Proof set of a sentence (error in textbook?)

Question

I'm reading the Chellas book on Modal Logic and in one place he defines the proof set of a sentence relative to a system $\Sigma$, denoted $|A|_\Sigma$ to be the set of a $\Sigma$-maximal sets of sentences containing $A$. In other words

$$|A|_\Sigma = \{\max_\Sigma\Gamma: A\in \Gamma\}$$

However, elsewhere he defines a canonical model for $\Sigma$ to be a model in which every world is $\Sigma$-maximal and for each $P_i$ (the set of worlds at which atomic sentence $\mathbb{P}_i$ is true) we have $P_i=|\mathbb{P}_i|_\Sigma$.

This last part makes little sense to me, though, because $P_i$ is supposed to be a set of worlds while $|\mathbb{P}_i|_\Sigma$ is supposed to be a set of sets of sentences. I guess we could resolve this by taking worlds to themselves be sets of sentences, even though as far as I've read Chellas leaves the notion of a world un-analyzed. Anyone know how to resolve this?

Answer 1

Yes, we can think at a possible world as the set of atomic sentences : the set of atomic sentences true in it.

If you are more "accustomed" to approach propositional logic in term of valuations, you have to consider that each valuation $v$ partition the set of atomic sentences into two disjoint sets: the set $\{ \mathbb{P}_i : v(\mathbb{P}_i)= \text {TRUE} \}$ and the set $\{ \mathbb{P}_i : v(\mathbb{P}_i)= \text {FALSE} \}$.

But, by bivalence, we can simply assume the first one : we will call it "a possible world".

See page 4 :

The picture is of a collection of possible worlds - including our own, the real world - at which sentences of the language are variously true and false. Our purpose is to model this, and we do so by means of an an infinite sequence of sets of possible worlds,

$$P_0, P_1, P_2, \ldots$$

The intuition behind this modeling is that, for each natural number $n$, the set $P_n$ collects just those possible worlds at which the corresponding atomic sentence $\mathbb{P}_n$ is true.

The formal definition of "$A$ is true at $\alpha$ in $\mathcal M$" is in page 5 :

• $$\vDash^{\mathcal M}_{\alpha} \mathbb{P}_n \ \text{ iff } \ \alpha \in P_n$$

• $$\vDash^{\mathcal M}_{\alpha} \top$$

• ...

Consider the second case : it can be "read" as $\top \notin \alpha$, for any $\alpha$.

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Maximality is a property of sets of sentences [page 53] and the proof set of a sentence $A$ is the set of $Sigma$-maximal sets of sentences containing $A$.

See pages 60 :

Thus in a canonical model for a system $\Sigma$, each world is a $\Sigma$-maximal set of sentences (and each such set is a world), and for each such world $\alpha$ and each natural number $n$,

$$\alpha \in P_n \ \text{ iff } \ \mathbb{P}_n \in \alpha.$$

As you can see, in this case it is stated explicitly that a world is a set of sentences.