Today s Lecture 2/9/10. Symbolizations continued Chapter 7.2: Truth conditions for our logical operators

Transcription

1 Today s Lecture 2/9/10 Symbolizations continued Chapter 7.2: Truth conditions for our logical operators

2 Announcements Exam #1 next Tuesday (Feb 16th)! HW: Study for exam The text book is on Library Reserve Pre-Law Information Session (see me for flyer)

3 A Study Guide! Validity, Invalidity, Soundness (e.g. be able to: define what a valid, invalid and sound argument is; recognize whether an argument is valid or invalid; answer the T/F questions we did in class)! Logical Form (e.g. be able to: recognize and label the five famous valid forms; recognize the two invalid forms we talked about in class; give a substitution instance for any argument form; explain the difference between a valid argument form and an invalid argument form)! Symbolizations (e.g. be ready to symbolize English statements; be sure and know the stylistic variants; be sure you know the conditions for a WFF).! Truth Tables (know the truth tables for our five compound statements p. 308; be able to determine the truth value of compound statements when certain conditions are given).

4 Review -- Biconditional P!Q says P if and only if Q P just in case Q P is both necessary and sufficient for Q Thus, A biconditional is a conjunction of two conditionals: (P"Q) (Q"P)

5 Review--Negation of a conjunction ~(p q) is not equivalent to ~p ~q Saying it's not the case that Peyton Manning is both a human and a cat, ~(p q) -- this is true is not thereby to say that: Peyton Manning is not a human and Peyton Manning is not a cat, ~p ~q -- this is false

6 Review--Negation of a disjunction ~(p # q) is not equivalent to ~p # ~q It helps to see a disjunction as "saying": 'at least one of the disjuncts is true'.

7 Review-- Negation of a disjunction Saying that at least one of the following disjuncts is true: Clinton is not President; Michelle Obama is not the First Lady (i.e. saying Clinton is not President or Michelle Obama is not First lady, ~p # ~q ) -- this is true. is not to thereby say: it's not the case that at least one of the following disjuncts is true: Clinton is President; Michelle Obama is First Lady [i.e. it's not the case that (Clinton is President or Michelle Obama is First Lady), ~(p # q)] -- this is false.

8 From Last Time Assuming that you lost my dog, I will forgive you only if you apologize to me; but if you don t apologize, I will not forgive you. D: You lost my Dog. F: I will forgive you A: You apologize to me (1) (D" (F " A)) (~A " ~F). Is (1) equivalent to: (2) D " (F! A)?

9 From Last Time (1) (D" (F " A)) (~A " ~F). (1) is equivalent to: (1$) (D" (F " A)) (F " A) (2) D " (F! A). (2) is equivalent to: (2$) D " [(F " A) (A " F)] Notice how in (1$) (F " A) follows D. But in (2$) (F " A) and (A " F) follows D. Thus (1$) and (2$) are not equivalent, and hence (1) and (2) are not either. On the assumption that D, having F sufficient for A does not thereby mean that A is also sufficient for F.

10 Quiz -- Symbolize the following: 1. Max is a dog only if he is a mammal (D: Max is a dog; M: Max is a mammal) 2. Socrates neither argued well nor spoke persuasively. (A: Socrates argued well; P: Socrates spoke persuasively) 3. Julia Roberts being a movie star is a sufficient condition, but not a necessary condition, for her being famous. (M: Julia Roberts is a movie star; F: Julia Roberts is famous)

11 Answers D " M ~A ~P or ~(A # P) (M " F) ~(F " M)

12 Pgs (Part D 1-20 All)

13 1. S "R 2. ~C " ~S 3. P"(B ~F)

14 4. (W # F) ~(W F) 5. B " (D # P) 6. ~(P # O) or (~P ~O)

15 7. B " (F " ~P) 8. [T " ~(S# J)] (S " ~J) 9. (C H) ~J

16 10. (B " F) " ~B 11. P # M or (~M " P) 12. (T # H) ~(T H)

17 13. A " U 14. C! H 15. (W " S) ~(S " W)

18 16. P " D 17. K " S 18. ~(S " C)

19 19. ~K! ~S 20. (R A)! H

20 Pages (Part E 2-14 even)

21 2. C " M 4. (M " N) ~N 6. (G " B) " (~B " ~G) 8. ~(M " G) 10. [(R A) " H] ~R 12. E # S or ~S " E 14. G " (~E # R) or G " (~R " ~E)

22 Truth Tables -- Preliminaries Up to this point we've been translating English statements into our symbolic language. Since statements comprise arguments, we've in essence been developing an ability to translate English arguments into our symbolic language. Our longer-range goal is to take arguments in symbolic form and test them for validity or invalidity by means of a truth table.

23 Preliminaries But before we do that we need to accomplish the shorterrange goal of determining the possible truth-values of our five logical operators (i.e. our five types of compound statements). Given that our operators are truth functional, we'll do that by examining the truth-values of the statements that make-up our five types of compound statements. And we'll do that by constructing a certain kind of truth table: a 'primitive truth table'.

24 Truth Functionality Our logical operators (~,, #, ",!) are all 'truth 'functional'. This means that first, compound statements (e.g. negations, conjunctions, disjunctions, conditionals, and biconditionals) can be either true or false. Secondly, and most importantly, the truth or falsity of any compound statement (i.e. its truth value) is entirely determined by (i.e. entirely a function of) the truth or falsity of its constituent statement(s).

25 Negations The truth-value of any negation (i.e. whether it's true or false) is determined by the truth-value of the relevant statement that is negated. e.g. "It's not the case that 2+2=4". To determine whether this negation is true or false we look to the statement negated: "2+2=4". This statement is true, so the claim that this statement is false, is false. "It's false that the moon is made of cheese". The moon is made of cheese is false; so the statement that claims it's false is true.

26 p ~ p T?

27 A negation has the opposite truth value of the statement negated. p ~ p T

28 Conjunctions The truth value of a conjunction is determined by the truth value of its conjuncts. A conjunction in effect claims: both of the conjuncts are true. So a conjunction will be true or false depending on whether both of its conjuncts are indeed true. E.g. Obama is President and the earth is flat. --this conjunction is false. E.g. Airplanes exist and cats exist. -- this conjunction is true.

29 Conjunctions p q p q T T? T F? F T? F F?

30 A conjunction is true if both of its conjuncts are true; it s false otherwise. p q p q T T T T F F F T F F F F

31 Disjunctions The truth value of a disjunction is determined by the truth value of its disjuncts. A disjunction in effect claims: at least one of the disjuncts is true. So a disjunction will be true or false depending on whether at least one of its disjuncts is indeed true. E.g. Either 2 is an odd number or 3 is an even number --this disjunction is false E.g. Either 2 is an even number or 3 is an even number -- this disjunction is true.

32 Disjunctions p q p # q T T? T F? F T? F F?

33 A disjunction is false if both of its disjuncts are false; it is true otherwise. p q p # q T T T T F T F T T F F F

34 Exercises! Suppose P is true and Q is false! What is the truth-value of these compound statements? ~ Q P Q ~Q P ~(P # Q) P # Q ~Q # (P Q)

35 Exercises! Suppose P is true and Q is false! What is the truth-value of these compound statements? ~ Q -- True P Q -- False ~Q P -- True ~(P # Q) -- False P # Q -- True ~Q # (P Q) -- True