Friday, February 23, 2018

More on wobbling of priors

In two recent posts (here and here), I made arguments based on the idea that wobbliness in priors translates to wobbliness in posteriors. The posts while mathematically correct neglect an epistemologically important fact: a wobble in a prior may be offset be a countervailing wobble in a Bayes’ factor, resulting in a steady posterior.

Here is an example of this phenomenon. Either a fair coin or a two-headed coin was tossed by Carl. Alice thinks Carl is a normally pretty honest guy, and so she thinks it’s 90% likely that a fair coin was tossed. Bob thinks Carl is tricky, and so he thinks there is only a 50% chance that Carl tossed the fair coin. So:

  • Alice’s prior for heads is (0.9)(0.5)+(0.1)(1.0) = 0.55

  • Carl’s prior for heads is (0.5)(0.5)+(0.5)(1.0) = 0.75.

But now Carl picks up the coin, mixes up which side was at the top, and both Alice and Bob have a look at it. It sure looks to them like there is a head on one side of it. As a result, they both come to believe that the coin is very, very likely to be fair, and when they update their credences on their observation of the coin, they both come to have credence 0.5 that the coin landed heads.

But a difference in priors should translate to a corresponding difference in posteriors given the same evidence, since the force of evidence is just the addition of the logarithm of the Bayes’ factor to the logarithm of the prior odds ratio. How could they both have had such very different priors for heads, and yet a very similar posterior, given the same evidence?

The answer is this. If the only relevant difference between Alice’s and Carl’s beliefs were their priors for heads, then indeed they couldn’t get the same evidence and both end up very close to 0.5. But their Bayes’ factors also differ.

  • For Alice: P(looks fair | heads)≈0.82; P(looks fair | tails)≈1; Bayes’ factor for heads vs. tails ≈0.82

  • For Bob: P(looks fair | heads)≈0.33; P(looks fair | tails)≈1; Bayes’ factor for heads vs. tails ≈0.33.

Thus, for Alice, that the coin looks fair is pretty weak evidence against heads, lowering her credence from 0.55 to around 0.5, while for Bob, that the coin looks fair is moderate evidence against heads, lowing his credence from 0.75 to around 0.5. Both end up at roughly the same point.

Thus, we cannot assume that a difference with respect to a proposition in the priors translates to a corresponding difference in the posteriors. For there may also be a corresponding difference in the Bayes’ factors.

I don’t know if the puzzling phenomena in my two posts can be explained away in this way. But I don’t know that they can’t.

A slightly different causal finitist approach to finitude

The existence of non-standard models of arithmetic makes defining finitude problematic. A finite set is normally defined as one that can be numbered by a natural number, but what is a natural number? The Peano axioms sadly underdetermine the answer: there are non-standard models.

Now, causal finitism is the metaphysical doctrine that nothing can have an infinite causal history. Causal finitism allows for a very neat and pretty intuitive metaphysical account of what a natural number is:

  • A natural number is a number one can causally count to starting with zero.

Causal counting is counting where each step is causally dependent on the preceding one. Thus, you say “one” because you remember saying “zero”, and so on. The causal part of causal counting excludes a case where monkeys are typing at random and by chance type up 0, 1, 2, 3, 4. If causal finitism is false, the above account is apt to fail: it may be possible to count to infinite numbers, given infinite causal sequences.

While we can then plug this into the standard definition of a finite set, we can also define finitude directly:

  • A finite set or plurality is one whose elements can be causally counted.

One of the reasons we want an account of the finite is so we get an account of proof. Imagine that every day of a past eternity I said: “And thus I am the Queen of England.” Each day my statement followed from what I said before, by reiteration. And trivially all premises were true, since there were no premises. Yet the conclusion is false. How can that be? Well, because what I gave wasn’t a proof, as proofs need to be finite. (I expect we often don’t bother to mention this point explicitly in logic classes.)

The above account of finitude gives an account of the finitude of proof. But interestingly, given causal finitism, we can give an account of proof that doesn’t make use of finitude:

  • To causally prove a conclusion from some assumptions is to utter a sequence of steps, where each step’s being uttered is causally dependent on its being in accordance with the rules of the logical system.

