question about Herbert B. Enderton's book : A mathematical introduction to logic

Question

I hope someone can help me. My question arises on page 114 of the second edition of the book. Here the notion of 'prime formula' is introduced to enable one to view a formula as a formula of sentential logic and as a formula of first order logic at the same time. One way to do this is to let the prime formulas be the sentential symbols. So far so good - now to my question: In order to guarantee unique readability of strings of symbols it is demanded in the book that the symbols of sentential logic (including of course the sentential symbols) be such that no symbol is a finite string of other symbols. This is explained in detail on page 14. Now if I choose prime formulas a my 'new' sentential symbols I have no idea how to show that this holds for the 'new' set of symbols containing:

Negation symbol, implication symbol, brackets and prime formulas (which themselves are strings of symbols potentially containing for example the negation symbol)

The problem arises mainly because tuples are treated in a set-theoretic way (as sets) in the book as explained in the preliminary chapter 0 on page five.

I hope I managed to make clear what exactly the question is - I did my best to do so...

Answer 1

I think that this is not a real issue.

A formula of first-order logic is not made of propositional symbols.

The correct syntactical rules are "managed" in page 105-on, with a "new" proof of the UNIQUE READABILITY THEOREMs for terms and formulae.

Is only in showing that we can "reuse" the tautologies as valid f-o formulae (and thus as axioms) [see page 114] :

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.

that we use the "subterfuge" of :

[taking] the sentence symbols to be the prime formulas of our first-order language. Then any tautology of sentential logic (that uses only the connectives $¬,→$) is in axiom group 1. There is no need to replace sentence symbols here by first-order wffs; they already are first-order wffs.

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Added

My understanding is this :

(i) Review the sintax of First-Order Languages (page 69) and compare with The Language of Sentential Logic (page 13).

The following :

$(A_1 \rightarrow A_1)$ --- (A)

is a well-formed formula of sentential logic but not of first-order logic.

And it is a tautology.

(ii) The following :

$(A_1^1v_1 \rightarrow A_1^1v_1)$

is a well-formed formula of first-order logic (specifically, of Pure predicate language (see page 70).

According to the definition of logical axiom (page 112), the following

$\forall v_1(A_1^1v_1 \rightarrow A_1^1v_1)$ --- (B)

in an instance of a logical axiom group 1, i.e. the generalization of a (first-order) instance of a tautology, because - accdording to the explanation on page 114 :

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.

Formula (B) is the generalization of the formula obtainable from (A) by replacing the sentence symbol $A_1$ by a the wff of the first-order language $A_1^1v_1$, which is an atomic formula, and so a prime one.

(iii) I think that Enderton's note (page 114) is an euristic suggestion; instad of starting from a tautology and find the suitable substitution which gives us the "corresponding" f-o formula, we can start directly from the f-o formula "reading it" as a sentential one, where in place of sentential letters we have prime formulas. In this way we can take benefit of the truth-table device to check if the f-o formula is an instance of an axiom group 1.