# First order Logic deducing theorems

We are in First Order Logic and I have a set of rules and clauses ( = axioms).

I use a sound and complete deductive system and the rules of inference in order to derive some formulas from the initial set.

Is it correct to call "theorems" these derived formulas?

Yes. To provide more context:

• A theory is a deductively closed collection of formulas
• We call the individual formulas the theorems of the theory

Any collection of formulas can be used to generate a theory, which is the smallest theory containing those axioms. When we do this, we call the individual formulas from the starting collection "axioms".

The theorems of the theory generated from a set of axioms are precisely the formulas that can be derived from the rules of inference.

(I assume throughout this post a fixed collection of rules of inference)

• Soundness and completeness belongs only to "theorems of the theory" or also to theorems derived from rules/facts ? – Qwerto Mar 1 '18 at 18:45
• @Qwerto: I'm not sure I understand your question; as I read it, they're completely unrelated. The notions of soundness and completeness are properties of the deductive system itself (or more precisely its relation to some other means of judging formula) -- they have nothing to do with any particular theory or its theorems. E.g. the "soundness and completeness" theorems of classical logic say that syntactic entailment (you can derive the consequence from the premise) is the same relation as semantic entailment (in every interpretation where the premise holds, so does the consequence). – Hurkyl Mar 2 '18 at 0:24

It would probably be better to call them theorems of the theory.

For example, $x(x^2)=(x^2)x$ is a theorem of first-order group theory - calling it just a "theorem" seems dubious.

Yes: theorems are true statements that you can prove from inside the system.

• Thanks. I asked this because my prof called theorems the formulas derived only from an axiom scheme ( that is , when the set of rules and facts is empty) – Qwerto Mar 1 '18 at 17:52
• There is no notion of truth here. – Hurkyl Mar 1 '18 at 17:59

'Theorems' are those things that you can derive from axioms.

The axioms specify the subject matter, and hence what the theorems are theorems 'of' or 'about'

Thus, if the axioms are basic logical tautologies, then the theorems are theorems of logic .. which are further tautologies.

If the axioms are about set theory, or number theory, then the theorems are about those areas. Etc.