Vtyjkyuk

Taken from P. Suppes "Introduction to logic" pp16

Starting from the sentence "If $P$ tautologically implies $Q$, then.."

Question: Is it true that, at least within propositional logic, we are able to prove only tautologies or do I mis-understand something?

If yes, is propositional logic special in some way? Is the notion of tautology meaningful only in propositional logic?

1. Suppose we are given the premiss '$P land Q$', then we can of course trivially derive '$Q$'. Further, the premiss $P land Q$' is said to tautologically entail the conclusion '$Q$' -- essentially because the corresponding conditional '$(P land Q) to Q$' is a tautology.


2. But note, '$P land Q$' and '$Q$' themselves may not even true let alone be tautologies (perhaps '$P$' for example says 'Trump is a philosopher' and '$Q$' says 'Trump is a modest man'). That's fine. In propositional logic, we can prove a non-tautology, given one or more non-tautologies as premisses!


3. However, if we are given no premisses to work with, if we have to rely on propositional logic reasoning alone, then yes, all we can prove in that case are tautologies.


4. Is the notion of tautology meaningful only in propositional logic? Terminology varies. But I think mainstream usage is to reserve "tautology" primarily for sentences of the language of propositional logic that come out true on every line of the appropriate truth-table, and hence are provable from no premisses in an appropriate classical propositional logic.

It's not difficult to see that the following assertions are equivalent.

1. The statement $mathcal B$ follows from the statements $mathcal A_1, mathcal A_2,ldots ,mathcal A_n$.


2. The implication $mathcal A_1landldotslandmathcal A_nimplies mathcal B$ is a tautology.

Not as easy to suggest this might be the case, though, without any a priori intuition.