Clicker: Silly starter questions added September 4

1. Clicker: Silly starter questions added September 4 • According to www.fbi.gov, How many FBI special agents are there? • 572 • 8,192 • 13,890 • 127,736

2. Clicker: Silly starter questions added September 4 Approximately how many people apply for employment at the FBI each year 2500 25000 250,000 2,500,000

3. Law of the excluded middle • For every proposition, either the proposition is true or its negation is true • Either “Socrates is a man” or “Socrates is not a man” • Either “It is true that Socrates is a man” or “It is true that Socrates is not a man” • What about “This sentence is neither true nor false” • Problem of “self-reference” or implied “it is true that…”

4. Problems with the excluded middle • Many statements have an element of uncertainty: • Either it is raining or it is not raining (virga) • Either Sophia Vergara is blonde or she is not blonde. • Either NMSU has a better football team or UTEP has a better football team…

5. Logical arguments I: The syllogism • Aristotle, Prior Analytics: a syllogism is "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.”

6. Syllogism cont. • A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion. • Major premise: All men are mortal. • Minor premise:Socrates is a man. • Conclusion: Socrates is mortal. • Major premise: All mortals die. • Minor premise: All men are mortals. • Conclusion: All men die.

7. Modus ponendo ponens("the way that affirms by affirming”) • If P, then Q. P. Therefore, Q • If Socrates is a man then Socrates is mortal • Socrates is a man • Socrates is mortal

8. Logic and causality

9. Causality • Socrates is a dog. • all dogs are green • Socrates is green.

10. Universe of discourse

11. Logic and symbol of propositional calculus • P, Q, R etc: propositional variables • Substitute for statements, e.g., P: Plato is a dog, Q: Plato is Green • Logical connectives: ∧,∨,,− • Proposition: If Plato is a dog then Plato is green: • P->Q

12. Truth tables

13. Clicker question • P: Socrates is a man • Q: Socrates is mortal. • P->Q: If Socrates is a man then Socrates is mortal. • Suppose that Socrates is not a man. • Is the whole statement: P->Q true or false? • Clicker: True (A) or False (B)

14. Truth table for implication

15. Truth table for modus ponens • No matter what truth values are assigned to the statements p and q, the statement • (p->q)^p->q is true • p

16. Exercise: complete the truth table for modus tollens

17. For compound statements with conjunctions (∧) to be true, the elements on both sides of ∧ must have the value “T”so the fourth column is as follows:

18. For condition statements or “implications” with “ ->” to be true, either the statement to the left of the implication has to be false or the statement to the right of the implicationhas to be true. The statement (p->q)∧-q is false in the first three cases and the statement –p is true in the last, so the fourth column has value “T” in all cases

19. Clicker questions: • First row: True (A) or False (B) • Second row: True (A) or False (B) • Third row: True (A) or False (B) • Last row: True (A) or False (B)

20. Logical equivalence • Two formulas are logically equivalent if they have the same truth values ones values are assigned to the atoms. • Ex: p->q is equivalent to –p∨q is equivalent to –(p∧-q) • How to check logical equivalence: verify that the statements always have the same values

21. Exercise: verify that the statementsp->q ,–p∨q , and–(p∧-q) are logically equivalent

22. Boole (1815-1864) and DeMorgan (1806–1871) • De Morgan’s laws: • not (P and Q) = (not P) or (not Q) • not (P or Q) = (not P) and (not Q)

23. Boolean algebra

24. Other logical deduction rules