A first-order sentence is (logically) valid iff it's true in every interpretation. And it's valid iff it can be deduced from the FO axioms alone.
One normal case of showing that a FO sentence is true is deducing it (syntactically).
I guess that indirect proofs have to be interpreted more "semantically":
"Assume that ~F is true in an interpretation. [Deduction of a contradiction] Thus ~F couldn't have been true. Thus F must be true in every interpretation." (But we are left without a deduction of F from the axioms.)
Is this reading of indirect proofs correct?
Are there also direct proofs that work "semantically", i.e. that show directly that a sentence is true in every interpretation?
Like truth tables do.