Let 𝜈(𝑥) mean 𝑥 is a vegetarian, 𝑚(𝑦) for 𝑦 is meat, and 𝑒(𝑥,𝑦) for 𝑥 eats 𝑦. Based on these,…

Question

Let 𝜈(𝑥) mean 𝑥 is a vegetarian, 𝑚(𝑦) for 𝑦 is meat, and 𝑒(𝑥,𝑦) for 𝑥 eats 𝑦. Based on these, consider the following sentences :

I. ∀𝑥∨(𝑥) ⇔ (∀𝑦 𝑒(𝑥,𝑦) ⟹ ¬𝑚(𝑦))

II.∀𝑥∨(𝑥) ⇔ (¬(∃𝑦 𝑚(𝑦) ∧ 𝑒(𝑥,𝑦)))

III.∀𝑥(∃𝑦 𝑚(𝑦) ∧ 𝑒(𝑥,𝑦)) ⇔ (𝑥 ) ⇔ ¬∨(𝑥)

One can determine that

Accepted Answer

Correct answer: D — Answer: All three sentences are equivalent. Reasoning (for an arbitrary individual x): Start with the inner implication: e(x,y) ⇒ ¬m(y) is logically equivalent to ¬(e(x,y) ∧ m(y)). Taking the universal quantifier gives ∀y (e(x,y) ⇒ ¬m(y)) ⇔ ∀y ¬(e(x,y) ∧ m(y)), and by quantifier negation this is equivalent to ¬∃y (e(x,y) ∧ m(y)). Therefore ν(x) ⇔ ∀y (e(x,y) ⇒ ¬m(y)) is equivalent to ν(x) ⇔ ¬∃y (e(x,y) ∧ m(y)). This shows sentence I is equivalent to sentence II. Finally, an equivalence of the form ν(x) ⇔ ¬E (where E stands for ∃y (e(x,y) ∧ m(y))) can be rewritten by symmetry of ⇔ as E ⇔ ¬ν(x). That is exactly the form of sentence III. Combining these steps, I ⇔ II ⇔ III for every x, so all three sentences state the same condition.

Answer

A. Only 𝐼 and 𝐼𝐼 are equivalent sentences

Answer

B. Only 𝐼𝐼 and 𝐼𝐼𝐼 are equivalent sentences.

Answer

C. Only 𝐼 and 𝐼𝐼𝐼 are equivalent sentence .

Answer

D. 𝐼,𝐼𝐼, and 𝐼𝐼𝐼 are equivalent sentences.

Let 𝜈(𝑥) mean 𝑥 is a vegetarian, 𝑚(𝑦) for 𝑦 is meat, and 𝑒(𝑥,𝑦) for…