Formal Logic: Quantifiers, Predicates, and Validity CS 130 – Discrete Structures Variables and Statements • Variables: A variable is a symbol that stands for an individual in a collection or set. For example, the variable x may stand for one of the days. We may let x = Monday, x = Tuesday, etc. • We normally use letters at the end of the alphabet as variables, such as x, y, z. • A collection of objects is called the domain of objects. For the above example, the days in the week is the domain of variable x. CS 130 – Discrete Structures 55 Quantifiers • Propositional wffs have rather limited expressive power. E.g., “For every x, x > 0”. • Quantifiers: Quantifiers are phrases that refer to given quantities, such as "for some" or "for all" or "for every", indicating how many objects have a certain property. • Two kinds of quantifiers: – Universal Quantifier: represented by , “for all”, “for every”, “for each”, or “for any”. – Existential Quantifier: represented by , “for some”, “there exists”, “there is a”, or “for at least one”. CS 130 – Discrete Structures 56 Predicates • Predicate: It is the verbal statement which describes the property of a variable. Usually represented by the letter P, the notation P(x) is used to represent some unspecified property or predicate that x may have. – P(x) = x has 30 days. – P(April) = April has 30 days. – What is the truth value of (x)P(x) where x is all the months and P(x) = x has less than 32 days • Combining the quantifier and the predicate, we get a complete statement of the form (x)P(x) or (x)P(x) • The collection of objects is called the domain of interpretation, and it must contain at least one object. CS 130 – Discrete Structures 57 Truth Values of the Expressions • What is the truth value of (x)P(x) in the following interpretations? – P(x): x is yellow; the domain of interpretation is the collection of all flowers. – P(x): x is a plant; the domain of interpretation is the collection of all flowers. – P(x): x is either positive or negative; the domain of interpretation consists of the integers. • Can you find one interpretation in which both (x)P(x) is true and (x)P(x) is false? • Can you find one interpretation in which both (x)P(x) is true and (x)P(x) is false? CS 130 – Discrete Structures 58 Unary, Binary,…, N-ary Predicates • Predicates involving properties of a single variable: unary predicates • Binary, ternary and n-ary predicates are also possible • (x) (y)Q(x,y) is a binary predicate. This expression reads as “for every x there exists a y such that Q(x,y)” • Assume Q(x,y) is the property that x < y, what’re the interpretations? – ( x)( y) Q(x, y), ( y) ( x) Q(x, y), and ( x)( x) Q(x, x) – We cannot collapse separate variables together into one without changing the nature of the expression • Constants are also allowed in expressions, such as a, b, c, 0, 1, 2, etc. CS 130 – Discrete Structures 59 Interpretation • Formal definition: An interpretation for an expression involving predicates consists of the following: – A collection of objects, called domain of interpretation, which must include at least one object. – An assignment of a property of the objects in the domain to each predicate in the expression. – An assignment of a particular object in the domain to each constant symbol in the expression. • Predicate wffs can be built similar to propositional wffs using logical connectives with predicates and quantifiers. • Must obey the rules of syntax to be considered a wff • Examples of predicate wffs – (x)[P(x) Q(x)] – (x) ((y)[P(x,y) V Q(x,y)] R(x)) – S(x,y) Λ R(x,y) CS 130 – Discrete Structures 60 Scope of a Variable in an Expression • The parentheses or brackets are used wisely to identify the scope of the variable. – (x) ((y)[P(x,y) V Q(x,y)] R(x)) • Scope of (y) is P(x,y) V Q(x,y) while the scope of (x) is the entire expression – (x)S(x) V (y)R(y) • Scope of (x) is S(x) while the scope of (y) is R(y) – (x)[P(x,y) (y) Q(x,y)] • Scope of variable y is not defined for P(x,y) hence y is called a free variable. Such expressions might not have a truth value at all. • P(x): x > 0; P(y)^ P(5), P(y) V P(5). – What is the truth of the wff (x)(A(x) Λ (y)[B(x,y) C(y)]) , where A(x) is “x > 0” , B(x, y) is “x > y”, C(y) is “y 0”, and x is the domain of positive integers and y is the domain of all integers? CS 130 – Discrete Structures 61 Translation of Verbal Statements to Symbolic Form Using Intermediate Statements • “Every person