Inference using Biconditional statements (For each of these sets of premises, what relevant conclusion(s) can be drawn? Rosen 8th Ed

Question

1. All foods that are healthy to eat do not taste good (Premise): $\forall x (H(x) \to \lnot G(x))$
2. Tofu is healthy to eat. (Premise): $H_{Tofu}$
3. You do not eat tofu (Premise): $\lnot E_{Tofu}$
4. Cheeseburgers are not healthy to eat (Premise): $\lnot H_{Cheeseburger}$

5. You only eat what tastes good. (Premise): $$\forall x (E(x) \to G(x)) $$ or? $$\forall x (E(x) \leftrightarrow G(x)) $$

Solving for conclusions is as follows:

$6.H_{Tofu} \to \lnot G_{Tofu}$ (Universal Instantiation 1.)

$7.\lnot G_{Tofu}$ (Modus ponens 2. & 6.) Conclusion 1: Tofu taste bad

$8.\lnot H_{Tofu} \lor \lnot G_{Tofu}$ (and Logical Equivalence 6.)

$9. E_{Tofu} \to G_{Tofu} \equiv \lnot E_{Tofu} \lor G_{Tofu}$ (Universal Instantiation 5. and Logical Equivalence)

$10.\lnot E_{Tofu} \lor \lnot H_{Tofu} \equiv E_{Tofu} \to \lnot H_{Tofu}$ (Resolution 8. & 9.)

$11. \forall x(E(x) \to \lnot H(x)) $ (Universal Generalization of 10. with 4.) Conclusion 2: All foods you eat are not healthy

Am I correct in assuming premise 5. is not a biconditional statement.

Also is 11. the best conclusion that I can draw, should I look to an Existential Generalization?

Answer 1

Hint

Your translation in symbols of 5 :

You only eat what tastes good,

is correct.

Premise 5 means that you do not eat something that does not taste good, i.e. $\lnot \exists x (E(x) \land \lnot G(x))$.

This, in turn, is equivalent to : $\forall x (E(x) \to G(x))$.