The a priori and the a posteriori

I've been thinking about Chalmers' Constructing the World. It is absolutely crucial for Chalmers to have a distinction between what is a priori knowable and what is a posteriori knowable.

Now, imagine that we have evolved to believe that the number one has a successor (call this proposition "Two") as well as that many snakes are dangerous ("Snake"). In both cases, let us suppose, we evolved to believe the true claim because believing it conduced to our survival. (We may add that the reason belief in the claim conduced to our survival was explained by the truth of the claim, if we are worried about debunking arguments.) It now looks like Two and Snake are on par: either both beliefs are a priori or both beliefs are a posteriori. However, Chalmers cannot afford to say that Snake could be a priori—that will destroy much of his story.

So it seems that if Chalmers' story is to work, we will have to say that Snake and Two are a posteriori. However, it is also important for Chalmers to say that fundamental principles like Two are knowable a priori. The story doesn't destroy this: after all, something can be both a posteriori known and a priori knowable (say, a result of a calculation done with a calculator). Nonetheless, there is a problem here. It may well be that all fundamental principles like Two are known by us through something like this evolutionary mechanism (and "something like" includes the theistic variant where God instills this belief in us on account of its truth). And if so, then what reason do we have to think that they are a priori knowable, given that we don't know them a priori? One might have some conviction that some hypothetical or actual non-human reasoner knows them a priori, but it is difficult to see that conviction as justified.

However, I think there is a speculative non-naturalist story one could give that would help make the distinction. Suppose that all our knowledge is at base perceptual. However, sometimes what we perceive are abstracta and sometimes what we perceive are concreta. Knowledge that is grounded only in perception of abstracta is a priori, while knowledge that is grounded at least in part in perception of concreta is a posteriori. It may be that we can just see the number two as the successor of the number one. There is some phenomenological plausibility to this.

This would be really nice for Chalmers. For it is plausible that if any abstracta and their abstract relations are observable, they all are. If so, then all facts about the realm of abstracta are a priori knowable.

Granted, on this story knowledge of abstracta is observational and hence empirical. But while this does mean that the above use of "a priori" and "a posteriori" is idiosyncratic, the above use nonetheless may help recover at least some of Chalmers' story.

Some, but perhaps not the two-dimensionalism? For the account above is apt to make it a priori that wateriness is H₂O-ness, since that seems to just be a fact about abstracta.

Inequality

1. There is nothing intrinsically bad in heaven.
2. There is inequality in heaven.
3. So, inequality is not intrinsically bad.

Some reasons for accepting (2):

• God is in heaven and humans are in heaven, and there is infinite inequality there.
• There are angels in heaven and humans are in heaven, and there is inequality there.
• Heavenly bliss is proportional to merit (though of course the merit is the work of God's grace), and merit is unequal.

Even more on simplicity and theism

Some naturalists say that theism needlessly complicates our view of the world by positing that

• in addition to the material concrete contingent things, there is something immaterial, necessary and concrete.

But the naturalist needs to say that the naturalist needlessly complicates our view of the world by being committed to the claim that

• in addition to the dependent concrete contingent things, there is something independent, concrete and contingent.

(Say, the Big Bang or the universe as a whole.)

Does talking of needless complication get us ahead here?

Simplicity and theism

I have argued elsewhere, as my colleague Trent Dougherty also has and earlier, that when we understand simplicity rightly, theism makes for a simpler theory than naturalism. However, suppose I am wrong, and naturalism is the simpler theory. Is that a reason to think naturalism true? I suspect not. For it is theism that explains how simplicity can be a guide to truth (say, because of God's beauty and God's desire to produce an elegant universe), while on naturalism we should not think of simplicity as a guide to truth, but at most as a pragmatic benefit of a theory. Thus to accept naturalism for the sake of simplicity is to cut the branch one is sitting on.

Illocutionary force and propositions

Suppose I say to Bill: "Make all of your papers be between two and four pages." Bill hands in an eight page paper for his first assignment. I rebuke him and he apologizes. He then hands in another eight page paper for his second. When I rebuke him, he says: "You told me to bring it about that all my papers be between two and four pages. With my first paper I ensured that the proposition that all my papers are between two and four pages is false. Sorry! By the time of my second paper, it was too late to undo this: no matter what length of paper I wrote, that proposition would still be false. So I might as well write the length that I like."

Bill's mistake was thinking that the content of my command was the proposition that all his papers be between two and four pages. I didn't command that proposition. Rather, I commanded distributively of each of his papers that it be between two and four pages.

