Logic symbols 4 Wikipedia

Contents

1 Atomic formula 11.1 Atomic formula in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Atomic sentence 32.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Atomic sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Atomic formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.4 Compound sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.5 Compound formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Translating sentences from a natural language into an articial language . . . . . . . . . . . . . . . 52.4 Philosophical signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Categorical proposition 73.1 Translating statements into standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Properties of categorical propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Quantity and quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Operations on categorical statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.1 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.2 Obversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.3 Contraposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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4 Clause (logic) 114.1 Empty clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Implicative form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Conditioned disjunction 135.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Contingency (philosophy) 146.1 Contingency and relativism in rhetoric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Contradiction 167.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Contradiction in formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.2.1 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2.2 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency . . . . . . 18

7.3 Contradictions and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.1 Pragmatic contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.2 Dialectical materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.4 Contradiction outside formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Converse implication 228.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.5 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Converse nonimplication 249.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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9.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.5.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10 CornishFisher expansion 2710.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11 Deductive closure 2911.1 Epistemic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Exclusive or 3012.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.2 Equivalencies, elimination, and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.3 Relation to modern algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.4 Exclusive or in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Alternative symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.7 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

12.7.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.8 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13 Expression (mathematics) 3713.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3 Syntax versus semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.3 Formal languages and lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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13.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 False (logic) 4014.1 In classical logic and Boolean logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2 False, negation and contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15 Formation rule 4215.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.2 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.3 Propositional and predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

16 Frege system 4416.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

16.4.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

17 Freges propositional calculus 4617.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

18 Functional completeness 4918.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.2 Informal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918.3 Characterization of functional completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.4 Minimal functionally complete operator sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.6 In other domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.7 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

19 Ground expression 5319.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5319.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

19.2.1 Ground terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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19.2.2 Ground atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5419.2.3 Ground formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

19.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

20 Material equivalence 5520.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5520.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

20.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5520.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

20.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

21 Implicational propositional calculus 5821.1 Virtual completeness as an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5821.2 Axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5821.3 Basic properties of derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5921.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

21.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5921.5 The BernaysTarski axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6121.6 Testing whether a formula of the implicational propositional calculus is a tautology . . . . . . . . . 6121.7 Adding an axiom schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.8 An alternative axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

22 Indicative conditional 6522.1 Distinctions between the material conditional and the indicative conditional . . . . . . . . . . . . . 6522.2 Psychology and indicative conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

23 Intermediate logic 6723.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.4 Relation to modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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24 List of logic systems 7024.1 Classical propositional calculus systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

24.1.1 Implication and negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7024.1.2 Implication and falsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.1.3 Negation and disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.1.4 Sheers stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

24.2 Implicational propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.3 Intuitionistic and intermediate logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4 Positive implicational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.5 Positive propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.6 Equivalential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

25 Literal (mathematical logic) 8125.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8125.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

26 Logical biconditional 8226.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

26.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.1.2 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

26.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

26.3.1 Biconditional introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.3.2 Biconditional elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

26.4 Colloquial usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

27 Logical conjunction 8727.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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28 Logical connective 9328.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28.1.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.1.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

28.2 Common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.2.1 List of common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.2.2 History of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.2.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

28.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

29 Logical consequence 9929.1 Formal accounts of logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.2 A priori property of logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.3 Proofs and models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

29.3.1 Syntactic consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.3.2 Semantic consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

29.4 Modal accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10029.4.1 Modal-formal accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.4.2 Warrant-based accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.4.3 Non-monotonic logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

29.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10129.7 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10229.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

30 Logical disjunction 10430.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10530.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

30.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10630.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10630.4 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.5 Applications in computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

30.5.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.5.2 Logical operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.5.3 Constructive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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30.6 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31 Logical equality 11031.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.2 Alternative descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11231.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

32 Logical NOR 11332.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

32.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.3 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

