The definition of truth in First Order Logic

Question

I was reading Mathematical Logic by Chiswell Hodges and there was a definition written in Chapter on Quantifier Free Logic where the the definition for an atomic formula to be true was given by,

(a) If $\chi$ is $R(t_1, . . . , t_n)$, where R is an n-ary relation symbol of $\sigma$ and t1, . . . , tn are terms (necessarily closed since $\chi$ is closed), then $\models_A$ $\chi$ $\quad$ if and only if
$$R_A((t1)_A, . . . , (tn)_A).$$ Here ‘$R_A((t1)_A, . . . , (tn)_A)$’ means that the n-tuple $((t1)_A, . . . , (tn)_A)$ is in the relation $R_A$.

(b) If $\chi$ is $(s = t)$ where $s$ and $t$ are terms (again necessarily closed), then $\models_A \chi$ if and only if $s_A = t_A$.

where $A$ is a $\sigma$-structure.

Then the definition of satisfaction was given by,

Definition 5.7.6. If $\phi$ is atomic then $(a_1, . . . , a_n)$ satisfies $\phi$ in A if and only if $$ \models_A \phi[t_1/y_1, . . . , t_n/y_n]$$ where $t_1, . . . , t_n$ are closed terms such that for each i, $(t_i)_A = a_i$.

Then in the chapter of First Order Logic the author says the following statement,

There is a twist, first noticed by Alfred Tarski. We explained ‘satisfaction’ in terms of ‘true’. But in order to give a formal definition of truth for LR, it seems that we need to go in the opposite direction: first we define satisfaction by recursion on complexity, and then we come back to truth as a special case.

But then immediately the author says the following words,

Let $\sigma$ be a signature, $\phi(y_1, . . . , y_n)$ a formula of $LR(\phi)$, A a $\sigma$-structure and $(a_1, . . . , a_n)$ an n-tuple of elements of A. Then we define ‘$(a_1, . . . , a_n)$ satisfies $\phi$ in A’ by recursion on the complexity of $\phi$. Clauses $(a)–(f)$ of the definition are exactly as Definition 5.7.6. We add two more clauses for quantifiers.

But (a),(b) ( (c)-(f) are just clauses for $\implies, \lor, \land,\iff, \neg$) points to the definition given above and that itself is dependent on the definition on truth given above. I am asking is this definition circular or the author is talking about different kinds of truth in the given definition. If the definition is indeed circular so what is it's correct form?

Thank you in advance.

Answer 1

We have a language (a signature plus logical constants) and a structure $A$ (a "piece of the (mathematical) world").

The first step is to define a "meaning" for the terms (the "names"): Def.5.6.2. The meaning is relative to the structure $A$, i.e. the constant $c$ will refer to an element $c^{A}$ of $A$.

Then the authors define what does it mean for s sentence $\phi$ to be true in the structure $A$: $A \vDash \phi$ (reading: "structure $A$ makes sentence $\phi$ true" or "$A$ is a model of $\phi$", Def.5.6.4 above).

Up to now, no circularity...

The next step is to extend the above semantics to formulas that are not sentences, i.e. with free occurrences of variables.

This is done in Def.5.7.6 with the new relation:

"$(a_1,\ldots,a_n) \text { satisfies } \phi \text { in } A$".

This is a new definition that relies not on the previous one, but only of that regarding terms.

The last step is to move to full predicate logic; to do this, the authors have to manage quantifiers.

This is done in Def.7.3.1 that again relies only on the previous rulkes for managing terms.

In most textbook, this is the only semantics for first-order logic defined: only for "pedagogical reasons" the authors have decided to present the topic this way, with increasing levels of complexity of the language.