# Thread: Could anyone help me with this problem on the interpretation of first-order logic?

1. ## Could anyone help me with this problem on the interpretation of first-order logic?

I am a student of logic who just became familiar with first-order predicate logic. I got stuck in this problem. Although I was hinted to translate→into ¬,∨ to solve the problem, I still have no idea what to do next. Here is the problem:
Use semantics under the substitution to check whether 1> ∃x (Pa→Qx) ⇔Pa →∃x(Qx)
2> ∃x (Px→Qa) ⇔∃x(Px) →Qa. If either 1> or 2> fails, can you suggest a rectification?

2. ## Re: Could anyone help me with this problem on the interpretation of first-order logic

Originally Posted by yytsau
I am a student of logic who just became familiar with first-order predicate logic. I got stuck in this problem. Although I was hinted to translate→into ¬,∨ to solve the problem, I still have no idea what to do next. Here is the problem:
Use semantics under the substitution to check whether 1> ∃x (Pa→Qx) ⇔Pa →∃x(Qx)
2> ∃x (Px→Qa) ⇔∃x(Px) →Qa. If either 1> or 2> fails, can you suggest a rectification?
$\begin{array}{l} \left( {\exists x} \right)\left[ {P(a) \to Q(x)} \right]\\ \left( {\exists x} \right)\left[ {\neg P(a) \vee Q(x)} \right]\\ \neg P(a) \vee \left( {\exists x} \right)\left[ {Q(x)} \right]\\ \left( {P(a) \to \left( {\exists x} \right)\left[ {Q(x)} \right]} \right) \end{array}$