2004/9/15fuzzy set theory chap02.ppt1 Classical Logic the forms of correct reasoning - formal logic

2004/9/15fuzzy set theory chap02.ppt2 Symbolic logic Definition –Language represented by a small set of symbols reflecting the fundamental structure of reasoning with full precision. Propositional logic Predicate logic premise conclusion

2004/9/15fuzzy set theory chap02.ppt3 Forms of reasoning

2004/9/15fuzzy set theory chap02.ppt4 The structure of propositional logic Simple proposition –A proposition that does not contain any other proposition. (atomic proposition) Affirmative proposition –A proposition that contains no negating words or prefixes. A dog has four legs and tomorrow is Sunday. Proposition pProposition q Complex proposition

2004/9/15fuzzy set theory chap02.ppt5 Logic Operations

2004/9/15fuzzy set theory chap02.ppt6 Negation p = 『 a dog has four legs 』 q = 『 Elvis is mortal 』 Truth table

2004/9/15fuzzy set theory chap02.ppt7 Conjunction

2004/9/15fuzzy set theory chap02.ppt8 Disjunction

2004/9/15fuzzy set theory chap02.ppt9 Implication antecedentconsequent

2004/9/15fuzzy set theory chap02.ppt10 Equivalence

2004/9/15fuzzy set theory chap02.ppt11 Truth values of complex propositions

2004/9/15fuzzy set theory chap02.ppt12 Table of a complex proposition

2004/9/15fuzzy set theory chap02.ppt13 Contradictions and Tautologies ContradictionsTautologies Tautology:logical implicationTautology:logical equivalence

2004/9/15fuzzy set theory chap02.ppt14 Logic functions

2004/9/15fuzzy set theory chap02.ppt15 Valid inference

2004/9/15fuzzy set theory chap02.ppt16 Invalid inference error

2004/9/15fuzzy set theory chap02.ppt17 Basic Inference forms

2004/9/15fuzzy set theory chap02.ppt18 Rules of Replacement

2004/9/15fuzzy set theory chap02.ppt19 Predicate Logic Singular Proposition General Proposition Subject termPredicate term

2004/9/15fuzzy set theory chap02.ppt20 Singular Propositions Lassie is a dog Dl Individual constant l Predicate variable D Fido is a dog Df Buster is a dog Db Ginger is a dog Dg

2004/9/15fuzzy set theory chap02.ppt21 Generalization Lassie is a dog Dl Individual constant l Predicate variable D Fido is a dog Df Buster is a dog Db Ginger is a dog Dg DxDx x: Individual variable Dx: propositional function Df: substitution instances DxDx Dl instantiation generalization

2004/9/15fuzzy set theory chap02.ppt22 General Propositions ( ∃ x)Dx : There exists at least one x, such that the x is a dog ( ∃ x)( Dx ∧ Qx) : There exists at least one thing, such that it is both a dog and a quadruped. ( ∀ x) Dx : For any x, x is a dog ( ∀ x) Dx  Qx : for any x, if x is a dog, then x is a quadruped Existential generalization ∃ x : Existential quantifier universal generalization ∀ x : universal quantifier

2004/9/15fuzzy set theory chap02.ppt23 Relations represented by predicate logic John loves Mary --- Ljm L ： relation j,m ： individual constant Everything is attracted by something --- ( ∀ x )( ∃ y)Ayx x y

2004/9/15fuzzy set theory chap02.ppt24 Quantifier Negation It is false that everything is square --- ¬ ( ∀ x )Sx There is something which is not square --- ( ∃ x) ¬ Sx Quantifier negation equivalences

2004/9/15fuzzy set theory chap02.ppt25 The Square of Opposition