This means that we should not analyze my speech act as having a propositional content plus an illocutionary force. The content of the speech act wasn't a proposition, but something else. Perhaps the content of the speech act was an ordered pair of properties, the property P of being one of Bill's papers, and the property L of being between two and four pages in length. And the illocutionary force was of something one might call distributive command. Successful distributive command in respect of a pair of properties P and L creates for each instance x of P a reason to make x have L.

There are, I think, assertion-like speech acts that also have such a non-propositional content. For instance, assertoric endorsement. A paradigm case: I endorse what you are about to assert. The content of assertoric endorsement is a property which is supposed to be had by one or more propositions—say, the property of being soon asserted by you—and when successful, the assertoric endorsement makes you stand behind each of these propositions as if you asserted it. This kind of assertoric endorsement is distributive.

I wish I knew what kinds of entities can be contents of speech acts. The above suggests that some speech acts have propositions as contents, some have pairs of properties, some have single properties. There must be many other options.

Simplicity as a sign of design

It seems hard to deny that simplicity is a guide to truth in science. But the best account of the simplicity of a theory is brevity of expression in a language whose terms cleave nature at the joints. But that brevity of expression in a language is a guide to truth is a sign that a rational being is behind our universe.

A theory of contingency and an argument for a causal Principle of Sufficient Reason

Consider this theory, a modification of my causal power account of possibility:

• A proposition p is contingent provided that something has a power for p and something has a power for not-p.

Here, I say that x has a power for p if and only if x has the power to bring p about, or x has the power to bring it about that something has the power to bring p about, or ....

It follows from this theory that every contingent true propositions has a causal explanation.

For suppose for reductio ad absurdum that p is contingently true and has no causal explanation. Let q be the conjunction of p with the claim that p has no causal explanation. Then q is true, and it is not necessarily true since p is not necessarily true, so q is contingent. It follows from our account of contingency that something has the power to ... bring q about (where the "..." is a possible chain of causal power claims). But that's absurd, since something that brings q about thereby also brings p about, and then p isn't bereft of causal explanation!

Love of truth

Let's say I am grading final exams and am very curious how a student who had been struggling all semester will do in the course. So I forthwith submit a B+ for her to our grading system, without bothering with any more calculations, and my curiosity is satisfied.

There is something perverse here. Of course, there is a perversion of justice—that's clear. But I think there may also be a perversion of the intellectual life. Genuine love of truth is not satisfied by making a proposition true or false. Genuine love of truth, at least as proper to creatures, seeks to make the mind reflect the world, not to make the world reflect the mind. If this line of thought is wrong, then the counterexample to evidentialism in my previous post fails.

The issue comes also comes up in third-person cases. My friend thinks that I will be wearing a long-sleeved shirt today. Does a loving desire to promote his intellectual goods give me any reason to wear such a shirt? I doubt it. But if not, then this is very puzzling. For surely my friend is intrinsically the better off for getting right what I will wear.

Maybe the case is a bit like throwing a game. My daughter wants to beat me at chess. But she wants to beat me by her own powers, rather than because I didn't try hard. Is there any value to beating me if I don't try at all?

This example suggests that when I wear a long-sleeved shirt to make my friend be right, his being right is not an achievement of his, and hence it's not much of a victory. But maybe this makes the epistemic life sound too proud. Maybe we should rather see it humbly as a comformation of our minds to reality.

Maybe the direction-of-fit issue here is parallel to one with desires. I bought my friend a trinket for his birthday. I then slip him a pill that induces a desire for the trinket. Surely that's perverse—it gets things the wrong way around. I should make the world conform to my friend's (reasonable) desires, not the other way around. And to make my friend's beliefs conform to the world, not the other way around.

It might be different in the case of God. Aquinas says that God knows creation by creating. Maybe here is a crucial difference between God and creatures.

A counterexample to evidentialism?

Consider Williamson-style beliefs that obviously have the property that they have to be correct if they are believed. For instance, if I believe that I have a belief, then that belief is guaranteed to be correct. Call beliefs like this obviously self-guaranteeing.

