tag:blogger.com,1999:blog-3891434218564545511.post4126416894283316400..comments2021-05-13T12:47:57.847-05:00Comments on Alexander Pruss's Blog: Grounding and category theoryAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-3891434218564545511.post-38027138296642968142013-04-21T04:03:24.656-05:002013-04-21T04:03:24.656-05:00Logics via category theory - ToposesLogics via category theory - ToposesAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-65905412404137733592013-04-17T22:06:17.698-05:002013-04-17T22:06:17.698-05:00My account of full grounding in terms of partial g...My account of full grounding in terms of partial grounding has this counterexample. Let RLJ be the proposition that Romeo loves Juliet. Let R be the proposition that Romeo exists. Let J be the proposition that Juliet exists. Suppose love is a fundamental relation. Then RLJ is partially grounded in R as well as in J. But it is not fully grounded in {R,J}, of course.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-50023243467428055192013-04-10T15:34:13.173-05:002013-04-10T15:34:13.173-05:00How should we interpret the identity arrows that g...How should we interpret the identity arrows that go from p to p? One way is that a special case of the concept of weak partial grounding is the concept of grounding equivalence. For instance (p and q) is grounding-equivalent to (q and p). There will be a grounding arrow from (p and q) to (q and p) and one going back. Moreover, these two arrows will be inverses of each other.<br /><br />We can now say that f:p→q is a grounding-equivalence iff f is an isomorphism, i.e., there is a g:p→q such that fg and gf are both identity arrows.<br /><br />Giving grounding-equivalences, we will be able to say that when f is the obvious grounding by p of (p and q) and g is the obvious grounding by p of (q and p), these aren't completely separate groundings. Rather, they are intimately related: f = hg, where h is the grounding-equivalence by (q and p) of (p and q). Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-38796240159163792782013-04-09T23:36:24.132-05:002013-04-09T23:36:24.132-05:00By the way, I think these infinite cases might rat...By the way, I think these infinite cases might rather neatly correspond to the concept of a limit in a category.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-8677339501561261322013-04-09T23:23:47.239-05:002013-04-09T23:23:47.239-05:00The height case doesn't bother me, because I d...The height case doesn't bother me, because I don't think the less determinate height fact is grounded in a more determinate height fact, unless the latter is maximally determinate.<br />But we can come up with a case where we have an infinite conjunction p1 and p2 and ...., where pk is grounded in qk. Then the whole conjunction is grounded in q1 and p2 and ..., which is grounded in q1 and q2 and p3 and .... But then we want all of this sequence to be grounded in q1 and q2 and q3 and ....<br />Yes, the truth case is another case in the vicinity, but it's also next door to Patrick Grim style paradoxes about "all propositions" so I am cautious. But then again my own motivating cases are related to truth glut paradoxes like the truthteller, so I am not in much better shape.<br />If you think these approaches are worthwhile, maybe we can write something together. You know the grounding literature better than I. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-31405182053128010642013-04-09T23:04:01.850-05:002013-04-09T23:04:01.850-05:00Very interesting. A lot here to think about. I'...Very interesting. A lot here to think about. I'm pretty satisfied with the graph-relative definition of full grounding in terms of partial grounding. (Though I'm also worried about defining partial grounding in terms of direct partial grounding, perhaps for the same reasons you are. A lot of folks in the grounding literature think that where F is a determinate of G, and a is F, then [Fa] grounds [Ga]. Where we have a determinable associated with an infinite quality space, e.g. height, we might get collections of propositions, e.g. a collection of increasingly determinable true propositions about my height, which can be densely ordered by the grounding relation, where none of the propositions in the collection can be said to "directly" ground any of the others.) <br />I also like the second point in the last comment as a way of dealing with grounding loops. Another case which comes to mind in which we might have a similar grounding loop, which might be handled similarly: Assume two (apparently widely accepted) principles: (i) If a is F, then [a is F] grounds [something is F]. (ii) Given any true proposition p, p grounds [p is true]. Now, let Q be the proposition that something is true. Q grounds [Q is true] by the second principle. And [Q is true] grounds Q by the first principle.Brian Cutterhttps://www.blogger.com/profile/17059155559949747916noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-71912829762976349752013-04-09T15:34:32.482-05:002013-04-09T15:34:32.482-05:001. In my long comment, <= will only be a partia...1. In my long comment, <= will only be a partial ordering if there are no loops.<br /><br />2. Another option for a category-based approach is to make the objects in the category be <em>labeled by</em> propositions, without the objects actually being propositions. This would mean there is a map from objects to propositions. We could then suppose the categories to have no loops, which would give us something like irreflexivity. The objects would then be "grounding nodes" rather than propositions. The same proposition can, perhaps, be found at more than one grounding node. For instance, consider the grounding loop: <I should respect you> → <I should keep promises to you> → <I should respect you>, when I promised to respect you. We might want to have two separate nodes for <I should respect you>--one, a node that grounds the duty to keep promises to you, and the other the node that is grounded by the duty to keep promises to you.