Universal generalization From Wikipedia, the free encyclopedia Contents 1 Absorption (logic) 1 1.1 Formal notation ............................................ 1 1.2 Examples ............................................... 1 1.3 Proof by truth table .......................................... 2 1.4 Formal proof ............................................. 2 1.5 References ............................................... 2 2 Associative property 3 2.1 Definition ............................................... 3 2.2 Generalized associative law ...................................... 4 2.3 Examples ............................................... 4 2.4 Propositional logic .......................................... 7 2.4.1 Rule of replacement ..................................... 7 2.4.2 Truth functional connectives ................................. 7 2.5 Non-associativity ........................................... 7 2.5.1 Nonassociativity of floating point calculation ......................... 8 2.5.2 Notation for non-associative operations ........................... 8 2.6 See also ................................................ 10 2.7 References ............................................... 10 3 Axiom 11 3.1 Etymology ............................................... 11 3.2 Historical development ........................................ 12 3.2.1 Early Greeks ......................................... 12 3.2.2 Modern development ..................................... 12 3.2.3 Other sciences ......................................... 13 3.3 Mathematical logic .......................................... 14 3.3.1 Logical axioms ........................................ 14 3.3.2 Non-logical axioms ...................................... 15 3.3.3 Role in mathematical logic .................................. 16 3.3.4 Further discussion ....................................... 17 3.4 See also ................................................ 17 3.5 References ............................................... 17 i ii CONTENTS 3.6 Further reading ............................................ 17 3.7 External links ............................................. 18 4 Axiom schema 19 4.1 Formal definition ........................................... 19 4.2 Finite axiomatization ......................................... 19 4.3 Examples ............................................... 19 4.4 Finitely axiomatized theoreies ..................................... 19 4.5 In higher-order logic .......................................... 20 4.6 See also ................................................ 20 4.7 References .............................................. 20 5 Axiomatic system 21 5.1 Properties ............................................... 21 5.2 Relative consistency .......................................... 21 5.3 Models ................................................. 21 5.4 Axiomatic method ........................................... 22 5.4.1 History ............................................ 22 5.4.2 Issues ............................................. 22 5.4.3 Example: The Peano axiomatization of natural numbers ................... 23 5.4.4 Axiomatization ........................................ 23 5.5 See also ................................................ 23 5.6 References .............................................. 23 6 Biconditional elimination 24 6.1 Formal notation ............................................ 24 6.2 See also ................................................ 25 6.3 References ............................................... 25 7 Biconditional introduction 26 7.1 Formal notation ............................................ 26 7.2 References ............................................... 26 8 Commutative property 27 8.1 Common uses ............................................. 27 8.2 Mathematical definitions ........................................ 28 8.3 Examples ............................................... 28 8.3.1 Commutative operations in everyday life ........................... 28 8.3.2 Commutative operations in mathematics ........................... 28 8.3.3 Noncommutative operations in everyday life ......................... 29 8.3.4 Noncommutative operations in mathematics ......................... 30 8.4 History and etymology ......................................... 30 8.5 Propositional logic .......................................... 31 CONTENTS iii 8.5.1 Rule of replacement ..................................... 31 8.5.2 Truth functional connectives ................................. 31 8.6 Set theory ............................................... 31 8.7 Mathematical structures and commutativity .............................. 32 8.8 Related properties ........................................... 32 8.8.1 Associativity ......................................... 32 8.8.2 Symmetry ........................................... 32 8.9 Non-commuting operators in quantum mechanics ........................... 32 8.10 See also ................................................ 33 8.11 Notes ................................................. 34 8.12 References ............................................... 34 8.12.1 Books ............................................. 34 8.12.2 Articles ............................................ 35 8.12.3 Online resources ....................................... 35 9 Conjunction elimination 36 9.1 Formal notation ............................................ 36 9.2 References .............................................. 37 10 Conjunction introduction 38 10.1 Formal notation ............................................ 38 10.2 References ............................................... 38 11 Constructive dilemma 39 11.1 Formal notation ............................................ 39 11.2 Variable English ........................................... 39 11.3 Natural language example ...................................... 39 11.4 References ............................................... 40 12 De Morgan’s laws 41 12.1 Formal notation ............................................ 41 12.1.1 Substitution form ....................................... 43 12.1.2 Set theory and Boolean algebra ................................ 43 12.1.3 Engineering .......................................... 44 12.1.4 Text searching ......................................... 44 12.2 History ................................................. 44 12.3 Informal proof ............................................. 45 12.3.1 Negation of a disjunction ................................... 45 12.3.2 Negation of a conjunction ................................... 45 12.4 Formal proof ............................................. 46 12.5 Extensions ............................................... 46 12.6 See also ................................................ 47 12.7 References ............................................... 48 iv CONTENTS 12.8 External links ............................................. 48 13 Deduction theorem 49 13.1 Examples of deduction ........................................ 49 13.2 Virtual rules of inference ....................................... 50 13.3 Conversion from proof using the deduction meta-theorem to axiomatic proof ............ 50 13.4 The deduction theorem in predicate logic ............................... 51 13.5 Example of conversion ........................................ 52 13.6 Paraconsistent deduction theorem ................................... 53 13.7 The resolution theorem ........................................ 53 13.8 See also ................................................ 53 13.9 Notes ................................................. 53 13.10References ............................................... 53 13.11External links ............................................. 54 14 Destructive dilemma 55 14.1 Formal notation ............................................ 55 14.2 Natural language example ....................................... 55 14.3 Proof ................................................. 56 14.4 Example proof ............................................. 56 14.5 References ............................................... 56 14.6 Bibliography ............................................. 56 14.7 External links ............................................. 56 15 Disjunction elimination 57 15.1 Formal notation ............................................ 57 15.2 See also ................................................ 58 15.3 References ............................................... 58 16 Disjunction introduction 59 16.1 Formal notation ............................................ 59 16.2 References .............................................. 59 17 Disjunctive syllogism 60 17.1 Formal notation ............................................ 60 17.2 Natural language examples ...................................... 61 17.3 Inclusive and exclusive disjunction .................................. 61 17.4 Related argument forms ........................................ 61 17.5 References ............................................... 62 18 Distributive property 63 18.1 Definition ............................................... 63 18.2 Meaning ................................................ 63 18.3 Examples ............................................... 64 CONTENTS v 18.3.1 Real numbers ......................................... 64 18.3.2 Matrices ............................................ 65 18.3.3 Other examples ........................................ 65 18.4 Propositional logic