  • A proof is a sequence of steps that could be uttered in causally proving.

My infinite “proof” that I am the Queen of England cannot be causally given if causal finitism is true, because then each day’s utterance will be causally dependent on the previous day’s utterance, in violation of causal finitism. However, interestingly, the above account of proof does not guarantee that a proof is finite. A proof could contain an infinite number of steps. For instance, uttering an axiom or stating a premise does not need to causally depend on previous steps, but only on one’s knowledge of what the axioms and premises are, and so causal finitism does not preclude having written down an infinite number of axioms or premises. However, what causal finitism does guarantee is that the conclusion will only depend on a finite number of the steps—and that’s all we need to make the proof be a good one.

What is particularly nice about this approach is that the restriction of proofs to being finite can sound ad hoc. But it is very natural to think of the process of proving as a causal process, and of proofs as abstractions from the process of proving. And given causal finitism, that’s all we need.

Wobbly priors and posteriors

Here’s a problem for Bayesianism and/or our rationality that I am not sure what exactly to do about.

Take a proposition that we are now pretty confident of, but which was highly counterintuitive so our priors were tiny. This will be a case where we were really surprised. Examples:

  1. Simultaneity is relative

  2. Physical reality is indeterministic.

Let’s say our current level of credence is 0.95, but our priors were 0.001. Now, here is the problem. Currently we (let’s assume) believe the proposition. But if our priors were 0.0001, our credence would have been only 0.65, given the same evidence, and so we wouldn’t believe the claim. (Whatever the cut-off for belief is, it’s clearly higher than 2/3: nobody should believe on tossing a die that they will get 4 or less.)

Here is the problem. It’s really hard for us to tell the difference in counterintuitiveness between 0.001 and 0.0001. Such differences are psychologically wobbly. If we just squint a little differently when looking mentally a priori at (1) and (2), our credence can go up or down by an order of magnitude. And when our priors are even lower, say 0.00001, then an order of magnitude difference in counterintuitiveness is even harder to distinguish—yet an order of magnitude difference in priors is what makes the difference between a believable 0.95 posterior and an unbelievable 0.65 posterior. And yet our posteriors, I assume, don’t wobble between the two.

In other words, the problem is this: it seems that the tiny priors have an order of magnitude wobble, but our moderate posteriors don’t exhibit a correspnding wobble.

If our posteriors were higher, this wouldn’t be a problem. At a posterior of 0.9999, an order of magnitude wobble in priors results in a wobble between 0.9999 and 0.999, and that isn’t very psychologically noticeable (except maybe when we have really high payoffs).

There is a solution to this problem. Perhaps our priors in claims aren’t tiny just because the claims are counterintuitive. It makes perfect sense to have tiny priors for reasons of indifference. My prior in winning a lottery with a million tickets and one winner is about one in a million, but my intuitive wobbliness on the prior is less than an order of magnitude (I might have some uncertainty about whether the lottery is fair, etc.) But mere counterintuitiveness should not lead to such tiny priors. The counterintuitive happens all too often! So, perhaps, our priors in (1) and (2) were, or should have been, more like 0.10. And now perhaps the wobble in the priors will probably be rather less: it might vary between 0.05 and 0.15, which will result in a less noticeable wobble, namely between 0.90 and 0.97.

Simple hypotheses like (1) and (2), thus, will have at worst moderately low priors, even if they are quite counterintuitive.

And here is an interesting corollary. The God hypothesis is a simple hypothesis—it says that there is something that has all perfections. Thus even if it is counterintuitive (as it is to many atheists), it still doesn’t have really tiny priors.

But perhaps we are irrational in not having our posteriors wobble in cases like (1) and (2).

Objection: When we apply our intuitions, we generate posteriors, not priors. So our priors in (1) and (2) can be moderate, maybe even 1/2, but then when we updated on the counterintuitiveness of (1) and (2), we got something small. And then when we updated on the physics data, we got to 0.95.

Response: This objection is based on a merely verbal disagreement. For whatever wobble there is in the priors on the account I gave in the post will correspond to a similar wobble in the counterintuitiveness-based update in the objection.

