Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant.
-Which of the following is the best translation into M of this argument?
A) (∀x)(Px ⊃ ∼Cx) (∀x)(Cx ⊃ Nx) / (∃x)(∼Nx • Px)
B) (∀x)(Px ≡ ∼Cx) (∀x)(Cx ⊃ Nx) / ∼(∃x)Nx ⊃ (∃x)Px
C) (∀x)(Px ⊃ ∼Cx) (∀x)(Cx ⊃ Nx) / (∃x)∼Nx ⊃ (∃x)Px
D) (∀x)(Px ≡ ∼Cx) (∀x)(Cx ⊃ Nx) / (∃x)∼Nx ⊃ (∃x)Px
E) (∀x)(Px ≡ ∼Cx) (∀x)(Cx ⊃ Nx) / (∃x)(∼Nx • Px)

Question

Quizplus · Accepted Answer

The argument structure is best captured by choice D, which translates the premises and conclusion accurately into logical form. The first premise states that things are pleasant if and only if they are not too crowded, which is represented as (∀x)(Px ≡ ∼Cx). The second premise states that everything too crowded is noisy, represented as (∀x)(Cx ⊃ Nx). The conclusion drawn is that if something isn't noisy, then something is pleasant, which is logically represented as (∃x)∼Nx ⊃ (∃x)Px. This choice correctly captures the bi-conditional relationship in the first premise and the conditional relationship in the conclusion.

Things Are Pleasant If, and Only If, They Are Not