Logic symbols 3Wikipedia

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

i

ii CONTENTS

4 Conditioned disjunction 114.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Contingency (philosophy) 125.1 Contingency and relativism in rhetoric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Contradiction 146.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Contradiction in formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.2.1 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.2 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency . . . . . . 16

6.3 Contradictions and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3.1 Pragmatic contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3.2 Dialectical materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.4 Contradiction outside formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Converse implication 207.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Converse nonimplication 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CONTENTS iii

8.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.5.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 CornishFisher expansion 259.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10 Exclusive or 2710.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.2 Equivalencies, elimination, and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3 Relation to modern algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.4 Exclusive or in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.5 Alternative symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.7 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.7.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.8 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11 Expression (mathematics) 3411.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.3 Syntax versus semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.3.3 Formal languages and lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

11.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

12 False (logic) 3712.1 In classical logic and Boolean logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.2 False, negation and contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iv CONTENTS

13 Ground expression 3913.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.2.1 Ground terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.2 Ground atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.2.3 Ground formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

13.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

14 Material equivalence 4114.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

14.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15 Indicative conditional 4415.1 Distinctions between the material conditional and the indicative conditional . . . . . . . . . . . . . 4415.2 Psychology and indicative conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

16 Logical biconditional 4616.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

16.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.1.2 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

16.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16.3.1 Biconditional introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.3.2 Biconditional elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16.4 Colloquial usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

17 Logical conjunction 5117.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

CONTENTS v

17.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 Logical connective 5718.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

18.1.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.1.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18.2 Common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.2.1 List of common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.2.2 History of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.4 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.5 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

19 Logical disjunction 6319.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

19.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.4 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5 Applications in computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

19.5.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5.2 Logical operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.5.3 Constructive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

19.6 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vi CONTENTS

19.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

20 Logical equality 6920.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.2 Alternative descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

21 Logical NOR 7221.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

21.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.3 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

22 Material conditional 7422.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

22.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7522.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

22.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7622.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 7622.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

22.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7722.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7722.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7722.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

23 Material equivalence 7923.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

23.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

23.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

24 Material nonimplication 8224.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

CONTENTS vii

24.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

25 Modal operator 8425.1 Modality interpreted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

25.1.1 Alethic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.2 Deontic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.3 Axiological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.4 Epistemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.5 Doxastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

26 Negation 8526.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

26.3.1 Double negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.3.2 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.3.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.3.4 Self dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

26.4 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.5 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.6 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27 Open sentence 8927.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

28 Polish notation 9128.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.2 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.3 Order of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

viii CONTENTS

28.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.5 Polish notation for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

29 Probabilistic proposition 95

30 Proposition 9630.1 Historical usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

30.1.1 By Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.1.2 By the logical positivists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9630.1.3 By Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

30.2 Relation to the mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9730.3 Treatment in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9730.4 Objections to propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9730.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9830.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9830.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

31 Propositional formula 9931.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

31.1.1 Relationship between propositional and predicate formulas . . . . . . . . . . . . . . . . . 10031.1.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

31.2 An algebra of propositions, the propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 10031.2.1 Usefulness of propositional formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10131.2.2 Propositional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10131.2.3 Truth-value assignments, formula evaluations . . . . . . . . . . . . . . . . . . . . . . . . 101

31.3 Propositional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10231.3.1 Connectives of rhetoric, philosophy and mathematics . . . . . . . . . . . . . . . . . . . . 10231.3.2 Engineering connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10231.3.3 CASE connective: IF ... THEN ... ELSE ... . . . . . . . . . . . . . . . . . . . . . . . . . 10231.3.4 IDENTITY and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

31.4 More complex formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.4.2 Axiom and denition schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.4.3 Substitution versus replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

31.5 Inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.6 Parsing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

31.6.1 Connective seniority (symbol rank) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10631.6.2 Commutative and associative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10731.6.3 Distributive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

