# Mendelson, Logic, Lemma 2.16: a possible typo or flawed understanding?

The book is Mendelson's Introduction to Logic, 6th edition.
Here is the statement of Lemma 2.16:

Let $$J$$ be a consistent, complete scapegoat theory. Then $$J$$ has a model $$M$$ whose domain is the set $$D$$ of closed terms of $$J$$.

Definition of scapegoat theory in his book is:

A theory $$K$$ is a scapegoat theory if, for any wf $$\mathscr B(x)$$ that has $$x$$ as its only free variable, there is a closed term $$t$$ such that $$\vdash_K (\exists x) \neg \mathscr B(x) \implies \neg \mathscr B(t)$$.

Here is how Mendelson starts proving it:

For any individual constant $$a_i$$ of $$J$$, let $$(a_i)^M = a_i$$. For any function letter $$f^k_n$$ of $$J$$ and for any closed terms $$t_1,\dots,t_n$$ of $$J$$, let $$(f^k_n)^M (t_1,...,t_n) = f^k_n(t_1,...,t_n)$$. (Notice that $$f^k_n(t_1,...,t_n)$$ is a closed term. Hence, $$(f^k_n)^M$$ is an $$n$$-ary operation on $$D$$.) For any predicate letter $$A^k_n$$ of $$J$$, let $$(A^k_n)^M$$ consist of all $$n$$-tuples $$(t_1,...,t_n)$$ of closed terms $$t_1,\dots,t_n$$ of $$J$$ such that $$\vdash_J A^k_n(t_1,...,t_n)$$ . It now suffices to show that, for any closed wf $$\mathscr C$$ of $$J$$: \begin{align*} (*) \vDash_M \mathscr C \text{ if and only if } \vdash_J \mathscr C. \end{align*}

He proves it with induction on the number of connectives and quantifiers. I don't type steps that are not needed for my questions.

Case 4. $$\mathscr C$$ is $$(\forall x_m ) \mathscr D$$.

Case 4b. $$\mathscr D$$ is not a closed wf. Since $$\mathscr C$$ is closed, $$\mathscr D$$ has $$x_m$$ as its only free variable, say $$\mathscr D$$ is $$F(x_m)$$. Then $$\mathscr C$$ is $$(\forall x_m)F(x_m)$$.

(I don't type one of the directions here.)

Assume $$\vdash_J \mathscr C$$ and not $$\vDash_M \mathscr C$$. Thus, \begin{align} &\text{(1)} \vdash_J (\forall x_m)F(x_m),\\ &\text{(2) not} \vDash_M (\forall x_m)F(x_m). \end{align} By (2), some sequence of elements of the domain $$D$$ does not satisfy $$(\forall x_m)F(x_m)$$. Hence, some sequence $$s$$ does not satisfy $$F(x_m)$$. (Following sentence has a possible typo.) Let $$t$$ be the $$i$$'th component of $$s$$. Notice that $$s^*(u) = u$$ (Note: this is a function mapping terms of the language to elements of the domain) for all closed terms $$u$$ of $$J$$ (by the definition of $$(a_i)^M$$ and $$(f^k_n)^M$$). Observe also that $$F(t)$$ has fewer connectives and quantifiers than $$\mathscr C$$ and, therefore, the inductive hypothesis applies to $$F(t)$$, that is, $$(*)$$ holds for $$F(t)$$. Hence, by Lemma 2(a) on page 60, $$s$$ does not satisfy $$F(t)$$. So, $$F(t)$$ is false for $$M$$. But, by (1) and particularization rule, $$\vdash_J F(t)$$, and so, by $$(*)$$ for $$F(t)$$, $$\vDash_M F(t)$$. This contradiction shows that, if $$\vdash_J \mathscr C$$, then $$\vDash_M \mathscr C$$.

Here is Lemma 2 (a):

Let $$t$$ be free for $$x_i$$ in $$\mathscr B(x_i)$$. Then $$s$$ satisfies $$\mathscr B(t)$$ if and only if $$s'$$, obtained by replacing the $$i$$th element of $$s$$ with $$s^*(t)$$ satisfies $$\mathscr B(x_i)$$.

## A possible typo

I baffled my head around the picked $$i$$'th element. It doesn't make sense to me and it seems that Mendelson meant $$m$$'th element. Here is why: pick $$t$$ as $$m$$'th element, then $$s$$ is a sequence that suits the hypothesis of Lemma 2 (a), since $$s^*(t)=t$$, and so it does not satisfy $$F(t)$$, but $$t$$ is closed, and since $$x_m$$ was the only free variable, the formula is closed, hence $$F(t)$$ is false. Everything from his proof follows.

## Questions

1. Is it a typo?
2. If it is not, could you provide some clues, why $$i$$'th element makes sense here and why the proof follows from it?
• Scapegoat theory is not a standard term, you should define it in your question. (It may be what is sometimes called Henkin theory, but the precise definition may be relevant.) Jun 4 at 19:10
• @PrimoPetri, now edited. But it seems that in this particular direction this hypothesis is not used. Jun 4 at 19:28
• It's a typo that can be fixed as you suggest. Jun 4 at 20:37