Saturday, October 25, 2014

Propositions that never become true

According to open future views, the proposition that in 2015 a fair and indeterministic coin lands heads has some probability but is not true. However, that proposition is apt to become true in 2015. So the probability of the proposition isn't the same as the probability of the proposition being true, since it's certainly not true now, but might well become true in 2015.
So far so good (or bad). Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads. Now note that on open future views Q can never possibly become true. For on any date, the proposition requires for its truth that there will be infinitely fair and indeterministic heads results still past that date, and on open future views a proposition that requires an undetermined future event won't be true.
So, open future views have to say that it's impossible for Q to ever to be true. But a proposition such that it's impossible for it ever to be true should get probability zero. But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory. So open future views should be rejected.
Here's another argument in the same vein. Suppose I know I will have an eternal afterlife, and I promise you that I will freely pray for you every day, ad infinitum, starting November 1, 2014. On open future views, the object of my promise is a proposition that can never be true. But it's clearly a bad thing to promise something that can never be true. Yet what I promised wasn't a bad thing to promise. So open future views are false.
One might even have the direct intuition that one could keep the promise. That intuition is incompatible with open future views.

Friday, October 24, 2014

Yet yet another probability paradox

Start with a set M of countably infinitely many people, and a set D of countably infinitely many fair dice. Suppose that there are no natural orderings on the set D, and that each person in M has exactly one of the dice in D assigned to her. (Or if you prefer, these are sets of unique names of people and coins respectively.) You are a person in M, and you know what all the members of D are but have no information whatsoever on which member of D is yours. Now all the dice are simultaneously and independently tossed. Obviously, your probability that your die showed sixes is 1/6.

Then the set of all the dice that landed sixes is revealed to you. Call the revealed set D6.

Suppose—this will be no surprise, as it had probability one—that the set of six-landing dice is infinite and the set of non-six-landing dice is infinite as well. Before it was revealed to you which dice landed sixes, your probability that your die yielded a six was 1/6. Did that probability change after you learned which set was the set of dice that landed sixes?

There are three options:

  1. No, it didn't change at all—it stayed at 1/6.
  2. Yes, it changed to an undefined value.
  3. Yes, it changed to some other defined value.
To choose between the options, observe first that your current probability that your die landed six must now be exactly the same as the probability that your die is a member of D6. But the fact that D6 is in fact the set of the six-showing dice carries no information as whether your die is in D6. Since all the dice are independent and fair, learning which dice landed sixes is completely irrelevant to finding your die. So whatever probability you assign to your die being among the members of D6 after the revelation must be the same as the probability you assigned to it before the revelation.

So, if we choose option (1), then already before you found out that the double-six rollers were the members of D6, you would have already assigned probability 1/6 to your die being in D6. But there was no natural ordering on the set D of dice, so the set D6 will be epistemically on par with its complement WW6. Both are simply countably infinite sets with countably infinite complements, and we can easily define an isomorphism of D onto itself that swaps the two sets. So if prior to learning the dice results you assigned 1/6 to your die being in D, you should have equally assigned 1/6 to your die being in DD6. But that's incoherent, since it's a given that the die is in D or DD6 but 1/6+1/6=1/3<1. So it seems that (1) is not an option.

That leaves (2) and (3). But those options are very strange. They imply that in such infinite die rolling scenarios, more data can always destroy your reasonable initial probability assignments.

Now, you might think that the above scenario only works when you don't know which die is yours, and that's kind of a strange scenario. But one can modify the scenario to work even when you do know which die is yours, but there is some other unique feature you don't know about your die, say, which of infinitely many (metaphysically) possible exotic particles is hidden inside the die, which of infinitely many angels has your die as a personal favorite, or what an independent sequence of rolls of the die yielded. Then the set D will be set of these unique features, and D6 will be the set of these features among the dice that landed six.

