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Predicate Logic … predicate logic … a predicate (relation) – is a function that maps its arguments to the truth values 0 or 1 known example: less then (symbol <), for arguments 5 and 8 we have 5<8 is true, and 8<5 is false they can be written as infix like <, or letter(s), for lecture 14: predicate logic example R(x,y) or mother(x,y) of#45# ece#627,#winter#‘13# 2# of#45# predicate logic predicate logic … … in propositional logic, the preposition Every peach is fuzzy predicate logic is not a replacement for propositional is represented by a single symbol p, logic but an extension or refinement of it in predicate logic, the statement is shown in finer detail - with universal quantifier ( ∀ x)(peach(x) ⊃ fuzzy(x)) ece#627,#winter#‘13# 3# of#45# ece#627,#winter#‘13#€ € 4# of#45# predicate logic predicate logic … … - with existential quantifier - with both quantifiers ~( ∃ x)(peach(x) ∧ ~fuzzy(x)) ( ∀ x)( ∃ y)(integer(x) ⊃ (prime(y) ∧ x<y) € € € € € € ece#627,#winter#‘13# 5# of#45# ece#627,#winter#‘13# 6# of#45# predicate logic predicate logic … … order of quantifiers while … ( ∀ x)( ∃ y) ( man(x) ∧ dept(x,account) ⊃ ( ∃ y)( ∀ x) ( man(x) ∧ dept(x,account) ⊃ (woman(y) ∧ hometown(y,Boston) ∧ married(x,y)) ) (woman(y) ∧ hometown(y,Boston) ∧ married(x,y)) ) € € € € € € € € € € € € ece#627,#winter#‘13# 7# of#45# ece#627,#winter#‘13# 8# of#45# predicate logic predicate logic formation rules formation rules vocabulary contains symbols for constants and a term is either a constant (2 or a, b, c, …), a variables, parenthesis, Boolean operators, and variable (x, y or x0, x1, x2, …), or a function or an symbols for quantifiers, functions and predicates operator symbol applied to its arguments, each of which is itself a term all of them are combined according to three rules for example: f(x), 2+2 ece#627,#winter#‘13# 9# of#45# ece#627,#winter#‘13# 10# of#45# predicate logic predicate logic formation rules formation rules an atom is either a single letter (p) representing a formula is either an atom, a formula preceded by preposition or a predicate symbol (P, Q, R, …) applied ~, any two formulas A and B together with any two- to its arguments, each of which is itself a term place Boolean operator op in the combination (A op B), or any formula A and any variable x in either of for example: P(f(x), 2+2), Q(7) the combinations ( ∃ x)A or ( ∀ x)A € € ece#627,#winter#‘13# 11# of#45# ece#627,#winter#‘13# 12# of#45# predicate logic predicate logic formulas - examples formulas - examples (P(f(x),2+2) ⊃ Q(7)) John is tall ~(P(f(x),2+2) ⊃ Q(7)) T(j) ( ∀ y)~(P(f(x),2+2) ⊃ Q(7)) John is taller than Bill TR(j,b) ( ∃ x)( ∀ y)~(P(f(x),2+2)€ ⊃ Q(7)) Everybody sleeps € ∀ x (S(x)) the occurrence€ of x in f(x) is€ bound by the quantifier ( ∃ x), Somebody likes David the quantifier€ € ( ∀ y) has not effect€ on the formula, y does not occur as an argument of any function/predicate ∃ x [L(x,d)] € € ece#627,#winter#‘13#€ 13# of#45# ece#627,#winter#‘13# 14# of#45# € predicate logic predicate logic formulas - examples rules of inference There are happy people ∃ x H(x) rule of inference - Some books are interesting is to preserve truth, if we start with formulas that are ∃x [B(x) ∧ I(x)] true, the result of performing a rule of inference on Some books are€ interesting and some are easy to read them must also be true ∃ x [B(x) ∧ I(x)] ∧ ∃ x [B(x) ∧ E(x)] No books €are good € ∀x [B(x)→ ¬ G(x)] € € € € € ece#627,#winter#‘13# 15# of#45# ece#627,#winter#‘13# 16# of#45# € € predicate logic predicate logic rules of inference rules of inference issue of equivalence: the rule for relating the quantifiers ( ∀ x)(peach(x) ⊃ fuzzy(x)) ~( ∃ x)(peach(x) ∧ ~fuzzy(x)) ( ∃ x)A is equivalent to ~( ∀ x)~A ( ∀ x)A is equivalent to ~( ∃ x)~A if these€ formulas were€ represented by p and q, there would€ be no way to €prove p ≡ q, but the rules of € € predicate logic can show the equivalence € € € ece#627,#winter#‘13# 17# of#45# ece#627,#winter#‘13# 18# of#45# predicate logic predicate logic rules of inference rules of inference so, the first rule: and ~( ∃ x)~(peach(x) ⊃ fuzzy(x)) ~( ∃ x)(peach(x) ∧ ~fuzzy(x)) knowing, that ~(p ∧ ~q), then this shows 1st formula implies 2nd, if we use the inverse € € of rules€ to show the €2nd implies 1st – then both formulas are equivalent ~( ∃ x)~~(peach(x) ∧ ~fuzzy(x)) € € € ece#627,#winter#‘13# 19# of#45# ece#627,#winter#‘13# 20# of#45# predicate logic predicate logic rules of inference rules of inference Modus ponens: Hypothetical syllogism: from p and p ⊃ q, derive q from p ⊃ q and q ⊃ r, derive p ⊃ r Modus tollens: Disjunctive syllogism: from€ ~q and p ⊃ q, derive ~p € from p ∨ € q and ~p, derive€ p € € ece#627,#winter#‘13# 21# of#45# ece#627,#winter#‘13# 22# of#45# predicate logic predicate logic rules of inference rules of inference Conjunction: Subtraction: from p and q, derive p ∧ q from p ∧ q, derive p Addition: (extra conjuncts may be thrown away) from p, derive€ p ∨ q € (any formula may be added to a disjunction) € ece#627,#winter#‘13# 23# of#45# ece#627,#winter#‘13# 24# of#45# predicate logic predicate logic equivalences equivalences Idempotency: Associativity: p ∧ p is equivalent to p, and p ∧ (q ∧ r) is equivalent to (p ∧ q) ∧ r, and p ∨ p is equivalent to p p ∨ (q ∨ r) is equivalent to (p ∨ q) ∨ r Commutativity:€ Distributivity:€ € € € € p ∧ q is equivalent to q ∧ p, and € € p ∧ (q ∨ r) is equivalent€ to€ (p ∧ q) ∨ (p ∧ r), and p ∨ q is equivalent to q ∨ p p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r) € € € € € € € ece#627,#winter#‘13# 25# of#45# ece#627,#winter#‘13# 26# of#45# € € € € € € € predicate logic predicate logic equivalences equivalences Absorption: De Morgans laws: p ∧ (p ∨ q) is equivalent to p, and ~(p ∧ q) is equivalent to ~p ∨ ~q, and p ∨ (p ∧ q) is equivalent to p ~(p ∨ q) is equivalent to ~p ∧ ~q Double€ negation:€ € € € € p is equivalent to ~~p € € ece#627,#winter#‘13# 27# of#45# ece#627,#winter#‘13# 28# of#45# predicate logic predicate logic rules for quantifiers rules for quantifiers if A is an atom, then all occurrences of a variable in A are said to be free if a formula C was derived form a formula A by preceding A with either ( ∀ x) or ( ∃ x), then all free if a formula C was derived from formulas A and B by occurrences of x in A are said to be bound in C, combining them with Boolean operators, then all occurrences of variables that are free in A and B are all free occurrences€ of other€ variables in A remain free also free in C in C ece#627,#winter#‘13# 29# of#45# ece#627,#winter#‘13# 30# of#45# predicate logic predicate logic rules for quantifiers rules for quantifiers rules for dealing with variables depend on which let Φ(x) be a formula with one or more free occurrences are free and bound and which variables occurrences of a variable