Translating english to FOL

Question

Note that ‘X’ and ‘Y’ are variables here, they are not particular sentences of TFL.

a) Suppose X is contingent and Y is a contradiction. Is it the case that Y entails X? Explain your answer, using a mock truth table and the definitions of contingent, contradiction, and entailment.

b) Suppose that X ⊢ Y. Does it follow that ¬Y, X is a provable contradiction? Explain your answer, making reference to particular rules of inference.

Answer 1

There's a variety of notations and methods of explanation that can be used in any FOL course, so my answers here may not match exactly what's expected in your particular course, but here are the basic ideas for you.

a) When we ask if Y entails X, we're asking if there's any truth assignment that would make Y true and X false. If there is such a truth assignment, then Y does not entail X. If there is no such truth assignment, then Y does entail X.

Since Y is a contradiction, there is no truth assignment that would make Y true at all. So then there is no truth assignment that would make Y true and X false. So Y entails X.

b) Different classes and texts use different proof notations for derivations and represent their rules differently, so I can't give you the precise answer, but generally, there will be a series of allowed steps according to the rules of inference that you can reference that will be something like what I'll include, Note that the question specifically says that X and Y are not particular sentences in the language, and so you seem to be warned away from trying to make this an actual derivation in the formal system, so an explanation along these lines should work.

1. X ⊢ Y is given. That means whatever sentences in the language X and Y are, X materially implies Y

2. We know then, that ¬Y ⊢ ¬X follows because of Modus Tollens: likely not an inference rule, but likely something you've proved already in your class and so are allowed to appeal to. No matter what X and Y are, if X ⊢ Y is a statement in the language, we'll be able to derive ¬Y ⊢ ¬X from it.

3. ¬Y is given

4. So we can derive from 2 and 3, ¬X. If we have ¬Y ⊢ ¬X, and we have ¬Y, no matter what X and Y are, we can derive ¬X from the standard conditional derivation rule.

5. But we were also given X as a precondition.

6. So now we have our ⊥ (contradiction) because we have both X (5) and not ¬X (4), and we'll have a derivation rule that lets us derive a contradiction from that, no matter what X and ¬X are.

I laid this out in a form that's close to what you'd use for a formal derivation, but also is somewhat chatty. Hope that helps! Feel free to schedule a session with me, if you like!