HON 102 - HW 5

Construct truth tables and indicate which of the following LSL-sentences are equivalent to each other: (a) (¬Q ∨ P) (b) (¬P → ¬Q) (c) Q → (Q → P) (d) ¬(Q ∨ ¬P) (e) ¬(¬P & Q) (f) ¬(¬(P ↔ ¬Q) ∨ (Q & ¬P))

(a), (b), (c) and (e); (d) and (f)

Assume you are creating a list of LSL-sentences which contain no sentence letters other than P, Q, and R (where each of these three can be used as many times as you want). What is the largest number of sentences which can be placed on the list without there being any two equivalent sentences?

256

Which of the following strings of symbols is a grammatically well-formed LSL-sentence?

B. and C.

"If John goes to the party, he will enjoy himself." The negation of this sentence (which is true when the given sentence is false and false when the given sentence is true) is:

John goes to the party, but he will not enjoy himself.

"Either Jack or Jill ride a bicycle." The negation of this sentence (which is true when the given sentence is false and false when the given sentence is true) is:

Neither Jack nor Jill ride a bicycle.

Relative to the interpretation I (P) = T, I (Q) = F, I (R) = F, I (S) = T, I (T) = F, and I (U) = F, the following LSL-sentence is: ((((R → ¬S) ∨ T) → (P & (U ∨ ¬Q))) & (P ∨ S))

True

Suppose, for the sake of the argument, that it is true that Jim wins the lottery. Now, it is true that 2 + 2 = 4. This shows that the unary connective "it is very likely that P" is not truth-functional.

True

The negation of a tautology is always a contradiction.

True

"If the budget deficit goes up, it will be due to tax cuts. On the other hand, if the deficit goes down, it will be due to spending cuts. Since it must either go up or come down, it is clear that either taxes or spending will be cut." This argument is:

deductively valid and unsound

"No number that is evenly divisible by two is prime. So, four cannot be prime, since it is evenly divisible by two." This argument is:

deductively valid and unsound

Determine the main connective of the following LSL-sentence: ¬((P → Q) ↔ (¬(R & ¬S) & T))

negation

It is possible to construct a contradictory sentence in LSL using no sentential connectives other than conjunction and disjunction.

False

Relative to the interpretation I (P) = F, I (Q) = T, and I (R) = T, the following LSL-sentence is: ¬(P → ¬(Q ∨ R))

False

Suppose, for the sake of the argument, that it is true that the number of planets in our solar system is identical with the number 8 (poor Pluto!). Now, it is true that the number eight is identical with itself. This shows that the unary connective "it is necessary that P" is truth-functional.

False

The following string of symbols is a grammatically well-formed LSL-sentence: (P ¬→ (Q & R))

False

The following string of symbols is a grammatically well-formed LSL-sentence: (¬(((P → Q) ∨ ¬R) & (Q ↔ ¬Q)))

False

The negation of a contingent sentence is always a tautology.

False

Without constructing a truth table, determine the interpretation that makes the following LSL-sentence true: (((A → ¬B) ∨ (B → ¬A)) → C)

I (A) = T, I (B) = T, I (C) = F

Without constructing a truth table, determine the interpretation that makes the following LSL-sentence true: ((¬P & ¬Q) & ¬((R → S) ∨ (¬Q → S)))

I (P) = F, I (Q) = F, I (R) = T, I (S) = F

Determine the main connective of the following LSL-sentence: (((¬P → Q) ↔ (¬(A ∨ B) → (Q → B))) → C)

Conditional