Thursday, October 23, 2014

Yet another probability paradox

You know for sure that infinitely many people, including yourself, each are independently tossing fair coins. You don't see your coin's result. But then you learn for sure something amazing: only finitely many of the coins came up heads. This is extremely unlikely—indeed, by the Law of Large Numbers it has zero probability—but it seems nonetheless possible. What probability should you now assign to your coin being heads?

Intuition: Very small, maybe zero, maybe infinitesimal.

Here's an argument, however, that you should stick to your guns and continue to assign 1/2. Let F be the proposition that only finitely many of the coins landed heads. Let G be the proposition that of the coins other than yours, only finitely many of the coins landed heads. Learning G does not affect your probability that your coin landed heads. The coins are all independent, so no information about the other tosses tells you about yours. But, now, necessarily (given the setup that you toss only one coin) F is true if and only if G is true. For your coin won't make the difference between infinitely and finitely many heads. So learning F does not affect your probability that your coin landed heads.

To make sticking to your guns even more amazing, note that this works for any infinity of people, even a very high uncountable infinity. Wow!

Wednesday, October 22, 2014

Scoring rules and epistemic rationality

Scoring rules measures the inaccuracy of one's credences. Roughly, when p is true, and one assigns credence r to p, then a scoring rule measures the distance between r and 1, while when p is false, the scoring rule measures the distance between r and 0. The smaller the score, the better.

Some scoring rules are better than others. Let's suppose some scoring rules are right. Then this thesis seems to be implicit in some applications of scoring rules (e.g., here):

  1. If S is the right scoring rule, then a credence-assignment policy is epistemically rational only if following the policy minimizes expected total or average S-scores.
(And there will be a debate about whether we should have "total" or "average"—see link.)

But (1) is false. Here's a simple counterexample that works for most reasonable scoring rules. Consider a situation like this: A fair coin is flipped. If you assign credence 0.51 to heads, a mindreader who knows your credence assignments will immediately reveal to you how the coin landed. Otherwise, you will never have any information on how the coin landed.

Obviously, the epistemically rational thing to do is to assign 0.5 to heads. But this leads to higher expected total and average scores on most reasonable scoring rules. For if you assign 0.51, then once the mindreader tells you how the coin landed, you will update your credence to be very close to 0 or 1, and your score will be very low. And the only cost of this scenario is the slight inoptimality from briefly having score 0.51 instead of the optimal score of 0.5. So the epistemically rational policy for dealing with situations like this, namely assigning 0.5, does less well in expected scores than the epistemically irrational policy of assigning 0.51.

The case may seem farfetched. But there are real-life cases that may be similar. It may be that for psychological reasons when you are a bit more sure, or a bit less sure (depending on your character and the thesis), of a thesis than rationality calls for, you will be better able to investigate whether the thesis is true. Thus it may be better for your long term epistemic score that you do what is epistemically irrational.

Tuesday, October 21, 2014

Hair

For a while, I've thought that:

  1. Hair is not alive.
  2. Every part of me is alive.
  3. So, hair is not a part of me.
This goes against the wisdom embodied in court precedent which has, I understand, held that cutting someone's hair without consent is battery rather than, say, theft.

Interestingly, in L'usage de la Raison, Mersenne talks of the human as a microcosm and mentions that humans, like the universe, have non-living parts, and gives hair as an example. So Mersenne denies (2). And on further reflection, I don't think I really had much reason to accept (2). Indeed, there seem to be other clear counterexamples to (2), such as the electrons in my heart (they are parts of my heart, and parthood seems transitive, at least in this case). Maybe one could argue that while the electrons are at least a part of a living part of me, hair isn't a part of a living part of me. But that would beg the question. For if my hair is a part of me, it's also a part of my head, and my head is surely a living part of me.

So I don't see much ground for denying that hair is a part of me. It's just one of my many nonliving parts.

Of course, speaking fundamentally, there is no such thing as hair (just as there are no hearts, chairs, stones, etc.). There is only I, who am hirsute.

