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): $\lnot H_{Cheeseburger}$
  1. 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?



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 : $\forall x (E(x) \to G(x))$.


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