Molinism and prophecy

Suppose Bob offers a cookie to Alice who is free to accept or reject it. God would like to manifest his omniscience to Alice by announcing to her whether she will accept or reject the cookie. To that end, God uses his middle knowledge to consider the following conditionals:

1. Were God to announce that Alice will accept, Alice would accept.

2. Were God to announce that Alice will reject, Alice would reject.

God then follows the following algorithm:

1. If (i) is true and regardless of (ii), God announces that Alice will accept.

2. If (i) is false but (ii) is true, God announces that Alice will reject.

3. If both (i) and (ii) are false, God refrains from making an announcement.

But there is a problem. Both (i) and (ii) seem to be necessary truths! For God cannot lie and cannot be mistaken. Thus, following the algorithm, God will always announce that Alice will accept, and indeed she will. Hence God can with certainty get Alice to freely accept the cookie—which seems wrong! So there is something wrong with Molinism.

However, I wonder whether the Molinist has to say that (i) and (ii) both have to be true. Granted, their necessity follows from the following plausible rule for subjunctive conditionals:

4. Necessarily: If p is possible, and p entails q, then were p to hold, q would hold,

assuming that the antecedents of (i) and (ii) are possible and given S5. (And one can avoid S5 by assuming that necessarily the antecedents of (i) and (ii) are possible, and that the entailment between God announcing something and its being true is not only necessary but necessarily necessary.)

But perhaps (4) is not as obviously true as it seems. Consider a case where free will is not involved. There is a perfectly reliable sound-activated light. God is debating whether to announce “There will be no miracle and yet the light will not turn on.” It seems that it would be reasonable for God to reason that were he to make that announcement, the light would turn on (since it’s perfectly reliable, it would take a miracle for it not to), and hence the announcement would be false. Yet God’s making the announcement is possible (there is no problem about God making the announcement in a world where the lightbulb is broken).

So perhaps the Molinist can say that there can be subjunctive conditionals of free that are true even though the antecedent entails the negation of the consequent. If so, then (i) and (ii) could still be contingent.

I am still dubious of a logic of subjunctives where the subjunctive can violate an entailment. But perhaps it’s not as bad as I once thought it (compare this old paper of mine).

A variant of Molinism

The biggest problem with Molinism is the grounding problem: there is nothing in reality wherein the subjunctive conditionals of free will (“middle truths”) could be grounded. As Reuben perceptively noted in a recent comment, the problem is especially big if God is simple, as then the middle truths can’t even be grounded in God.

Anyway, it seems to me that there might be some benefit to Molinists to think about what one might call Individually Grounded Weak Molinism (IGWM). On IGWM, there are no middle truths about the free actions of non-actual persons, but there are middle truths about the free actions of actual persons, which are grounded in properties of these very persons. Thus, “when” (in the explanatory and not temporal sense) God is deciding whether to create Curley, there is no fact of the matter whether Curley would accept a $5000 bribe in circumstances C. But “as soon as” Curley is actual (on eternalism, this will be the case even before Curley temporally exists—maybe as soon as God decides to create Curley), there will be a fact of the matter whether he would accept that bribe in C.

On IGWM, there is an “ontological home” for the middle truths. Moreover, IGWM ameliorates another problem for standard Molinism: the problem of evil. On standard Molinism, God presumably has a vast amount of knowledge that he can use in optimizing the world. E.g., God knows that Adam and Eve would sin, but there is a vast number of couples almost exactly like Adam and Eve, and it seems very unlikely that they would all have sinned in Adam and Eve’s circumstances, so why didn’t God create them? But on IGWM, God only knows that Adam and Eve would sin in the Garden of Even once he decides to create them. One might objet that are presumably a vast number of gardens almost like Eden that God could put Adam and Eve in, and on IGWM God middle-knows how they would behave in each garden, and it would seem surprising if Adam and Eve would have sinned in all of them. Not so. For it might be that there are close correlations between middle-truths about how Eve (say) would act in circumstance C₁ and how she would act in C₂ if C₁ and C₂ are very similar.

But IGWM still has enough bite to yield some of the advantages of Molinism. First, we have the biblical cases. Granted, these could be interpreted in a non-Molinist way, but a Molinist way seems particularly neat. For instance, in the parable of Lazarus and the rich man, God tells the rich man that his brothers wouldn’t repent even if someone came back from the dead to talk to them. On IGWM, this conditional makes sense, and is grounded in the rich man’s brothers. (Granted, this is a parable, so it’s not a prooftext for Molinism.) Second, we still have the nice story about how God can work in our lives and achieve his gracious goals without undercutting our freedom: God knows what we would choose in which circumstances, and can control the circumstances we are in. Third, the Molinist account of efficacious grace works on IGWM with no changes. Fourth, and this is the immediate motivation for my post, my Molinist defense of hell works just fine on this story.

