Infinite dependence regresses and set theory

Given the Axiom of Dependent Choice, the Axiom of Regularity in set theory is equivalent to the statement that there are no backwards infinite membership regresses, i.e., no cases where we have a backwards infinite sequence of sets ...,A₋₃,A₋₂,A₋₁,A_-0, where each set is a member of the next. Why think this is true? Well, intuitively, a set depends on its members. That suggests that the reason to believe the Axiom of Regularity is that there cannot be an infinite dependency regress. And that in turn has all sorts of other consequences (including that there is a first cause).

Kant and Lewis on our freedom

Kant (on one reading) holds that the initial conditions of the universe and the laws of nature depend on us (noumenally speaking). This reconciles determinism with freedom: sure, our actions are determined by the laws and initial conditions, but the laws and initial conditions are up to us. Kant also thinks that a further merit of this view is that one can blame people whose misdeeds come from a bad upbringing, because noumenally speaking they were responsible for their own upbringing.

Lewis holds that freedom is compatible with determinism, and in a deterministic world had one acted otherwise, the laws would have been different.

Everybody agrees that the view I ascribe to Kant is crazy (though not everybody agrees that the ascription is correct). But Lewis's view is supposed to be much saner than Kant's.

How? The obvious suggestion is that Lewis only makes the laws depend counterfactually on our actions (assuming determinism) while Kant makes the laws depend explanatorily on our actions. But that suggestion doesn't work, since Lewis's best-systems account of laws makes the laws depend on the law-governed events, and so it makes the laws depend not just counterfactually on our actions but also explanatorily: the laws' being as they are is grounded in part in our actions. So both accounts make the laws explanatorily depend on us.

Admittedly, Kant also makes the past, not just the laws, depend on our actions. But that's also true for Lewis, albeit to a smaller degree, because of his doctrine of small miracles...

Dependence

Is being dependent an intrinsic property of an entity?

Suppose we say that it is intrinsic. Then we have the following interesting consequence. Assuming there are dependent entities, it is possible to have an intrinsic property, D, whose possession entails the obtaining of a genuine relation (a dependence relation) to another entity, but where D is, nonetheless, not relational. This would force us to deny strong recombination principles in accounts of modality. And that would be a good thing from my point of view. For one, it would force a humility in the move from apparent conceivability to possibility. (The modal problem of evil is one place where this matters.)

Could we say that being dependent is not an intrinsic property? That, I think, would be odd. If being dependent is not an intrinsic property, or at least is not entailed by the intrinsic properties of the entity (all I need for the arguments of the previous paragraph is that being dependent is entailed by the intrinsic properties of a thing), then being dependent is not a matter of some kind of inner need or lack in the entity. If George could survive without water, and without any substitute (natural or supernatural) for water, and without without any intrinsic difference in him, then he is not really dependent on water for his existence. My intuition is that the notion of a dependence that does not supervene on the intrinsic properties of a thing and that (therefore) is merely accidental is a sham dependence. I don't yet have a very good argument here, beyond just restatements of the intuition.

If I am right, then Hume has no conceptual resources to affirm that any entity is genuinely causally dependent. For on his view, "causal dependence" would have to be an extrinsic property of an entity, and hence, if I am right, would at best be just a sham dependence.

Metaphysically light existence

In the previous post, I argued that artefacts do not exist in a "metaphysically serious way". The notion of metaphysically serious existence is a foggy one. But I think I can give two sufficient conditions for it. The trivial one is that Fs don't exist in a metaphysically serious way if there just are no Fs. The non-trivial one is that Fs don't exist in a metaphysically serious way provided that whenever "x" is the name of an existent F, the proposition that x exists is a proposition that holds in virtue of the truth of some proposition that does not make reference to the x.

For instance, if x is a particular hole in a wall, the claim "x exists" holds in virtue of a proposition reporting a certain area's being surrounded by the wall but not itself containing any parts of the wall. Likewise, if x is a waltz that George and Sally are dancing, then the claim "x exists" holds in virtue of George and Sally waltzing at a certain time in a certain way (the "certain" encode the amount of precision to ensure that we're talking of this waltz rather than another waltz). Thus, holes and waltzes don't exist in a metaphysically serious way.

Note that the "in virtue of" relation here is more than just "being entailed by."

On my view claims like "This table exists" may be true in virtue of facts about arrangements of particles and/or fields as well as the intentions and/or practices of agents and/or communities. Or, alternately, such claims are false as they stand, but they are close approximations to true claims which hold in virtue of facts about arrangements of particles and/or fields as well as the intentions and/or practices of agents and/or communities.