Predicate logic - Symbolizing sentences **K_ = _ is a Kiwi, M_ = _ is a Moa, F_ = _is flightless** **If something is a moa only if it's flightless then if all kiwis are flightless, some kiwis are moas.** $$A_x( (M_x \to F_x) \to ( A_x(K_x \ \& \ F_x) \to E_x(K_x \ \& \ M_x) )$$ **a = Anja P_ = _ is a philosopher W_ = _ is wise** **If all philosophers are wise, then Anja is a philosopher iff she's wise.** $$A_xP_x \to (P_a \leftrightarrow W_a)$$ **H_ = _is a Historian** **If only historians are wise and no philosophers are wise then no historians are philosophers.** $$A_x( (H_x \ \& \ W_x) \ \& \ (P_x \to \ \backsim W_x)) \to (H_x \to \ \backsim P_x) )$$ _Are these symbolizations correct? Thanks in advance! :)_

First two are close!

First one:

$$\forall x( (Mx \rightarrow Fx)\color{red}{)} \rightarrow ( \forall x(Kx \color{red}{\rightarrow} Fx) \rightarrow \exists x(Kx \land Mx) )$$

... actually, rather than adding a close parenthesis, you can just remove the first open parenthesis:

$$\forall x( Mx \rightarrow Fx) \rightarrow ( \forall x(Kx \rightarrow Fx) \rightarrow \exists x(Kx \land Mx) )$$

Second one:

$$\forall x (Px \color{red}{\rightarrow Wx)} \rightarrow (Pa \leftrightarrow Wa)$$

For the third one:

Notice that this is a conditional statement .. so you need a quantifier for the antecedent, but also a quantifier for the consequent ... try again!