determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
-1. (∀x)(Ax ⊃ Dex)
2) (∃x)(Bx • Dxe)
3) (∀x)(Bx ⊃ Ax) / (∃x)[Ax • (Dxe • Dex)]
A) Valid
B) Invalid. Counterexample in a domain of one member, in which: Ae: True Be: False Dee: False
C) Invalid. Counterexample in a domain of two members, in which:
Aa: True Ba: True Daa: True Dea: False
Ae: True Be: True Dae: False Dee: False
D) Invalid. Counterexample in a domain of two members, in which:
Aa: True Ba: False Daa: True Dea: True
Ae: False Be: False Dae: False Dee: True
E) Invalid. Counterexample in a domain of two members, in which: Aa: False Ba: True Daa: False Dea: False
Ae: True Be: True Dae: True Dee: False

Question

Quizplus · Accepted Answer

The argument is valid because the premises logically lead to the conclusion. Given that for all x, if Ax then Dex (1), and there exists an x such that Bx and Dxe (2), and for all x, if Bx then Ax (3), it logically follows that there exists an x such that Ax and (Dxe and Dex), satisfying the conclusion. The premises ensure that any x that is Bx is also Ax, and since there exists at least one Bx that is Dxe, and all Ax are Dex, the conclusion necessarily follows.

Determine Whether the Given Argument Is Valid or Invalid