Monday, October 20, 2014

Limiting frequencies and probabilities

You are one of infinitely many blindfolded people arranged in a line with a beginning and no end. Some people have a red hat and others have a white hat. The process by which hat colors were assigned took no account of the order of people. You don't know where you were in the line. Suppose you learn the exact sequence of hat colors, say, RWRRRRWRWRWWWWRWWWR.... But you still don't know your position. What should your probability be that your hat is red?

A natural way to answer this is to compute the limiting frequency of reds. Let R(n) be the number of red hats among the first n people, and then see if R(n)/n converges to some number. If so, then that number, call it r, seems to be a reasonable value for the probability. Call the assignment of r to the probability when the limit r exists the frequency rule.

Here's a curious and simple thing I hadn't noticed before. If you think the frequency rule is always the right rule, then for all integers n, you are committed to being almost certain that your position is greater than n. Here's why. Suppose that the sequence that comes up is n white hats followed by just red hats. The limiting frequency of R(n)/n is 1. So by the frequency rule, you're committed to assigning probability 1 to having a red hat. But since you have a red hat if and only if your position is greater than n, you are committed to assigning probability 1 to your position being greater than n. And since there is no connection between the hat color arrangement and the order of people on the line, if you have this commitment after learning the sequence of hat colors, you also had it before. The argument applies for all n, so for all n you must have been almost certain that your position in the sequence is greater than n.

And this in turn leads to the paradoxes of nonconglomerability. For instance, suppose that I flip a fair coin. If it's heads, I let N be your position number. If it's tails, I choose a number N at random such that P(N=n)=2n. In either case, I reveal to you the value of N, but not how the flip went. For any number n, the probability that N=n is zero given heads (since you're almost certain that your position is greater than n), and the probability that N=n is greater than zero given tails, so by Bayes' Theorem you will be almost certain that the coin landed tails. So I can make you be sure that a coin landed tails, and thereby exploit you in paradoxical ways.

So the frequency rule isn't as innocent as it seems. It commits one to something like an infinite fair lottery.

Friday, October 17, 2014

Too late!

Let's say that something very good will happen to you if and only if the universe is in state S at midnight today. You labor mightily up until midnight to make the universe be in S. But then, surely, you stop and relax. There is no point to anything you may do after midnight with respect to the universe being in S at midnight, except for prayer or research on time machines or some other method of affecting the past. It's too late for anything else!

This line of thought immediately implies two-boxing in the Newcomb's Paradox. For suppose that the predictor will decide on the contents of the boxes on the basis of her predictions tonight at midnight about your actions tomorrow at noon when you will be shown the two boxes. Her predictions are based on the state of the universe at midnight. Let S be the state of the universe being such as to make her predict that you will engage in one-boxing. Then until midnight you will labor mightily to make the universe be in S. You will read the works of epistemic decision theorists, and shut out from your mind the two-boxers' responses. But then midnight strikes. And then, surely, you stop and relax. There is no point to anything you may do after midnight with respect to whether the universe was in S at midnight or not, except for prayer or research on time machines or some other method of affecting the past, and in the Newcomb paradox one normally stipulates that such techniques are not relevant. In particular, with respect to the universe being in S at midnight tonight, it makes no sense to choose a single box tomorrow at noon. So you might as well choose two. Though, if you're lucky, by midnight tonight you will have got yourself into such a firm spirit of one-boxing that by noon tomorrow you will be blind to this thought and will choose only one box.

Thursday, October 16, 2014

Continuous Sleeping Beauty

A coin is tossed without the result being shown to you. If it's heads, you are put in a sensory deprivation chamber for 61 minutes. If it's tails, you are put in it for 121 minutes. Data from your past sensory deprivation chamber visits shows that after about a minute, you will lose all track of how long you've been in the chamber. So now you find yourself in the chamber, and realize that you've lost track of how long you've been there. What should your credence be that the coin landed heads?

Why is this a Sleeping Beauty case? Well, take the following discretized version. If it's heads, you get woken up 1,001,000 times and if's tails, you get woken up 2,001,000 times. There is no memory wiping, but empirical data from past experiments shows that you completely stop keeping track of wake-up counts after you've been woken up a thousand times. So now you've been woken up, and you know you've stopped counting. What should your credence be? This is clearly a version of Sleeping Beauty, except that instead of memory-wiping we have a cessation of keeping count, which plays the same role of being a non-rational process disturbing normal rational processes.

