Consider three propositions:

- (2) or (3) is true.
- (1) or (3) is true.
- The sky is blue.

What should we say about (1)-(3)? It was plausible to say that (3) grounds (1) and (2). But the line of thought that (3) grounds (2) and (2) grounds (1) was also plausible. We might say that there are three pathways to grounding among (1)-(3):

- (3) to both (1) and (2)
- (3) to (2) to (1)
- (3) to (1) to (2)

There are multiple grounding pathways. Here is one way to formalize this. Take as the primitive notion that of a *grounding graph*. A grounding graph encodes a particular mutually compatible grounding pathway. Each grounding graph is a directed graph whose vertices are propositions. It will often be a contingent matter whether a given graph is or is not a grounding graph: the same graph can be a grounding graph in one world but not in another. The notion is not a formal one. Moreover, grounding graphs will be backwards-complete: they will go as far back as possible. But their futures may be incomplete.

Say that a *parent* of a vertex *b* in a directed graph *G* is any vertex *a* such that *a*→*b* is an arrow of *G*, and then *b* is called a child of *a*. An *ancestor* is then a parent, or a parent of a parent, or .... An *initial* vertex is one that has no vertices.

We can say that *a* *partly grounds* *b* in *G* if and only if *a* is an ancestor of *b* in *G* and that *a* is *fundamental* in *G* if and only if *a* is initial in *G*. We say that a proposition *a* partly grounds *b* provided that there is a grounding graph *G* such that *a* partly grounds *b* in *G*, and that a proposition *p* is fundamental if and only if there is a grounding graph *G* such that *p* is fundamental in *G*. We say that the *a* partly grounds *b* *compatibly* with *c* partly grounding *a* provided that there is a single grounding graph in which both partial grounding relations hold.

We say that a finite or infinite sequence of vertices is a *chain* in *G* provided that there is an arrow from each element of the sequence to the next. We say that *b* is the *terminus* of a chain *C* provided that *b* is the last element of *C*.

We stipulate that a set *S* of vertices *grounds* *b* in *G* provided that (a) every vertex in *S* is an ancestor of *b* and (b) every chain whose terminus is *b* can be extended to a chain still with terminus *b* and that contains at least one member of *S*. In particular, the set of all the parents of *b* grounds *b* if it is non-empty.

We now have some *bridge axioms* that interface between the notion of a grounding graph and other notions:

- Truth: Every vertex of a grounding graph
*G*is*true*. - Explanation: Every non-initial vertex is
*explained by*its parents. - Partial Explanation: Every parent
*partly explains*each of its children.

We add this very metaphysical axiom, which is a kind of Principle of Sufficient Reason:

- Universality: Every true proposition is a vertex of some grounding graph.

Now we add some structural axioms:

- Noncircularity: There is no grounding graph
*G*in which*a*is a parent of*b*and*b*is a parent of*a*. - Lower Bound: If
*C*is a chain in a grounding graph*G*, then there is a vertex*p*of*G*which is the ancestor of all the vertices in*C*, other than*p*itself if*p*is in*C*. - Wellfoundedness: No vertex of a grounding graph is the terminus of an infinite chain.
- Absoluteness of Fundamentality: No vertex is initial in one grounding graph and non-initial in another.
- Truncation: If
*G*_{1}is a grounding graph and*G*_{2}is a subgraph of*G*_{1}relatively closed under the parent relation (if*b*is in*G*_{2}and*a*is a parent of*b*in*G*_{1}then*a*is in*G*_{2}and*a*is a parent of*b*in*G*_{2}), then*G*_{2}is a grounding graph.

Absoluteness of Fundamentality says that if a proposition is fundamental, it is fundamental in every grounding graph where it is found. Of course Wellfoundedness entails Noncircularity and Lower Bound. And Noncircularity plus Absoluteness of Fundamentality entails that if *a* partly grounds *b* and *b* partly grounds *a*, then (a) these two grounding relations do not hold in the same grounding graph and (b) in every grounding graph where one of these relations holds, at least one of *a* and *b* is grounded in something other than *a* and *b*, so that there are no fundamental circles.

We can now add some "logical axioms". These are just a sampling.

- Disjunction Introduction: If a grounding graph
*G*contains a vertex <*p*> but not the vertex <*p*or*q*>, then the graph formed by appending <*p*or*q*> to*G*together with an arrow from <*p*> to it is also a grounding graph. - Conjunction Introduction: If a grounding graph
*G*contains vertices <*p*> and <*q*> but not the vertex <*p*&*q*>, then the graph formed by appending <*p*&*q*> to*G*toegther with arrows from <*p*> and <*q*> to it is also a grounding graph. - Existential Introduction: If a grounding graph
*G*contains a vertex <*F**a*> but no vertex <(∃*x*)*F**x*>, then the graph formed by appending <(∃*x*)*F**x*> together with an arrow from <*F**a*> to <(∃*x*)*F**x*> is a grounding graph. - Conjunctive Concentration: If a grounding graph
*G*contains a vertex*b*with distinct parents <*p*> and <*q*> but no vertex <*p*&*q*>, then the graph formed by removing the arrows from <*p*> and <*q*> to*b*, adding the vertex <*p*&*q*> and inserting arrows from <*p*> and <*q*> to <*p*&*q*>, and from <*p*&*q*> to*b*is a grounding graph. - No Disjunctive Overdetermination: If a grounding graph contains <
*p*or*q*>, then it contains at most one of the arrows <*p*>→<*p*or*q*> and <*q*>→<*p*or*q*>.

Go back to our original example. There will be at least three distinct grounding graphs corresponding to the different grounding pathways. There will be a grounding graph where we have (3)→(2)→(1), and another where we have (2)→(3)→(1), and a third which contains (3)→(1) and (2)→(1). But there won't be a graph that contains both (2)→(1) and (1)→(2).

I don't really insist on this list of axioms. Probably the "logical axioms" are incomplete. Nor am I completely sure of all the axioms. But the point here is to indicate a way to structure further discussion.

[Definition of universality edited to fix problem pointed out in discussion.]

## 3 comments:

Perhaps the problems with (1) and (2) come from their nature as statements that other truth statements are true: they are meta-grounding statements.

Most paradoxes of truth values come from using statements about truth to speak about truth value itself, rather than data.

Unless I'm mistaken-- and I very well may be-- it appears to be impossible for a graph to satisfy these axioms. Suppose there is such a graph g. Then by well-foundedness, there will have to be some proposition-- call it p-- which is fundamental (initial) in g; p will be true (by Truth), and by Absoluteness of Fundamentality, there is no graph in which p is non-initial. But by Universality, given that p is true, there is a graph in which p is non-initial. Contradiction.

Yeah, I shouldn't have said "non-initial" in Universality. Thanks for catching that.

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