Is there a sentence $\phi$ that is entailed by all sentences with the same properties as $\phi$? Does $\phi$ entail all of these sentences?

Question

I'm in an introductory logic class, and I have both of these problems.

1. Is there a sentence $\phi$ with the following properties? If so, what is it? If not, why?

   • $\phi$ together with Fa entails Fb
   • $\phi$ is entailed by all the sentences that, together with Fa, entail Fb

2. Is there a sentence $\phi$ with the following properties? If so, what is it? If not, why?

   • $\phi$ together with Fa entails Fb
   • $\phi$ entails all the sentences that, together with Fa, entail Fb.

For (1), I've reached the conclusion that we want to see if $\phi$ can be derived from all the sentences $\psi$ in the set $\Gamma$ of sentences that, together with Fa, entail Fb.

For (2), I've reached the conclusion that we want to see if every $\psi$ can be derived from one $\phi$.

I reached these conclusions by noting that, by Completeness theorem and Soundness theorem, a sentence is a logical consequence of a set of premises if and only if it is derivable from those premises.

However, I am totally lost on how to prove if my conclusions can be satisfied. On one hand, I feel like the sentence $Fa \rightarrow Fb$ should be able to derive all other satisfactory sentences and vice versa. But on the other hand, the juxtaposition of these two problems leads me to believe that only one of them has a satisfactory sentence $\phi$, and I have no clue how I would determine such a thing if that is the case.

Answer 1

For 1, you can indeed take $Fa \to Fb$:

1. Together with $Fa$, it implies $Fb$

2. It is entailed by any $\psi$ that, together with $Fa$ implies $Fb$

Proof: take any such a $\psi$ and any interpretation that sets $\psi$ to true. Now, if this interpretation does not set $Fa \to Fb$ to True, then that means that it sets $Fa$ to True but $Fb$ to false. So, this interpretation would set $\psi$ and $Fa$ to True but $Fb$ to False, which contradicts that $\psi$ together with $Fa$ implies $Fb$. So, any interpretation that sets any such $\psi$ to True must also set $Fa \to Fb$ to True.

For 2, you can simply take $\bot$: this statement by itself implies everything, and so will:

1. Together with $Fa$ imply $Fb$ ... because $\bot$ implies $Fb$

2. Imply any $\psi$ for which it is true that $\psi$ and $Fa$ together imply $Fb$ ... because $\bot$ implies any $\psi$