Oddly, though, in the sensory deprivation chamber case, I have the intuition that you should go for 1/2, even though in the original Sleeping Beauty case I've argued for 1/3. I don't have much intuition about my discretized version of the sensory deprivation chamber case.

P.s. I was thinking of blogging another Sleeping Beauty case, but it looks like LessWrong has beaten me to essentially it. (There may be a published version somewhere, too.)

Tuesday, October 14, 2014

Clumps and continuity

Our backyard had been free of black cats for as long as we've lived in this house, well over 400 days, except that over the last two nights, a black cat has visited our yard, meowing at the doors and windows. It's reasonable to think that it will visit again tonight. Yet 99.5% of evenings have been free of black cats. So how can it be inductively reasonable to think a black cat will visit tonight?

Presumably, it is because the data from the last two days is more relevant than the data from the earlier days, even though there are two orders of magnitude more black-cat-free days. But why is that data more relevant?

Granted, yesterday and the day before are more temporally similar to today than the other days. But why should temporal similarity override other kinds of similarity? No doubt there are many features (say, temperature, lunar phase, etc.) in respect of which today is more like some other day in the past 400 than like yesterday or the day before—after all, the earlier 398 days have a wide diversity of properties. But temporal similarity seems particularly important.

Maybe it is because we expect clumping, both in time and in space. Two black-cat evenings suggest the beginning of a clump.

I am curious: Is our expectation of clumping a priori justified or only a posteriori? Clumping seems to be a kind of
continuity. Is an expectation of continuity a priori justified or only a posteriori?

Monday, October 13, 2014

Not a finetuning argument

In The Impiety... (1624), as part of the 6th argument for the existence of God, Mersenne writes:

The proportion found between all the bodies of the world also shows that there is a God who has made all the universe in weight, in number and in measure: for the earth has no other ratio with the sun than 1:140, with the moon than 40:1, ... (pp. 98-99)
(I don't know off hand what the ratios are exactly meant to be; if they are ratios of volume, the moon is within 25% of the truth but the sun is off several orders of magnitude; if they are ratios of diameter, the sun is within an order of magnitude of the truth but the moon is an order of magnitude off.)

Mersenne's argument is full of such numerical (claimed) facts (the sun goes around the earth in 365.241 days, the moon traverses the Zodiac in 27 days, etc., etc.) and claims that God is needed to explain these facts. Now, I'm right now teaching on the fine-tuning argument, so I am sensitized to seeing such numbers in an argument for the existence of God. But it's striking that nowhere can I see Mersenne saying why these numbers are at all better than others, especially since surely some tuning facts seem very close at hand--surely, for instance, if the sun were much bigger or much smaller than it is, it would be too hot or too cold for life.

Mersenne explicitly insists that the numbers aren't explained by the essential natures of the objects, just before the above quote:

For the sun wouldn't be any the less the sun if it were closer or further from the earth, just as the stars could still be stars if they absented themselves from us by more than 14,000 earth radii.
Mersenne's argument seems to be a pure application of the idea that all contingent facts need explanation, and the arbitrariness of the numbers in the numerical statements seems to be cited precisely in order to show the contingency of the numerical statements. The argument suggests a strikingly strong commitment to a Principle of Sufficient Reason for contingent facts: all he needs to argue for a cosmic cause is to argue that there are contingent cosmic facts. Mersenne is confident that God has "many reasons" (as he says in the case of one of the numerical claims) for making the numbers be what they are, but these are reasons "which we aren't going to know except in Paradise" (101-102).

Mersenne's argument isn't a design argument--it doesn't advert to value-laden features that a God would have good reason to actualize. I think it's a kind of cosmological argument, but an eccentric one. Rather than arguing from generic features like motion or causation as Aquinas did, it focuses on very particular features.

