Wednesday, July 7, 2010
Reducing sets to propositions
Lewis tried to reduce propositions to sets. I think that doesn't work. But maybe the other way around does. Plausibly, for any xs there is a proposition that those xs exist. One can then identify collections with those propositions that affirm the existence of one or more things. Then x is a member of a collection c if and only if c represents, perhaps inter alia, x’s existing. We can then pick out our favorite axioms of set theory, and stipulate that any collection of collections that satisfies these axioms counts as a universe of sets. We will have to make a decision when we defined a collection as a proposition that affirms the existence of one or more things whether nonexistent things are permitted. If they are, and if propositions are all necessary beings, then we will have sets of nonexistent things. If they are not, then we won’t.
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4 comments:
But I wonder why you would want to reduce sets to propositions. True propositions assert facts, but surely they should be reduced to how things stand, and there may well be some natural way that things coexist in reality, some natural kind of collection, whose properties might be more general than those of collections of propositions.
What are propositions? Do they depend upon language, or v.v., or is there some more complex interrelationship? A similar problem arises for collections. Events in history and objects on my table seem to coexist in different ways. Is there some basic way in which any conceivable things coexist, and is it dependent upon our conceptual powers, our semantics?
But collections are a basic concept in mathematics and logic, so it seems reasonable to address them seperately from propositions in general. Your basic collections depend upon how propositions are, but arguably we can theorise about how objective general collections would behave without making a prior decision about the objectivity of our linguistic frameworks.
And if you are right, could we not reduce anything to propositions, in a similar way? So I'm wondering why sets. Is it because you think that collections are not anything over and above the things collected? But in many ways that is captured by extensionality.
And are true propositions much more than facts? And what of false propositions? What if a proposition falsely affirms the existence of things that are really too numerous to exist? Is there then a collection of so many things after all? And finally, I wonder about collections of propositions, as in your "those propositions"...
I find sets weirder and more mysterious than propositions. I find it hard to believe in them, unless I can reduce them.
Hmm... you take sets to be a sort of collection though, are you saying that you find collections hard to believe in?
"Collection" is my weasel word for whatever in the right ontology helps the argument get off the ground. :-) For instance, if the reduction here works, collections could be conjunctive existential propositions. Or, in some contexts (not all), they could be mereological sums, though I am dubious. Or they could simply be a marker calling for a clever paraphrase.
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