i. Something that can be proven without premises.
In shorthand: "... one zilch here ..." |- "Some Snazzy Greek Letter Here"
But that's a "COS261 answer", so I need to look up the new version.
ii. OK I go for the earlier style answer first, and then I'll look up the book.
A tautology is something whose truth table evaluates to T for every possible valuation of the atomic propositions of its premises.
Then we have T whenever all T: That's "entails" (falls short of a tautology).
So now I look and see how wildly wrong I am...
Tautology ... almost exactly right. Neat shorthand answer would be " ... zilch ..." "|=" Phi. So a tautology is to semantic entailment as a theorem is to proof-by-rules.
And I'm not too terribly wrong with the "entails" story either.
Phi. Rho. Chi. Phi2. More Premises Here. |= Poseidon.
The symbol |= is expressed in English as "semantically entails", meaning those premises semantically entail some conclusion. (Taut has no prem).
But what does "semantically entails" mean?? ... Whenever ... (see up there somewhere, above).
iii. These are logically equivalent things. They express exactly the same relationship in different ways.
So one could expand on that and say that Soundness and Completeness are equivalent.
Outside the exam room I find saying it several different ways sometimes helps it stick a bit better. Yes, in an exam one would want to be a lot more terse. Sadly, sometimes I get distracted by interesting questions during exams ...