A test case for explanationism

Here is a test case for explanationist stories about initial priors, on which a more explanatory theory has a higher initial prior. Consider these hypotheses:

1. The universe is not created or governed by reason, has a finite lifetime, and has low initial entropy.

2. The universe is not created or governed by reason, has a finite lifetime, and has low final entropy.

Causation goes from past to future, so absent some kind of foresight involved in the initial conditions, initial conditions are more explanatory than final conditions. Now, presumably there is a nice one-to-one correspondence between worlds where (1) and (2) hold, so absent an explanationist bias in the priors, we shouldn’t have a preference between (1) and (2).

This is just a test case, but I have to confess I don’t have any direct intuition comparing the probabilities of (1) and (2). I like explanationism, so I have a theory-laden reason to assign a higher probability to (1) than to (2).

How has Aquinas not proved the Trinity?

St Thomas holds that we cannot know by natural reason that God is a Trinity. However, he also endorses a version of St Augustine’s account of the Trinity, on which God has a mental Word or concept of himself that he generates mentally, and then there is a Love that is responsive to that Word. Why is this not a proof by natural reason?

On this point, Thomas is sadly brief in the Summa Theologiae: “Again, the similarity with our intellect does not sufficiently prove anything about God, since the intellect is not found univocally in God and in us.”

But Aquinas thinks that we can literally talk about God, even if we do so by means of analogy. He has argued, by reason, that God exists and has all perfections. One of these perfections is knowledge of everything, and thus of himself. Does this not commit Aquinas to being able to say on the basis of natural reason that God generates his self-knowledge, a self-knowledge that, by divine simplicity, must then be God himself rather than a mere creature? And does not God love himself under the description that this knowledge provides?

I think Aquinas would (with perhaps minor cavils) endorse all the claims in my previous paragraph as knowable by natural reason. How is this not a proof of the Trinity? I think the answer is this. Focus on the generation of the Son—the case of the Holy Spirit will presumably be similar. We do not know by reason whether the relation of generating self-knowledge is a real relation in God. If it is a real relation it will individuate the knower from the knowledge, thereby ensuring that there are at least two persons in God. But it needs to be a real relation in God, rather than a merely conceptual one. And here Thomas’s brief remark about univocity seems apposite. Aquinas thinks that what makes our language about God analogical rather than univocal is that the grounding of claims about God is radically different from the grounding of similar claims about creatures. Thus, when we say Socrates is wise, this is grounded by the inherence in Socrates of an accident of wisdom, while when we say God is wise, this is grounded by the identity between God and his wisdom. Because of this, even if it turns out that in us the generation of self-knowledge is a real relation (but not one between persons, since we are not simple, and hence our self-knowledge does not need to be of one essence with us), since the grounding structure of claims about God is radically different, as far as unaided human reason goes, it may not be grounded in a real relation in God—though it might be. We need God’s revelation to know if it is a real relation or not. And indeed God has revealed that it is.

Optimalism and mediocritism

We can think of the optimalist theory of ultimate explanation as the claim:

1. Necessarily, that reality is for the best explains everything.

(I won’t worry in this post about two details. First, whether “reality” in (1) includes the principle of optimality itself—Rescher has suggested that it does, since it’s for the best that everything be for the best. Second, whether “reality” is all the detail of the world, or just the “core” of the world—the aspects not set by indeterministic causation.)

Given that only truths explain, (1) entails:

2. Necessarily, reality is for the best.

Notice that one could accept (2) without accepting (1). One might, for instance, be a Leibnizian and think that there is a two-fold structure to ultimate explanation: first, God’s existence is explained by the ontological argument and, second, God creates the best contingent reality. On this account everything is for the best, but that everything is for the best is not the ultimate explanation, because it does not explain why God exists. Or one might think that reality is necessary and brute, and it brutely has to be like it is. And as a very suprising but non-explanatory matter of fact the way it is is in fact optimal.

