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Genuine Realists about modality typically understand propositional content to be a function of the set of worlds where that proposition is true (the set of worlds might include impossible ones). Actualist Realists take the dependence to go in the other direction, taking a world to be a function of the set of propositions true at that world. Since this function is almost always identity,* let's treat it as such in what follows.
Kaplan established a cardinality paradox against Genuine Realism analogous to an earlier paradox about the set of all propositions put forward by Russell. Russell's paradox** is now taken analogically to present a problem for Actual Realists.
Here's how Kaplan's paradox goes. Assume the set of all possible worlds has the cardinality K. Then, by Cantor's Theorem, the powerset of the set of possible worlds has a greater cardinality. But if a proposition is a set of worlds, then the cardinality of the set of propositions is greater than the cardinality of the set of worlds. O.K. so far. But let's consider for each proposition a world where one being is thinking that proposition.**** But then the set of worlds has at least the cardinality as that of the set of propositions. Contradiction.
Here's my question. As an inferentialist, do I have to bother about this? We now have fully normalizable versions of standard modal systems such as K, S4, and S5.***** From the link it is clear how such Alex Simpson style systems employ eigenvariables as connected by a two place relation, and also how these are most easily interpreted as being about possible worlds. But I'm not sure that there's any reason to think that the possible worlds have anything to do with content individuation.
First, I'm not sure they should be interpreted by the inferentialist as being about possible worlds, but showing convincingly that they need not is a very long term project. So let's assume that they should be interpreted as denoting arbitrary possible worlds. Then we can still ask if the inferentialist needs to worry about the Kaplan-Russell paradoxes. For the inferentialist, canonical inferential role is what individuates content, so there is no need to identify propositions with sets of possible worlds or possible worlds with sets of propositions. Assume, perversely perhaps, that the inferentialist is a Lewisian Realist and thinks that there is nothing distinguished about the actual world. When talking about the eigenvariables in the meta-language she does quantify over the set of possible worlds, which in the spirit of Lewis is just sort of factically given.
But since inferential role (some aspects of of which are made explicit by the modal logic) determine issues of content individuation****** there is no need to posit a functional relation between the set of possible worlds and the set of propositions. But then at least standard versions of the Kaplan-Russell paradoxes don't get off the ground.
Maybe I'm missing something and there's some kind of obvious cardinality paradox for inferential role which is analogous to Russell's with respect to propositions. I would be disappointed if there were, but not completely surprised.********
[Notes:
*Apologies to Jessica Wilson– Given that the functions are usually the identity relation how is it that standard Genuine Realists takes propositions to depend on worlds and standard Actualist Realists take worlds to depend on propositions? Identity is thus too coarse grained a notion to capture the full metaphysical intent. But then what notion will work? A modal one such as supervenience? But modality is what's at issue. Determinable/determinates? But I've been convinced (follow the link and you will be too) that determinables aren't always grounded in determinates. We're clearly not talking about causal dependence. Will any small g grounding notion do the trick here, or do we need something more grand? If one will, which one?
**Not to be confused with Russell's Paradox, which (as Graham Priest has taught us) requires diagonalizing on the identity function applied to all subsets of a given set (including the set itself). As far as I can tell, neither the Kaplan/Russell paradoxes, nor the related Forrest/Armstrong paradox of recombination, do, since these paradoxes merely hinge only on a set's cardinality being less than the cardinality of its powerset, the proof of which just involves diagonalizing on an arbitrary imagined bijection. Because of this you don't get strict versions of Priest's Existence, Transcendence, and Closure for this family of paradoxes. But you do get analogues may be close enough for Priest's Priciple of Uniformed Solution to have dialectical sway.***
***Joshua Heller and I are explaining this in a paper on some of Quentin Meillassoux's modal arguments that we are presenting in a few weeks at the University of Nevada, Las Vegas. Even if the Principle of Uniform Solution doesn't sway the Meillassouxian, the Cantor/Priest's Domain Principle should. Part of why I'm thinking through this with respect to inferentialism is because David Beisecker is there, and I would like to be able to talk with him sensibly about this issue.
****This is how John Divers presents the paradox in his fantastic book, Possible Worlds.
*****Somewhere Greg Restall shows convincingly why a Brandomian about logic must care about normalizability. I need to get that and reread it.
******These kinds of appeals can be made in a respectably holistic way. See Lance and Hawthorne. Nothing they say is inconsistent with Brandom's philosophy of logic, though Restallian reasons for normativity do provide the dialectical werewithal to make Brandomian criticisms of the intended uses of some of the interesting formal frameworks that Brandom has developed with students. I think Simpson style systems, such as the one linked to above, do the work fine, though clearly Restall would prefer sequent calculus versions as opposed to Simpson's Prawitz type trees or my Fitch style proofs.
********Since the above involve paradoxes of totality I briefly thought of categorizing this post under all of the categories you can see at right if you scroll down. Luckily, it took me under a second to realize how thoroughly obnoxious that would have been. I think in Bukowski's Factotum the narrator (Henry Chinanski) tells his boss he's writing a novel about everything, and so the boss proceeds to go through a set of random things asking if those things were in the book. At each one, Chinanski says "Yep, that's in there too." I wonder if Graham Priest gets this kind of thing from well-meaning elderly relatives while writing his books, and if he responds like Bukowski/Chinanski?]
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