Thursday, August 16, 2012

Grounding graphs, new take

In a previous post, I looked at the idea of grounding graphs as global entities. But I think there is a more natural way of looking at them. There are two main views about grounding. On the truthmaker view, true propositions are grounded in entities that make them true. On the propositional view, true propositions are grounded in other true propositions. But I think a more natural approach is to say that propositions are grounded in graphs.

A candidate grounding graph for a proposition p is a directed graph G satisfying the following properties:

  1. all the vertices of G are true propositions
  2. p is a vertex of G
  3. all the vertices of G other than p are ancestors of p
  4. p is not the only vertex of G.
The grounding relation is then a relation between a proposition p and a candidate grounding graph for p. For instance, the proposition <The sky is blue or (roses are red and violets are blue)> is grounded in a graph with two vertices, one of which is <The sky is blue> and the other being the target proposition, with one arrow from the former to the latter. But it is also grounded in a more complex graph with four vertices: <Roses are red>, <Violets are blue>, <Roses are red and violets are blue>, and the target propositions, with arrows from the first two propositions to the third, and an arrow from the third to the target.

Define a proposition as fundamental provided that it is true but has no grounding graph. Say that a grounding graph for p is a candidate grounding graph for p that in fact grounds p. A vertex is initial provided that it has no ancestors and is final provided it has no children. A candidate grounding graph has exactly one final vertex. Say that G* extends G provided that (a) G* has the same final vertex as G and (b) every vertex of G that has a parent in G has exactly the same parents in G* as in G. The following are important properties of grounding graphs: Say that a graph where every initial vertex is fundamental is a fundamental graph.

  • Acyclicity: Every grounding graph is acyclic.
  • Extensibility: If G is a grounding graph for p, then there is a fundamental extension of G that is also a grounding graph for p.
  • Adjoining: If G1 is a grounding graph for p, and G2 is a grounding graph for some initial vertex q of G1 such that G2 has a fundamental extension whose only vertex in common with G1 is q, then the graph whose vertex collection is the union of the vertex collections of G1 and G2 and whose arrow collection is the union of the arrow collections of G1 and G2 is also a grounding graph for p.
  • Truncation: If G is a grounding graph for p, then any subgraph of G that is a candidate grounding graph and that has the property that if it contains any one of G's arrows to q then it contains all of G's arrows to q is a grounding graph.

The following is very controversial but very helpful:

  • Well-foundedness: No grounding graph contains an infinite chain of arrows.
This is compatible with some grounding graphs being infinite. For instance, we could have a fundamental grounding graph for an infinite conjunction. There, the infinite conjunction will have infinitely many parents. Moreover, there may be arbitrarily long chains in the graph—the first parent might be fundamental, the second might have a chain of length two to a fundamental ancestor, and so on.

I think that if we reject well-foundedness, we should reject acyclicity. For the most plausible putative counterexamples will be infinitely nested propositions like p1&(p2&(p3&...)). But if we accept such propositions, we will also accept p&(p&(p&...)), and these will be cyclically grounded if the former will be non-well-foundedly grounded. But we shouldn't reject acyclicity, so we should accept well-foundedness, and if there are such infinitely nested propositions, we should ground them all at once in the symmetric conjunction p1&p2&... which then is grounded in each of its conjuncts.

Finally, we want to say something about how this interacts with logic. Say that p is free of q provided that is a fundamental grounding graph for p that does not contain q.

  • Disjunction introduction: If p is free of (p or q), then the following is a grounding graph: p→(p or q).


Heath White said...

I think this approach could be helpful with some of the Liar et al paradoxes, and I hope you keep working on it. Two comments:

- At present, I don't see that acyclicity is required by your definition of a grounding graph. Seems important!

- the arrows are just entailments, right?

Alexander R Pruss said...


Right: acyclicity is an additional assumption.

The arrows are "grounding arrows". :-) I think that one might add as an axiom that if a grounding graph contains p→q, then p entails q. I can't think of a plausible counterexample to the thesis that grounding entails entailment.

Heath White said...

Well, here is a potential counterexample to the entailment idea.

From "S" we might have an arrow to "'S' is true". But if we were speaking a different language, the inference would be no good. Maybe you have to say " true in English" or implicitly hold the language constant.

Alexander R Pruss said...

I shouldn't have agreed given that I think of each individual arrow as only partial grounding. For a full grounding, we need all the parents of a vertex.

And perhaps the conjunction of the parents of a vertex entails the vertex?

In your case, from <S> we might put an arrow to <'S' is true>, but the latter will have other parents beside <S>, namely parents that specify facts about the language.