Suppose now that I am unable to introspect my beliefs and am not a sufficiently good observer to gain evidence as to what I believe on the basis of my behavior. Unsure whether I have any beliefs, but thinking that true beliefs are valuable to have although false ones are valuable to avoid, I try to will myself to believe that I have a belief, because it is clear to me that that claim will be true if I believe it. (You might ask: If I do that, don't I already believe something, namely that the belief will be true if I believe it? Maybe, but that's beside the point, since I am unable to tell that I believe it.) I don't know if I will succeed—and even if I do succeed, I won't know that I have succeeded—since willing myself to have a belief is a notoriously shaky thing. There seems to be nothing incompatible with the love of truth in willing myself to believe that I have a belief, indeed there seems to be nothing epistemically bad. But I am (a) willing myself to believe something I now do not have evidence for, and (b) if I do come to believe it, I will believe it without any evidence for it. If indeed there is nothing epistemically bad here, then (b) gives a counterexample to synchronic evidentialism and (a) gives a counterexample to diachronic evidentialism.

But perhaps there is something perverse here. See tomorrow's post.

Explaining the necessary with the contingent

It may seem initially slightly surprising, but there are necessary truths that are explained by contingent ones. For instance, it is a necessary truth that Obama is president or 2+2=4. And this necessary truth can be explained by the fact that the majority of the electoral college voted for Obama, or, perhaps even better, by facts about the way the Democrats and Republicans campaigned. Another necessary truth that can be explained in the same way is that it is or is not the case that Obama is president.

That the necessary can sometimes be explained with the contingent is, I think, a rather more trivial claim than that the contingent can be explained with the necessary.

An interesting epistemic scoring rule

A forecast p is an assignment of probabilities to events in some space Ω. A proper score is an assignment of a random variable s_p to each forecast on that space, with the property that E_ps_p≤E_ps_q whenever p is a consistent forecast (one that satisfies the axioms of probability) and q is any other forecast. Here, E_p is expectation with respect to the probability function p. Propriety basically says that if we have a consistent forecast, then by our own lights no other forecast is expected to have a better score. The scores are thought of as penalties or distances from truth—smaller is better.

One thing proper scoring rules have been used for is to argue that our credences should be consistent. For instance, under a simplifying assumption, Predd et al. have basically shown that the proper score for an inconsistent forecast is always dominated (from below) by a proper score for some consistent forecast. The simplifying assumption is that scores are computed for individual events and added.

Now, here is a curious proper score that does not satisfy this simplifying assumption. Suppose we're working with a finite space Ω with n points. Suppose p is consistent. Let m(p) be a point of Ω where p is maximized for a forecast p. (Use any tie-breaking method you like if that point isn't unique.) Then let s_p be 0 at m(p) and 1 everywhere else. Then if p and q are consistent, E_ps_q=1−p(m(q)) (where p(ω)=p({ω})). Since p(m(p))≥p(m(q)) by definition of m, it follows that E_ps_p≤E_ps_q. Observe that p(m(p))≥1/n. Thus, E_ps_p≤1−1/n. Finally, if p is inconsistent, let s_p be 1−1/n everywhere. Then s is a proper score.

For consistent forecasts, our s is a best guess score: a forecast's maximum point (with whatever tie breaker one likes) counts as the forecast's "best guess", and we get the perfect score 0 if we guessed right, and we get 1 otherwise. And for inconsistent forecasts, I just assigned a value that makes the score proper and, well, that makes what I am about to say true.

Namely: the above score s does not have the domination property that I talked about earlier. Let q be any inconsistent forecast. Then s_q is 1−1/n everywhere. If p is any consistent forecast, however, then s_p is 1 at all but one point, and so s_p does not dominate s_q from below.

Now, our score s is not a strictly proper score (Predd et al. actually work with strictly proper scores): for a strictly proper score s, whenever q differs from p and p is consistent, we will have E_ps_p<E_ps_q. But we can make our score strictly proper. Fix a small constant c. Then s+cb, where b is the standard Brier score, will be strictly proper. But if c is small enough, s+cb will also fail to have the domination property.

We should already have been suspicious of the argument for consistency based on proper scores and domination when proper scores were defined: the definition treated consistent forecasts in a special way (i.e., E_ps_p≤E_ps_q was only required when p is consistent—of course, it's hard to define E_p when p is inconsistent, so there is some excuse). But now we have even more reason to be suspicious: it is only some proper scores that have the property that scores of inconsistent forecasts are dominated by scores of consistent ones. Now, if we had some philosophical reason to think that the right way to score forecasts is by adding up scores for individual events, this would be better. But I don't know of such a philosophical reason.

