Multiple Questions in Philosophy

1. Using the letters provided below, symbolize the claim: “The only way we can avoid
threatening wildlife is to avoid increasing agricultural production.”
W = Wildlife are threatened.
A = Agricultural production is increased.
A. (W → A)
B. W → A
C. A → W
D. (A → W)
2. Using the letters provided below, symbolize the claim: “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.”
W = Wildlife are threatened.
A = Agricultural production is increased.
P = The use of pesticides is continued.
A. (A → P)  (P → W)
B. (A → P)  (P → W)
C. (P → A)  (W → P)
D. (A → P)  (P → W)
3. Using the letters provided below, symbolize the claim: “Together, the continued use of
pesticides and the increase in agricultural production will guarantee that wildlife will be
threatened.”
W = Wildlife are threatened.
A = Agricultural production is increased.
P = The use of pesticides is continued.
A. W → (P & A)
B. (P & W) → A
C. (A & W) → P
D. (P  A) → W

4. Using the letters provided below, symbolize the claim: “Wildlife will not be threatened
provided we do not continue the use of pesticides.”
W = Wildlife are threatened.
P = The use of pesticides is continued.
A. P → W
B. (P → W)
C.( P → W)
D. (W → P)
5. Using the letters provided below, symbolize the claim: “Agricultural production will not
increase even though the use of pesticides will continue.”
A = Agricultural production is increased.
P = The use of pesticides is continued.
A. P  A
B. (A  P)
C. (A  P)
D. P → A
6. Determine the interpretation showing that the following implication claim is false:
P  Q, (Q  R) → S, (P → R) = R → S
A. I (P) = T, I (Q) = F, I (R) = F, I (S) = F
B. I (P) = T, I (Q) = T, I (R) = T, I (S) = F
C. I (P) = T, I (Q) = F, I (R) = T, I (S) = F
D. I (P) = T, I (Q) = F, I (R) = T, I (S) = T
The general procedure here is:
1. Assume that the claim is false – by making all premises true and the conclusion false.
2. Typically starting with the conclusion, develop the truth values for all sentence letters.
3. As you go along, eliminate all interpretations that don’t fit.

7. Determine whether the following implication claim is true or false:
P  (Q → R), S → (P  R) = S → Q
A. True
B. False
The general procedure to determine the truth value of an implication claim is this:
1. Assume (for reductio) that the implication claim is false – by making all premises true
and the conclusion false.
2. Typically starting with the conclusion, develop the truth values for all sentence letters.
3. (a) If you obtain a contradictory evaluation T/F somewhere, then the overall assumption
is false; so, the implication claim is true [contradiction => implication].
(b) If you do not obtain a contradictory evaluation T/F, then the overall assumption is
true; so, the implication claim is false [no contradiction => no implication].
8. Determine the interpretation showing that the following implication claim is false:
(L  S), (P  Q) → L = Q  S
A. I (L) = F, I (S) = T, I (P) = F, I (Q) = F
B. I (L) = F, I (S) = F, I (P) = F, I (Q) = T
C. I (L) = F, I (S) = F, I (P) = F, I (Q) = F
D. I (L) = F, I (S) = F, I (P) = T, I (Q) = T
9. Determine the interpretation showing that the following implication claim is false:
P → (T  R), (R → S)  T, (S  Q) = Q → P
A. I (P) = F, I (T) = T, I (R) = T, I (S) = F, I (Q) = T
B. I (P) = F, I (T) = T, I (R) = T, I (S) = F, I (Q) = F
C. I (P) = T, I (T) = T, I (R) = T, I (S) = F, I (Q) = F
D. I (P) = T, I (T) = T, I (R) = T, I (S) = F, I (Q) = T
10. Determine the interpretation showing that the following implication claim is false:
P → (Q  R), R  S, (S  T) = (P  T)
A. I (P) = T, I (Q) = T, I (R) = T, I (S) = T, I (T) = F
B. I (P) = F, I (Q) = T, I (R) = T, I (S) = T, I (T) = T
C. I (P) = T, I (Q) = T, I (R) = T, I (S) = T, I (T) = T
D. I (P) = T, I (Q) = T, I (R) = F, I (S) = T, I (T) = T

11. Determine which of the lettered claims below is equivalent to the following: “Steve can give
blood if he has been tested.”
A. If Steve can give blood, then he has been tested.
B. If Steve has been tested, then he can give blood.
C. Steve cannot give blood, and he has not been tested.
D. Steve has not been tested, but he can give blood.
12. Determine which of the lettered claims below is equivalent to the following: “The gun cannot
be sold unless it has a trigger lock.”
A. Only if the gun has a trigger lock can it be sold.
B. The gun has no trigger lock, but it can be sold anyway.
C. If the gun cannot be sold, then it has no trigger lock.
D. If the gun has no trigger lock, then it can be sold.
13. Determine which of the lettered claims below is equivalent to the following: “The gun can be
sold even though it has no trigger lock.”
A. Only if the gun has a trigger lock can it be sold.
B. The gun has no trigger lock, but it can be sold anyway.
C. If the gun cannot be sold, then it has no trigger lock.
D. If the gun has no trigger lock, then it can be sold.

14. Determine the inference rule justifying the truth of the following derivation claim:
P → Q, P -NK Q
A. modus tollens
B. chain argument
C. disjunctive argument
D. modus ponens
Note:
Moore&Parker do not use the symbol “-NK “. Instead, they use the symbol “/” to indicate
provability (or derivability).
In our notation, the symbol “-NK “ (that is, the single turnstile symbol subscripted by “NK”
indicates syntactic consequence, whereas the double turnstile symbol “=” indicates semantic
consequence (which involves truth and falsehood).
If s -NK t (read: s proves t), then t “follows” from s merely syntactically, i.e. because of the
shape of the used symbols and the proof rules allowing to manipulate them in certain ways.
On the other hand, if s = t (read: s implies t), then t “follows” from s semantically, i.e. because
there is no interpretation (or row on a truth table) that makes s true and t false.
For the purposes of this course, all Group 1 and some Group 2 inference rules in Moore&Parker
are relevant. (cf. p.318, 11th ed.)
15. Determine the inference rule justifying the truth of the following derivation claim:
P  Q - NK (P  Q)
A. double negation
B. contraposition
C. DeMorgan law
D. association
16. Determine the inference rule justifying the truth of the following derivation claim:
(P  Q) - NK (P  Q)
A. double negation
B. commutation
C. contraposition
D. DeMorgan law

17. Determine the inference rule justifying the truth of the following derivation claim:
P  (Q  R) - NK (P  Q)  R
A. commutation
B. contraposition
C. DeMorgan law
D. association
18. Using the letters provided below, symbolize the claim: “If we plant from seed, we’ll have to
plant annuals.”
A = We plant annuals.
S = We plant from seed.
A. S → S
B. A → S
C. S → A
D. S  A
19. Using the letters provided below, symbolize the claim: “We cannot plant perennials unless
we plant from cuttings.”
P = We plant perennials.
C = We plant from cuttings.
A. P → ¬C
B. (¬P  C )
C. P  ¬C
D. P  C
20. Determine the inference rule justifying the truth of the following derivation claim:
A → B, B → A - NK A → A
A. constructive dilemma
B. chain argument
C. destructive dilemma
D. contraposition