33 Material conditional 11533.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

33.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11633.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

33.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11733.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 11733.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

33.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11833.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

34 Material equivalence 12034.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12034.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

34.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12034.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12134.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

34.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12134.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12234.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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34.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12234.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

35 Material nonimplication 12335.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

35.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

35.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

35.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12435.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

36 Modal operator 12536.1 Modality interpreted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

36.1.1 Alethic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12536.1.2 Deontic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12536.1.3 Axiological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12536.1.4 Epistemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12536.1.5 Doxastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

37 Negation 12637.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12637.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12637.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

37.3.1 Double negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.3.2 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.3.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.3.4 Self dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

37.4 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.5 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12737.6 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12837.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12837.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12937.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12937.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

38 Negation introduction 13038.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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38.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13038.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

39 Negation normal form 13139.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13139.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13239.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

40 Nicods axiom 133

41 Open sentence 13441.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

42 Polish notation 13642.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13642.2 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13742.3 Order of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13742.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13842.5 Polish notation for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13842.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13842.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13942.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

43 Predicate (mathematical logic) 14043.1 Simplied overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14043.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14043.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14143.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14143.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

44 Principle of distributivity 142

45 Probabilistic proposition 143

46 Proof by contrapositive 14446.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14446.2 Relation to proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14446.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14446.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

47 Proposition 14647.1 Historical usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

47.1.1 By Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14647.1.2 By the logical positivists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14647.1.3 By Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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47.2 Relation to the mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14747.3 Treatment in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14747.4 Objections to propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14747.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14847.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14847.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

48 Propositional calculus 14948.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15048.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15048.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

48.3.1 Closure under operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15248.3.2 Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

48.4 Generic description of a propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15348.5 Example 1. Simple axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15448.6 Example 2. Natural deduction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15548.7 Basic and derived argument forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15648.8 Proofs in propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

48.8.1 Example of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15648.9 Soundness and completeness of the rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

48.9.1 Sketch of a soundness proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.9.2 Sketch of completeness proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15848.9.3 Another outline for a completeness proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

48.10Interpretation of a truth-functional propositional calculus . . . . . . . . . . . . . . . . . . . . . . . 15948.10.1 Interpretation of a sentence of truth-functional propositional logic . . . . . . . . . . . . . . 159

48.11Alternative calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.11.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.11.2 Inference rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.11.3 Meta-inference rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.11.4 Example of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

48.12Equivalence to equational logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16248.13Graphical calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16348.14Other logical calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16348.15Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16448.16See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48.16.1 Higher logical levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16448.16.2 Related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48.17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16448.18Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48.18.1 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16548.19External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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49 Propositional formula 16649.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

49.1.1 Relationship between propositional and predicate formulas . . . . . . . . . . . . . . . . . 16749.1.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

49.2 An algebra of propositions, the propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 16749.2.1 Usefulness of propositional formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16849.2.2 Propositional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16849.2.3 Truth-value assignments, formula evaluations . . . . . . . . . . . . . . . . . . . . . . . . 168

49.3 Propositional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16949.3.1 Connectives of rhetoric, philosophy and mathematics . . . . . . . . . . . . . . . . . . . . 16949.3.2 Engineering connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16949.3.3 CASE connective: IF ... THEN ... ELSE ... . . . . . . . . . . . . . . . . . . . . . . . . . 16949.3.4 IDENTITY and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

49.4 More complex formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17149.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17149.4.2 Axiom and denition schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17249.4.3 Substitution versus replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

49.5 Inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17249.6 Parsing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

49.6.1 Connective seniority (symbol rank) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17349.6.2 Commutative and associative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17449.6.3 Distributive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17449.6.4 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17449.6.5 Laws of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17549.6.6 Laws of evaluation: Identity, nullity, and complement . . . . . . . . . . . . . . . . . . . . 17549.6.7 Double negative (Involution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

49.7 Well-formed formulas (ws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17549.7.1 Ws versus valid formulas in inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

