I. Determine the Truth Values of the Following Symbolized Statements

PHL 218 Homework1

I. Determine the truth values of the following symbolized statements.

A=T X=T Y=T B=F C=F Z=F

1. ~{~[(C & ~B) & (Z v ~A)]  ~[~(B ↔ Y) & (~X & Z)]}

2. ~{[ ~(Z  C)  (~X  B)]  ~[(C & Y) v ~ (Z  Y)]}  ~ (~Y Z)

II. Use the regular truth-table method to determine whether these are valid or invalid.

1． A v (B  C)/B & ~C// ~A2. ~A & (~B ~A)/ (C ~B)//~C

V. Use the short truth-table method to determine whether these are valid or invalid.

1.~K(L & G)/M  (J & ~K)/B & M//B & G2.(D & ~G) H/M & (H P)/M~G)// D & P

III. Construct truth-table to determine which of the following are tautologies, which are contradictions, and which are contingency.

1. {[(A & ~B) v ~ (B v C)] & ~ (A & ~B)}  ~ (B v C) 2. [(~P & R) v (~P & ~R)] ↔~ [(~P & R) v (~P & ~R)]

3. ~ [(M v ~S) ↔ (~M & T)]  [(S & T)  ~ (~T v M)]

IV. Which of the following sentences are derivable from the sentence form ~(p & ~q)  (q v r) ? For those which are, specific the substitutions for “p”, “q”, and “r” (2 points each).

1. ~(A & ~B)  (B v C)2. ~ (A & ~ ~B)  (~ B v C)

3. ~[(A v B) & ~ (C & ~D)]  ~ [(C & ~D) v (B ↔ A)]4. ~ (A & ~ ~B) (C v D)

VIII. Is the sentence “~ (~A v B)  [~ (~B & C)  (~A v B)] ” derivable from any of the following? (2 points each)

1. ~p  q2. ~p  ~ (~q  p)3. P  ~ (~q  p)

V. Symbolize the following claims, using the letters indicated.

P = We plant perennials; A = We plant annuals; S = We plant from seed; C = We plant from cuttings.

1.If we plant from seed, we’ll have to plant annuals.

2.We can plant perennials only if we plant from cuttings.

3.The only way we can plant from seed is to plant annuals.

4.If we plant both annuals and perennials, then we can plant from both seed and cuttings.

5.We cannot plant perennials if we plant from either seed or from cuttings.

6.If we don’t plant from seed, then we can’t plant either annuals or perennials.

7.We can’t plant perennials unless we plant from cuttings.

8.The only way we can plant both annuals and perennials is by planting from both cuttings and seed.

9.Either we will plant from cuttings, or, if we don’t plant perennials, we can plant from seed.

10.We can plant neither perennials nor annuals if we don’t plant from both cuttings and seed.

VI. Follow the same directions given in the preceding one.

W= Wildlife are (or will be) threatened; A= Agricultural production is increased; P= The use of pesticides is continued.

1.The only way we can avoid threatening wildlife is to avoid increasing agricultural production.

2.We cannot both increase agricultural production and avoid threatening wildlife.

3.If we are to increase agricultural production, we’ll have to continue the use of pesticides, but if we do that wildlife will be threatened.

4.Wildlife will not be threatened provided we do not continue the use of pesticides.

5.Wildlife will be threatened if either agricultural production is increased or pesticide use is continued.

6.The continuation of pesticide use will be sufficient to ensure that wildlife will be threatened.

7.The continued use of pesticides is necessary for increased agricultural production.

8.Together, the continued use of pesticides and the increase in agricultural production will guarantee that wildlife will be threatened.

9.Agricultural production will not increase even though the use of pesticides will continue.

10.While pesticide use will continue, agricultural production will not increase.

VII. For each of the following argument symbolizations, assign truth values to the letters to show the argument’s invalidity. (There is only one such assignment for each—one counterexample—so these are easy to grade.)

1.Q v P/~Q  ~R//R  P

2.~Q  P/R v S/Q  ~S// R

3.P v Q/P  R// R  Q

4.~R  ~Q/~P  (R & Q)// P

5.(Q & P)  R/S  ~R// S  ~Q

6.S  (P v R)/Q  S// Q  P

7.T  ~S/S v ~Q/~T  (Q v R)// ~Q  R

8.(Q & S)  (P v R)/T  (~T v S)// T  R

9.P  (Q  S)/Q v R// (P & R)  S

10.P  (Q v R)/~(Q  R)/S  P// ~S

11.P v Q/(Q & R)  S/~P  ~R// R  S

12.P v (Q  R)/S  ~(P v R)// S  Q

13.~L & ~S/(P v Q)  L// Q v S

14.P  (T & R)/(R  S) v T/~(S & Q)// Q  ~P

15.~P v (Q  R)/Q  (R v S)// Q  (~P v S)

VIII. Use the short truth-table method to determine whether these are valid or invalid.

1.P  Q/Q  R/R  S/P// S v T

2.A  B/C  D/B v D// A v C

3.A  B/~D  ~C/~D/~C  ~B// ~A

4.A  B/~C v D/E  F/G  H/A v C/E v G// (B v D) v (F v H)

5.A  B/C  D/~(~A & ~C)// ~B  (D & ~B)

6.~E v ~D/~(A  C)/~(~B v ~E)/~(A & B) v (C  D)// ~A

7.P  Q/~Q v R/~(R & ~S)/~P// ~S

8.P  (Q & R)/R  (T v S)/~T v (U & ~W)/P// ~W

9.M  (N  O) /(~N v O)  ~P /P// ~M

10.P  Q/~(Q & ~R)/~R v S/~(S & ~T)/T// P

IX. Display the following arguments by symbolizing them; then use the short truth table method to prove them valid or invalid. Use the letters provided (10 points each).

1. If it’s cold, Dale’s motorcycle won’t start. If Dale is not late for work, then his motorcycle must have started. Therefore, if it’s cold, Dale is late for work. (C, S, L)

2. If profits depend on unsound environmental practices, then either the quality of the environment will deteriorate, or profits will drop. Jobs will be plentiful only if profits do not drop. So, either Jobs will not be plentiful, or the quality of the environment will deteriorate. (U, Q, D, J)

3. If the senator votes against this bill, then he is opposed to penalties against tax evaders. Also, if the senator is a tax evader himself, then he is opposed to penalties against tax evaders. Therefore, if the senator votes against this bill, he is a tax evader himself. (V, O, T)

4. The creation story in the Book of Genesis is compatiblewith theory of evolution, but only if the creation story is not taken literally. If there is plenty of evidence for the theory of evolution, which there is, Genesis story cannot be true if it is not compatible with evolution theory. Therefore, if the Genesis story is taken literally, it cannot be true. (C, L, E, T)

5. If you had gone to class, taken good notes, and studied the text, you’d have done well on the exam. And, if you’d done well on the exam, you’d have passed the course. Since you did not pass the course and you go to class, you must not have taken good notes and not studied the text. (C, N, T, E, P)

6. Either John will go to class, or he’ll miss the review session. If John misses the review session, he’ll foul up the exam. If he goes to class, however, he’ll miss his ride home for the weekend. So John’s either going to miss his ride home or foul up the exam. (C, S, R, E)

7. If there was no murder committed, then the victim must have been killed by the horse. But the victim could have been killed by the horse only if he, the victim, was trying no injure the horse before the rice; and, in that case, there certainly was a crime committed. So, if there was no murder, there was still a crime committed. (M, H, I, C)

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