If the premises are necessary, is the conclusion necessary?
June 6, 2004 — 20:15

Author: Jon Kvanvig  Category: Divine Providence  Comments: 5

Here’s a question related to van Inwagen’s consequence argument and its implications for an Edwardsian position on the compatibility of strong sovereignty and (compatibilist) free will, I think, but I’m not here commenting on his argument but rather a general point. (Those of you more up-to-speed on van Inwagen and free will may be able to educate me here…)
The argument I’m thinking about begins with the usual understanding of determinism:
Premise 1: some specification of initial conditions
Premise 2: a listing of the true laws of nature
Conclusion: the entirety of the future
The idea of the argument is that determinism allows one to infer the entirety of the future given only the laws of nature and some specification of initial conditions.
(There is a caveat here that I will ignore below. We need also to insist that determinism involves the claim that nothing ever happens except what can be explained in the above fashion. Otherwise the possibility of miracles, in terms not of contradicting laws of nature, but of contravening them in some other way, changes the status of the above argument. It changes it in such a way that the conclusion follows from the premises, not of logical necessity, but only of nomological necessity. This difference won’t matter below, so I ignore it in what follows.)
Now for my version of a consequence argument. It employs three premises:
Premise 1: In the above argument, the first premise is necessary.
Premise 2: In the above argument, the second premise is necessary.
Premise 3: In the above argument, the connection between the premises and the conclusion is necessary.
Conclusion: Therefore, the conclusion of the above argument is itself necessary.
I’m interested in the general principle that “from necessary premises, necessary conclusions follow.” Call this principle “NPNC”. This principle is fine when the kinds of necessity in the premises are the same, but note that in my version of the consequence argument, there are 3 kinds of necessity. In premise 1, the necessity is accidental necessity; in premise 2, nomological necessity; and in premise 3, logical necessity. NPNC is also fine in some cases where the kinds of necessity differ. If, for example, the necessities can be nested in terms of strength, the principle is fine as long as the necessity attributed to the conclusion is the weakest kind. So is there any defensible version of NPNC for this argument?


Here’s a question related to van Inwagen’s consequence argument and its implications for an Edwardsian position on the compatibility of strong sovreignty and (compatibilist) free will, I think, but I’m not here commenting on his argument but rather a general point. (Those of you more up-to-speed on van Inwagen and free will may be able to educate me here…)
The argument I’m thinking about begins with the usual understanding of determinism:
Premise 1: some specification of initial conditions
Premise 2: a listing of the true laws of nature
Conclusion: the entirety of the future
The idea of the argument is that determinism allows one to infer the entirety of the future given only the laws of nature and some specification of initial conditions.
(There is a caveat here that I will ignore below. We need also to insist that determinism involves the claim that nothing ever happens except what can be explained in the above fashion. Otherwise the possibility of miracles, in terms not of contradicting laws of nature, but of contravening them in some other way, changes the status of the above argument. It changes it in such a way that the conclusion follows from the premises, not of logical necessity, but only of nomological necessity. This difference won’t matter below, so I ignore it in what follows.)
Now for my version of a consequence argument. It employs three premises:
Premise 1: In the above argument, the first premise is necessary.
Premise 2: In the above argument, the second premise is necessary.
Premise 3: In the above argument, the connection between the premises and the conclusion is necessary.
Conclusion: Therefore, the conclusion of the above argument is itself necessary.
I’m interested in the general principle that “from necessary premises, necessary conclusions follow.” Call this principle “NPNC”. This principle is fine when the kinds of necessity in the premises are the same, but note that in my version of the consequence argument, there are 3 kinds of necessity. In premise 1, the necessity is accidental necessity; in premise 2, nomological necessity; and in premise 3, logical necessity. NPNC is also fine in some cases where the kinds of necessity differ. If, for example, the necessities can be nested in terms of strength, the principle is fine as long as the necessity attributed to the conclusion is the weakest kind. So is there any defensible version of NPNC for this argument?
So, can the necessities in question be nested in terms of strength? Nomological and logical necessity can be: the former is weaker than the latter, since a nomologically necessary claim is true in some subset of the set of all logically possible worlds (in particular, in those with the same laws of nature as our world). But what of accidental necessity? Accidental necessity is the sort of necessity that attaches to the past–the past is, in some important sense, fixed (if you don’t think this, then the above argument is hopeless, of course). Thus, an accidentally necessary truth is one that is true in every world with the same history as our world, up to the moment of evaluation, in this case the present moment. As such, these worlds compose a subset of the logically possible worlds, so accidental necessity is weaker than logical necessity.
The crucial question, however, is the relationship between nomological and accidental necessity. Does every world with the same history as ours also share the same laws with our world (assuming ours is a deterministic world) or vice-versa? The answer seems to be “no”: two worlds could have the same history but different laws (the difference in the laws will be reflected only in some future differences in the two worlds), and two worlds could have the same laws but different histories (this is trivial–just imagine different initial conditions).
But, if the two kinds of necessity cannot be nested in terms of strength, I don’t see how to get any defensible reading of NPNC. Any suggestions?