49.8 Reduced sets of connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17649.8.1 The stroke (NAND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17649.8.2 IF ... THEN ... ELSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

49.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17849.9.1 Reduction to normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17849.9.2 Reduction by use of the map method (Veitch, Karnaugh) . . . . . . . . . . . . . . . . . . 179

49.10Impredicative propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18049.11Propositional formula with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

49.11.1 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18149.11.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

49.12Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18249.13Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18449.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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50 Propositional function 19250.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

51 Propositional proof system 19351.1 Mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.2 Algorithmic interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19451.4 Relation with computational complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19451.5 Examples of propositional proof systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19451.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

52 Propositional representation 19752.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19752.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

53 Propositional variable 20053.1 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.2 In rst order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

54 Resolution inference 20154.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20154.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

55 Rule of inference 20355.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20355.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20455.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 20455.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20555.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20555.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

56 Rule of replacement 20756.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

57 Second-order propositional logic 20857.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20857.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

58 Sentence (logic) 20958.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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58.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20958.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

59 Sequent 21159.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

59.1.1 The form and semantics of sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21159.1.2 Syntax details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21259.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21259.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21359.1.5 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

59.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21359.2.1 History of the meaning of sequent assertions . . . . . . . . . . . . . . . . . . . . . . . . . 21359.2.2 Intuitive meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

59.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21459.4 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21459.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21559.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21559.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21659.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

60 Sheer stroke 21760.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

60.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21860.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21860.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21860.4 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21860.5 Formal system based on the Sheer stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

60.5.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21860.5.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21960.5.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21960.5.4 Simplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

60.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22060.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22060.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22160.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

61 Strict conditional 22261.1 Avoiding paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22261.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22261.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22361.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22361.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

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62 Substitution (logic) 22562.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

62.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22562.1.2 Tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

62.2 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22662.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22662.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22762.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22762.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

63 Syncategorematic term 22863.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22863.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

64 System L 22964.1 Description of the deductive system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22964.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23064.3 History of tabular natural deduction systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23064.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23064.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23064.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23164.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

65 T-schema 23265.1 The inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23265.2 Natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23265.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23365.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23365.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

66 Tautology (logic) 23466.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23466.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23566.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23566.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23666.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23666.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23666.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 23766.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23766.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

66.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23866.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

66.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

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66.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

67 Theorem 23967.1 Informal account of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23967.2 Provability and theoremhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24067.3 Relation with scientic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24067.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24067.5 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24167.6 Lore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24267.7 Theorems in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

67.7.1 Syntax and semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24367.7.2 Derivation of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24367.7.3 Interpretation of a formal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24467.7.4 Theorems and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

67.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24467.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24467.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24567.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

68 Theory (mathematical logic) 25068.1 Theories expressed in formal language generally . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

68.1.1 Subtheories and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25068.1.2 Deductive theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25068.1.3 Consistency and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25068.1.4 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25168.1.5 Theories associated with a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

68.2 First-order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25168.2.1 Derivation in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25168.2.2 Syntactic consequence in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . 25268.2.3 Interpretation of a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25268.2.4 First order theories with identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25268.2.5 Topics related to rst order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

68.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25268.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25368.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25368.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

69 Truth table 25469.1 Unary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

69.1.1 Logical false . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25469.1.2 Logical identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25469.1.3 Logical negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

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69.1.4 Logical true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25469.2 Binary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

69.2.1 Truth table for all binary logical operators . . . . . . . . . . . . . . . . . . . . . . . . . . 25569.2.2 Logical conjunction (AND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25569.2.3 Logical disjunction (OR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25569.2.4 Logical implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25569.2.5 Logical equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25669.2.6 Exclusive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25669.2.7 Logical NAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25669.2.8 Logical NOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

