Understanding Universal Instantiation in Logical Arguments

1. Arguments with Quantified Statements M260 3.4

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

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

4. Simplify rk+1 r • rk+1 r = rk+1 r1 • = r(k+1)+1 • = rk+2 • Uses universal truths

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

6. Example • If a number is even, then its square is even. • K is a particular number that is even • Therefore k2 is even.

7. Formal version • hummm

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

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

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

11. 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.

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

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

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

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

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

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