Molinism and Powersets

Jacobus

Erasmus

(Kobus)

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November 22, 2021

Objection:

Here’s an objection to Molinism that was shared on the Mere Molinism Facebook group:

“By the way, Molinism is probably self-contradictory if we define a “possible world” as a “maximally consistent set of propositions,” so we need to be clear what we mean by “possible worlds” when discussing Molinism.

Here’s why:

1. Let’s call the set of possible worlds P. A subset of those is the set of feasible worlds. It’s a tenet of Molinism that the set of feasible worlds could have been different. So, if a subset of P could have been the set of feasible worlds, then we’ll call that a ‘set of possibly feasible worlds.’ Let’s call the set of all sets of possibly feasible worlds F.

2. The powerset of P is the set of all subsets of possible worlds. F looks like it would have the same cardinality as the powerset of P, since F is just a huge chunk of subsets of possible worlds contained in a set. The powerset of a set is always of greater cardinality than the set, so |𝒫(P)| is greater than |P|. So, since |𝒫(P)| = |F|, |F| is greater than |P|.

3. But something has gone wrong! For every set of possibly feasible worlds in F, there is a possible world where God finds that just those worlds are feasible. This shows that |F| equals |P|!”

How would FreeThinking Ministries respond?

– Will

Response:

Contra the objector, there is no self-contradiction.

First, it is false that “the powerset of a set is always of greater cardinality than the set”, since the powerset of an infinite set has the same cardinality as the latter, e.g.  |ℕ| = |powerset of ℕ|. So, if F and P are both infinite sets, then who cares if they have the same cardinality (e.g., א)? There is simply no self-contradiction here, and it seems the objector is troubled by set theory and not by the notion of possible and feasible worlds per se.

Second, there are different ways to talk about modality. For example, modality may be understood in an unrestricted or broadest metaphysical sense, or in a relative sense. If we are talking about unrestricted modality, then we are not concerned about (im)possibility and necessity relative to certain possible worlds but, rather, we are concerned with all metaphysically possible worlds.

So, for example, suppose there are four possible worlds in the broadest metaphysical sense: W1, W2, W3, and W4. Then that is it; that is our set of possible worlds. Suppose further that only W1 and W4 are feasible:

• Possible worlds = {W1, W2, W3, W4}
• Feasible worlds = {W1, W4}

Since we are talking about unrestricted modality, our set of feasible worlds is not relative to certain possible worlds and, hence, it makes no sense to say there are non-feasible but possibly feasible worlds. In this sense of modality, it is not true that “it’s a tenet of Molinism that the set of feasible worlds could have been different” and, thus, the objector’s critique is groundless.

On the other hand, we may introduce the notion of relative or restricted modality, which is to introduce consistency-relationships between the possible worlds. In this case, we may say, loosely speaking, that each possible world has its own sets of possible and feasible worlds. For example, suppose that some proposition (e.g., “Joe Biden is the 46th president of the US”) is true in W1 but false in W2. Then, W2 is impossible relative to W1, or, in other words, W2 is inconsistent with W1 and, hence, if W1 is actualised, then W2 is impossible (in the relative sense). Accordingly, W2 is a possible world in terms of broad, unrestricted modality, but W2 is impossible relative to W1.

Feasible worlds may also be discussed in terms of relative modality. For example, suppose that

• W1 and W2 are feasible relative to W1,
• W2 and W3 are feasible relative to W2,
• W1 and W3 are feasible relative to W3, and
• W1, W2, and W4 are feasible relative to W4.

In this case, and in the sense of relative modality, we may say “that the set of feasible worlds could have been different”.

We can now see that the objector is muddling and conflating unrestricted modality with relative modality: he/she talks about “all possible and feasible worlds” (unrestricted modality) and “possibly feasible worlds” (relative modality) in the same befuddled context. This is why “something has gone wrong” and why he/she arrives at the conclusion that “|F| equals |P|”.

– Jacobus Erasmus

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