IGWM does not solve all the problems with Molinism. The Adams circularity argument continues to be problematic. Suppose I freely choose A in circumstances C, and God put me in C because he middle knew that I would freely choose A in C. It’s surely note a coincidence that both (a) I freely choose A in C and (b) I would freely choose A in C. There needs to be some kind of explanatory connection between (a) and (b). If (b) is explanatorily prior to (a), then it seems that the middle facts about me explain my actions, and that seems to undercut my freedom. But if (a) is explanatorily prior to (b), then we have a circularity. For (b) is prior to C, and C is prior to (a).

And there may be a way in which the freedom worry about Molinism feels bigger on IGWM. If it’s a property of me that I would freely choose A in C, rather than just being an ungrounded truth, then it looks a bit more like that property constrains me and takes away my freedom.

Anselm and Brouwer

I was reading Anselm’s replies to Gaunilo, and was struck by this:

Furthermore: if it can be conceived at all, it must exist. For no one who denies or doubts the existence of a being than which a greater is inconceivable, denies or doubts that if it did exist, its non-existence, either in reality or in the understanding, would be impossible. For otherwise it would not be a being than which a greater cannot be conceived. But as to whatever can be conceived, but does not exist – if there were such a being, its non-existence, either in reality or in the understanding, would be possible. Therefore if a being than which a greater is inconceivable can be even conceived, it cannot be nonexistent.

Let → indicate subjunctive conditionals. Let E!(x) say that x exists.

1. E!(God) → □E!(God).

2. ∀x[if Conceivable(x) and  ∼ E!(x), then: E!(x) → ⋄ ∼ E!(x)].

3. So, not: (Conceivable(God) and  ∼ E!(God)).

4. So, if Conceivable(God), then E!(God).

The ∀x quantifier in (2) is problematic, since it ranges over beings that don’t exist. Perhaps we can read it substitutionally. Let’s suppose we can finesse this issue.

What interests me in (2) is that big conditional in it is most plausibly as seen as a special case of:

5. If C(p) and q, then p → ⋄q,

where C(p) says “conceivably p”, which may or may not be the same as “possibly p”.

We can prove (5) from the Brouwer Axiom

6. If q, then □⋄q,

where L is necessity, and the following principle about subjunctive conditionals:

7. If C(p) and Lr, then p → r,

namely that necessities would still hold no matter what conceivable things happened (to get (5) from (6) and (7), let r be ⋄q). Principle (7) is very plausible if conceivability is possibility: if a possible thing happened, anything necessary would still be true. It’s less plausible if conceivability is not possibility.

And, of course, the Brouwer Axiom is controversial, albeit not quite as much as S5. I initially hoped that the use of the subjunctive conditional in (2) allowed Anselm to get by with something weaker than Brouwer. But not so if the route to (2) goes through (5) and possibility implies conceivability (PIC). For we get Brouwer from (5), PIC and the very plausible principle:

8. If p is possible, then we do not have p → r and p →  ∼ r.

For suppose that contrary to Brouwer we are at a world where q is true, but ⋄q is false at some accessible world. By PIC,  ∼ ⋄q is conceivable. Let p be  ∼ ⋄q. Then C(p) and  ∼ p. But clearly  ∼ ⋄q →  ∼ ⋄q. If we had (5), we would have  ∼ ⋄q → ⋄q, and contradict (8).

So, if Anselm’s argument for (2) goes through (5), we don’t have an improvement over Brouwer. But we can still get (2), given some very plausible assumptions, and the following special case of (5):

9. If C(∼q) and q, then  ∼ q → ⋄q.

And I feel that (9) has some plausibility above (6), at least if conceivability is the same as possibility. For suppose q is true and  ∼ q is possible. Then it seems somewhat plausible that q is possible in the  ∼ q-worlds that are closest to the actual world. Maybe. But maybe there is some way to derive Brouwer from (9) and additional plausible premises.

Towards a counterexample to Weak Transitivity for subjunctives

Transitivity for a conditional → says that if A→B and B→C, then A→C. For subjunctive conditionals this rule is generally taken to be invalid. If I ate squash (B), I would be miserable eating squash (C). If I liked squash (A), I'd eat squash (B). But it doesn't follow that if I liked squash, I'd be miserable eating squash.

Weak Transitivity says that if A→B, B→A and A→C, then A→C. The squash counterexample fails, for it's false that if I were eating squash (B), I'd like squash (A).