x, then Φ(t) is the result of must be renamed to avoid name clashes with other substituting every free occurrence of x in Φ with t variables ece#627,#winter#‘13# 31# of#45# ece#627,#winter#‘13# 32# of#45# predicate logic predicate logic rules of quantifier negation rules of quantifier (in)dependence ( ∃ x)A ⇔ ~( ∀ x)~A (∀ x)( ∀ y)A(x,y) ⇔ ( ∀ y)( ∀ x)A(x,y) (∀ x)A ⇔ ~( ∃ x)~A ( ∃ x)( ∃ y)A(x,y) ⇔ ( ∃ y)( ∃ x)A(x,y) ~( ∃ x)A ⇔ ( ∀ x)~A ( ∃ x)(∀ y)A(x,y) ⇒ ( ∀ y)( ∃ x)A(x,y) € € ~(∀ €x)A ⇔ ( ∃ x)~A € € € € € € € € € € € € € € € € € € € € € € € € ece#627,#winter#‘13# 33# of#45# ece#627,#winter#‘13# 34# of#45# predicate logic predicate logic rules of quantifier movement rules of quantifier movement example: A→ (∀ x)(B(x)) ⇔ (∀ x)(A→ B(x)) A→ ( ∃ x)(B(x)) ⇔ ( ∃ x)(A→ B(x)) ( ∃ x)(P(x)) → ( ∀ y)(Q(y)) ⇔ ( ∀ y)[( ∃ x)(P(x)) → (Q(y))] ( ∀ x)(B(x))→ A ⇔ ( ∃ x)(B(x)→ A) ⇔ ( ∀ y)( ∀ x)[(P(x)) → (Q(y))] € € ( ∃ x)(B(x))€ €→ A ⇔ € (∀ x)(B(x)→ A) € € € € € € € € € € € € € € € € € € € € € € € € € € ece#627,#winter#‘13# 35# of#45# ece#627,#winter#‘13# 36# of#45# predicate logic predicate logic rules for quantifiers: permissible substitutions rules for quantifiers: permissible substitutions universal instantiation: dropping quantifiers: from ( ∀ x)Φ(x), derive Φ(c) where c is any constant if the variable x does not occur free in Φ, then from ( ∀ x)Φ(x) derive Φ, and from ( ∃ x)Φ(x) derive existential generalization: € from Φ(c), where c is any constant, derive ( ∃ x)Φ(x) adding quantifiers: provided that every occurrence of x in Φ(x) is free € from Φ derive ( ∀ x)Φ €or derive ( ∃ x)Φ, where x is any variable € € € ece#627,#winter#‘13# 37# of#45# ece#627,#winter#‘13# 38# of#45# predicate logic typed predicate logic rules for quantifiers: permissible substitutions … substituting equal for equals: this form is a purely syntactic extension of untyped for any terms s and t where s=t, derive Φ(t) from logic – its semantic identical to untyped logic, as well Φ(s), provided that all free occurrences of variables in as ever theorem and proof t remain free in Φ(t) the only difference – addition of a type label after the quantifier – x:N - (label is a monadic predicate - n(x)) ece#627,#winter#‘13# 39# of#45# ece#627,#winter#‘13# 40# of#45# typed predicate logic typed predicate logic … … for knowledge representation typed logic has the Universal: ( ∀ x:N)Φ(x) ≡ ( ∀ x)(n(x) ⊃ Φ(x)) advantage of being more concise and readable it can support rules of inference based on inheritance Existential: ( ∃ x:N)Φ(x) ≡ ( ∃ x)(n(x) ∧ Φ(x)) (they do not make logic more expressive, they € € € € shorten some proofs) € € € € ece#627,#winter#‘13# 41# of#45# ece#627,#winter#‘13# 42# of#45# typed predicate logic typed predicate logic … … with a string of multiple quantifiers of the same kind for untyped … and with the same type label, it is permissible to factor out the common quantifier and type label ( ∀ x)(number(x) ⊃ ( ∀ y)(number(y) ⊃ ( ∀ x,y,x:Number) ((x < y ∧ y < z) ⊃ x < z) ( ∀ z)(number(z) ⊃ € ((x€ < y ∧ y < z) ⊃ x < z) ))) € € € € € € € € € ece#627,#winter#‘13# 43# of#45# ece#627,#winter#‘13# 44# of#45# typed predicate logic … Untyped formula as a special case of a typed one Universal: ( ∀ x:T)Φ(x) ≡ ( ∀ x)(T(x) ⊃ Φ(x)) ≡ ( ∀ x) Φ(x) Existential:€ ( ∃ x:T)€ €Φ (x) ≡ ( ∃€ x)(T(x) ∧ Φ(x)) € € ≡ ( ∃ x) Φ(x) € € € € ece#627,#winter#‘13# 45# of#45# € € .
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