In propositional logic, is this L the set of simple statements, and the compound statements or just the set of simple statements?

Question

On p4 in Chang/Keisler's Model Theory,

Classical sentential logic is designed to study a set L of simple statements, and the compound statements built up from them. At the most intuitive level, an intended interpretation of these statements is a ‘possible world’, in which each statement is either true or false. We wish to replace these intuitive interpretations by a collection of precise mathematical objects which we may use as our models. The first thing which comes to mind is a function F which associates with each simple statement S one of the truth values ‘true’ or ‘false’. Stripping away the inessentials, we shall instead take a model to be a subset A of L the idea is that S $\in$ A indicates that the simple statement S is true, and S $\notin$ A indicates that the simple statement S is false.

Which one is L:

• the set of simple statements, and the compound statements built up from them, (I guess so from the first sentence)

• the set of simple statements? (I guess so from the last sentence)

Thanks.

Answer 1

$L$ is the set of simple statements - or, more commonly, of atomic propositions or propositional atoms.

(The comma indicates this: "a set $L$ of simple statements, and the compound statements built up from them" versus "a set $L$ of simple statements and the compound statements built up from them." The latter is ambiguous, but the former isn't.)

---

To quickly describe what's going on (since I think the quoted passage is a bit opaque):

The semantics of propositional logic amounts to assigning a truth value from $\{\top,\perp\}$ to each propositional sentence in a "coherent" way (e.g. we can't say that $a\wedge b$ gets $\top$ but $a$ and $b$ each get $\perp$). Such assignments are in fact equivalent to maps from $L$ to $\{\top,\perp\}$: every coherent assignment restricts to such a map, and every such map extends to exactly one coherent assignment. So we can replace the "full valuation" semantcs with this "atomic valuation" semantics, which is simpler in many contexts.

And we can go further and say that maps $L\rightarrow\{\top,\perp\}$ are themselves equivalent to subsets of $L$ - via the bijection assigning $m:L\rightarrow\{\top,\perp\}$ to the set $m^{-1}(\top)$, that is, the set of atomic propositions (= members of $L$) which $m$ declares to be true.

This shift, from "the set of maps from all sentences to $\{\top,\perp\}$ satisfying some complicated conditions" to "the set of subsets of $L$," is Chang/Keisler's "stripping away the inessentials."