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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?

Zuhair
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    Interesting. Is the author a non-logician philosopher? (Based on the name "Wittgenstein.") All I can think of is that one can view a proposition as representing the set of worlds in which that proposition is true. It may be that such a representation is useful in some manner, but this is way beyond my pay grade. – Brian Tung Jun 19 '22 at 17:24
  • @BrianTung, I've linked his homepage. He is there on the Wikipedia also see https://en.wikipedia.org/wiki/Peter_Simons_(academic). Do you mean a proposition is represented as a set\class of all models that satisfy it?1 – Zuhair Jun 19 '22 at 18:43
  • I think the paper you are citing is not talking about mathematics or about the philosophy of mathematics, but about philosophy in general. The "rich panoply of entities that the worlds throws at us" is outside the scope of set theory. – Rob Arthan Jun 19 '22 at 21:31
  • One thing I'm curious about is whether set theory is actually pushing out other tools in the philosophy. The impression I get from reading some mathematical literature about nonclassical logics here and there is that there are other approaches. Algebraic logic is the one I know best, but there are computational ones as well. – Greg Nisbet Jun 20 '22 at 05:36
  • @RobArthan, It speaks about set theory in general and its effect on mathematics and on philosophy in general, and the passage I've cited was under topic "false idols in philosophy", so it's about its impact on philosophy yes. Yes, it appears that the author is alluding to exploiting an interpretation about what propositions are that originated in SET THEORY (as in foundations of mathematics) then exported to other areas or in some sense generalized. My question here is about this original interpretation of what propositions are that Set theory or foundations (of mathematics) employed? – Zuhair Jun 20 '22 at 08:23
  • @GregNisbet, yes, that's what appears to me, but the author is even suggesting a bad effect of set theory on mathematics itself, appearantly he thinks that it filters a way important aspects about implemented mathematical notions when re-formalized as sets, and also he speaks about a bad effect when generalized outside of math to the arena of philosophical thought especially when involved with collective entities, it appears that he is claiming that set theory hijacked all sorts of collective entities and in doing so throwing away particulars involved in each collective scenario. – Zuhair Jun 20 '22 at 08:30
  • One last comment, is that I don't think set theory itself should take the blame for any misuse of it. There is a precise role attributed to set theory in foundations and as such it is fruitful and beneficial. There is no need to bar contemplating other sorts of collective entities, this is not in any sense a role attributed to set theory; nor there is any need to restrict pondering mathematical entities outside of set theoretic re-interpretation, and again such a restriction is not the job of set theory. – Zuhair Jun 20 '22 at 11:46
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    It is not so "uncommon"... See Chen Chung Chang & HJerome Keisler, Model Theory (North Holland, 3rd ed. 1990), page 4: 1.2 Model theory for sentential logic: "Classical sentential logic is designed to study a set $\mathscr S$ 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. [...] By a model for $\mathscr S$ we simply mean a subset $A$ of $\mathscr S$." – Mauro ALLEGRANZA Jun 20 '22 at 12:08
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    See also Carnap's Semantics. – Mauro ALLEGRANZA Jun 20 '22 at 13:35

2 Answers2

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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.

I hope this helps ^_^

Greg Nisbet
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HallaSurvivor
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    I think it's more likely they identify each proposition with its set of models. Of course, for any fixed proposition $p$ we can define a unary predicate $\varphi_p$ on models so, for any model $x$, $\varphi_p(x)$ means $x$ is a model of $p$. – J.G. Jun 19 '22 at 21:13
  • @J.G. that's another really great idea. It's hard to say exactly what the author meant, but thankfully it sounds like the exact meaning isn't relevant to whatever larger point the author is making. – HallaSurvivor Jun 19 '22 at 21:16
  • Yes, the word "possible" makes one think of modal logic. But, the idea of sets as properties, sets of tuples as Relations (and functions), is obvious and quite customarily implemented (almost ubiquitous) in mathematical logic. But, I'm not familier with implementing propositions as sets of say models satisfying them, or in possible worlds, or so..; I'm asking about if there is a well known robust almost ubiquitous way of capturing propositions as kind of sets? And I'm asking within the customary mathematical logic field and not elsewhere. – Zuhair Jun 20 '22 at 08:44
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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.

Greg Nisbet
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