CONTENTS ix

31.6.4 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10731.6.5 Laws of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.6.6 Laws of evaluation: Identity, nullity, and complement . . . . . . . . . . . . . . . . . . . . 10831.6.7 Double negative (Involution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

31.7 Well-formed formulas (ws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10831.7.1 Ws versus valid formulas in inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31.8 Reduced sets of connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.8.1 The stroke (NAND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.8.2 IF ... THEN ... ELSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

31.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.9.1 Reduction to normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.9.2 Reduction by use of the map method (Veitch, Karnaugh) . . . . . . . . . . . . . . . . . . 112

31.10Impredicative propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11331.11Propositional formula with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

31.11.1 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11431.11.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

31.12Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11531.13Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11731.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

32 Propositional function 12532.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

33 Propositional representation 12633.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

34 Rule of inference 12934.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13034.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 13034.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13134.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13134.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

35 Rule of replacement 13335.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

36 Sentence (logic) 13436.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

37 Sequent 136

x CONTENTS

37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.1.1 The form and semantics of sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.1.2 Syntax details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.1.5 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

37.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.2.1 History of the meaning of sequent assertions . . . . . . . . . . . . . . . . . . . . . . . . . 13837.2.2 Intuitive meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

37.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.4 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14137.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

38 Sheer stroke 14238.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.4 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.5 Formal system based on the Sheer stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

38.5.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14338.5.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.5.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438.5.4 Simplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

38.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14538.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14638.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

39 Strict conditional 14739.1 Avoiding paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14739.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14739.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14839.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14839.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

40 T-schema 15040.1 The inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15040.2 Natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

CONTENTS xi

40.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

41 Tautology (logic) 15241.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15241.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15341.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15341.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 15541.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15541.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

41.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15641.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

41.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15641.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

42 Theorem 15742.1 Informal account of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15742.2 Provability and theoremhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.3 Relation with scientic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.5 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.6 Lore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16042.7 Theorems in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

42.7.1 Syntax and semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.7.2 Derivation of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142.7.3 Interpretation of a formal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.7.4 Theorems and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

42.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

43 Theory (mathematical logic) 16843.1 Theories expressed in formal language generally . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

43.1.1 Subtheories and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.2 Deductive theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.3 Consistency and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.1.4 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

xii CONTENTS

43.1.5 Theories associated with a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.2 First-order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

43.2.1 Derivation in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16943.2.2 Syntactic consequence in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . 17043.2.3 Interpretation of a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.2.4 First order theories with identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.2.5 Topics related to rst order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

43.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17043.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17143.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17143.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

44 Truth value 17244.1 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2 Intuitionistic and constructive logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.3 Multi-valued logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.4 Algebraic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.5 In other theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

45 Truth-bearer 17545.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17545.2 Sentences in natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17645.3 Sentences in languages of classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17845.4 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17845.5 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17945.6 Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

46 Unity of the proposition 18346.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.2 Russell, Frege, Wittgenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18446.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18446.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

47 Universal quantication 18547.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

47.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

CONTENTS xiii

47.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18847.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

47.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18847.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

48 Well-formed formula 19048.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19148.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19148.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19248.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19248.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19448.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19448.12Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 195

48.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19548.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20148.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

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.

1

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)

3

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.

7

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 equivalent to the original. [In the followingtables that illustrate such operations, rows with equivalent statement shall be marked in green, while those withinequivalent statements shall be marked in red.]Some operations require the notion of the class complement. This refers to every element under consideration whichis not an element of the class. Class complements are very similar to set complements. The class complement of aset P will be called non-P.

3.3.1 Conversion

Main article: Converse (logic)

The simplest operation is conversion where the subject and predicate terms are interchanged.From a statement in E or I form, it is valid to conclude its converse. This is not the case for the A and O forms.

3.3.2 Obversion

Main article: Obversion

Obversion changes the quality (that is the armativity or negativity) of the statement and the predicate term.[9] Forexample, a universal armative statement would become a universal negative statement.Categorical statements are logically equivalent to their obverse. As such, a Venn diagram illustrating any one of theforms would be identical to the Venn diagram illustrating its obverse.