is nice” can be rephrased as “For any thing, if it is a person, then it is nice”. So, if P(x) is “x is a person” and Q(x) be “x is nice”, the statement can be symbolized as – (x)[P(x) Q(x)] • Variations: “All persons are nice” or “Each person is nice”. • “There is a nice person” can be rewritten as “There exists something that is both a person and nice” in symbolic form – (x)[P(x) Λ Q(x)] • Variations: “Some persons are nice” or “There are nice persons” • What would the following forms mean for the example above? – (x) [P(x) Λ Q(x)] is too strong – (x)[P(x) Q(x)] will be true if there are no persons in the world but that is not the case. • So almost always, goes with Λ (conjunction) and goes with (implication) • Can be confusing, so remember to frame the statement in different forms as possible CS 130 – Discrete Structures 62 More On Translation • The word “only” can be tricky depending on its presence in the statement: – X loves only Y If X loves anything, then that thing is Y – Only X loves Y If anything loves Y, then it is X – X only loves Y If X does anything to Y, then it is love • Example for forming symbolic forms from predicate symbols: – D(x) is “x is dog” – R(x) is “x is a rabbit” – C(x,y) is “x chases y” • All dogs chase all rabbits – For anything, if it is a dog, then for any other thing, if it is a rabbit, then the dog chases it (x)[D(x) (y)(R(y) C(x,y))] • Some dogs chase all rabbits – There is something that is a dog and for any other thing, if that thing is a rabbit, then the dog chases it (x)[D(x) Λ (y)(R(y) C(x,y))] • Only dogs chase rabbits – For anything, if it is a rabbit then, if anything chases it, that thing is a dog (y) [R(y) (x) (C(x, y) D(x))] – Or, for any two things, if one is a rabbit and the other chases it, then the other is a dog (y) (x)[R(y) Λ C(x,y) D(x)] CS 130 – Discrete Structures 63 Negation of Statements • A(x): Everything is beautiful – Negation will be “it is false that everything is beautiful”, i.e. “something is not beautiful” – In symbolic form, [(x)A(x)]’ (x)[A(x)]’ – Similarly, negation of “Something is beautiful” is “Nothing is beautiful” or “Nothing is beautiful” – Hence, [(x)A(x)]’ (x)[A(x)]’ • What is the negation of “Everybody loves somebody sometime” – Everybody hates somebody sometime – Somebody loves everybody all the time – Everybody hates everybody all the time – Somebody hates everybody all the time CS 130 – Discrete Structures 64 More Examples on Negation • What is the negation of the following statements? – Some pictures are old and faded. • Every picture is not old or not faded. – All people are tall and thin. • Someone is short or fat. – Some students eat only pizza. • Every student eats something which is not a pizza – Only students eat pizza. • There is a non-student who eats pizza. CS 130 – Discrete Structures 65 Class Exercises • S(x): x is a student For anything, if it is a student, then it is • I(x): x is intelligent intelligent (x)[S(x) I (x)] • M(x): x likes music There is something that is intelligent and it is a student and it likes • Write wffs than express music (x)[I(x) Λ S(x) Λ M(x)] the following statements: – All students are For anything, if that thing likes music, intelligent. then it is a student and it is not – Some intelligent intelligent students like music. (x)(M(x) S(x) Λ [I (x)]´) – Everyone who likes music is a stupid For any thing, if it likes music, then it student. is a student and it is intelligent – Only intelligent students (x)(M(x) S(x) Λ I(x)) like music. CS 130 – Discrete Structures 66 Validity • Analogous to a tautology of propositional logic • Truth of a predicate wff depends on the interpretation • A predicate wff is valid if it is true in all possible interpretations just like a propositional wff is true if it is true for all rows of the truth table • A valid predicate wff is intrinsically true Propositional Wffs Predicate Wffs Truth Values True or false – depends on True, false or neither(if the the truth value of statement wff has a free variable) letters Intrinsic truth Tautology- true for all truth Valid wff- true for all values of its statements interpretations Methodology Truth table to determine if it No algorithm to determine is a tautology validity CS 130 – Discrete Structures 67 Validity Examples • (x)P(x) (x)P(x) – This is valid because if every object of the domain has a certain property, then there exists an object of the domain that has the same property.