Comments:
  • One question in all this involves whether God is included within the possible world, as a combinatorial Lewis-style view would require. Leibniz wasn’t thinking this way at all when he discussed possible worlds. For him, God just looks at all the possible worlds and picks one. Thus accidental necessity is irrelevant to issues about what God is responsible for. However, if God’s choices are included in the history of the world, then that changes things, particularly if God’s thoughts are ordered temporally. Within the actual world’s history is a choice made by God to begin with certain conditions, but it’s consistent with things before that to have different initial conditions in the created world. To take the step back and say that God’s choices are predetermined raises questions about God’s freedom, which I plan to address in a post of my own (where I’ll defend Leibniz).
    So I don’t have a criticism here, but that’s because I haven’t really figured out which view of God’s relation to modality that you’re working with here.

    June 7, 2004 — 10:53
  • Jon,
    The nub of the problem with Van Inwagen’s argument is, as you know, that not all modalilties follow your principle. If they did, we’d have no dispute over the truth of beta.
    Cheers,
    Matt

    June 8, 2004 — 18:40
  • Matt, these comments are very helpful, both here and on my other post about free will and logical consequences. Tell me what K and beta are, and also, can you give me an example of a type of necessity that does not obey the principle “from necessary premises necessary conclusions follow”?

    June 8, 2004 — 20:07
  • Matt, these comments are very helpful, both here and on my other post about free will and logical consequences. Tell me what K and beta are, and also, can you give me an example of a type of necessity that does not obey the principle “from necessary premises necessary conclusions follow”?
    Jon,
    Beta is the transfer principle in the van Inwagen-style argument for incompatibilism. Let N be “it is power necessary that”, let p be a proposition detailing the entire state of the world at some past moment in time (along with the laws of nature), and let q be some future action. Then, the argument roughly is
    1. Np
    2. N(p->q)
    3. [beta] (Np&N(p->q))->Nq
    4. Therefore, Nq
    So beta is the “transfer principle,” it moves the power necessity (or no control) from the past and the conditional connecting the past and the future action to the future action.
    K is your principle, the axiom in the most basic modal system (Let “L” be a box): (Lp & L(p?q))?Lq
    K is fine for broad logical necessity. But, with power necessity, K (or its analogue) isn’t clearly correct.
    If you let the operator stand for “knows” in epistemic logic, it’s not at all clear that K holds for it (I can know that p, know that it implies q and not know that q because of some defect in my noetic structure–I haven’t put my beliefs together in the right sort of way).
    Matt
    p.s. I don’t know if you’ve been away from College Station long enough for this to mean anything, but I’ll try anyway: GO CUBS!

    June 8, 2004 — 22:30
  • Van Inwagen has given up on beta given the McKay-Johnson counterexample that shows its invalidity. See here for a summary of this. I’m unfamiliar with how the debate has proceeded since then, but I believe that was when van Inwagen started emphasizing that free will is just a mystery, and his revision in his Metaphysics book’s second edition didn’t seem to me to get out of the original problem McKay and Johnson presented.

    June 9, 2004 — 9:25