1. (∀x)[Ax ⊃ (Bx ⊃ Cx)]
2. ∼(∀x)(Bx ⊃ Dx)
-Which of the following propositions is derivable in M from the given premises?
A) (∀x)(Ax ⊃ Dx)
B) (∀x)Ax ⊃ (∃x)(Cx • ∼Dx)
C) (∀x)Ax ⊃ (∀x)Dx
D) (∀x)[Ax ⊃ (Cx • ∼Dx)]
E) (∀x)Ax ⊃ (∀x)(Cx • ∼Dx)

Question

Quizplus · Accepted Answer

From the premises, we cannot derive that for all x, if A then D (A), or that for all x, if A then for all x, D (C), directly. The negation of the universal conditional in premise 2 suggests there is at least one x for which Bx does not imply Dx, which can be interpreted as there exists an x such that Bx and not Dx. Given the first premise, if Ax then Bx implies Cx, it follows that if Ax, then it's possible to have Cx and not Dx for at least one x, leading to the correct answer (B) (∀x)Ax ⊃ (∃x)(Cx • ∼Dx), which correctly captures the logical relationship implied by the premises.

1. (∀X)[Ax ⊃ (Bx ⊃ Cx)]