I have been reading “Introduction to Mathematical Logic” by Elliot Mendelson (free pdf here) and I have found the way the author defines satisfaction, particularly a function $s^*$ that helps us define satisfaction, to be a little puzzling. Here is the relevant section of the text, on page 59.
So the important concept we need on our way to defining satisfaction is to define this function $s^*$ that assigns terms of $\mathscr{L}$ to members of $D$. But what I don’t understand is why this function is defined on every type of term except for predicate letters. It seems natural to define $s^*(A^n_k(t_1,…,t_n))$ as $(A^n_k)^M(s^*(t_1),…,s^*(t_n))$, just like we did with function letters; it even seems necessary to do so because how else are we supposed to talk about a sequence $(s_1,s_2,…)$ satisfying the wf $A^n_k(t_1,…,t_n)$ without a means of converting the terms in the wf to the relevant domain members in $s$?
Then what confuses me even further is that on the next page the situation is reversed; when Mendelson gets around to defining satisfaction, he defines what it is for a sequence to satisfy atomic wfs with predicate letters, but not what it is for a sequence to satisfy atomic wfs with function letters.
So I guess what I really have is two questions: 1) why have we not defined $s^*$ on predicate letters and how can we speak of $s$ satisfying a predicate letter in spite of that? 2) Why did we go through the trouble of defining $s^*$ for function letters if we don’t define satisfaction for function letters - and why don’t we do that?
