacy theorem, 80 Convince, 11 AEIO, 38 Corallary, 8 Agreement, 98 Cryptography, 9 Algebra, 7 Culture of proving, 4 Argument, 4, 9 Artificial Intelligence, 118 Assertions, 4, 19, 26 D Atomic, 60, 95 Data Analyst, 3 Axioms, 15, 93 Data Scientist, 3 Davis, M., 119 Decision Support Systems, 3 B Deduction, 4 Biconditional, 29 Deduction rules, 30, 93 Bill Nye, 11 Deduction theorem, 79, 106 Biology, 3 De Morgan, 29, 39 Bound variables, 98 Derivation, 30 Direct method, 43 Disjunction, 26 C Disjunctive Normal Form (DNF), 83 Calculus, 9 Distribution, 39 Cardinality, 44 Domain, 36, 99 Categorical forms, 38 Double Negation, 29 Categorical syllogism, 36 Dysjunctive syllogism, 32 Combinatorics, 22 Commutativity, 39 Complete, 77 E Completeness, 104 Econometrics, 3 Completeness theorem, 81, 82 Emojies, 16 Conjunction, 26, 33 Engineering, 9 Conjunctive Normal Form (CNF), 83 Entry, 59 Connectives, 26, 62 Euclid, 5 Consequences, 69 Euclidean Division Theorem, 48 Consistency, 77, 104 Euclid’s Theorem, 9 Constants, 36, 94 Ex falso quodlibet, 34 © Springer Nature Switzerland AG 2021 131 L. P. Cruz, Theoremus, https://doi.org/10.1007/978-3-030-68375-7 132 Index Existential generalization, 41 Law of Indentity (LOI), 30 Existential instantiation, 41 Law of Non-Contradiction (LNC), 30 Exists, 23 Leibnitz principle, 103 Exportation, 29 Lemma, 7 Logical equivalence, 98 Logic, equivalence, 29 F Logic, higher order, 93 Fallacy, 12, 42 Logic, non-classical, 30 Fallacy, logic, 25 Logic, propositional, 25 FALSE, 27 Logic, rules, 15 FALSITY, 66 Financial Quant, 3 First-Order Logic (FOL), 36, 39, 93 M First-order theory, 122 Marketing Quants, 3 Fitch Style, 30 Material Implication, 29 FOL, semantics, 99 Mathematical Induction (MI), 49, 51 FOL, syntax, 94 Mathematical induction, strong form, 50 Formulas, 29 Mathematical induction, weak form, 49 Formulas, well-formed, 60 Mathematical logic, 123 Free variables, 98 Model, 82 Functions, 94 Modus ponens, 31, 51 Modus tollens, 32 G Gentzen, Gerhard, 73 N Geometry, 5, 7 Natural deduction, 30, 73 Gödel, Kurt, 106 Notation, 16 Greek, 4 Number theory, 9 Groups, 123 O H Operations Research, 3 Higher Order Logic (HOL), 93 P I Philosophy, 4 Iff, 21 Physics, 9 If-Then, 20 PL, formal system, 59 Implication, 26 PL, semantics, 66 Indirect method, 44 PL, syntax, 60 Individuals, 36 Predicate Logic, 35, 36 Induction Hypothesis (I.H.), 49, 51 Predicates, 37, 94 Inference rules, 30, 41 Premise, 4 Inference rules, Gentzen, 74 Prenex form, 109 Initial formulas, 122 Primitive, 60, 95 Interpretation, 66 Prolog, 122 Proof, 4, 11 Proof, by cases, 48 K Proof, by construction, 54 Knowledge base, 122 Proof by contradiction, 45 Knowledge representation, 122 Proof, left-right, 46 Proof, rigorous, 14 Proof, sound, 15 L Proof, valid, 15 Law of Excluded Middle (LEM), 30 Proper symbols, 122 Index 133 Propositional Logic, 35 Standardizing variables, 108 Propositions, 7, 19 Statement, 4, 19, 98 Psychology, 3 Statements, equational, 21 Putnam, H., 119 Statements, relational, 21 Pythagorean Theorem, 8, 22 Statistics, 3 Structural induction, 64 Substitution, 98, 114 Q Syntax diagram, 61 Quantification rules, 102 Quantifiers, 22, 37, 108 T Tarski, A., 102 R Tautology, 70 RAA, 33, 45 Terms, 94 Rectified form, 108 Theorems, 3 Reduction ad absurdum, 33 Theories, 60, 70, 78 Refutation, 84 Thinking Skills, 5 Relations, 37 Tradition, 4 Resolution, 81, 107 Transposition, 29 Resolution calculus, 84 TRUE, 4, 10, 27 Resolution theorem, 87 TRUTHFULNESS, 66 Resolvents, 84 Truth table, 27 Rings, 124 Robinson, J. A., 91 U Unification, 114 S Universal generalization, 41 Satisfiability, 69 Universal instantiation, 41 Satisfiable, 38, 101 Unsatisfiable, 69, 101 Schemas, 29 Sentence, 19, 98 Sequent calculus, 73 V Simplification, 32 Validity, 11, 69 Skolem normal form, 113 Valuations, 26, 38 Skolem, T., 111 Variables, 36, 93 Skolemization, 113 Variables, propositional, 60 Skolemizing, 111 Velani, C., 127 Socrates, 9, 10, 35 Soundness, 12, 77, 104 Soundness theorem, 78, 104, 106 W Square of Opposition (SoO), 39 Willem de Vlamigh, 11.