10 CHAPTER 3. CATEGORICAL PROPOSITION

3.3.3 ContrapositionMain article: Contraposition

3.4 See also Square of opposition term logic

3.5 Notes[1] Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martins Press. p. 143. ISBN 0-312-

02353-7. OCLC 21216829. A categorical statement is an assertion or a denial that all or some members of the subjectclass are included in the predicate class.

[2] Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martins Press. p. 144. ISBN 0-312-02353-7. OCLC 21216829. During the Middle Ages, logicians gave the four categorical forms the special names of A,E, I, and O. These four letters came from the rst two vowels in the Latin word 'airmo' ('I arm') and the vowels in theLatin 'nego' ('I deny').

[3] Copi, Irving M.; Cohen, Carl (2002). Introduction to Logic (11th ed.). Upper Saddle River, NJ: Prentice-Hall. p. 185.ISBN 0-13-033735-8. Every standard-form categorical proposition is said to have a quality, either armative or negative.

[4] Damer 2008, p. 82.

[5] Lagerlund, Henrik (2010-01-21). Medieval Theories of the Syllogism. Stanford Encyclopedia of Philosophy. Retrieved2010-12-10.

[6] Murphree, Wallace A. (Summer 1994). The Irrelevance of Distribution for the Syllogism. Notre Dame Journal of FormalLogic 35 (3).

[7] Geach 1980, pp. 6264.

[8] Parsons, Terence (2006-10-01). The Traditional Square of Opposition. Stanford Encyclopedia of Philosophy. Retrieved2010-12-10.

[9] Hausman, Alan; Kahane, Howard; Tidman, Paul (2010). Logic and Philosophy: A Modern Introduction (11th ed.). Aus-tralia: Thomson Wadsworth/Cengage learning. p. 326. ISBN 9780495601586. Retrieved 26 February 2013. In theprocess of obversion, we change the quality of a proposition (from armative to negative or from negative to armative),and then replace its predicate with the negation or complement of the predicate.

3.6 References Copi, Irving M.; Cohen, Carl (2009). Introduction to Logic. Prentice Hall. ISBN 978-0-13-136419-6. Damer, T. Edward (2008). Attacking Faulty Reasoning. Cengage Learning. ISBN 978-0-495-09506-4. Geach, Peter (1980). Logic Matters. University of California Press. ISBN 978-0-520-03847-9. Baum, Robert (1989). Logic. Holt, Rinehart and Winston, Inc. ISBN 0-03-014078-1.

3.7 External links ChangingMinds.org: Categorical propositions Catlogic: An open source computer script written in Ruby to construct, investigate, and compute categoricalpropositions and syllogisms

Chapter 4

Conditioned disjunction

In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective in-troduced by Church.[1] Given operands p, q, and r, which represent truth-valued propositions, the meaning of theconditioned disjunction [p, q, r] is given by:

[p; q; r] $ (q ! p) ^ (:q ! r)

In words, [p, q, r] is equivalent to: if q then p, else r", or "p or r, according as q or not q". This may also be statedas "q implies p and, not q implies r". So, for any values of p, q, and r, the value of [p, q, r] is the value of p when qis true, and is the value of r otherwise.The conditioned disjunction is also equivalent to:

(q ^ p) _ (:q ^ r)

and has the same truth table as the ternary (?:) operator in many programming languages.In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally completefor classical logic.[2] Its truth table is the following:There are other truth-functionally complete ternary connectives.

4.1 References[1] Church, Alonzo (1956). Introduction to Mathematical Logic. Princeton University Press.

[2] Wesselkamper, T., A sole sucient operator, Notre Dame Journal of Formal Logic, Vol. XVI, No. 1 (1975), pp. 86-88.

11

Chapter 5

Contingency (philosophy)

Contingency and contingent redirect here. For other uses, see Contingency (disambiguation).

