tag:blogger.com,1999:blog-4987609114415205593.post6509483948850166575..comments2024-03-28T13:40:26.497+00:00Comments on M-Phi: What is Abstract Structure?Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-4987609114415205593.post-6368535320760589192013-03-12T18:12:17.174+00:002013-03-12T18:12:17.174+00:00Hi Sam,
Yes, right on both accounts.
Given some...Hi Sam,<br /><br />Yes, right on both accounts. <br /><br />Given some (unary) prop function $\hat{\Phi}$ and relation $R$, the proposition $\hat{\Phi}[R]$ is true iff $\Phi$ is true of $R$. The propositional function here is meant to be language-independent (like an unsaturated sense, in Frege's terminology). There will be similarities with the isomorphism-class identification, as you point out.<br /><br />There no special domain on either approach, but one can add a "labelling" on both. The formula $\Phi$ expressing the propositional function itself has all its first-order variables "ramsified". $\exists y_0 \exists y_1 \dots \theta$. One can remove these, leaving just the subformula $\theta$ with the free first-order variables. This expresses an unsaturated propositional function, with N many argument places, where N is the cardinality of the original model. So, the nodes in some sense correspond to the saturated argument positions in the propositional function.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-24731835320351975262013-03-12T17:23:32.409+00:002013-03-12T17:23:32.409+00:00Hi Jeff,
That's really helpful, thanks!
So ...Hi Jeff,<br /><br />That's really helpful, thanks! <br /><br />So I'd initially thought that the propositional-function view was something like the following: the abstract structure of X is the propositional function F such that $F(Y) \leftrightarrow \Phi(Y)$, where $\Phi$ is a particular categorical description for X. More explicitly, we just get F from a single instance of third-order comprehension (on the possibly infinite formula $\Phi$). But then the view is not much different from the one I outlined. The reason is that $\Phi(Y)\leftrightarrow Y\cong X$. Since equivalence is the analogue of identity for higher-order entities, the abstract structure of X in this sense just is the abstract structure on the isomorphism-class view. The important difference is that the isomorphism-class view doesn't have to saddle itself with an unwieldily proper class sized object language, and attendant logic. <br /><br />That's why I explicated the propositional-function account in terms of an equivalence class of formulas. But I see now that the idea is to have something like a primitive 'content abstraction' operator which takes formulas, and delivers a function which maps classes to propositions. However, if we do this, it still seems very natural to lay down some axiom connecting this content to the formula abstracted on; something like:<br /><br />(*) the functional content of S applies to Y just in case S is true of Y. <br /><br />But then, again, the abstract structure on this view looks awfully similar to the isomorphism-class view (as far as I can see). <br /><br />Re. the domain. Strictly speaking the abstract structure on the isomorphism-class view doesn't have a domain, since it's not itself a structure. And the nodes on the propositional-function approach can also be changed - it seems just as arbitrary which variables we pick as the canonical ones as it is which domain we pick to be the domain of the structures in the abstract structure on the isomorphism-class approach. Are you thinking of the propositional function as coming equipped with some non-arbitrary class of variables? It's not clear to me yet how one would go about doing that.<br /><br />All the best,<br />Sam<br /><br /><br /><br /><br /><br /><br /><br />Sam Robertshttps://www.blogger.com/profile/08367603167787877999noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-50929657334780509702013-03-11T21:20:12.963+00:002013-03-11T21:20:12.963+00:00Hi Sam,
Great - thanks. So, I guess we have two a...Hi Sam,<br /><br />Great - thanks. So, I guess we have two approaches for identifying an abstract structure $\hat{X}$ for $X$:<br /><br />i. The isomorphism-class approach<br />ii. The propositional-function approach.<br /><br />(Both will incur set/class issues, but let's pretend we can get round that!!) <br /><br />So, in going propositional, we don't identify $\hat{X}$ with the equivalence class, like {Phi: \Phi is a categorical description of X}. Rather, $\hat{X}$ is the second-order propositional content itself. Hopefully, this gets a unique thing to be $\hat{X}$. This is then expressed, linguistically, by one of these Phi in some sufficiently large language; and that can be skolemized, by introducing constants. So, properly speaking, nodes are not components of the abstract structure, but rather additional components (i.e., existentially bound first-order variables) of some (implementation-dependent) description of the structure (i.e., a formula Phi).