Thursday, February 22, 2018

In practice priors do not wash out often enough

Bayesian reasoning starts with prior probabilities and gathers evidence that leads to posterior probabilities. It is occasionally said that prior probabilities do not matter much, because they wash out as evidence comes in.

It is true that in the cases where there is convergence of probability to 0 or to 1, the priors do wash out. But much of our life—scientific, philosophical and practical—deals with cases where our probabilities are not that close to 0 or 1. And in those cases priors matter.

Let’s take a case which clearly matters: climate change. (I am not doing this to make any first-order comment on climate change.) The 2013 IPCC report defines several confidence levels:

  • virtually certain: 99-100%

  • very likely: 90-100%

  • likely: 66-100%

  • about as likely as not: 33-66%

  • unlikely: 0-33%

  • very unlikely: 0-10%

  • exceptionally unlikely: 0-1%.

They then assess that a human contribution to warmer and/or more frequent warm days over most land areas was “very likely”, and no higher confidence level occurs in their policymaker summary table SPM.1. Let’s suppose that this “very likely” corresponds to the middle of its confidence range, namely a credence of 0.95. How sensitive is this “very likely” to priors?

On a Bayesian reconstruction, there was some actual prior probability p0 for the claim, which, given the evidence, led to the posterior of (we’re assuming) 0.95. If that prior probability had been lower, the posterior would have been lower as well. So we can ask questions like this: How much lower would the prior had to have been than p0 for…

  • …the posterior to no longer be in the “very likely” range?

  • …the posterior to fall into the “about as likely as not range”?

These are precise and pretty simple mathematical questions. The Bayesian effect of evidence is purely additive when we work with log likelihood ratios instead of probabilities, i.e., with log p/(1 − p) in place of p, so a difference in prior log likelihood ratios generates an equal difference in posterior ones. We can thus get a formula for what kinds of changes of priors translate to what kinds of changes in posteriors. Given an actual posterior of q0 and an actual prior of p0, to have got a posterior of q1, the prior would have to have been (1 − q0)p0q0/[(q1 − q0)p0 + (1 − q1)q0], or so says Derive.

We can now plug in a few numbers, all assuming that our actual confidence is 0.95:

  • If our actual prior was 0.10, to leave the “very likely” range, our prior would have needed to be below 0.05.

  • If our actual prior was 0.50, to leave the “very likely” range, our prior would have needed to be below 0.32.

  • If our actual prior was 0.10, to get to the “about as likely as not range”, our prior would have needed to be below 0.01.

  • If our actual prior was 0.50, to get to the “about as likely as not range”, our prior would have needed to be below 0.09.

Now, we don’t know what our actual prior was, but we can see from the above that variation of priors well within an order of magnitude can push us out of the “very likely” range and into the merely “likely”. And it seems quite plausible that the difference between the “very likely” and merely “likely” matters practically, given the costs involved. And a variation in priors of about one order of magnitude moves us from “very likely” to “about as likely as not”.

Thus, as an empirical matter of fact, priors have not washed out in the case of global warming. Of course, if we observe long enough, eventually our evidence about global warming is likely to converge to 1. But by then it will be too late for us to act on that evidence!

And there is nothing special about global warming here. Plausibly, many scientific and ordinary beliefs that we need to act on have a confidence level of no more than about 0.95. And so priors matter, and can matter a lot.

We can give a rough estimate of how differences in priors make a difference regarding posteriors using the IPCC likelihood classifications. Roughly speaking, a change between one category and the next (e.g., “exceptionally unlikely” to “unlikely”) in the priors results in a change between a category and the next (e.g., “likely” to “very likely”) in the posteriors.

The only time priors have washed out is cases where our credences have converged very close to 0 or to 1. There are many scientific and ordinary claims in this category. But not nearly enough for us to be satisfied. We do need to worry about priors, and we better not be subjective Bayesians.

Yet another life-based argument against thinking machines

Here’s yet another variant on a life-based argument against machine consciousness. All of these arguments depend on related intuitions about life. I am not super convinced by them, but they have some evidential force I think.