The focus on these very particular features seems to have two benefits. The first is that it makes any appeal to necessity as the explanation implausible. Maybe it's necessary that there is motion, but it is incredible that it be necessary that the ratio of the diameter of the earth to that of the moon have to be 3.665:1 (to use modern numbers). So we get contingency very easily. The second feature is one I didn't notice right away. The astronomical features cited by Mersenne are ones that would reasonably be thought to be permanent features. They are thus prime candidates to be dismissed by it is so, as it has always been so. Mersenne's focus on the seeming arbitrariness of these features makes it very clear that would be no explanation. Thus Mersenne's cosmological argument works whether or not the past is finite. It is not disturbed by an infinite regress but does not need one either.

Of course, we no longer think that these particular features are permanent in the same way--the earth and sun changed in size in the formation of the solar system. But impermanent features are no better explained by an infinite regress than permanent ones--the permanence of the features in Mersenne's argument is only heuristic (and I don't see him explicitly drawing the reader's attention to the permanence). Plus we could run the argument on the basis of the apparently permanent but seemingly arbitrary elements in the laws of nature, such as precise values of constants.

The downside of Mersenne's argument, however, is that unless it is explained why the features are desirable, it is difficult to show that the cause of these features of the universe must be intelligent.

Thursday, October 9, 2014

I am not a panentheist

In the past, while reading about panentheism I've had a hard time seeing how it's not simply a version of orthodox theism. That we are in God, that all our existence is a participation in God and that God is present in everything by his omnipresence and universal sustenance and causal concommittance are all perfectly orthodox ideas, ones that, for instance, any Thomist will embrace. Thus I thought that I probably am a panentheist if panentheism is understood as "All things are in God and God is in all things" (Matthew Fox). Of course, many panentheists would add to the above various unorthodox doctrines, but these did not seem to me to be a part of panentheism. However, I am now thinking, after a careful look at the rather confusing panentheism article in SEP that a core doctrine of panentheism is divine dependence on the the non-divine. And that's not compatible with orthodox theism, I think, and certainly not with what I think. So I guess I'm not a panentheist after all.

Wednesday, October 8, 2014

Another argument from Mersenne

In The Impiety of Deists, etc., Mersenne also gives this theistic argument:

And if there is no God, no independent being, it would be impossible that one exist, and thus our imagination would exceed all the beings of the word: and the being of our thoughts and our fantasies [phantasies] would infinitely exceed all real beings, and what would be imaginary would surpass the true, which cannot be. (p. 75)

There are a couple of interesting things. First, often Leibniz gets credited with noticing that if God possibly exists, then God actually exists. But here we see Mersenne claiming the contrapositive, almost two decades before Descartes' Meditations (in an objection to Descartes' Meditations, Mersenne also makes the point in the Leibniz form).

Second, we get an interesting argument:

  1. If God doesn't exist, our imagination exceeds reality.
  2. Our imagination does not exceed reality.
  3. So, God exists.
We can also replace "exceed(s)" with "infinitely exceed(s)", which makes (2) even more plausible and (1) is still true. There are obvious connections between this and Descartes' infinity argument in the Meditations.

When thinking about this argument, I was initially puzzled why Mersenne starts the argument by arguing that if there is no God then the existence of God is impossible. After all, (1)-(3) doesn't seem to require the impossibility of God, just the non-actuality of God. My tentative interpretation is that Mersenne has in mind a fairly strong notion of "exceeds". Possibility has a certain foot in reality, and so for imagination to fully exceed reality, one would have to not only imagine something greater than what actually exists, but greater than what is possible. Now, God is greater than all non-divine possibilities. So if God is impossible, then the content of our thoughts outruns not just actuality but possibility, and that's what makes that content strongly outrun reality.