In philosophy and logic, contingency is the status of propositions that are neither true under every possible valuation(i.e. tautologies) nor false under every possible valuation (i.e. contradictions). A contingent proposition is neithernecessarily true nor necessarily false. Propositions that are contingent may be so because they contain logical connec-tives which, along with the truth value of any of its atomic parts, determine the truth value of the proposition. Thisis to say that the truth value of the proposition is contingent upon the truth values of the sentences which comprise it.Contingent propositions depend on the facts, whereas analytic propositions are true without regard to any facts aboutwhich they speak.Along with contingent propositions, there are at least three other classes of propositions, some of which overlap:

Tautological propositions, which must be true, no matter what the circumstances are or could be (example:It is the case that the sky is blue or it is not the case that the sky is blue.).

Contradictions which must necessarily be untrue, no matter what the circumstances are or could be (example:Its raining and its not raining.).

Possible propositions, which are true or could have been true given certain circumstances (examples: x + y =4; There are only three planets; There are more than three planets). All necessarily true propositions, and allcontingent propositions, are also possible propositions.

5.1 Contingency and relativism in rhetoricAttempts in the past by philosophers and rhetoricians to allocate to rhetoric its own realm have ended with attemptingto contain rhetoric within the domain of contingent and relative matters. Aristotle explained in Rhetoric, The duty ofrhetoric is to deal with such matters as we deliberate upon without arts or systems to guide us" [1] Aristotle stressesthe contingent because no one deliberates on the necessary or impossible. He believed that the unavoidable andpotentially unmanageable presence of multiple possibilities or the complex nature of decisions creates and invitesrhetoric.[1] Aristotles view challenges the view of Plato, who said that rhetoric had no subject matter except for deceit,and gives rhetoric its position at the pinnacle of political debate.Contemporary scholars argue that if rhetoric is merely about the contingent, it automatically excludes that which iseither necessary or impossible. The necessary is that which either must be done or will inevitably be done. Theimpossible is that which will never be done; therefore, it will not be deliberated over. For example, the UnitedStates Congress will not convene tomorrow to discuss something necessary, such as whether or not to hold elections,or something impossible, such as outlawing death. Congress convenes to discuss problems, dierent solutions to thoseproblems, and the consequences of each solution.This again raises the question of contingency because that which is deemed necessary or impossible depends almostentirely on time and perspective. In United States history, there was a time when even a congressman who opposedslavery would conclude that its retraction would be impossible. The same held true for those who favored womenssurage. Today in the United States, slavery has been abolished and women have the right to vote. In this way,

12

5.2. REFERENCES 13

although rhetoric viewed across time is entirely contingent and includes a broader denition, rhetoric taken moment-by-moment is much more narrow and excludes both the necessary and the impossible. When faced with decisions,people will choose one option at the exclusion of the others.[2] This inevitably produces unforeseen consequences.Because of these consequences, decision makers must deliberate and choose. Another problem arises when one askswhere this knowledge of what issues are necessary and impossible originates and how the knowledge can beapplied to others.Rhetorician Robert L. Scott answers this problem by asserting that while rhetoric is indeed contingent and relative, it isalso epistemic.[3] Thus, for Scott, what should be debated is a matter of rhetoric, as individuals make meaning throughlanguage and determine what constitutes truth, and therefore, what is beyond question and debate. Theorist LloydBitzer makes ve assumptions about rhetoric in his book Rhetoric, Philosophy, and Literature: An Exploration.[4]

1. Rhetoric is a method for inquiring into and communicating about the contingent.2. This inquiry does not yield certain knowledge, but only opinion.3. The proper mode of working in this realm is deliberation that relies on reasonable judgment.4. This deliberation and decision making is audience centered.5. This engagement with the audience is constrained by time.The study of contingency and relativism as it pertains to rhetoric draws from poststructuralist and postfoundationalisttheory. Richard Rorty and Stanley Fish are leading theorists in this area of study at the intersection of rhetoric andcontingency .

5.2 References[1] Aristotle. Rhetoric. Trans. W. Rhys Roberts. New York: Random House, 1954.

[2] Gaonkar, Dilip Parameshwar. Contingency and Probability. Encyclopedia of Rhetoric. Ed. Thomas O. Sloane. NewYork: Oxford UP, 2001. 156.

