Are propositions sets of possible worlds?

Question

In article "Against Set Theory" by Peter Simons (Appeared in Johannes Marek and Maria Reicher, eds., Experience and Analysis. Proceedings of the 2004 Wittgenstein Symposium. Vienna: öbv&hpt, 2005, 143–152.). He wrote:

One effect of set theory in ontology has thus been to cripple the development of an adequate ontology of collective entities. This however is far from the worst of its effects. In general the employment of set theory, usually hand in hand with model-theoretic semantics, has been to persuade many philosophers that the rich panoply of entities the world throws at us can be reduced to individuals and sets of various sorts, for example sets as properties, sets of ordered tuples as relations, sets of possible worlds as propositions, and so on and so forth. (bold italics is mine)

I can understand properties being interpreted as sets and vise-verse, also relations being interpreted as sets of ordered tuples and vise-verse, but what I don't know of is interpreting propositions as sets of possible worlds. Is that a common way of interpreting propositions?

Can anybody clarify this point?

Answer 1

Based on the name of the book Against Set Theory and the general tenor of the paragraph, I think that the author is claiming that set theory, by virtue of its ubiquity and acceptance as a foundation of mathematics, is preventing research into alternative formalisms for describing collections.

The author's remarks remind me of this question on the Philosophy stack exchange. In the mid-19th century and before, Aristotlean logic, and some extensions of it like those of Ibn Sina, were the state of the art.

These systems were limited, but nothing better was known at the time. Looking beyond them would require abandoning formalism by the standards of the time. It was also known that there were valid arguments they could not account for, such as geometric arguments. So the limits could be seen at the time, even if a better general-purpose system was not known.

The author, if I had to guess, is describing one common technique for giving the semantics of propositions in modal logic. Also, even if this isn't what the author originally had in mind, it seems like a reasonable application of their criticism.

We might describe a modal model as a map $v : V \to 2^W$ where $V$ is the set of variable symbols and $W$ is the set of worlds, together with $R : 2^{W \times W}$, an accessibility relation on worlds, and a distinguished world $w$.

In this case we might have a rule like this for explaining $\square$.

$$ W, v, R, w \models \square \varphi \\ \textit{if and only if} \\ \text{for all $u$ in $W$, if $wRu$, then $W, v, R, u \models \varphi$} $$

And this rule for explaining the value of a primitive proposition $\alpha$.

$$ W, v, R, w \models \alpha \;\;\textit{if and only if}\;\; \text{$w$ is in $v(\alpha)$} $$

So, we're identifying the meaning of a primitive proposition with the worlds where it's true.

Philosophically this seems like an odd choice, or at least a reductive one, since we're stripping away all possible meaning from a primitive proposition.

Answer 2

It's hard to say for sure, but I think the author is referring to the fairly common practice of identifying a predicate $\varphi(x)$ with the set of all "worlds" in which it's true $\{ x \mid \varphi(x) \}$. This is a surprisingly robust identification, since logical and $\varphi \land \psi$ becomes the intersection $\{ x \mid \varphi(x) \} \cap \{ x \mid \psi(x) \}$, and similarly $\lor$ becomes $\cup$, and $\lnot$ becomes complementation. So (at least in a non-intuitionistic setting) we fully recover the boolean algebra of propositions inside the subsets of $X$. My one worry with this answer is that the author uses the word proposition, which (at least to me) has a connotation of a variable-free question, rather than a predicate $\varphi(x)$

Another thing the author might mean, where we would use the word proposition (and a setting where "possible worlds" gets thrown around a lot) is in modal logic. Here we have multiple possible worlds, and here again we identify a proposition $\varphi$ with the set of worlds where that proposition is true. Then we can form formulas $\square \varphi$ and $\lozenge \varphi$ which say that $\varphi$ is "necessary" or "possible". Again, this is nice because the boolean algebra of subsets of the possible worlds implements all the classical operations on formulas. Though to interpret $\square$ and $\lozenge$ we need some ~bonus information~ -- usually either a topology or a relation on the set of possible worlds.

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I hope this helps ^_^