  1. Only harm to a living thing can be a great intrinsic evil.

  2. If machines can be conscious, then a harm to a machine can be a great intrinsic evil.

  3. Machines cannot be alive.

  4. So, harm to a machine cannot be a great intrinsic evil. (1 and 3)

  5. So, machines cannot be conscious. (2 and 4)

Tuesday, February 20, 2018

Castigation

Mere criticism is a statement that something—an action, a thought, an object, etc.—falls short of an applicable standard. But sometimes instead of merely criticizing a person, we do something more, which I’ll call “castigation”. When we castigate people to their face, we are not merely asserting that they have fallen short of a standard, but we blame them for it in a way that is intended to sting. Mere criticism may sting, but stinging isn’t part of its intent. Mill’s “disapprobation” is an example of castigation:

If we see that ... enforcement by law would be inexpedient, we lament the impossibility, we consider the impunity given to injustice as an evil, and strive to make amends for it by bringing a strong expression of our own and the public disapprobation to bear upon the offender.

But now notice something:

  1. Castigation is a form of punishment.

  2. It is unjust and inappropriate punish someone who is not morally culpable.

  3. So, it is unjust and inappropriate to castigate someone who is not morally culpable.

In an extended sense of the word, we also castigate people behind their backs—we can call this third-person castigation. In doing so, we express the appropriateness of castigating them to their face even when that castigation is impractical or inadvisable. Such castigation is also a form of punishment, directed at reputation rather than the feelings of the individual. Thus, such castigation is also unjust and inappropriate in the case of someone lacking morally culpability.

I exclude here certain speech acts done in training animals or small children which have an overt similarity to castigation. Because the subject of the acts is not deemed to be a morally responsible person, the speech acts have a different significance from when they are directed at a responsible person, and I do not count them as castigation.

Thus, whether castigation is narrow (directed at the castigated person) or extended, it is unjust and inappropriate where there is no moral culpability. Mere criticism, on the other hand, does not require any moral culpability. Telling the difference between the castigation and mere criticism is sometimes difficult, but there is nonetheless a difference, often conveyed through the emotional load in the vocabulary.

In our society (and I suspect in most others), there is often little care to observe the rule that castigation is unjust absent moral culpability, especially in the case of third-person castigation. There is, for instance, little compunction about castigating people with abhorrent (e.g., racist) or merely silly (e.g., flat earth) views without investigation whether they are morally culpable for forming their beliefs. Politicians with policies that people disagree with are pilloried without investigation whether they are merely misguided. The phrase “dishonest or ignorant” which should be quite useful for criticism that avoids the risk of unjust castigation gets loaded to the point where it effectively castigates a person for possibly being ignorant. This is not to deny, of course, that one can be morally blameworthy for abhorrent, silly or ignorant views. But rarely do we know an individual to be morally culpable for their views, and without knowledge, castigation puts us at risk of doing injustice.

I hope I am not castigating anyone, but merely criticizing. :-)

Here is another interesting corollary.

  1. Sometimes it permissible to castigate friends for their imprudence.

  2. Hence, sometimes people are morally culpable for imprudence.

In the above, I took it that punishment is appropriate only in cases of moral wrongdoing. Mill actually thinks something stronger is the case: punishment is appropriate only in cases of injustice. If Mill is right, and yet if we can rightly castigate friends for imprudence, it follows that imprudence can be unjust, and the old view that one cannot do injustice to oneself is false.

Monday, February 19, 2018

Leibniz on PSR and necessary truths

I just came across a quote from Leibniz that I must have read before but it never impressed itself on my mind: “no reason can be given for the ratio of 2 to 4 being the same as that of 4 to 8, not even in the divine will” (letter to Wedderkopf, 1671).

This makes me feel better for defending only a Principle of Sufficient Reason restricted to contingent truths. :-)

Life, thought and artificial intelligence

I have an empirical hypothesis that one of the main reasons why a lot of ordinary people think a machine can’t be conscious is that they think life is a necessary condition for consciousness and machines can’t be alive.

The thesis that life is a necessary condition for consciousness generalizes to the thesis that life is a necessary condition for mental activity. And while the latter thesis is logically stronger, it seems to have exactly the same plausibility.

