Tuesday, April 9, 2013

Grounding and category theory

I've been searching for the right kind of mathematical structure to think about the phenomenon of grounding or partial grounding with. The orthodoxy is that the right structure is a partial ordering. That the axioms of partial ordering are satisfied by partial grounding has been challenged and defended. But even the critics have tended to take partial grounding to be something like[note 1] a single relation, or perhaps a small collection of related relations, between pairs of propositions or between a proposition and a set of propositions. I've offered two suggestions (first and second) on how to model grounding using graphs. But I now think all of these approaches abstract away too much of the structure of grounding and/or are unable to capture all the prima facie possibilities that a theory of grounding should recognize.

For instance, the relational approach loses sight of the structural fact that one can sometimes have two different grounding relations between a pair of propositions. Let W be the proposition that Smith is drinking water and let H be the proposition that Smith is drinking H2O. Let D be the disjunction of W and H: the proposition that Smith is drinking water or drinking H2O. If we think of grounding as a relation, we can certainly say that H grounds D. But we want to be able to say that there are two groundings between H and D: H grounds D directly by being one of its disjuncts and indirectly by grounding W which is another disjunct. And this structure is not captured by the orthodoxy. The graph approach nicely captures this sort of thing, but it does not adequately capture the compositional structure which is that the indirect grounding that H provides for D is a composition between a grounding by H of W and a grounding by W of D. There are ways to make it capture this, say by identifying composition of grounding with sequences of arrows in the directed graph, but this won't work for infinite sequences of arrows, something that we should not rule out in the formalism. I realized this when trying to finish a paper on grounding and fundamentality.

I am now wondering if the right structure isn't that of a category. Maybe the objects are true propositions. The arrows are groundings, i.e., token grounding-like relations. Every arrow is at least a partial weak grounding (weak, because there are identity arrows). Some arrows may be full groundings. There is a nice associative compositional structure.

There will be further structure in the category. For instance, perhaps, every family of true propositions will have a coproduct, which is the conjunction of the propositions in that family. The canonical injections are the partial groundings that conjuncts give to the conjunction, and the universal property of the coproduct basically says that when a proposition is weakly partially grounded by each member of a family of propositions, then there is a coproduct weak partial grounding arrow from the conjunction of that family to the proposition. This is very nice.

We might also consider a move to a category where the objects are all propositions, but that creates the challenge that we need to keep track of which propositions are true and which groundings are actual. For that p partially grounds q is, in general, a contingent matter.[note 2] Truth and actuality of grounding respects the category structure. Actual groundings only obtain between truths, and compositions of actual groundings are actual.

The category structure is going to yield transitivity of weak partial grounding. There are apparent counterexamples to transitivity. But it is my hope that when we keep track of the additional structure, and think of the token groundings as central rather than the relation of there being a token grounding, the result is not going to be problematic.

It is now interesting to investigate what different category theoretic phenomena occur in the grounding category, and how they connect up with metaphysical phenomena. One thing I'd like to see is if there is a neat category theoretic characterization of full, as opposed to partial, grounding.

I have an intuitive worry about the above approaches. Intuition would suggest that if conjunctions are coproducts, then disjunctions would be products. But they're not. For in general there is no grounding from a disjunction to disjuncts. This could be related to another worry, that because categories include identity arrows, I had to take the arrows to be weak groundings—i.e., I had to allow each proposition to count as having a grounding between itself and itself.

I do not know if category theory will in the end provide a good mathematical home to grounding structures, but I am hopeful.


Brian Cutter said...

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.)

Alexander R Pruss said...

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.

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.

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}.

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.)

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.

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.

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.

I may want to weaken the category axioms to remove the identity arrows in some cases.

Alexander R Pruss said...
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Alexander R Pruss said...

I've also been playing with the idea of using a similar framework for causation and for explanation.

Alexander R Pruss said...

1. In my long comment, <= will only be a partial ordering if there are no loops.

2. Another option for a category-based approach is to make the objects in the category be labeled by 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.

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.

This would get rid of the duplication between propositions and states of affairs on some ontologies, like Plantinga's.

Propositions could even be taken to be equivalence classes of states of affairs.

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.

This may be nuts.

Brian Cutter said...

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.)
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.

Alexander R Pruss said...

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.
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 ....
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.
If you think these approaches are worthwhile, maybe we can write something together. You know the grounding literature better than I.

Alexander R Pruss said...

By the way, I think these infinite cases might rather neatly correspond to the concept of a limit in a category.

Alexander R Pruss said...

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.

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.

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 Pruss said...

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.

Anonymous said...

Logics via category theory - Toposes