[3] Scott, Robert L. On Viewing Rhetoric As Epistemic. Central States Speech Journal 18 (1967), p. 9.

[4] Bitzer, Lloyd F. Rhetoric and Public Knowledge. Rhetoric, Philosophy and Literature: An Exploration. Ed. D.M. Burks,p.70. West Lafayette, IN, 1978.

5.3 Further reading

Chapter 6

Contradiction

For other uses, see Contradiction (disambiguation).In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurswhen the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions ofeach other. Illustrating a general tendency in applied logic, Aristotles law of noncontradiction states that One cannotsay of something that it is and that it is not in the same respect and at the same time.By extension, outside of classical logic, one can speak of contradictions between actions when one presumes that theirmotives contradict each other.

6.1 History

By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In theensuing dialogue Dionysodorus denies the existence of contradiction, all the while that Socrates is contradictinghim:

". . . I in my astonishment said: What do you mean Dionysodorus? I have often heard, and have beenamazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras andothers before them, and which to me appears to be quite wonderful, and suicidal as well as destructive,and I think that I am most likely to hear the truth about it from you. The dictum is that there is no suchthing as a falsehood; a man must either say what is true or say nothing. Is not that your position?"

Indeed, Dionysodorus agrees that there is no such thing as false opinion . . . there is no such thing as ignorance anddemands of Socrates to Refute me. Socrates responds But how can I refute you, if, as you say, to tell a falsehoodis impossible?".[1]

6.2 Contradiction in formal logic

Note: The symbol ? (falsum) represents an arbitrary contradiction, with the dual tee symbol > usedto denote an arbitrary tautology. Contradiction is sometimes symbolized by Opq", and tautology byVpq". The turnstile symbol, ` is often read as yields or proves.

In classical logic, particularly in propositional and rst-order logic, a proposition ' is a contradiction if and only if' ` ? . Since for contradictory ' it is true that ` '! for all (because? ! ), one may prove any propositionfrom a set of axioms which contains contradictions. This is called the "principle of explosion" or ex falso quodlibet(from falsity, whatever you like).In a complete logic, a formula is contradictory if and only if it is unsatisable.

14

6.2. CONTRADICTION IN FORMAL LOGIC 15

This diagram shows the contradictory relationships between categorical propositions in the square of opposition of Aristotelian logic.

6.2.1 Proof by contradiction

Main article: Proof by contradiction

For a proposition ' it is true that ` ' , i. e. that ' is a tautology, i. e. that it is always true, if and only if :' ` ?, i. e. if the negation of ' is a contradiction. Therefore, a proof that :' ` ? also proves that ' is true. The use ofthis fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This appliesonly in a logic using the excluded middle A _ :A as an axiom.

16 CHAPTER 6. CONTRADICTION

6.2.2 Symbolic representation

In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may beused to represent a contradiction include , Opq, )( , , , and ; in any symbolism, a contradiction may besubstituted for the truth value "false, as symbolized, for instance, by 0. It is not uncommon to see Q.E.D. or somevariant immediately after a contradiction symbol; this occurs in a proof by contradiction, to indicate that the originalassumption was false and that its negation must therefore be true.

6.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency

A consistency proof requires (i) an axiomatic system (ii) a demonstration that it is not the case that both the formula pand its negation ~p can derived in the system. But by whatever method one goes about it, all consistency proofs wouldseem to necessitate the primitive notion of contradiction; moreover, it seems as if this notion would simultaneouslyhave to be outside the formal system in the denition of tautology.When Emil Post in his 1921 Introduction to a general theory of elementary propositions extended his proof of theconsistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM) he observed thatwith respect to a generalized set of postulates (i.e. axioms) he would no longer be able to automatically invoke thenotion of contradiction such a notion might not be contained in the postulates:

The prime requisite of a set of postulates is that it be consistent. Since the ordinary notion of consistencyinvolves that of contradiction, which again involves negation, and since this function does not appear ingeneral as a primitive in [the generalized set of postulates] a new denition must be given.[2]