<br /><br />On the isomorphism-class approach, one picks a class X* of variables, and yes, this can play a sort of role as a domain for $\hat{X}$. But I don't really want a domain! Rather I want to explain it away, by going propositional. For one can replace X* be any other set, Z, of the same cardinality, and that is the kind of implementation-dependence I want to avoid. On the prop-function approach, the domain of nodes is a kind of artifact of the labelled description of the structure. The structure is "domainless".<br /><br />I think you're right though that the isomorphism-class approach can introduce a labelling as you suggest. So, possibly, whatever I say is the advantage of the categorical prop function approach may be mimicked on the isomorphism class approach. <br /><br />The third point is, oddly enough, what is driving all this (in a paper I'm writing on "Leibniz equivalence"). I want to get an analysis of possible worlds in which worlds are *not* equivalence classes. So, it's a form of propositional "ersatzism". A possible world $w$ is $\hat{X}[\vec{R}]$, where $\hat{X}$ is a categorical second-order propositional function, and $R_i$ is a sequence of relations on concreta. So, e.g., an example of a possible world might be the proposition that there are exactly three concreta x,y,z and they bear relations ... blah ... blah. This world is the image of applying the corresponding second-order propositional function to the sequence of relations. (And this second-order propositional function is the "abstract structure" of that possible world, relative to those relations.)<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-58696394816934239152013-03-11T09:18:20.472+00:002013-03-11T09:18:20.472+00:00Hi Jeff,
Thanks! So there are two proposals on th...Hi Jeff,<br /><br />Thanks! So there are two proposals on the table (as I see it). On the first, we just go third-order in the usual way. Then we say that the abstract structure of X is {Y: Y is isomorphic to X}. On the second, we also ascend to a third-order language, but we add a theory of possibly non-set-sized formulas, and let the abstract structure of X be {Phi: \Phi is a categorical description of X}. <br /><br />In order to have a unique class of variables which can serve as nodes on the second approach, we'd need to associate each X with a class of variables in such a way that if X and Y are isomorphic, they are associated with the same class. Call this class of variables X*. <br /><br />If X* is available, then there's no reason why the advocate of the first approach can't avail themselves of it. They could then associate X with {Y: Y is isomorphic to X & the domain of Y is X*}. And the nodes of the abstract structure could be identified with X*. Permutation invariance is explained as it would be on the second approach - if you apply a permutation of the nodes to any model in the abstract structure you get another model in the same abstract structure. <br /><br />On the second point, I'm not seeing clearly what the desiderata are for labeling. If it's not too much hassle could you say where you think the approach I've outlined would be unsatisfactory re. labeling. <br /><br />I need to think more about the third point. <br /><br />Thanks for the discussion!<br /><br />Best<br />SamSam Robertshttps://www.blogger.com/profile/08367603167787877999noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-45098165804261828972013-03-10T17:24:52.408+00:002013-03-10T17:24:52.408+00:00Hi Sam,
Yes, all that sounds right.
But the iso...Hi Sam,<br /><br />Yes, all that sounds right. <br /><br />But the isomorphism class (collection) approach is something I have a kind of dislike for. I like three things about the propositional function approach,<br /><br />1. The identification of the nodes with variables (in a sense, it explains the permutation invariance, or "lack of individuality" of the nodes); <br /><br />2. The connection between labelling and skolemization and specific implementations.<br /><br />3. I can apply this to get an account of possible worlds, by applying a second-order propositional function $\hat{\Phi}$ of this kind to a sequence of concretum relations. <br /><br />So, e.g., a possible world $w$ in which there are exactly six spacetime points, related in such-and-such a way, can be thought of as the image of a certain propositional function $\hat{\Phi}$ on the properties/relations Spacetime-Point and Between. (Worlds are then categorical propositions, "built up" from concretum relations.) I think that this then ensures that "isomorphic worlds" (worlds with no qualitative differences) are in fact identical. This is what Leibniz equivalence in physics (spacetime theory) amounts to.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-28026634530559248762013-03-10T16:54:09.716+00:002013-03-10T16:54:09.716+00:00Hi Jeff,