Now, the claim that life is a necessary condition for mental activity (I keep on wanting to say that life is a necessary condition for mental life, but that introduces the confusing false appearance of tautology!) can be understood in two ways:

  1. Life is a prerequisite for mental activity.

  2. Mental activity is in itself a form of life.

On 1, I think we have an argument that computers can’t have mental activity. For imagine that we’re setting up a computer that has mental activity, but we stop short of making it engage in the computations that would make it engage in mental activity. I think it’s very plausible that the resulting computer doesn’t have any life. The only thing that would make us think that a computer has life is the computational activity that underlies supposed mental activity. But that would be a case of 2, rather than 1: life wouldn’t be a prerequisite for mental activity, but mental activity would constitute life.

All that said, while I find the thesis that life is a necessary condition for mental activity, I am more drawn to 2 than to 1. It seems intuitively correct to say that angels are alive, but it is not clear that we need anything more than mental activity on the part of angels to make them be alive. And from 2, it is much harder to argue that computers can’t think.

Thursday, February 15, 2018

Physicalism and ethical significance

I find the following line of thought to have a lot of intuitive pull:

  1. Some mental states have great non-instrumental ethical significance.

  2. No physical brain states have that kind of non-instrumental ethical significance.

  3. So, some mental states are not physical brain states.

When I think about (2), I think in terms similar to Leibniz’s mill. Leibniz basically says that if physical systems could think, so could a mill with giant gears (remember that Leibniz invented a mechanical calculator running on gears), but we wouldn’t find consciousness anywhere in such a mill. Similarly, it is plausible that the giant gears of a mill could accomplish something important (grind wheat and save people from starvation or simulate protein folding leading to a cure for cancer), and hence their state could have great instrumental ethical significance, but their state isn’t going to have the kind of non-instrumental ethical significance that mental states do.

I worry, though, whether the intuitive evidence for (2) doesn’t rely on one’s already accepting the conclusion of the argument.

Beyond binary mereological relations

Weak supplementation says that if x is a proper part of y, then y has a proper part that doesn’t overlap x.

Suppose that we are impressed by standard counterexamples to weak supplementation like the following. Tibbles the cat loses everything but its head, which is put in a vat. Then Head is a part of Tibbles, but obviously Head is not the same thing as Tibbles by Leibniz’s Law (since Tibbles used to have a tail as a part, but Head did not use to have a tail as a part), so Head is a proper part of Tibbles—yet, Head does not seem to be weakly supplemented.

But suppose also that we don’t believe in unrestricted fusions, because we have a common-sense notion of what things have a fusion and what parts a thing has. Thus, while we are willing to admit that Tibbles, prior to its injury, has parts like a head, lungs, heart and legs, we deny that there is any such thing as Tibbles’ front half minus the left lung—i.e., the fusion of all the molecules in Tibbles that are in the front half but not in its left lung.

Imagine, then, that there is a finite collection of parts of Tibbles, the Ts, such that there is no fusion of the Ts. Suppose next that due to an accident Tibbles is reduced to the Ts. Observe a curious thing. By all the standard definitions of a fusion (see SEP, with obvious extensions to a larger number of parts), after the accident Tibbles is a fusion of the Ts.

So we get one surprising conclusion from the above thoughts: whether the Ts have a fusion depends on extrinsic features of them, namely on whether they are embedded in a larger cat (in which case they don’t have a fusion) or whether they are standalone (in which case their fusion is the cat). This may seem counterintuitive, but artefactual examples should make us more comfortable with that. Imagine that on the floor of a store there are hundreds of chess pieces spilled and a dozen chess boards. By picking out—perhaps only through pointing—a particular 32 pieces and a particular board, and paying for them, one will have bought a chess set. But perhaps that particular chess set did not exist before, at least on a common-sense notion of what things have a fusion. So, one will have brought it into existence by paying for it. The pieces and the board now seem to have a fusion—the newly purchased chess set—while previously they did not.

Back to Tibbles, then. I think the story I have just told shows that if we deny weak supplementation and unrestricted fusions also suggests something else that’s really interesting: that the standard mereological relations—whether parthood or overlap—do not capture all the mereological facts about a thing. Here’s why. When Tibbles is reduced to a head, we want to be able to say that Tibbles is more than its head. And we can say that. We say that by saying that Head is a proper part of Tibbles (albeit one that is not weakly supplemented). But if Tibbles is more than his head even after being reduced to a head, then by the same token Tibbles is more than the sum of the Ts even after being reduced to the Ts. But we have no way of saying this in mereological vocabulary. Tibbles is the fusion or sum of the Ts when that fusion is understood in the standard ways. Moreover, we have no way of using the binary parthood or overlap relations to distinguish the how Tibbles is related to the Ts from relationships that are “a mere sum” relationship.

