Predicate Logic (PL) 1.Syntax: The language of PL Formula VS. Sentence Symbolize A-, E, I-, and O-sentences 2.Semantic: a sentence compare two sentences.

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Presentation transcript:

Predicate Logic (PL) 1.Syntax: The language of PL Formula VS. Sentence Symbolize A-, E, I-, and O-sentences 2.Semantic: a sentence compare two sentences compare a set of sentences 3.Derivation

Syntax: The language of PL SSingular term: a, b, c, …v PPredicate term: A’, B’, B’’, C’’,D…X, Y, Z,.. VVariables: w, x, y, z QQuantifier:  (universal quantifier)  (existential quantifier)  Main logical operator: &, ~, , ,  FFormula: A, B, C, Ax, Ba, Cxy, Cb, (  x)Gx

Formula VS. Sentence Formula atomicRabz, Hxy, Hat, Gxy… truth-functional ~(Rabz & Hxy), Rabz & Hxy… quantificational (  x) (Hay  (  x) Gxy), (  x) Gxy… Sentence atomicRabz, Ga, Az, Le… truth-functional Rz & Ga, ~Ba, (Le  Ca)  Ba… quantificational (  x) (Gx & Sax), (  x) Gxy, … 一個句子不能含有自由變元 (free variable)

A recursive definition of formula of PL 1.Every atomic formula of PL is a formula of PL 2.If P is a formula of PL, so is ~ P 3.If P and Q are formula of PL. so are ( P & Q ), ( P  Q ), ( P  Q ), and ( P  Q ) 4.If P is a formula of PL that contains at least one occurrence of x and no x-quantifier, then (  x) P and (  x) P are both formula of PL. 5.Nothing is a formula of PL unless it can be formed by repeated applications of clauses 1-4.

Symbolize (1) UD: all people Lxy: x likes y (1) Every likes someone. (  x) (  y) Lxy (2) Someone likes everyone. (  x) (  y)Lxy (3) Everyone is liked by someone. (  x) (  y) Lyx

Symbolize (2) UD: everything Lxy: x likes y Px: x is a person (1) Everyone likes someone. (  x) (Px  (  y) (Py & Lxy) (2) Someone likes everyone. (  x) (Px & (  y) (Px  Lxy) (3) Everyone is liked by someone. (  x) (Px  (  y) (Px & Lyx)

A-, E, I-, and O-sentences A: All S are PE: No S are P I: Some S are P I: Some S are not P (  x) (Sx  Px) =~(  x) (Sx & ~Px) (  x) (Sx  ~Px) =~(  x) (Sx & Px) (  x) (Sx & Px) =~(  x) (Sx  ~Px) (  x) (Sx & ~Px) =~(  x) (Sx  Px) contradictory