Arguments with Quantified Statements

Arguments with Quantified Statements Lecture 10 Section 2.4 Thu, Jan 27, 2005

Universal Modus Ponens • The universal modus ponens argument form: x  S, P(x)  Q(x) P(a) for a particular a  S Q(a)

Example • Let F be the set of all functions from R to R. • f  F, if f is differentiable, then f is continuous. • The function f(x) = x2 + 1 is differentiable. • Therefore, f(x) is continuous.

An Argument within in Argument • f  F, if f is differentiable, then f is continuous. • f  F, if f is a polynomial, then f is differentiable. • The function f(x) = x2 + 1 is a polynomial. • Therefore, f(x) is differentiable. • Therefore, f(x) is continuous.

Universal Transitivity • The previous example could have been handled differently using the argument form of universal transitivity: x  S, P(x)  Q(x) x  S, Q(x)  R(x)  x  S, P(x)  R(x)

Universal Transitivity • Equivalently, P(x)  Q(x) Q(x)  R(x)  P(x)  R(x)

Universal Modus Tollens • The universal modus tollens argument form: x  S, P(x)  Q(x) ~Q(a) for a particular a  S  ~P(a)

Diagrams • The statement x  S, P(x)  Q(x) means that the truth set of P is a subset of the truth set of Q. • The statement P(a) means that a is in the truth set of P. • Therefore, a must be in the truth set of Q.

Truth set of Q Truth set of P a Diagrams • Therefore, we can represent these statements by using Venn diagrams.

Truth set of Q Truth set of P a A Diagram for Universal Modus Ponens x  S, P(x)  Q(x) P(a) for a particular a  S Q(a)

Continuous functions Example

Continuous functions Differentiable functions Example

Continuous functions Differentiable functions Polynomial functions Example

Example Continuous functions Differentiable functions Polynomial functions f(x) = x2 + 1

Diagrams • Recall the example that showed that x  R, x/(x + 2)  3  x  -3.

{xR | x -3} {xR | x/(x + 2)  3} Example

0 -4 1 -3 2 -2 3 -1 4 Example • A better representation:

Statements with “No” • Rewrite the statement “No HSC student would ever lie” using quantifiers. • ~(x  {HSC students}, x would lie) • x  {HSC students}, ~(x would lie) • x  {HSC students}, x would not lie • Thus, this is a universal statement.

Arguments with “No” • Which arguments are valid? • No HSC student would ever lie. Joe is an HSC student. Therefore, Joe would never lie. • No HSC student would ever lie. Buffy is an RMC student. Therefore, Buffy would lie.

Joe Arguments with “No” • The diagram shows that Joe cannot be a liar. People HSC Students Liars

Statements with “No” • Note that the following two statements are equivalent. • No HSC student is a liar. • No liar is an HSC student.

Joe Arguments with “No” • Where would we place the oval for RMC students? People HSC Students Liars

Arguments with “No” • Where would we place the oval for RMC students? People RMC Students ? HSC Students Liars

Arguments with “No” • Where would we place the oval for RMC students? People HSC Students Liars RMC Students ?

Arguments with “No” • Where would we place the oval for RMC students? RMC Students ? People HSC Students Liars

Arguments with “No” • Where would we place Buffy? People HSC Students Liars RMC Students

Buffy Arguments with “No” • Where would we place Buffy? People HSC Students Liars RMC Students

Arguments with “No” • Where would we place Buffy? People HSC Students Liars RMC Students Buffy

Arguments with “No” • Which fallacy is committed in the “Buffy” argument?

A Logical Conclusion? • Is the following argument valid? x, y, z, if x is better than y and y is better than z, then x is better than z. A peanut butter sandwich is better than nothing. Nothing is better than sex.  A peanut butter sandwich is better than sex.

Arguments with Quantified Statements