Posts solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Con-sistency oered by Ernest Nagel and James R. Newman in their 1958 Gdel's Proof. They too observe a problemwith respect to the notion of contradiction with its usual truth values of truth and falsity. They observe that:

The property of being a tautology has been dened in notions of truth and falsity. Yet these notionsobviously involve a reference to something outside the formula calculus. Therefore, the procedure men-tioned in the text in eect oers an interpretation of the calculus, by supplying a model for the system.This being so, the authors have not done what they promised, namely, 'to dene a property of for-mulas in terms of purely structural features of the formulas themselves. [Indeed] . . . proofs ofconsistency which are based on models, and which argue from the truth of axioms to their consistency,merely shift the problem.[3]

Given some primitive formulas such as PMs primitives S1 V S2 [inclusive OR], ~S (negation) one is forced todene the axioms in terms of these primitive notions. In a thorough manner Post demonstrates in PM, and denes(as do Nagel and Newman, see below), that the property of tautologous as yet to be dened is inherited": ifone begins with a set of tautologous axioms (postulates) and a deduction system that contains substitution and modusponens then a consistent system will yield only tautologous formulas.So what will be the denition of tautologous?Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2 into which fall (the outcome of)the axioms when their variables e.g. S1 and S2 are assigned from these classes. This also applies to the primitiveformulas. For example: A formula having the form S1 V S2 is placed into class K2 if both S1 and S2 are in K2;otherwise it is placed in K1", and A formula having the form ~S is placed in K2, if S is in K1; otherwise it is placedin K1".[4]

Nagel and Newman can now dene the notion of tautologous: a formula is a tautology if, and only if, it falls in theclass K1 no matter in which of the two classes its elements are placed.[5] Now the property of being tautologousis described without reference to a model or an interpretation.

For example, given a formula such as ~S1 V S2 and an assignment of K1 to S1 and K2 to S2 one canevaluate the formula and place its outcome in one or the other of the classes. The assignment of K1 to~S1 places ~S1 in K2, and now we can see that our assignment causes the formula to fall into class K2.Thus by denition our formula is not a tautology.

6.3. CONTRADICTIONS AND PHILOSOPHY 17

Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulasderived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from eitherclass K1 or K2, the deduction violates the inheritance characteristic of tautology, i.e. the derivation must yield an(evaluation of a formula) that will fall into class K1. From this, Post was able to derive the following denition ofinconsistency without the use of the notion of contradiction:

Denition. A system will be said to be inconsistent if it yields the assertion of the unmodied variable p[S in the Newman and Nagel examples].

In other words, the notion of contradiction can be dispensed when constructing a proof of consistency; what replacesit is the notion of mutually exclusive and exhaustive classes. More interestingly, an axiomatic system need notinclude the notion of contradiction.

6.3 Contradictions and philosophyAdherents of the epistemological theory of coherentism typically claim that as a necessary condition of the justi-cation of a belief, that belief must form a part of a logically non-contradictory (consistent) system of beliefs. Somedialetheists, including Graham Priest, have argued that coherence may not require consistency.[6]

6.3.1 Pragmatic contradictionsA pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. Aninconsistency arises, in this case, because the act of utterance, rather than the content of what is said, underminesits conclusion.[7] For examples, arguably, Nietzsche's statement that one should not obey others, or Moores paradox.Within the analytic tradition, these are seen as self-refuting statements and performative contradictions. Other tra-ditions may read them more like zen koans, in which the author purposes makes a contradiction using the traditionalmeaning, but then implies a new meaning of the word which does not contradict the statement.

6.3.2 Dialectical materialismIn dialectical materialism, contradiction, as derived by Karl Marx from Hegelianism, usually refers to an oppositioninherently existing within one realm, one unied force or object. This contradiction, as opposed to metaphysicalthinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not can-celling each other out, but actually dening each others existence. According to Marxist theory, such a contradictioncan be found, for example, in the fact that:

(a) enormous wealth and productive powers coexist alongside:(b) extreme poverty and misery;(c) the existence of (a) being