<br /><br />I don't know what the nodes would correspond to in the metaphysics of the world. Maybe states of affairs? For maybe there is more than one state of affairs of my being obligated to respect you. One such state of affairs grounds my duty to keep my promises and the other is grounded by my duty to keep my promises.<br /><br />This would get rid of the duplication between propositions and states of affairs on some ontologies, like Plantinga's.<br /><br />Propositions could even be taken to be equivalence classes of states of affairs.<br /><br />And we could then make sense of the odd locution "It is doubly true that p." It is doubly true that p iff there are two states of affairs, each labeled with p.<br /><br />This may be nuts.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-20192657326191358652013-04-09T14:07:57.805-05:002013-04-09T14:07:57.805-05:00I've also been playing with the idea of using ...I've also been playing with the idea of using a similar framework for causation and for explanation.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-62137769605837944352013-04-09T14:03:49.569-05:002013-04-09T14:03:49.569-05:00This comment has been removed by the author.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-51490127062421113032013-04-09T14:01:09.703-05:002013-04-09T14:01:09.703-05:00I was hoping for this to be an alternative to my r...I was hoping for this to be an alternative to my relativization of grounding to graphs. But if it's not an alternative to it but an extension, then I will say that grounding is always relative to a grounding-category. <br /><br />I can then define full grounding more or less as I do on graphs. Here's how I do on graphs. These are directed graphs, and each arrow means "directly partially grounds". (I am worried about the "directly", which was why I switched to categories.) A set S of nodes fully grounds a node x relative to a graph G provided that every maximal ancestral lineage of x meets S. S is an ancestral lineage of x iff S is a set of ancestors of x and S is totally ordered under the transitive closure (in the full graph) of the arrow relation.<br /><br />Here's how this takes care of the disjunction problem. There is no grounding overdetermination within a single grounding graph. Thus, A or B, in the case where both A and B are true, has two grounding graphs, one of them ending with A→(A or B) and the other ending with B→(A or B). In the first graph every maximal ancestral lineage of (A or B) intersects {A}. In the second graph every maximal ancestral lineage of (A or B) intersects {B}.<br /><br />On the other hand, any grounding graph for (A and B) ends with two arrows going to (A and B), one from A and the other from B. Thus, {A,B} will be a full grounding for (A and B) (as will {A,B1,B2,B3} if B1, B2 and B3 are the only three nodes that have arrows pointing to B, etc.) <br /><br />Now, back to the Category Theory version. (Which I am very rusty on. About two decades ago, I passed a comp on Category Theory, but have not used it almost at all in between.) Define a partial ordering on the category by y<=x iff there is an arrow from y to x. An ancestral lineage of x is a set of objects less than x that are totally ordered by <=. A set S of objects is then a full grounding of x provided that S intersects every maximal ancestral lineage of x.<br /><br />The primitive concept on this view is: C is a grounding category. Intuitively, a grounding category represents a complete, consistent, correct and non-redundant grounding story for every object (i.e., proposition) in the category. The arrows are weak partial groundings relative to those stories.<br /><br />There is a lot to work out. I've only been thinking about categories in this context since this morning, in light of technical troubles with the graph-theoretic approach.<br /><br />I may want to weaken the category axioms to remove the identity arrows in some cases.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-35944735624934738442013-04-09T13:21:39.780-05:002013-04-09T13:21:39.780-05:00Very cool! I'd also be interested to see wheth...Very cool! I'd also be interested to see whether full grounding, as opposed to partial grounding, could be given a neat category-theoretic treatment. (I don't know enough about category theory to even speculate about this.) If not, then that may be a problem for the whole category-theoretic approach to grounding. As others have pointed out, it does not seem possible to take partial grounding as primitive and define full grounding in terms of it. The worry stems from the fact that two truths might have the same collection of partial grounds while differing in their full grounds. (So, e.g., if A and B are each fundamental truths, then A&B and AvB are both partially grounded in A and partially grounded in B, and not partially grounded in anything else. But AvB, unlike A&B, is fully grounded in A, as well as in B.) On the other hand, it is straightforward to define partial grounding in terms of full grounding, at least on the standard conception of full grounding as a relation between a set/plurality of facts/propositions and a fact/proposition. (q is a partial ground of p iff q is a member of some set/plurality which is a full ground of p.)Brian Cutterhttps://www.blogger.com/profile/17059155559949747916noreply@blogger.com