Chapter 2

1. Chapter 2 The Logic of Quantified Statements

2. Section 2.4 Arguments with Quantified Statements

3. Universal Instantiation • Universal instantiation “If some property is true of everything in a domain, then it is true of any particular thing in the domain” All men are mortal. Socrates is man. ∴ Socrates is mortal. • Universal instantiation is the fundamental tool of deductive reasoning.

4. Example • Example: rk+1 * r • where r is a particular real number and k is a particular integer. • For all real numbers x and all integers m and n, xm * xn = xm+n. • For all real numbers x, x1 = x. • rk+1 * r = rk+1 * r1 • = r(k+1)+1 • = rk+2

5. Example • rk+1 * r = rk+1 * r1 • For all real numbers x, x1 = x. (universal truth) • r is a particular real number. (particular instance) • ∴ r1 = r. • = r(k+1)+1 • For all real numbers x and all integers m and n, xm * xn = xm+n • r is a particular real number and k + 1 and 1 are particular integers. • ∴rk+1 * r1 = r(k+1)+1

6. Universal Modus Ponens • Universal modus ponens is a combination of universal instantiation and modus ponens. • Universal Modus Ponens • If x makes P(x) true, then x makes Q(x) true. • a makes P(a) true. • ∴ a makes Q(a) true.

7. Example • Rewrite the argument using quantifiers, variables, and predicate symbols. • If a number is even, then its square is even. • k is a particular number that is even. • ∴ k2 is even • major premise • ∀x, if x is even then x2 is even. • E(x) be “x is even”, S(x) be “x2 is even”, and let k stand for a particular number that is even. The argument becomes: • ∀x, if E(x) then S(x). • E(k), for a particular k. • ∴ S(k) (argument has the form of universal modus ponens)

8. Example • Write the conclusion inferred using universal modus ponens. • If T is any right triangle with hypotenuse c and legs a and b, then c2 = a2 + b2. • The right triangle shown has both legs = 1 and hypotenuse c. • ∴____________________

9. Universal Modus Tollens • Universal modus tollens combines universal instantiation with modus tollens. • Universal Modus Tollens (formal) • ∀x, if P(x) then Q(x). • ~Q(a), for a particular a. • ∴ ~P(a)

10. Example • Rewrite the following argument using quantifiers, variables, and predicate symbols. Write the major premise. Is this argument valid. • All human beings are mortal. • Zeus is not mortal. • ∴ Zeus is not human. • major premise • ∀x, if x is human then x is mortal. • H(x) be “x is human.”, M(x) be “x is mortal” and Z be Zeus • ∀x, if H(x) then M(x) • ~M(Z) • ∴ ~H(Z) (argument form of universal modus tollens; valid)

11. Example • Write the conclusion that can be inferred using universal modus tollens. • All professors are absent minded. • Tom Jones is not absent-minded. • ∴______________________

12. Validity of Arguments with Quantified Statements • An argument is valid if, and only if, the truth of its conclusion follows necessarily from the truth of its premises.

13. Diagrams to Test for Validity • Example • (informal) All integers are rational numbers. • (formal) ∀integers n, n is a rational number.

14. Example • Using a diagram show validity • All human beings are mortal. • Zeus is not mortal. • ∴ Zeus is not a human being.

15. Diagrams for Invalidity • Show invalidity of the following argument • All human beings are mortal. • Felix is mortal. • ∴ Felix is a human being.

16. Diagrams for Invalidity

17. Converse & Inverse Error • Converse Error (Quantified) • ∀x, if P(x) then Q(x). • Q(a) for a particular a. • ∴ P(a) (invalid conclusion) • Inverse Error (Quantified) • If x makes P(x) true, then x makes Q(x) true. • a does not make P(a) true. • ∴ a does not make Q(a) true. (invalid conclusion)

18. Argument with “No” • Example: Use diagrams to test the following argument for validity. • No polynomial functions have horizontal asymptotes. • This function has a horizontal asymptote. • ∴This function is not a polynomial function.

19. Argument with “No” • Rewrite the previous problem in universal modus tollens (formal form) • P(x) be “x is a polynomial function” • Q(x) be “x does not have a horizontal asymptote” • a be “this function” • Formal • ∀x, if P(x) then Q(x). • ~Q(a), for a particular a. • ∴~P(a) (universal modus tollens)

20. Universal Transitivity • Formal Version • ∀x(P(x) → Q(x)). • ∀x(Q(x) → R(x)). • ∴∀x(P(x) → R(x)).

21. Example • Rewrite the informal to formal and show validity through Universal Transitivity. • All the triangles are blue. • If an object is to the right of all the squares, then it is above all the circles. • If an object is not to the right of all the squares, then it is not blue. ∴ All the triangles are above all the circles.

22. Example • ∀x, if x is a triangle, then x is blue. • ∀x, if x is to the right of all the squares, then x is above all circles. • ∀x, if x is not to the right of all the squares, then x is not blue. ∴ ∀x, if x is a triangle, then x is above all the circles. • Reorder (premises such that the conclusion of each is the hypothesis of the next) 1. ∀x, if x is a triangle, then x is blue. 3. ∀x, if x is blue, then x is to the right of all the squares. (contrapositive form, if not q then not p) 2. ∀x, if x is to the right of all the squares, then x is above all circles. ∴ ∀x, if x is a triangle, then x is above all the circles.