69.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25669.3.1 Truth table for most commonly used logical operators . . . . . . . . . . . . . . . . . . . . 25769.3.2 Condensed truth tables for binary operators . . . . . . . . . . . . . . . . . . . . . . . . . 25769.3.3 Truth tables in digital logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25769.3.4 Applications of truth tables in digital electronics . . . . . . . . . . . . . . . . . . . . . . . 257

69.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25869.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25869.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25869.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25969.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25969.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

70 Truth value 26070.1 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26070.2 Intuitionistic and constructive logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26070.3 Multi-valued logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170.4 Algebraic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170.5 In other theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26170.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

71 Truth-bearer 26371.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26371.2 Sentences in natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26471.3 Sentences in languages of classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26671.4 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26671.5 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26771.6 Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26871.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26871.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26971.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

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72 Unity of the proposition 27172.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27172.2 Russell, Frege, Wittgenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27172.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27272.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27272.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

73 Universal quantication 27373.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

73.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27473.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

73.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27473.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27573.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

73.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27673.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27773.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27773.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27773.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

74 Unsatisable core 27874.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

75 Well-formed formula 27975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28075.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28075.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28175.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28175.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28275.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28275.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28275.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28275.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28275.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28375.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

76 Wolfram axiom 28476.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28476.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28476.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

77 Zeroth-order logic 286

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77.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28677.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 287

77.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28777.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29577.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Chapter 1

Atomic formula

In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositionalstructure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas.Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining theatomic formulas using the logical connectives.The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example,the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols togetherwith their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbolswith a given signature, which may or may not be satisable with respect to a given model.[1]

1.1 Atomic formula in rst-order logicThe well-formed terms and propositions of ordinary rst-order logic have the following syntax:Terms:

t c j x j f(t1; :::; tn) ,

that is, a term is recursively dened to be a constant c (a named object from the domain of discourse), or a variable x(ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk. Functionsmap tuples of objects to objects.Propositions:

A;B; ::: P (t1; :::; tn) j A ^B j > j A _B j ? j A B j 8x: A j 9x: A ,

that is, a proposition is recursively dened to be an n-ary predicate P whose arguments are terms tk, or an expressioncomposed of logical connectives (and, or) and quantiers (for-all, there-exists) used with other propositions.An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formulaof the form P (t1, , tn) for P a predicate, and the tk terms.All other well-formed formulae are obtained by composing atoms with logical connectives and quantiers.For example, the formula x. P (x) y. Q (y, f (x)) z. R (z) contains the atoms

P (x) Q(y; f(x)) R(z)

When all of the terms in an atom are ground terms, then the atom is called a ground atom or ground predicate.

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2 CHAPTER 1. ATOMIC FORMULA

1.2 See also In model theory, structures assign an interpretation to the atomic formulas. In proof theory, polarity assignment for atomic formulas is an essential component of focusing. Atomic sentence

1.3 References[1] Wilfrid Hodges (1997). A Shorter Model Theory. Cambridge University Press. pp. 1114. ISBN 0-521-58713-1.

1.4 Further reading Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.

Chapter 2

Atomic sentence

In logic, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to asa proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For exampleThe dog ran is an atomic sentence in natural language, whereas The dog ran and the cat hid. is a molecularsentence in natural language.From a logical analysis, the truth or falsity of sentences in general is determined by only two things: the logical formof the sentence and the truth or falsity of its simple sentences. This is to say, for example, that the truth of the sentenceJohn is Greek and John is happy is a function of the meaning of "and", and the truth values of the atomic sentencesJohn is Greek and John is happy. However, the truth or falsity of an atomic sentence is not a matter that is withinthe scope of logic itself, but rather whatever art or science the content of the atomic sentence happens to be talkingabout.[1]

