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Enderton's An Mathematical Introduction to Logic says on p114

Tautologies

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 $\neg$ and $\to$ ) by replacing each sentence symbol by a wff of the first-order language. ...

There is another, more direct, way of looking at axiom group 1. Divide the wffs into two groups:

  1. The prime formulas are the atomic formulas and those of the form $\forall x \alpha$ .

  2. The nonprime formulas are the others, i.e., those of the form $\neg{\alpha}$ or $\alpha \to \beta$.

Thus any formula is built up from prime formulas by the operations $E_{\neg}$ and $E_{\to}$.

Does "any formula" here mean any wff in FOL, or any formula in the axiom group 1?

Now go back to sentential logic, but take the sentence symbols to be the prime formulas of our first-order language. Then any tautology of sentential logic (that uses only the connectives $\neg$, $\to$ ) is in axiom group 1. There is no need to replace sentence symbols here by first-order wffs; they already are first-order wffs.

It says "There is no need to replace sentence symbols here by first-order wffs". Is there still a need to replace sentence symbols here by the prime wffs?

Conversely, anything in axiom group 1 is a generalization of a tautology of sentential logic. (The proof of this uses Exercise 8 of Section 1.2.)

Does "a tautology of sentential logic" above need to have its sentence symbols replaced with the prime wffs of FOL?

Are there two ways of defining axiom group 1:

  • at the end of the quote: "anything in axiom group 1 is a generalization of a tautology of sentential logic." (with sentence symbols replaced by the prime wffs?)

  • at the beginning of the quote: "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 $\neg$ and $\to$ ) by replacing each sentence symbol by a wff of the first-order language."

?

Do the two definitions define the same axiom group 1? Or is the one defined at the end of the quote is a proper subset of the one defined at the beginning?

Tim
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1 Answers1

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Does "any formula" here mean any wff in FOL, or any formula in the axiom group 1?

Yes, any wff in FOL.

It says "There is no need to replace sentence symbols here by first-order wffs". Is there still a need to replace sentence symbols here by the prime wffs?

No, because the sentence symbols are the prime wffs. As Enderton says "they already are first-order wffs".

Does "a tautology of sentential logic" above need to have its sentence symbols replaced with the prime wffs of FOL?

Same answer. In our context, the sentence symbols of sentential logic are prime wffs of FOL.

Are there two ways of defining axiom group 1?

Yes. (1) As originally defined: Sentences of axiom group 1 are generalizations of wffs obtainable from tautologies of sentential logic by replacing each sentence symbol by a wff of FOL. (2) The reformulation: Sentences of axiom group 1 are generalizations of tautologies of the sentential logic in which the sentence symbols are prime formulas of FOL.

Do the two definitions define the same axiom group 1?

Yes. Why do you think otherwise?

It's clear that this is what Enderton is asserting. He writes "Then any tautology of sentential logic is in axiom group 1... Conversely, anything in axiom group 1 is a generalization of a tautology of sentential logic."

I do think Enderton should have written "Then any generalization of a tautology of sentential logic is in axiom group 1... Conversely, anything in axiom group 1 is a generalization of a tautology of sentential logic."

Alex Kruckman
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  • Thanks. "the sentence symbols are the prime wffs." Does FOL have sentence symbol in its syntax? (By the way, could anyone upvote my post? I have haters on this site) – Tim Oct 09 '23 at 01:29
  • @Tim No, "sentence symbol" (aka "proposition symbol" or "propositional variable") is a notion of propositional/sentential logic. Some presentations of FOL allow $0$-ary relation symbols, which play the same role. But this is not standard: see the discussion here. – Alex Kruckman Oct 09 '23 at 01:35
  • For any set $X$, we can build sentential logic using the elements of $X$ as sentence symbols. Call the resulting sentential language $\mathrm{Sent}(X)$. Enderton's point is that if the elements of the set $X$ are first-order formulas, then the sentences of $\mathrm{Sent}(X)$ are also first-order formulas (in addition to being sentences of sentential logic). – Alex Kruckman Oct 09 '23 at 01:37
  • More on $0$-ary relation symbols here. – Alex Kruckman Oct 09 '23 at 01:47
  • thanks. "Why do you think otherwise?" Because one replaces sentence symbols with any FOL wff, and the other with only prime wff of FOL. I don't know why they can define the same axiom group 1. – Tim Oct 09 '23 at 02:35
  • Consider the sentential tautology $P\lor \lnot P$. Subsituting $R(x,y)\to R(y,x)$ for $P$ and generalizing, we see that axiom group $1$ contains the sentence $\forall x\forall y, (R(x,y)\to R(y,x))\lor \lnot (R(x,y)\to R(y,x))$. But this is also a generalization of a tautology in the sentential logic built from the prime formulas, since whenever $Q$ and $Q'$ are sentence symbols, $(Q\to Q')\lor \lnot (Q\to Q')$ is a tautology [apply this with $Q = R(x,y)$ and $Q' = R(y,x)$]. The fact that this reasoning always works follows from Exercise 8 of Section 1.2., as Enderton has told you. @Tim – Alex Kruckman Oct 09 '23 at 02:47