Van Inwagen thinks that the very same particles cannot have their activities constitute two life events. I am not so sure.
Pretend for the sake of simplicity that our world is made of fundamental particles with well-defined positions in three-dimensional space, with n basic quantities Q1, ..., Qn (mass, charge, etc.), otherwise interchangeable, and with no particles coming in and out of existence. And suppose we have dogs and cats (or something like them).
Suppose, further, that our world contains Rover the dog and Felix the cat. No particle that is ever a part of one of them is a part of the other. Finally, we have a very odd coincidence. There is a one-to-one correspondence ϕ between the particles that are ever a part of Rover and those that are ever a part of Felix such that particle c is a part of Rover at time t if and only if ϕ(c) is a part of Felix at that time. This last condition can be assured if we assume that Rover and Felix don’t change in what particles they have and they have the same number, but we don’t need such a strong assumption.
Now imagine a six-dimensional world made of fundamental particles, with 2n basic quantities U1, ..., Un, V1, ..., Vn. Now suppose:
- The first three coordinates of the particle positions and the quantities U1, ..., Un over time are physically related by laws exactly like the ones that govern the three coordinates of the particle positions and (respectively) the quantities Q1, ..., Qn in our world are.
Thus, if we project the six-dimensional space of this world to our three-dimensional space by ignoring coordinates 4–6, and relabel Ui as Qi, the behavior of the particles will look just like the familiar behavior of particles in our world.
Then add:
- The last three coordinates of the particle positions and the quantities V1, ..., Vn over time are physically related by laws exactly like the ones that govern the three coordinates of the particle positions and (respectively) the quantities Q1, ..., Qn in our world are.
In other words, likewise if we ignore coordinates 1–3, and relabel Vi as Qi, the behavior of the particles will be like that of our world’s particles.
Note that the Ui and Qj quantities don’t interact, and the particle positions with respect to coordinates 1–3 and coordinates 4–6 don’t interact. This is like two non-interacting universes in one, except that the particles are shared between them.
Now suppose that there is a one-to-one correspondence f between the particles of our world and those of the six-dimensional world such that:
- If at time t, particle c is at (x,y,z) and has basic quantity values Q1 = a1, ..., Qn = an, then at t in the six-dimensional world, particle f(c) is at (x,y,z,u,v,w) for some u, v and w, and has basic quantity values U1 = a1, ..., Un = an.
And there is also also another one-to-one correspondence g such that:
- If at time t, particle c is at (x,y,z) and has basic quantity values Q1 = a1, ..., Qn = an, then at t in the six-dimensional world, particle g(c) is at (u,v,w,x,y,z) for some u, v and w, and has basic quantity values V1 = a1, ..., Vn = an.
In theory, these correspondences f and g could be the same. But they are not. There is a twist:
- If c is ever a particle of Rover, then g(c) = f(ϕ(c)) and if c is ever a particle of Felix, then g(c) = f(ϕ−1(c))
where ϕ is our correspondence between Rover and Felix’s particles. This twist implies that g maps Felix’s and Rover’s particles to particles that f respectively maps Rover’s and Felix’s to.
We can mathematically arrange all this.
Now, fix a time t. If the cs are Rover’s particles at t, let the es consist of the particles corresponding to them under the mapping f. Note that the es are also the particles corresponding to Felix’s particles under the mapping g.
Here’s the fun thing. With respect to their first three coordinates and the quantities U1, ..., Un, the es behave just like Rover’s particles. With respect to their last three coordinates and the quantities V1, ..., Vn, the es behave just like Felix’s particles.
It seems right to say this: the es in our six-dimensional world have two lives, a canine-style life just like Rover’s and a feline-style life just like Felix’s. There is no interaction or unity between these two lives other than due to their being realized in the very same particles.
Suppose someone insists that there must be only one canine-feline hybrid life here because the particles are the same. Modify the mappings slightly, so that the Rover-like life is realized by particles e1, ..., eN while the Felix-like life is realized by particles e2, ..., eN + 1. Now the particles are not the same, but they are mostly the same. Now we have the same life event but no longer the same particles (just an overlap, like in the case of conjoint twins). But two particles should make little difference here. The slight change in mapping shouldn’t make a difference between have a canine-type and a feline-type life on the one hand, and having a hybrid canine-feline life.
