Given the argument: (A • B) ≡ (C ∨ D) / A ⊃ E / B ⊃ F // C ⊃ (E • F).
This argument is:
A) Cogent.
B) Invalid.
C) Sound.
D) Valid.
E) Uncogent.

Question

Quizplus · Accepted Answer

The argument is valid. 
To prove this, we can use the method of logical equivalences:
(A • B) ≡ (C ∨ D)    (given)
(A ⊃ E)               (given)
(B ⊃ F)               (given)
Therefore:
(A ⊃ E) ≡ (~A ∨ E)         (material implication)
(B ⊃ F) ≡ (~B ∨ F)         (material implication)
(A • B) ≡ (~( ~A ∨ E) ∨ ~( ~B ∨ F))    (substituting equivalent forms)
(A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F)          (De Morgan's law)

Now, we want to show that C ⊃ (E • F) is a logical consequence of the three premises:
(A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F)
(A ⊃ E)
(B ⊃ F)

Assume C for conditional proof:
We need to prove E • F
From (A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F) and (A ⊃ E), we can infer ~A ∨ E.
Similarly, from (A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F) and (B ⊃ F), we can infer ~B ∨ F.
Since C ≡ (A • B) ∨ D by the first premise, we know one of two things:
1. A • B is true, in which case either ~A ∨ E or ~B ∨ F must be true. Consequently, E • F must be true because E and F are the only two remaining options.
2. D is true, in which case the conclusion still holds since the conditional is vacuously true when the antecedent is false.

In either case, we have shown that C ⊃ (E • F) is a logical consequence of the premises. Therefore, the argument is valid.

Given the Argument: (A • B) ≡ (C ∨ D)