# How are tautology axioms in a Hilbert system for FOL obtained from tautologies in Sentential Logic?

SECTION 2.4 A Deductive Calculus In Enderton's A Mathematical Introduction to Logic divides the set of axioms into several groups. The first group is called "tautologies" on p114, which are obtained from tautologies in Sentential Logic:

In the beginning:

Axiom group 1 consists of generalizations of formulas to be called tautologies. These are the wffs obtainable from tautologies of sentential logic (having only the connectives ¬ and → ) by replacing each sentence symbol by a wff of the ﬁrst-order language.

• Is "tautologies" (for FOL) defined as "the wffs obtainable from tautologies of sentential logic (having only the connectives ¬ and → ) by replacing each sentence symbol by a wff of the ﬁrst-order language"?

• Do both "formulas" before "tautologies" and "These" after "tautologies" refer to "tautologies" (for FOL) defined above?

Near the end:

Then any tautology of sentential logic (that uses only the connectives ¬ , → ) is in axiom group 1.

• Is "tautology of sentential logic" exactly "tautology" for FOL?

• Since "Axiom group 1 consists of generalizations of formulas to be called tautologies" at the beginning, isn't it that "any tautology of sentential logic" is not in axiom group 1", but any generalization of "any tautology of sentential logic (that uses only the connectives ¬ , → ) is in axiom group 1"?

Thanks.

How are tautology axioms in a Hilbert system for FOL obtained from tautologies in Sentential Logic?

This is a tautology of sentential logic:

$$(A_1 \to A_1)$$.

These are examples of Axiom group 1 formulas, obtained according to Enderton's specification from the tautology above:

$$\forall x (Px \to Px), (\forall xPx \to \forall xPx)$$.

Is "tautology of sentential logic" exactly "tautology" for FOL?

Obviously not. $$A_1 \to A_1$$ is a formula of sentential logic [see syntactical specifications page 14] but it is not a FOL formula, while $$\forall x (Px \to Px)$$ is a FOL formula [see syntactical specifications page 69] but it is not a sentential logic formula.

Regarding the question about Enderton's quote:

Do both "formulas" before "tautologies" and "These" after "tautologies" refer to "tautologies" (for FOL) defined above?

we have to note that the author is commenting the definition of axioms of FOL. Thus, "formulas" are FOL formulas; they are called "tautologies" because they are the formulas like $$(\forall xPx \to \forall xPx)$$ that can be "produced" form sentential logic tautologies.