I don't know whether Weak Transitivity is valid. But here's something that at least might be a counterexample. Suppose a heavy painting hangs on two strong nails. But if one nail were to fail, eventually--maybe several days later--the other would fail. The following seem to be all not unreasonable:

1. If the right nail failed (B), the left nail would fail because of the right's failure (C).
2. If the left nail failed (A), the right nail would fail because of the left's failure (D).

So, by Weakening (if P→Q and Q entails R, then P→R):

3. If the left nail failed (A), the right would fail (B).
4. If the right nail failed (B), the left would fail (A).

If Weak Transitivity holds, then:

5. If the left nail failed (A), the left nail would fail because of the right's failure (C).

But surely (2) and (5) aren't true together.

As I said, I am not sure if Weak Transitivity is valid. If it is, then there is something wrong with (1)-(4), probably with (1) and (2). Maybe there is. But the example should at least give one reason not to be very confident about Weak Transitivity. (There is another reason: Weak Transitivity is incompatible with the non-triviality of the Adams Thesis for subjunctives.)

From ease to counterfactuals?

Consider the concept of how easy it is for a proposition to be made true, given how things are. It is by far easiest for propositions that are already true: nothing more needs to happen. It is hardest for self-contradictory propositions, like that Socrates is not Socrates: there is no way at all for it to happen. Contingently false propositions that require changes that go far back in time are going to be harder to be made true than ones that don't. And we can talk of the ease of p being made false as just the ease of not-p being made true. So, we can offer this account of counterfactuals:

• p→q holds if and only if it is easier for p to be made true than for the material conditional p⊃q to be made false.

This yields the Lewis-Stalnaker account of counterfactuals provided that we stipulate that a is easier to be made true than b if and only if there is a world where a holds which is closer than every world where b holds.

But we need not make this stipulation. We might instead take the easier to be made true relation as more fundamental. (And while we might define a closeness relation in terms of it—say, by saying that w₁ is closer than w₂ iff <w₁ is actual> is easier to be made true than <w₂ is actual>—depending on which axioms easier to be made true satisfies, that might not yield an account equivalent to the Lewis-Stalnaker one.)

On some assumptions, this is a variant of the central idea in yesterday's post.

A sufficient condition for a subjunctive conditional

Start with the idea of grades of necessity. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: p<q if and only if p is less necessary than q.

Let → indicate subjunctive conditionals. Thus "p→q" says that were it that p, it would be that q. Let ⊃ be the material conditional. Thus "p⊃q basically says that p is false or q is true or both. Then, the following seems plausible:

1. If ~p<(p⊃q), then p→q.

I.e., if the material conditional has more necessity than the denial of its antecedent, the corresponding subjunctive conditional holds.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that if this glass is dropped, then it falls is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial per impossibile conditionals. Let p be the proposition that water is H₃O, and let q be the proposition that a water molecule has four atoms. Then ~p<(p⊃q), because p⊃q is a definitional truth while ~p is a merely metaphysical necessity. Hence were p to hold, q would hold: were water to be H₃O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the non-trivial holding of its right-hand-side.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.

Two fun counterfactuals

1. If I were a better football player than everybody else, I would be very strong.
2. If everyone else were a worse football player than I, nobody would be very strong.

Both of these conditionals are true. But their antecedents are logically equivalent. This shows[note 1] that one cannot substitute logical equivalents for logical equivalents in the antecedents of counterfactuals while preserving truth value, even when one restricts one's consideration to counterfactuals with possible antecedents—i.e., counterfactuals are hyyperintensional. And this, in turn, shows that possible worlds and probabilistic accounts of counterfactuals fail.

I am not happy with this argument. I want to say that the antecedents of (1) and (2) describe families of possible worlds. So we need a interpretation of the antecedents of (1) and (2) on which, although seeming logically equivalent, these antecedents rigidify different features. Thus, the antecedent of (1) rigidifies the range of others' abilities, while the antecedent of (2) rigidifies my abilities.

It is tempting to do this with the overused distinction between semantics and pragmatics: the antecedents of (1) and (2) implicate non-equivalent things, though their propositional content is the same. But if we did that, then either we need to depart from the possible worlds or probabilistic analysis (since that analysis is in terms of truth, not implicature), or we would have to say that although (1) is true and (2) is false, or (1) is false and (2) is true, the real communication goes on at the level of implicature. But the view that (1) is true and (2) is false is implausible, as is the view that (1) is false and (2) is true. (Lewis's closeness account forces one to keep everyone else's abilities constant, so I guess he has to say that (2) is false, but surely (2) is true—not just something that implicates truly.)