1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)]
3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz]
4. (∀x)(Ex ⊃ Bx)
-Which of the following propositions is derivable from the given premises in F?
A) (∀x){Ex ⊃ [(∃y)Cxy ⊃ (∃y)Cyx]}
B) (∀x)[Bx ⊃ (Ex $ \lor $ Ax)]
C) (∀x)[Ax ⊃ (∀y)(By ⊃ Cyx)]
D) (∀x)[Ax ⊃ (∀y)(Hy ⊃ Cxy)]
E) (∀x){Ax ⊃ (∀y)[(Hy • By) ≡ Cxy]}

Question

Quizplus · Accepted Answer

From premise 1, we can use Universal Instantiation to get:
(Aa ⊃ (∀y)(By ⊃ Cay))
Then, we can use Existential Instantiation on premise 2 to obtain:
Ea and (∀y)(Hy ⊃ Cay)
We can then use Simplification to get:
Hy ⊃ Cay
Using Universal Instantiation on premise 3, we can obtain:
(Cab • Cbc) ⊃ Cac
We can then use Universal Instantiation on our previous results to get:
(Cay • Cby) ⊃ Cax
Using this and our simplified version of premise 1, we can use Universal Generalization to get:
Ax ⊃ (∀y)(Hy ⊃ Cxy)
which matches option D.

1 (∀X)[Ax ⊃ (∀Y)(By ⊃ Cxy)] 2 ∨ \lor ∨ Ax)] C) (∀X)[Ax ⊃ (∀Y)(By ⊃ Cyx)] D) (∀X)[Ax ⊃

1 (∀X)[Ax ⊃ (∀Y)(By ⊃ Cxy)]
2 $\lor$ Ax)]
C) (∀X)[Ax ⊃ (∀Y)(By ⊃ Cyx)]
D) (∀X)[Ax ⊃