Logic has developed articial languages, for example sentential calculus and predicate calculus partly with the purposeof revealing the underlying logic of natural languages statements, the surface grammar of which may conceal theunderlying logical structure; see Analytic Philosophy. In these articial languages an Atomic Sentence is a string ofsymbols which can represent an elementary sentence in a natural language, and it can be dened as follows.In a formal language, a well-formed formula (or w) is a string of symbols constituted in accordance with the rules ofsyntax of the language. A term is a variable, an individual constant or a n-place function letter followed by n terms.An atomic formula is a w consisting of either a sentential letter or an n-place predicate letter followed by n terms. Asentence is a w in which any variables are bound. An atomic sentence is an atomic formula containing no variables.It follows that an atomic sentence contains no logical connectives, variables or quantiers. A sentence consisting ofone or more sentences and a logical connective is a compound (or molecular sentence). See vocabulary in First-orderlogic

2.1 Examples

2.1.1 Assumptions

In the following examples:* let F, G, H be predicate letters; * let a, b, c be individual constants; * let x, y, z be variables.

2.1.2 Atomic sentences

These ws are atomic sentences; they contain no variables or conjunctions:

F(a)

H(b, a, c)

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4 CHAPTER 2. ATOMIC SENTENCE

2.1.3 Atomic formulaeThese ws are atomic formulae, but are not sentences (atomic or otherwise) because they include free variables:

F(x) G(a, z) H(x, y, z)

2.1.4 Compound sentencesThese ws are compound sentences. They are sentences, but are not atomic sentences because they are not atomicformulae:

x (F(x)) z (G(a, z)) x y z (H(x, y, x)) x z (F(x) G(a, z)) x y z (G(a, z) H(x, y, z))

2.1.5 Compound formulaeThese ws are compound formulae. They are not atomic formulae but are built up from atomic formulae using logicalconnectives. They are also not sentences because they contain free variables:

F(x) G(a, z) G(a, z) H(x, y, z)

2.2 InterpretationsMain article: Interpretation (logic)

A sentence is either true or false under an interpretation which assigns values to the logical variables. We mightfor example make the following assignments:Individual Constants

a: Socrates b: Plato c: Aristotle

Predicates:

F: is sleeping G: hates H: made hit

Sentential variables:

2.3. TRANSLATING SENTENCES FROM A NATURAL LANGUAGE INTO AN ARTIFICIAL LANGUAGE 5

p: It is raining.

Under this interpretation the sentences discussed above would represent the following English statements:

p: It is raining.

F(a): Socrates is sleeping.

H(b, a, c): Plato made Socrates hit Aristotle.

x (F(x)): Everybody is sleeping.

z (G(a, z)): Socrates hates somebody.

x y z (H(x, y, z)): Somebody made everybody hit somebody. (They may not have all hit the same personz, but they all did so because of the same person x.)

x z (F(x) G(a, z)): Everybody is sleeping and Socrates hates somebody.

x y z (G(a, z) H(x, y, z)): Either Socrates hates somebody or somebody made everybody hit somebody.

2.3 Translating sentences from a natural language into an articial lan-guage

Sentences in natural languages can be ambiguous, whereas the languages of the sentential logic and predicate logicsare precise. Translation can reveal such ambiguities and express precisely the intended meaning.For example take the English sentence Father Ted married Jack and Jill. Does this mean Jack married Jill? Intranslating we might make the following assignments: Individual Constants

a: Father Ted

b: Jack

c: Jill

Predicates:

M: ociated at the marriage of to

Using these assignments the sentence above could be translated as follows:

M(a, b, c): Father Ted ociated at the marriage of Jack and Jill.

x y (M(a, b, x) M(a, c, y)): Father Ted ociated at the marriage of Jack to somebody and Father Tedociated at the marriage of Jill to somebody.

x y (M(x, a, b) M(y, a, c)): Somebody ociated at the marriage of Father Ted to Jack and somebodyociated at the marriage of Father Ted to Jill.

To establish which is the correct translation of Father Ted married Jack and Jill, it would be necessary to ask thespeaker exactly what was meant.

