does deductive completeness implies semantic completeness

Question

i wanted to understand godel's $ \ \bf completeness \ $ theorem, so while doing some research on google i found this wikipedia page " https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem#:~:text=G%C3%B6del's%20completeness%20theorem%20is%20a,simplest%208%20are%20shown%20left). " saying -

A first-order formula is called logically valid if it is true in every structure for the language of the formula (i.e. for any assignment of values to the variables of the formula). To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system. A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense. Thus, in a sense, there is a different completeness theorem for each deductive system. A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system.

Its the 2nd para under $ \bf Preliminaries \ $ section.

" A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense. " - does this also means if a formula has a formal deduction then it must be semantically valid ??

Answer 1

Yes, this is the soundness theorem. Technically this is a separate result from the completeness theorem, but it is - comparatively - so simple that it's often not mentioned explicitly.

The proof is by induction on the complexity of the deduction. Basically, you need to show that each of the rules of your system are "truth-preserving." Of course the details will depend on your system, but for every system I'm familiar with this is quite straightforward. (Especially in conjunction with the proof of completeness/compactness of propositional logic, this highlights the difficulty in proving the completeness of first-order logic.)

Answer 2

Complementing Noah's answer, to address the question in your title: Syntactic completeness implies "semantic completeness" (in the sense that every formula is either valid or its negation is) provided the deductive system is sound, i.e. that the deductibility of a formula guarantees its validity. Any useful deductive system has the soundness property, but it is not automatically baked into it just because it is a deductive system, and has to be proven separately.

Answer 3

Atleast according to godel's completeness theorem, if you can give a formal deduction to a formula, given that the deduction contains axioms or other formally deductable formulas ( thus logically valid ), then that formula too must be logically valid.. so in this condition you can say if the system is deductively complete, all valid formulas have deductions, thus all are logically valid, therefore the system is semantically complete..