I am emphasizing this, because I want to problematize (1). Grant (2). Why should we think that the fact that everything is for the best in fact explains everything?

Suppose that modal fatalism is true, and that it so happens that reality is exactly mid-way between the worst and the best possibility, and is in the only option mid-way between the worst and best. (I assume one can talk of options for reality even given modal fatalism. Otherwise, optimalism falls apart. The “options for reality” may be something like narrowly logically possible worlds.) Then:

3. Necessarily, reality is exactly middling.

Now suppose a “mediocritist” said: “And that reality is necessarily exactly middling explains why reality is what it is.” But why would we buy that? Or suppose that reality is necessarily the only one that is exactly 56.4% of the way up between the worst and the best (where worst would count as 0% of the way up and best as 100%)? Surely we wouldn’t conclude that its being exactly at 56.4% explains why it is the way it is. But if not, then why should its being at 50% explain it, as on mediocritism, or its being at 100% explain it, as on optimalism?

I think what the optimalist ought to say at this point is that analogously to non-Humean pushy laws of nature, there are non-Humean pushy laws of metaphysics. One of these laws is that everything is for the best. It is the pushiness of this metaphysical law that explains reality. But there is something rather odd about pushy laws prior to all beings—they seem really problematically ungrounded.

More on strong open-mindedness

For the last couple of days I have been exploring what I like to call strongly open-minded accuracy scoring rules. It’s well known that every proper scoring rule is open-minded in the sense that it never requires you to reject free information: the expected epistemic utility of updating on the free information is always at least as good as your current expected epistemic utility. It’s strictly open-minded provided that in non-trivial cases (i.e., when the information has a non-zero probability of having statistical relevance to the credences you are scoring) you are required to accept the free information.

Now there are two reasons why one might accept free information about some proposition q. First, you might be wrong about q: your credence may be high while q is false or your credence might be low while q is true. Second, even if you are right about q, the free information may boost your credence in the right direction. I say that a scoring rule is strongly open-minded provided that it licenses you to accept and update on the free information even if you disregard the first consideration. We can then tack on “strictly” if it requires you to do so in non-trivial cases. In the case of a strongly open-minded scoring rule, your acceptance of free information is not a sign of doubt in your propositions—it is not a way of hedging your bets—and thus is arguably compatible with faith in the propositions being evaluated.

A strongly open-minded scoring rule can also be characterized in the following way. There is a more ordinary kind of epistemic paternalism where I might have reason to block another from receiving free information on the grounds that this information could mislead due to the fact that the other has different likelihoods from the ones I think are right. For instance, if too many people have an unjustified mistrust of Dr. Smith such that they are likely to believe the opposite of what Dr. Smith’s experiments reveal, there is reason to give a grant to someone else, because Dr. Smith’s experiments are likely to lead people away from the truth, for no fault of Dr. Smith’s. Call this likelihood-based paternalism. But there is another kind of motivation of the refusal of free information for another, which we might call pure-risk-based paternalism. Even if someone else has the same likelihoods as you do—trusts Dr. Smith just as you do—perhaps the risk that Dr. Smith’s experiments will, by pure chance, provide evidence away from the truth is enough to justify not funding these experiments.

I’ve been collecting results about these issues. Here’s what I seem t have so far, though I have to emphasize that sometimes the proofs are just in my head and I might be wrong. I will specialize on scoring rules for a single proposition, given as a pair of functions T and F, where T(x) is the value of having credence x when the proposition is true and F(x) is the value of having credence x when the proposition is false.

1. A scoring rule sometimes calls for pure-risk-based paternalism if and only if it is not strongly open-minded.

2. A scoring rule that’s strongly open-minded is open-minded.

3. A scoring rule (T,F) is (strictly) strongly open-minded if and only if xT(x) and (1−x)F(1−x) are both (strictly) convex.

4. The logarithmic scoring rule is strictly strongly open-minded. The Brier and spherical rules are not strongly open-minded.