Note that I am not doing Aristotelian metaphysics here. By “life events”, I mean the kind of empirical event a biologist might talk about, not a form or anything like that. I doubt that the same matter could have two different substantial forms.
4 comments:
There is an argument by you which can be found in The Principle of Sufficient Reason: A Reassessment, pp. 242–243, section 13.4.2.
My objection is that the crucial step in your argument is not established merely by the Brouwer Axiom, S5, or weak PSR, because your argument requires a substantive principle governing counterfactual similarity that is stronger than the modal accessibility principles on which S5 operates. Let A be the actual world, let p be the proposition that there is nothing that causes E, and let q be the true proposition that E occurs. Your weak CP gives A ⊨ ◇¬p, while the actual world satisfies A ⊨ p ∧ q. The Brouwer Axiom, as you explain, concerns the symmetry of the accessibility relation: if A R W, then W R A. Thus, when we move to an accessible ¬p-world W, the actual world A remains accessible from W, so W ⊨ ◇q. However, your crucial principle (115), (q ∧ p ∧ ◇¬p) ⊃ (¬p □→ (p ◇→ q)), requires substantially more than this. It requires that at every relevant ¬p-world W, were p to hold, q might hold; that is, W ⊨ p ◇→ q. Under the Lewis-style counterfactual semantics you yourself invoke, this is not equivalent to W ⊨ ◇q. The former requires a sufficiently similar p ∧ q-world relative to W, whereas the latter requires only some accessible q-world. S5 constrains the accessibility relation R, but it does not determine the similarity ordering or selection function used to evaluate □→ and ◇→. Consequently, W R A and q(A) yield W ⊨ ◇q, but they do not by themselves yield W ⊨ p ◇→ q, because A's accessibility from W does not entail that A is among the relevant or sufficiently similar p-worlds selected relative to W, nor does it entail that some sufficiently similar p ∧ q-world exists. This becomes especially clear if the explanatory ¬p-world W is radically different from A: there may be a world W* much more similar to W than A is, such that W* ⊨ p ∧ ¬q, while A remains accessible from W and satisfies p ∧ q. Then W ⊨ ◇q can be true while W ⊭ p ◇→ q, because the closest p-worlds to W need not preserve q. Thus the accessibility claim W R A does not entail the counterfactual claim W ⊨ p ◇→ q. The same distinction applies to your premise (113): at a world W where E has a cause, (113) gives W ⊨ p □→ ¬q, whereas your proposed analogue (115) is needed to give W ⊨ p ◇→ q. The contradiction therefore depends precisely on having both W ⊨ p □→ ¬q and W ⊨ p ◇→ q. But (116) and (117) only govern the logical relations among counterfactuals once those counterfactuals have been established; they do not establish (115). Hence the decisive premise is (115), which you characterize only as an "analogue" of the Brouwer Axiom. My objection is that the analogy does not constitute a derivation: Brouwer/S5 establishes symmetry of metaphysical accessibility, A R W → W R A, whereas (115) requires a principle connecting that accessibility structure with the counterfactual similarity structure at W. In short, your argument establishes, at most, W ⊨ ◇q, but it needs W ⊨ p ◇→ q; and ◇q does not entail p ◇→ q. Therefore, unless an independent and adequately defended bridge principle is supplied, the inference from the Brouwer/S5 observation to (115) remains unsupported. Weak PSR supplies the existence of a possible ¬p-world, S5 supplies accessibility, and (113) supplies the counterfactual ¬q at such a world, but nothing so far guarantees that the actual q-world remains sufficiently similar to that world to make q counterfactually possible under p. The precise semantic gap is therefore: A ⊨ p ∧ q ∧ ◇¬p, A R W, W R A, and W ⊨ p □→ ¬q do not by themselves entail W ⊨ p ◇→ q; the latter requires the additional counterfactual principle expressed by (115).
Yes, the Brouwer analogue does not follow from the Brouwer axiom. It is just an analogue to it, and analogy is not derivation, but nonetheless the Brouwer analogue is plausible.
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