Arguments with Quantified Statements M260 3.4. Universal Instantiation If some property is true for everything in a domain, then it is true of any particular.

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Presentation transcript:

Arguments with Quantified Statements M

Universal Instantiation If some property is true for everything in a domain, then it is true of any particular thing in the domain.

Example All human beings are mortal. Socrates is a human being. Therefore Socrates is mortal.

Simplify r k+1 r r k+1 r = r k+1 r 1 = r (k+1)+1 = r k+2 Uses universal truths

Universal Modus Ponens For all x, if P(x) then Q(x) P(a) for a particular a Therefore Q(a)

Example If a number is even, then its square is even. K is a particular number that is even Therefore k 2 is even.

Formal version hummm

Prove that the sum of two even integers is even do it here

Universal Modus Tollens For all x, if P(x) then Q(x). ~Q(a) for some particular a Therefore ~P(a)

Example All human beings are mortal. Zeus is not mortal. Therefore Zeus is not a human being.

Valid Argument Form No matter what predicates are substituted for the predicate symbols in the premises, if the premise statements are all true then the conclusion is also true. An argument is valid if its form is valid.

Use Diagrams for Zeus and Felix All human beings are mortal. Zeus is not mortal. Therefore Zeus is not a human being.

Use Diagrams for Zeus and Felix All human beings are mortal. Felix is mortal. Therefore Felix is a human being.

Converse Error For all x, if P(x) then Q(x). Q(a) for a particular a. Therefore P(a) INVALID

Inverse Error For all x, if P(x) then Q(x). ~P(a) for a particular a. Therefore ~Q(a) INVALID

No No polynomials have horizontal asymptotes This function has a horizontal asymptote. Therefore this function is not a polynomial

Rewrite For all x, if x is a polynomial, then x does not have a horizontal asymptote. Use Universal Modus Tollens.