5. If a proper scoring rule is generated by the Schervisch-style integral representation T(x) = T(1/2) + ∫_x^1/2(1−t)b(t)dt and F(x) = F(1/2) + ∫_1/2^xtb(t)dt and b is sufficiently differentiable, then the scoring rule is strongly open-minded if and only if the derivative of log b(x) lies between (3x−2)/[x(1−x)] and (3x−1)/[x(1−x)].

6. A strongly open-minded scoring rule whose logarithm is sufficiently differentiable is unbounded.

7. If your credence in a hypothesis H is at least (at most) 1/2, then a proper scoring rule will not call for purely-risk-based epistemic paternalism with respect to someone whose credence is equal to or higher (lower) than yours.

8. If your credence in a hypothesis H is 1/2, then no proper scoring rule calls for purely-risk-based epistemic paternalism for that hypothesis.

9. For any credences p and r such that 1/2 < r and p < r, there is a strictly proper scoring rule and a situation where the scoring rule calls for the individual with credence r to have purely-risk-based epistemic paternalism for that hypothesis.

Algorithmic priors and human nature

One promising way to define priors is with algorithmic probability, such as Solomonoff priors. The idea is that we have a language L (say, one based on Turing machines), and we imagine generating random descriptions in L in a canonical way. E.g., add an end-of-string symbol to L and randomly and independently generate symbols until you hit the end-of-string symbol, and then conditionalize on the string uniquely describing a situation, and take the probability of a specific situation s to be the probability of that a random description so generated describes s.

These kinds of priors are rather appealing for science, since they appear to be induction-friendly, as they assign high probabilities to compressible—more briefly expressible—situations. Thus, if our situations are distributions of color among ravens, monochromatic distributions get much higher probability as they can be much more briefly described, like ∀x(B(x)) or ∀x(W(x)).

Philosophically, I think the big problem is with the choice of the language. It would be nice if we could let L be a language that cuts nature exactly at the joints. But we don’t know that language. And absent that language, we need something arbitrary.

Here is a particular version of the problem. Take Kuhn’s division of science into ordinary and revolutionary science. One aspect of this division is that in ordinary science, we have a scientific language, and are discovering things within it. In that case, it is reasonable to take L to be that language. However, when we are doing revolutionary science and creating new paradigms, we cannot do that. The new paradigms either cannot be described in the old language or their description is unwieldy in a way that does not do justice to the plausibility of the new paradigm. Indeed, much of the point of revolutionary science is to create a language within which the description of the world is simpler, and then argue that this language is therefore more likely to cut nature at the joints.

Another version of this problem is what language L we choose when we are generating fundamental priors. Practically speaking, we cannot use a scientific language that cuts nature at the joints, because we have not yet discovered it. If this was merely a practical concern, we could try to say that this doesn’t matter: the fundamental priors are ones that we ought to have rather than any that we actually have or could have—perhaps ought does not imply can. But the concern is not merely practical. For one of the main points of our inductive reasoning is to discover what concepts cut nature at the joints, and this is largely an empirical enterprise. If the right fundamental priors were to reflect the joints in nature, then the enterprise wouldn’t make much sense, as we would be obligated to have already completed much of the enterprise before we started it.

So, I think, we have to say L does not always cut nature at the joints, and yet this generating appropriate priors for us. But we still need a constraint on L. After all, we could imagine a language that thwarts our empirical enterprise, such as one where only fairies can be described briefly and anything else requires very long descriptions, so we have very high priors for fairies and very low priors for everything else. What will be the constraint? Practically, we pretty much have to start with some ordinary human language. I think our ideal should not be far from what is practical. Thus, I propose, if we are going to go with algorithmic priors, we should choose L to be a language that fits well with our human nature as communicators. This is an anthropocentric choice, and I think human epistemology is rightly anthropocentric.

But why think that the anthropocentric choice is apt to lead to truth? There are two stories to be told here. First, it may be that human nature requires a measure of trust in human nature. Second, that trust is vindicated if we are created by a good God who loves the truth.

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.