Clicker: Silly starter questions added January 24

1. Clicker: Silly starter questions added January 24 • 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 • Either Sophia Vergara is blonde or she is not blonde. • Either NMSU has a better basketball team or UTEP has a better basketball 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. Identify the major premise: • All dogs have four legs • Milo is a dog • ________________________ • People who solve problems can get jobs. • Students good in math can solve problems. • _______________________ • Women like a man with a prominent chin. • Robert Z’dar has a prominent chin.

8. 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 • Therefore, Socrates is mortal

9. Logic and causality

10. Causality • Plato is a dog. • all dogs are green • Plato is green.

11. Universe of discourse

12. 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

13. Truth tables

14. 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)

15. Truth table for implication

16. Why is “If p then q” true whenever p is false? • The ice cream analogy: if you clean your room then we can go for ice cream. • If you speak the truth then you can cross the bridge • If you study hard then you will get a good grade

17. 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

18. Exercise: complete the truth table for modus tollens

19. Simple and compound statements • A simple statement is sometimes called an atom. E.g., Milo is a dog; Socrates is a man; Men are mortal. • A compound statement is a string of atoms joined by logical connectives (and, or, then, not) • Logical equivalence: pq; -(p∧-q) • Truth value of a compound statement is inherited from the values of the atoms.

20. 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:

21. 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

22. 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)

23. Logical equivalence • Two formulas are logically equivalent if they have the same truth values once 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

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

25. Tautology and contradiction: T or C • A logical statement that is always true, independent of whether each of the symbols is true, is called a tautology. • p ∨ ~p = t, • A logical statement that is always true, independent of whether each of the symbols is true, is called a contradiction. • p ∧ ~p = c • Note: ~t = c and ~c = t

26. Logical equivalence laws • Commutative laws: p ∧ q = q ∧ p; p ∨ q = q ∨ p • Associative laws: (p ∧ q) ∧ r = p ∧ (q ∧ r), • (p ∨ q) ∨ r = p ∨ (q ∨ r) • Distributive laws: p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) • p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) • Identity, universal bound, idempotent, and absorption laws: • p ∧ t = p, p ∨ c = p • p ∨ t = t, p ∧ c = c • p ∧ p = p, p ∨ p = p • p ∨ (p ∧ q) = p, p ∧ (p ∨ q) = p • De Morgan’s laws: ~(p ∧ q) = ~p ∨ ~q, ~(p ∨ q) = ~p ∧ ~q

27. Show that the following are logically equivalent:(r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) and p ^ q • ( ~r V (p^q) ) ^ (r V q) • =( (~rVp) ^ (~rVq) )^(rVq) (DL2) • = (~rVp) ^( (~rVq) )^(rVq)) (AL1) • = (~rVp) ^( (~r^r)Vq) (DL1) • = (~rVp) ^(cVq) (definition of c) • = (~rVp) ^q (ID2) so • (r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) • =(r V p) ^ (~rVp) ^q =((r^~r)Vp)^q=(cVp)^q=p^q

28. 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)

29. Boolean algebra

30. Other logical deduction rules

31. Exercises

32. Exercise 1: Complex deduction • • Premises: • – If my glasses are on the kitchen table, then I saw them at breakfast • I was reading the newspaper in the living room or I was reading • the newspaper in the kitchen • – If I was reading the newspaper in the living room, then my glasses • are on the coffee table • – I did not see my glasses at breakfast • – If I was reading my book in bed, then my glasses are on the bed table • – If I was reading the newspaper in the kitchen, then my glasses are • on the kitchen table • • Where are the glasses?

33. Deduce the following using truth tables or deduction rules • ~(p V ~q) V (~p ^ ~q) ≡ ~p

34. Write each of the following three statements in the symbolic form and determine which pairsare logically equivalent • a. If it walks like a duck and it talks like a duck, then it is a duck • b. Either it does not walk like a duck or it does not talk like a duck, or it is a duck • c. If it does not walk like a duck and it does not talk like a duck, then it is not a duck