If this is right, then we can expand the argument as follows:

  1. If God doesn't exist, it is impossible for God to exist. (Premise)
  2. God is greater than all possibilities and actualities other than God. (Premise)
  3. We can think of God. (Premise)
  4. We cannot think of anything that exceeds all actualities and possibilities.
  5. God doesn't exist. (Supposition for reductio)
  6. God is not a possibility or actuality. (4 and 8)
  7. We can think of something that exceeds all actualities and possibilities. (5, 6 and 9)
  8. Contradiction! (7 and 10)
  9. So, God exists. (By reductio)

Finally, it is rather interesting how Mersenne argues for the thesis if God doesn't exist, he can't exist. In the context of another argument, he says:

He isn't a being, as we supposed, he can't exist: since who would make him, and who would give him being [qui luy donneroit estre]? (p. 119)
My first thought on this was that Mersenne subscribes to the causal theory of possibility that I've defended. My second thought, however, was that his argument may be broader. The "who would give him being?" rhetorical question may work on any view on which possibility is grounded in actuality given the plausibility that God's possibility couldn't be grounded in anything other than himself, or else he wouldn't truly be an independent being (and notice the focus on independence in the first Mersenne quote).

By the way, while I am relying on my own translations in the above (partly for fun), professional translations can be found here.

Tuesday, October 7, 2014

Two applications of 'ought implies can'?

First application:
1. God ought to exist.
2. So, God can exist. (Ought implies can)
3. So, God exists. (By Ontological Argument)

Second:
4. It's an evil for a person to die.
5. If naturalism is true, people can't live forever.
6. Evils ought not be there.
7. So, people ought not die.
8. So, people can live forever. (Ought implies can)
9. So, naturalism is not true.

In the second argument, for 5 to be true, the "can" must be stronger than metaphysical possibility. Thus that argument requires a stronger ought implies can principle.

All that said, I am sceptical of 1 and 7. An impersonal "ought" not addressed to any person is weird.

The core of the second argument, however, can perhaps be rescued:
10. It's an evil for a person to die.
11. What is normal for a kind of being is not an evil.
12. If naturalism is true, it's normal for people to die.
13. So, if naturalism is true, it's not an evil for a person to die.
14. So, naturalism is not true.

Mersenne's supreme bad vs. supreme good argument

In his 1624 The Impiety of Deists, Atheists and Libertines of This Time... (dedicated to Cardinal Richelieu, by the way), Mersenne gives this fascinating little argument:

Nobody fails to acknowledge that if there is a supremely good being [un estre souverainement bon], it merits the name of God, since we don't mean anything by that name other than that which has all [the] sorts of perfections, and which lacks nothing. Now I will show that this supreme good exists. If it didn't exist, its privation would exist, which would be a supreme bad [mal], and consequently the supreme non-being, since the bad and the non-being are the same thing: but it doesn't in the least seem that the privation exists more than its actuality, which must necessary precede it. Thus one must confess that there is a supreme goodness, and then that there cannot be a supreme badness. So we have a supreme being, since we deny a supreme non-being, it being necessary that the one or the other exist....
There is actually more than one argument here. There is an interesting and deeply metaphysical argument based on evil as the privation of a good. But there is also the kernel of a rather interesting and simple argument:
  1. It would be supremely bad if God doesn't exist.
  2. The world doesn't exemplify a supreme bad.
  3. So, God exists.
My son suggests using the goodness in the world to argue for (2). That would be an interesting hybrid design argument.

Monday, October 6, 2014

Desire as belief

David Lewis argues against an anti-Humean toy theory on which the value of a proposition A is the agent's credence in the proposition Av which says that A is valuable. Thus: V(A)=C(Av) where C is credence. His argument depends crucially on the independence assumption that the value of A doesn't depend on whether A is true:

  1. C(Av|A)=C(Av).

But independence understood this way is plainly false. Suppose my initial valuation of my daughter making dessert (D) is V(D)=C(Dv)=1/2. But suppose I completely trust her judgments of value, she had informed me that she made dessert if and only if making dessert was valuable, and this was all in the background knowledge. Then obviously my valuation of D had better change upon learning whether D is true. If D is true, then her making dessert was valuable; if not, it wasn't. Thus: C(Dv|D)=1 and C(Dv|~D)=0, contrary to (1).