Right. And if propositional functions ar...Hi Jeff,<br /><br />Right. And if propositional functions are supposed to exist for all formulas with one free second-order variable (as your descriptions are), then they'll have to be third-order entities. But then it looks like you might as well just have the abstract of a second-order structure be the third-order 'collection' of all isomorphic structures. <br /><br />All the best,<br />SamSam Robertshttps://www.blogger.com/profile/08367603167787877999noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-33880549312579528732013-03-10T16:19:10.758+00:002013-03-10T16:19:10.758+00:00Hi Sam,
Yes, I think that's right - Hodes'...Hi Sam,<br /><br />Yes, I think that's right - Hodes's point is in the ball-park of neo-logicist approach, so doesn't require ZF at all.<br /><br />But I think to get this "abstract structure = propositional function" view to work, one would have to have the propositions either in a distinct sort, or make some sort of class/set style distinction. <br /><br />To take, the most extreme example, we try to get the abstract structure $\hat{V}$ of the cumulative hierarchy V. Because we proceed by infinitarily axiomatizing V, with a distinct variable for each set (along with the elementary diagram of V), then $\hat{V}$ is initially going to be expressed by a formula of class size, and so it cannot be an element of V.<br /><br />So, it looks like the abstract structure of a given model is, in some general sense, at a higher level.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-38311006202039470222013-03-10T16:13:50.035+00:002013-03-10T16:13:50.035+00:00Hi Jeff,
Thanks, I'll look it up. My guess wa...Hi Jeff,<br /><br />Thanks, I'll look it up. My guess was that the second-order form of (ISO) is inconsistent independently of ZF. This assumes that the hat operator ^ is a type-lowering device. But I take it (?) your Fregean propositions are not objects, and so your version of (ISO) isn't susceptible to Burali-Forti. <br /><br />Best,<br />SamSam Robertshttps://www.blogger.com/profile/08367603167787877999noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-67294527138462865522013-03-10T15:58:41.694+00:002013-03-10T15:58:41.694+00:00Hi Sam,
Oh - they were good questions anyway! I l...Hi Sam,<br /><br />Oh - they were good questions anyway! I linked in the previous posts now.<br /><br />Yes, I was thinking of the second-order form of (ISO), and I think the argument goes via Burali-Forti style reasoning. It's in a paper by Hodes from the 80s.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-87290253841984098782013-03-10T15:52:17.695+00:002013-03-10T15:52:17.695+00:00Hi Jeff,
Thanks for this! I deleted my post after...Hi Jeff,<br /><br />Thanks for this! I deleted my post after searching for your previous posts on this (which I should have done before I posted!); sorry about that. <br /><br />Best,<br />Sam<br /><br />ps I'm much less sure that this is pertinent now, but if R_1 and R_2 are set-sized, then (ISO) can surely be modeled in ZF (indeed, for any equivalence relation on sets). If R_1 and R_2 are arbitrary second-order relations, then can't we just use Burali-Forti reasoning to show that (ISO) is inconsistent?Sam Robertshttps://www.blogger.com/profile/08367603167787877999noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-30336147918479724122013-03-10T15:29:49.480+00:002013-03-10T15:29:49.480+00:00Hi Sam,
On the first point, then, for cardinaliti...Hi Sam,<br /><br />On the first point, then, for cardinalities, one can use that, yes. But, for the corresponding principle for isomorphism, I think (!), by a result of Harold Hodes, that it leads to an inconsistency. I.e., let (ISO) be the abstraction principle,<br /><br />(ISO) ^R1 = ^R2 iff R1 is isomorphic to R2<br /><br />Then adding (ISO) to ZF is inconsistent. (I don't know the proof.)<br /><br />The example you mention involves two acceptable, though arbitrary, skolemizations of the structure-formula (so, a and b are skolem terms). The structural description has no constants - only bound variables. That's how I account for "labelling". (In a sense, the nodes correspond to first-order variables; skolemization corresponds to arbirtarily picking some node of the structure and labelling it.)<br /><br />Hopefully, the arbitrariness is removed in two ways:<br />i) M is first described categorically by the first-order ramsified description (no constants);<br />ii) Then we pass to the abstract (second-order) propositional content, assuming that logically equivalent formulas express the same abstract proposition. <br /><br />For the case of M, we have the second-order propositional function,<br /><br />The proposition that there are exactly two things x,y such that Xxx & Xyy & Xyx & -Xxy.<br /><br />As applied to M, this yields a true proposition.<br /><br />So, the hope it to have something categorical in the required sense and implementation-independent too.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.com