6 CHAPTER 2. ATOMIC SENTENCE

2.4 Philosophical signicanceAtomic sentences are of particular interest in philosophical logic and the theory of truth and, it has been argued, thereare corresponding atomic facts. An Atomic sentence (or possibly the meaning of an atomic sentence) is called anelementary proposition by Wittgenstein and an atomic proposition by Russell:

4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existenceof states of aairs. 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of astate of aairs.: Wittgenstein, Tractatus Logico-Philosophicus, s:Tractatus Logico-Philosophicus.

A proposition (true or false) asserting an atomic fact is called an atomic proposition.: Russell, Introduction toTractatus Logico-Philosophicus, s:Tractatus Logico-Philosophicus/Introduction

see also [2] and [3] especially regarding elementary proposition and atomic proposition as discussed by Russelland Wittgenstein

Note the distinction between an elementary/atomic proposition and an atomic factNo atomic sentence can be deduced from (is not entailed by) any other atomic sentence, no two atomic sentences areincompatible, and no sets of atomic sentences are self-contradictory. Wittgenstein made much of this in his TractatusLogico-Philosophicus. If there are any atomic sentences then there must be atomic facts which correspond to thosethat are true, and the conjunction of all true atomic sentences would say all that was the case, i.e. the world since,according toWittegenstein, The world is all that is the case. (TLP:1). Similarly the set of all sets of atomic sentencescorresponds to the set of all possible worlds (all that could be the case).The T-schema, which embodies the theory of truth proposed by Alfred Tarski, denes the truth of arbitrary sentencesfrom the truth of atomic sentences.

2.5 See also Logical atomism Logical constant Truthbearer

2.6 References Benson Mates, Elementary Logic, OUP, New York 1972 (Library of Congress Catalog Card no.74-166004) Elliot Mendelson, Introduction to Mathematical Logic, Van Nostran Reinholds Company, New York 1964 Wittgenstein, Tractatus_Logico-Philosophicus: s:Tractatus Logico-Philosophicus.]

[1] Philosophy of Logic, Willard Van Orman Quine

[2] http://plato.stanford.edu/entries/logical-atomism/

[3] http://plato.stanford.edu/entries/wittgenstein-atomism/

Chapter 3

Categorical proposition

In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some ofthe members of one category (the subject term) are included in another (the predicate term).[1] The study of argumentsusing categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with theAncient Greeks.The Ancient Greeks such as Aristotle identied four primary distinct types of categorical proposition and gave themstandard forms (now often called A, E, I, and O). If, abstractly, the subject category is named S and the predicatecategory is named P, the four standard forms are:

All S are P. (A form)

No S are P. (E form)

Some S are P. (I form)

Some S are not P. (O form)

A surprisingly large number of sentences may be translated into one of these canonical forms while retaining all ormost of the original meaning of the sentence. Greek investigations resulted in the so-called square of opposition,which codies the logical relations among the dierent forms; for example, that an A-statement is contradictory toan O-statement; that is to say, for example, if one believes All apples are red fruits, one cannot simultaneouslybelieve that Some apples are not red fruits. Thus the relationships of the square of opposition may allow immediateinference, whereby the truth or falsity of one of the forms may follow directly from the truth or falsity of a statementin another form.Modern understanding of categorical propositions (originating with the mid-19th century work of George Boole)requires one to consider if the subject category may be empty. If so, this is called the hypothetical viewpoint, inopposition to the existential viewpoint which requires the subject category to have at least one member. The existentialviewpoint is a stronger stance than the hypothetical and, when it is appropriate to take, it allows one to deduce moreresults than otherwise could be made. The hypothetical viewpoint, being the weaker view, has the eect of removingsome of the relations present in the traditional square of opposition.Arguments consisting of three categorical propositions two as premises and one as conclusion are known ascategorical syllogisms and were of paramount importance from the times of ancient Greek logicians through the Mid-dle Ages. Although formal arguments using categorical syllogisms have largely given way to the increased expressivepower of modern logic systems like the rst-order predicate calculus, they still retain practical value in addition totheir historic and pedagogical signicance.

