[Math] conclusions/rules of inference

logic

I need to find the relevant conclusion/conclusions from the following premises and explain the rules of inference used to obtain each conclusion from the promises:

Problem:
"All foods are healthy to eat do not taste good". "Tofu is healthy to eat" "You only eat what tastes good". "Yo do not eat tofu" "Cheeseburgers are not healthy to eat"

My answer(without explaining the details of how I derived the rules of inference):
Conclusion 1: Tofu does not taste good (universal modus ponens)
Conclusion 2: You don't eat Tofu since you only eat what tastes good (simplification and conjunction)
Conclusion 3: Cheeseburgers taste good (universal modus ponens)
Conclusion 4: You eat cheeseburgers since you only eat what tastes good (this one makes sense although I'm not sure how to derive it from a rule of inference, any help would be appreciated)

Are my answers correct or did i miss anything?
Thanks!

Best Answer

Given: "All foods [that] are healthy to eat do not taste good". "Tofu is healthy to eat." "You only eat what tastes good." "You do not eat tofu." "Cheeseburgers are not healthy to eat."

Comments on your conclusions:

Conclusion 1: Tofu does not taste good (universal modus ponens)

Let us say, universal instantiation of the first given statement, to “If Tofu is healthy, it does not taste good,” followed by modus ponens. So far, you are basically on track.

Conclusion 2: You don't eat Tofu since you only eat what tastes good (simplification and conjunction)

The statement "You don't eat Tofu" is apparently given, so there's no need to derive it. But suppose it is not given. Then read “You only eat what tastes good” as “For all things, if you eat it, it tastes good” or, equivalently, “For all things, if it doesn’t taste good you don’t eat it.” Then instantiate that for tofu, and then from “Tofu does not taste good” (Conclusion 1) this second conclusion follows by another round of modus ponens. (I'm sidestepping the treatment of "since" because I don't think that's easily handled at this level of logic.)

Conclusion 3: Cheeseburgers taste good (universal modus ponens)

This conclusion doesn’t follow. Basically we’re told that healthy foods do not taste good, but we're not told that all unhealthy foods do taste good. We're also not told that you eat cheeseburgers. So we have no information warranting the conclusion that cheeseburgers taste good.

Conclusion 4: You eat cheeseburgers since you only eat what tastes good (this one makes sense although I'm not sure how to derive it from a rule of inference, any help would be appreciated)

If we knew that you eat cheeseburgers (we do not), we would know they taste good, since you only eat good-tasting things. But from the fact that you eat only good things it does’t follow that you eat all good things. Perhaps you’re also a vegetarian, and eat only good foods that are meatless.

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