TeXlipse

I've been editing LaTeX files using an old version of WinEdt. But at least in the old version I was using, I was having a terrible time ensuring things like matching \begin{...} and \end{...} for frame and itemize blocks when editing Beamer presentations. But I think I finally found a better way to handle LaTeX: the TeXlipse plugin for Eclipse. I can now have background building (at least when I save, and I press ctrl-s quite often instinctually), syntax highlighting and indenting, autocomplete, a handy hierarchical view, and very nice handling of error messages.

The downsides are that you need Eclipse (but I have it already installed for Android software development, and it is free after all) and Eclipse is bloated and doesn't start fast, setting up a new project takes a few more clicks than before, and the PDF viewer that comes with TeXlipse doesn't show all the graphical elements in a Beamer file. The last is a nuisance, but the nice way that the PDF file is linked with the LaTeX source compensates for it, as does the fact that I can just set up SumatraPDF as a secondary PDF viewer, and SumatraPDF (unlike Acrobat) will automatically reload the pdf file when it is regenerated. All in all, it seems worthwhile.

Proper scoring rules and gambling

There are two ways of evaluating a credence assignment. There is the decision-theoretic method: you consider how you are going to do given this credence assignment when presented with some gambles. And there is the scoring rule method: you consider how far you are from "the truth", i.e., the credence assignment that assigns 0 to the falsehoods and 1 to the truths, and you measure this with respect to a proper scoring rule.

There are various parallel results for the two methods.

It turns out that there is a good reason why there are parallel results. The two methods are equivalent. Assume an underlying probability space Ω. To avoid measurability issues, suppose Ω is finite. Denote a scoring rule by a function s(p,q) where p is a consistent credence assignment for some family of sentences and q is a consistent extreme credence assignment (0/1 valued) for the same family. By "the truth", I mean the extreme credence assignment that assigns 1 to each true sentence and 0 to each false one. A scoring rule is proper provided that E_p(−s(p',T))≤E_p(−s(p,T)) where T is the random variable that assigns to each point ω of Ω a function T(ω) that in turn assigns to each sentence its truth value at ω (i.e., 1 if true, 0 if false), and where E_p is expectation with respect to the credence assignment p.

Theorem. For any proper scoring rule s(p,q), there exists a family F of gambles such that for any consistent credence assignment there is a gamble that maximizes the expected payoff, and when you choose that maximizing gamble your payoff will be −s(p,T). Conversely, suppose that F is a family of gambles such that for any credence assignment there is a gamble that maximizes the expected payoff. Let V(p,q) be the payoff of such a gamble for credence assignment p when q is the truth. Then −V(p,q) is a proper scoring rule.

The proof is actually very simple. (I had very complicated proofs of special cases of this Theorem in the past, but now I see it is all very simple, even trivial.) For the left-to-right direction, for any possible credence assignment p, define the gamble G_p as follows: at ω, you get paid −s(p,T(ω)). Then the propriety of the scoring rule guarantees that G_p maximizes the expected payoff when p is your credence assigment. Conversely, let s(p,q)=−V(p,q). Propriety is easy to check—it just follows from maximization.

Explaining the contingent via the necessary

It has been claimed that contingent truths cannot be explained by necessary ones. Indeed, Peter van Inwagen has contended that this shows that the Principle of Sufficient Reason is false. But it seems that here is a case of a necessary truth explaining a contingent one: That it's extremely unlikely that 30 fair die throws would be all sixes explains why nobody has tossed 30 sixes in a row with a fair die.

Pantheism and omnipresence

If a view falls short with respect to the main doctrine it's organized around, that view is seriously flawed. For instance, if Calvinism fell short with regard to sovereignty, it would be seriously flawed. For pantheism, the relevant doctrine is omnipresence. On its face, pantheism is designed to make omnipresence work out perfectly: if God is everything, then he is where anything is.

But is that enough for omnipresence? First, perhaps omnipresence should also imply that God is in the places where nothing other than God exists—in otherwise empty space. Whether pantheism can account for that perhaps depends on whether it's deflationary (God is nothing but everything) or inflationary (everything is God, in addition to what it ordinarily is, and there may be more to God than ordinary things—and hence in particular God might be where there is nothing ordinary). That said, perhaps this is not so serious. If substantivalism about space is false, then maybe there are no empty places, except in a manner of speaking.

More seriously, by making God be everything, God comes to be only partly present everywhere. Only a part of God is in this room where I am—a very small part and, at least on the deflationary variant, a very insignificant part. Yes, God is in the stone and the butterfly and the galaxy—but all of these are very small bits of God. Classical theists, however, have the doctrine of divine simplicity and so we can say that where God is, all of God is.