Predicate Logic Lecture 7

Limitations of Prepositional Logic The prepositional calculus has its limitations that you cannot deal properly with general statements of the form “All men are mortal” You can not derive from the conjunction of this and “Socrates is a man” that “Socrates is mortal”

Limitations If All men are mortal = P Socrates is a Man = Q Socrates is mortal = R Then (P & Q)  R is not valid To do this, you need to analyze propositions into predicates and arguments, and deal explicitly with quantification

Expressions of the Form For all x, x has the property F For some x, x has the property F And so on Predicate Logic provides a formalism for performing this analysis of prepositions and additional methods for reasoning with quantified expressions

Predicate Logic The term predicate logic derives from the fact that we analyze prepositions into predicate-argument compositions Thus instead of representing the preposition that an object, a has some property, F by the prepositional constant P; we typically represent it by placing the predicate in front of the thing that it is predicated of; Fa

Representation A similar relationship applies to the expression of a relationship R between n objects R a1,…,an A predicate is a function that returns either the value true or false for example Is_a_bird(x) Is a function of the variable x, which is true if x is a bird or false otherwise

Power of Predicate Logic Relations can take any number of arguments for example Lives(Ali, Islamabad) represents Ali lives in Islamabad The real power of predicate logic becomes apparent when we wish to represent more general statements such as Bashir gave Marium Everything The translation into predicate logic is (all x (gave Bashir Marium X)) I.e For any thing, lets call it x, Bashir gave it to Marium

Quantifiers The variable x is understood to range over some domain of appropriate objects The fact that it is governed by the operator “all” signifies that it is universally quantified The embedded expression, gave(Bashir Marium x) is said to be within the scope of the quantifier “all”

Quantifiers Contrast this with another kind of quantifier as in Bashir gave Marium something Translation into predicate logic is (some x(gave Bashir Marium x)) I.e there is something, let us call it x such that Bashir gave it to Marium Here, the variable x still ranges over a domain of appropriate objects but the fact that it is governed by the operator “some” signifies that it is “existentially quantified”

Example Consider the problem of representing the hopefully obvious fact that Every father is Male This can be done using prepositional Logic without writing out a preposition for each father asserting maleness What makes it difficult to convert the content of this sentence into prepositional logic is that it contains the idea that for each person it is possible to decide if they are a father, and therefore that they are male

Example This idea of a yes/no decision can be captured as a predicate function or simply a predicate A predicate is a function that returns either the value true or value false e.g Is_a_Father(x) Is a function of the variable x, which is true if x is a father and false otherwise The statement “All fathers are male can now be represented as” Is_a_father(x) is_a_male(x) That is “if x is a father then x is male”

Proof using Unification and Resolution Proof procedure in predicate logic are similar to those used in prepositional logic One complication is that it might be necessary to give variables particular values to deduce the conclusion

Example 1 Parent_of(x,y)child_of(y,x) If x is parent of y then y is a child of x 2 Child_of(y,x) AND Male(y) son(y,x) If y is a child of x and is male then y is a son of x 3 Child_of(y,x) AND female(y)Daughter(y,x) If y is a child of x and is female then y is a daughter of x 4 parent_of(Ahmad,Hina) Ahmad is Hina’s parent 5 female(Hina) Hina is female

Example We can deduce daughter(Hina,Ahmad) by way of the following steps If we substitute Ahmad for x and Hina for y in 1 and then use Modus Ponens with 4 Child(Hina,Ahmad) AND female(Hina)daughter(Hina, Ahmad)

Using Resolution Proof in predicate logic is composed of a variety of processes including modus ponens, chain rule, substitution, matching etc However we have to take variables into account e.g parent_of(Ahmad, Bilal) AND Not Parent_of(Ahmad, Khurram) can not be resolved In other words the resolution method has to look inside predicates to discover if they can be resolved

Unification This substitution of variables to make resolution possible is called Unification and proof in predicate logic can be based on this joint method of Unification and resolution Let us rework the proof of daughter(Hina,Ahmad)

Convert first to Clausal form 1 Not parent_of(x,y) or child_of(y,x) 2 Not child_of(y,x) or not male(y) or son(y,x) 3 Not child_of(y,x) or Not female(y) or daughter(y,x) 4 parent of (Ahmad,Hina) 5 female (Hina)

Example 7 not child(ahmad,hina) or female (hina) By substituting Ahmad for x and Hina for y in 3 and resolving with 6 8 Not Parent(Ahmad,Hina) or Not Female(Hina) by substituting Ahmad for x and Hina for y in 1 and resolving with 7 9 Not Parent(Ahmad,Hina) by resolving 8 with 5 10 null clause by resolving 9 with 4

Writing facts in Predicate Logic Marcus was a man

Fact 1 Man(Marcus) Doesn’t capture past tense Marcus was a Pompeian

Fact 2 Pompeian(Marcus) All Pompeians were Romans X:Pompeian(x)Roman(x) All Romans were either loyal to Caesar or hated him

Fact 3 Inclusive OR X:Roman(x) loyalto(x,Caesar) V hate(x, Caesar) Exclusive OR X:Roman(x)  [(loyalto(x,Caesar) V hate(x,Caesar)) AND ~(loyalto(x,Caesar) AND hate(x,Caesar))] Everyone is loyal to someone

Fact 4 X: y: loyalto(x,y) Meaning that for each person their exists someone to whom he or she is loyal, possibly a different someone for everyone Y: x: loyalto(x,y) There exists someone to whom everyone is loyal People only try to assassinate rulers they are not loyal to

Fact 5 X: y: person(x) AND ruler(y) AND tryassassinate(x,y) ~loyalto(x,y)