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8 CHAPTER 3. CATEGORICAL PROPOSITION

3.1 Translating statements into standard form

3.2 Properties of categorical propositionsCategorical propositions can be categorized into four types on the basis of their quality and quantity, or theirdistribution of terms. These four types have long been named A, E, I and O. This is based on the Latin airmo (Iarm), referring to the armative propositions A and I, and nego (I deny), referring to the negative propositions Eand O.[2]

3.2.1 Quantity and quality

Quantity refers to the amount of members of the subject class that are used in the proposition. If the propositionrefers to all members of the subject class, it is universal. If the proposition does not employ all members of the subjectclass, it is particular. For instance, an I-proposition (Some S are P) is particular since it only refers to some of themembers of the subject class.Quality refers to whether the proposition arms or denies the inclusion of a subject within the class of the predicate.The two possible qualities are called armative and negative.[3] For instance, an A-proposition (All S are P) isarmative since it states that the subject is contained within the predicate. On the other hand, an O-proposition(Some S are not P) is negative since it excludes the subject from the predicate.An important consideration is the denition of the word some. In logic, some refers to one or more, which couldmean all. Therefore, the statement Some S are P does not guarantee that the statement Some S are not P is alsotrue.

3.2.2 Distributivity

The two terms (subject and predicate) in a categorical proposition may each be classied as distributed or undis-tributed. If all members of the terms class are aected by the proposition, that class is distributed; otherwise it isundistributed. Every proposition therefore has one of four possible distribution of terms.Each of the four canonical forms will be examined in turn regarding its distribution of terms. Although not developedhere, Venn diagrams are sometimes helpful when trying to understand the distribution of terms for the four forms.

A form

An A-proposition distributes the subject to the predicate, but not the reverse. Consider the following categoricalproposition: All dogs are mammals. All dogs are indeed mammals but it would be false to say all mammals aredogs. Since all dogs are included in the class of mammals, dogs is said to be distributed to mammals. Since allmammals are not necessarily dogs, mammals is undistributed to dogs.

E form

An E-proposition distributes bidirectionally between the subject and predicate. From the categorical proposition Nobeetles are mammals, we can infer that no mammals are beetles. Since all beetles are dened not to be mammals,and all mammals are dened not to be beetles, both classes are distributed.

I form

Both terms in an I-proposition are undistributed. For example, Some Americans are conservatives. Neither termcan be entirely distributed to the other. From this proposition it is not possible to say that all Americans are conser-vatives or that all conservatives are Americans.

3.3. OPERATIONS ON CATEGORICAL STATEMENTS 9

O form

In an O-proposition only the predicate is distributed. Consider the following: Some politicians are not corrupt.Since not all politicians are dened by this rule, the subject is undistributed. The predicate, though, is distributedbecause all the members of corrupt people will not match the group of people dened as some politicians. Sincethe rule applies to every member of the corrupt people group, namely, all corrupt people are not some politicians,the predicate is distributed.The distribution of the predicate in an O-proposition is often confusing due to its ambiguity. When a statementlike Some politicians are not corrupt is said to distribute the corrupt people group to some politicians, theinformation seems of little value since the group some politicians is not dened. But if, as an example, this groupof some politicians were dened to contain a single person, Albert, the relationship becomes more clear. Thestatement would then mean, of every entry listed in the corrupt people group, not one of them will be Albert: allcorrupt people are not Albert. This is a denition that applies to every member of the corrupt people group, andis therefore distributed.

Summary

In short, for the subject to be distributed, the statement must be universal (e.g., all, no). For the predicate to bedistributed, the statement must be negative (e.g., no, not).[4]

Criticism

Peter Geach and others have criticized the use of distribution to determine the validity of an argument.[5][6] It hasbeen suggested that statements of the form Some A are not B would be less problematic if stated as Not every Ais B,[7] which is perhaps a closer translation to Aristotle's original form for this type of statement.[8]

3.3 Operations on categorical statementsThere are several operations (e.g., conversion, obversion, and contraposition) that can be performed on a categoricalstatement to change it into another. The new statement may or may not be