In An Essay on Free Will, van Inwagen tries to define his notion of the laws of nature (for his First Argument that follow), this is the concluding paragraph:

Thus, on our "sets of possible worlds" model for propositions, a law of nature is any set of worlds that has as a subset the set of all worlds in which the laws of nature are the same as those of the actual world, or, as we might say, are nomologically congruent with the actual world.

Does he mean that the sets of possible worlds (for his first argument) is limited to only worlds where the laws are the same as in the actual world?

I know nothing about Modal Logic and trying to dig through this.

  • Ontological modal logic attempts to finds things out about this world by considering the only truth that we can assert without qualification is logic; thus it expands out this actual world into many possible worlds. Van Inwagen identifies a world with a natural law as a world where a law hold like ours; basically he's saying he's saying he's only considering those worlds whose physics is the same as ours. Apr 29 '15 at 14:07

Not quite. It's a little easier to read if we make the sets explicit.

Let curly braces denote any set, where the name of that set is inside the braces. For instance, set x is {x}. The set's name, followed by a colon, indicates the set's members (separated by commas). Hence if a, b, c are members of set x, we can show this as {x: a,b,c}.

Subsets are members of a set. Hence if a law of nature is "any set of worlds that has a subset ๐•Ÿ", then this is {x: {๐•Ÿ}}.

Let's say that a law of nature at the actual world can be denoted by Li. Then the set of all laws that hold in the actual world (i.e., the world we currently live in) is { L: L1, L2 .., Ln }. Let {๐•ƒ} be the set of worlds in which the laws in L and only the laws in L are true.

By Van Inwagen's definition, a law of nature just is the set of worlds {Wa} for which ๐•ƒ is a subset, or {Wa: {๐•ƒ}}. That's all. It may still be possible that a different law not in L holds for some other possible world, but then that law would not be a natural one. For instance, say {๐•Š} is some set of possible worlds for which { S: S1, S2 .., Sn } are not in the actual world. Then S can still be in the set of possible worlds {Wb}.

All this means is that if { Wb: {๐•ƒ}, {๐•Š} }, then Wb would still count as a law of nature just like Wa would. { Wc: {๐•Š} } would be excluded as a law of nature, even if it overlaps with some members from ๐•ƒ.

That is, say a is the actual world, b is a possible world for which all and only L hold true, and c is a possible world where all L and some S holds true. Then any W with members a, b, and c can have the subsets { X: a, b }, { Y: b, c }, and { Z: a, c }. If these were all the possible worlds that exist, then W and X would be natural laws, but Y and Z would not.

All this is, is a way of stating very specifically that the laws of nature are just those laws which are our own laws of nature, for the actual world. If we're talking about possible other sets of laws, such as a possible world P where photons experience relativistic effects (e.g., in Greg Egan's The Clockwork Rocket), then our speed of light is a law of nature, and the laws governing light for P are not.

It's an astoundingly complicated way to make what may seem to be a trivial statement,ย but Van Inwagen is concerned with preserving substantive possible outcomes of human decisions. Putting this in the verbiage of modal realism is hard work, but what he wants to say is that determinism is false. If determinism was true, then any possible world could not have all and only the same laws as our world. It would require some other law of nature to obtain for it to be some world other than our actual world. Thus there must be a real set of natural laws which can "lead to" more than one possible world (loosely speaking), if determinism is false.

  • (I get goosebumps when someone explain things to me in such detail). As I understand from your conclusion in plain English, then what van Inwagen means as 'law' is simply the actual law as in our world. Then how can {Wb} still count as 'law'? By that definition I thought only {Wa} count?
    – onion
    Apr 29 '15 at 17:21
  • @onion As Inwagen states in your quote, a natural law is any law which has some subset of possible worlds for which all and only L hold true. Modal logic can treat the past and future as possible worlds as well. So, if someone makes a choice at some moment in time t1, then at a later point t2 it must still be the case that there are possible worlds which are not the actual world, but for which the same laws L hold true. Yet if we are truly free (for Inwagen), then it must be the case that we can "introduce" a new law of nature (or fact of the matter of what someone chose).
    – Ryder
    Apr 29 '15 at 22:27
  • In this context, {Wb} can be interpreted as t2, and {Wa} as t1. That is, we can add new laws of nature, but never take away, if it really is a law of nature and not some law like P's. I have no source, but I imagine Inwagen must have seen this kind of unidirectional attribute as a benefit to explaining the arrow of time as well.
    – Ryder
    Apr 29 '15 at 22:36

Propositions as sets of worlds

As van Inwagen says in the quoted passage, he talks in terms of a "sets of possible worlds" model for propositions. According to this model, a proposition is identified with a set of possible worlds, namely the set of worlds in which the proposition is true. (Alternatively but equivalently you can think of propositions as functions from possible worlds to truth values).

Cf. e.g. David Lewis: On the Plurality of Worlds, Blackwell: 1986, pp. 53-55.

Laws of natures as certain kinds of propositions

Let's call a world in which the laws of nature are as in the actual world, a natural world. Then if you define a law of nature in the way van Inwagen does, you say in effect that a law of nature L is a superset of the set N of natural worlds, i.e. LS, or in the expanded form: ∀ x ( xNxS). That is, a law of nature is considered to be any proposition which is (at least) true in every world that has the same laws of nature as our world. In more mundane terms: a proposition P is a law of nature if and only if any possible situation governed by the actual laws of nature is a situation, in which P is true.

Hope that helps a little bit in understanding his argument (which I do not know).

Afterthought: I doubt this is a good definition, since trivially, a proposition consisting of all the worlds physically like ours plus some randomly chosen bizarre alien worlds counts as a law of nature, too, according to this definition.

  • Does it demonstrate like this? i.imgur.com/oA5Q0tB.png (N: Natural worlds; A: Alien worlds); all of these L are valid law under this definition?
    – onion
    May 9 '15 at 16:32
  • yes. and since the actual world is an element of N it is an element of L1, L2, and L3, so, these three laws are all true, as are all laws of nature according to the definition.
    – r.b
    May 11 '15 at 7:16
  • then it's weird (that he allow such alien worlds to count), maybe he has some intention here...
    – onion
    May 17 '15 at 3:53
  • maybe, but as said, I do not know van Inwagens argument, so can not judge this. Probably this overgeneration of his definition does not matter for the argument to go trough, so he is not worried by it.
    – r.b
    May 25 '15 at 13:12

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