Extending a theory with the theory of mereological fusions
"Arithmetic with fusions" (draft) is a joint paper with my graduate student Thomas Schindler (MCMP). The abstract is:
In this article, the relationship between second-order comprehension and unrestricted mereological fusion (over atoms) is clarified. An extension $\mathsf{PAF}$ of Peano arithmetic with a new binary mereological notion of ``fusion'', and a scheme of unrestricted fusion, is introduced. It is shown that $\mathsf{PAF}$ interprets full second-order arithmetic, $Z_2$.Roughly this shows:
First-order arithmetic + mereology = second-order arithmetic.This implies that adding the theory of mereological fusions can be a very powerful, non-conservative, addition to a theory, perhaps casting doubt on the philosophical idea that once you have some objects, then having their fusion also is somehow "redundant". The additional fusions can in some cases behave like additional "infinite objects"; positing their existence allows one to prove more about the original objects.
Hi Jeff,
ReplyDeleteIt seems like there's a more radical application of the argument you give against the redundancy of fusions. In particular, the argument seems to show, mutatis mutandis, that pluralities are not redundant. As I understand it, it is a fairly marginal view (composition as identity) which takes fusions to be nothing over and above the things they fuse. It is a pretty common view, however, which takes some things to be nothing over and above the things among them.
That makes me worried about the argument. Are you happy with this application to pluralities, or is the argument more sensitive to fusions than I'm seeing?
Best,
Sam
Hi Sam, roughly, as you see, all (plurals, second-order, fusions) are interpretable in each other syntactically. But then it's still unclear whether any philosophical conclusion that to one applies to the other. Boolos argued that plurals is better than second-order because it seems to avoid "ontological commitment" to the second-order thingys - but I am sceptical on that front already. So, given their equivalence, I am inclined to draw similar conclusions in all three cases, concerning "innocence", "redundancy", etc. I suspect Quine would have gone further and treated them (plurals, second-order, fusion) as notational variants. But I wouldn't want to go that far.
ReplyDeleteJeff
Hi Jeff,
ReplyDeleteThanks for the response! Just to be clear, I take it that your argument is: because (the theories of) pluralities, fusions, subsets are non-conservative over PA, they are not redundant or innocent (in the relevant sense). It's not: because (the theories of) pluralities, fusions, subsets are mutually interpretable, either all or none are redundant (in the relevant sense); and, at least one of them (say subsets) isn't redundant. (The latter would be bad, I take it, because a theory of the first-order definable subsets over PA would be mutually interpretable with a suitable truth theory over PA. In this case, mutual interpretability does not entail an equivalence of redundancy -- one quantifies over new objects, namely sets, and the other doesn't).
If that's right, then I guess I'd need to hear more. It's not at all clear to me why the non-conservativity of plural quantification over PA, for instance, gives us reason to think plural quantification is not innocent (as far as I understand the notion of innocence here).
Best,
Sam
Hi Sam, yes, pretty much that's right. I think Quine would take the hardline view that plurals, second-order and fusions are notational variants, and probably pooh-pooh the idea that one (second-order and fusions) quantifies over new objects and the other (plurals) doesn't. But I think that may be going too far. But maybe one should bit the bullet: plurals is just second-order logic in sheep's clothing, and the alleged ontological innocence is an illusion ...
ReplyDeleteHere, though, "innocence" means "proof-theoretically conservative". Cf., Volker Halbach's "How Innocent is Deflationism?" (Synthese 2001) where he discusses the Shapiro/Ketland arguments about non-conservation of truth axioms. There's Quine's point, too, about the interplay between ontology/ideology. I.e., you may be able to eliminate ontology by enriching the "ideology". This is illustrated by the idea that one may eliminate definable sets of numbers by replacing them with (codes of) predicates and a truth predicate with appropriate axioms. This is attractive for predicativsts (like Feferman) who are sceptical about the concept "arbitrary subset of N".
Maybe, as you suggest though, the results make a bigger impact on the "composition as identity" thesis. Our paper very briefly quotes Lewis, and then avoids the topic!!
Jeff
Yes, I think the view you attribute to Quine is impossibly hardline. A truth theory over PA just doesn't quantify over anything new, even if a mutually interpretable theory of subsets does.
ReplyDeleteAh, I see; so the relevant notion of innocence here just is proof-theoretic conservativity. I was assuming it was something like ontological innocence (which is why I couldn't see how the argument was supposed to go). If "redundant" has a similar meaning, then I think your results show conclusively that fusions (assuming the relevant mereology) are not innocent or redundant. In that case, you could drop the qualifier "may" in the last line before section 2.
Best,
Sam
Sam, right, the Quine "notational variant" view would be hardline, but the more limited conclusion against Lewis seems to be justified. In any case, we try and keep clear of the philosophical stuff in the article, only mentioning it briefly at end of Sect 1. Two years ago, I did write a philosophical paper, much more general, discussing several different cases -- adding math objects/comprehension, adding fusions, adding a truth predicate and adding possible worlds -- but this was before Thomas helped me figure out how the mereological extension worked. I shortened that longer article to this,
ReplyDeletehttps://www.academia.edu/1867517/What_Difference_Does_it_Make
(Note - Smiths title.) I plan to go back to the earlier, longer version of this and rewrite it.
Jeff
Cool; do send me a copy when you get round to rewriting it!
ReplyDeleteYes, will do, if it materializes. One can also obtain a conservative extension result concerning adding a theory PW of possible worlds to a theory (in a modal language). For example, if one has a theory K of kangaroos, and it contains various modal claims (e.g., "each kangaroo might not have had a tail", etc.). One adds the theory PW of possible worlds to this theory K, and shows that if K+PW implies A, then K already implies A. So, one might conclude then that PW theory is just a "useful tool". Long ago, I tried to see how to do this - in 1997 - but didn't see how to do it. Now I know, thanks to a talk I heard at MCMP two years ago by Chuck Cross - the trick is to use hybrid logic. Of course, it would be extremely interesting to get a case of a (schematic) theory T such that T + PW is non-conservative over T.
ReplyDeleteJeff
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