Here is perhaps a more vivid, but even more controversial, way of seeing the above point. Suppose that we have a tree-like object whose mereological facts are like this. Any branch is a part. But there are no “shorn trunks” in the ontology, i.e., no trunk-minus-branches objects (unless the trunk in fact has no branches sticking out from it). This corresponds to the intuition that while I have arms and legs as parts, there is no part of me that is constituted by my head, neck and trunk. And (this is the really controversial bit) there are no other parts—there are no atoms, in particular. In this story, suppose Oaky is a tree with two branches, Lefty and Righty. Then Lefty and Righty are Oaky’s only two proper parts. Moreover, by the standard mereological definitions of sums, Oaky is the sum of Lefty and Righty. But it’s obvious that Oaky is more than the sum of Lefty and Righty!

And there is no way to distinguish Oaky using overlap and/or parthood from a more ordinary case where an object, say Blob, is constituted from two simple halves, say, Front and Back.

What should we do? I don’t know. My best thought right now is that we need a generalization of proper parthood to a relation between a plurality and an object: the As are jointly properly parts of B. We then define proper parthood as a special case of this when there is only one A. Using this generalization, we can say:

  • Head is a proper part of Tibbles before and after the first described accident.

  • The Ts are jointly properly parts of Tibbles before and after the second described accident.

  • Lefty and Righty are jointly properly parts of Oaky.

  • It is not the case that Front and Back are jointly properly parts of Blob.

Wednesday, February 14, 2018

Mereology and constituent ontology

I’ve just realized that one can motivate belief in bare particulars as follows:

  1. Constituent ontology of attribution: A thing has a quality if and only if that quality is a part of it.

  2. Universalism: Every plurality has a fusion.

  3. Weak supplementation: If x is a proper part of y, then y has a part that does not overlap x.

  4. Anti-bundleism: A substance (or at least a non-divine substance) is not the fusion of its qualities.

For, let S be a substance. If S has no qualities, it’s a bare particular, and the argument is done.

So, suppose S has qualities. By universalism, let Q be the fusion of the qualities that are parts of S. This is a part of S by uncontroversial mereology. By anti-bundleism, Q is a proper part of S. By weak supplementation, S has a part P that does not overlap Q. That part has no qualities as a part of it, since if it had any quality as a part of it, it would overlap Q. Hence, P is a bare particular. (And if we want a beefier bare particular, just form the fusion of all such Ps.)

It follows that every substance has a bare particular as a part.

[Bibliographic notes: Sider thinks that something like this argument means that the debate between constituent metaphysicians overlap bare particulars is merely verbal. Not all bare particularists find themselves motivated in this way (e.g., Smith denies 1).]

To me, universalism is the most clearly false claim. And someone who accepts constituent ontology of attribution can’t accept universalism: by universalism, there is fusion of Mt. Everest and my wedding ring, and given constituent ontology, the montaineity that is a part of Everest and the goldenness of my ring will both be qualities of EverestRing, so that EverestRing will be a golden mountain, which is absurd.

But universalism is not, I think, crucial to the argument. We use universalism only once in the argument, to generate the fusion of the qualities of S. But it seems plausible that even if universalism in general is false, there can be a substance S such that there is a fusion Q of its qualities. For instance, imagine a substance that has only one quality, or a substance that has a quality Q1 such that all its other qualities are parts of Q1. Applying the rest of the argument to that substance shows that it has a bare particular as a part of it. And if some substances have bare particular parts, plausibly so do all substances (or at least all non-divine substances, say).

If this is right, then we have an argument that:

  1. You shouldn’t accept all of: constituent ontology, weak supplementation, anti-bundleism and anti-bare-particularism.

I am an anti-bundleist and an anti-bare-particularist, but constituent ontology seems to have some plausibility to me. So I want to deny weak supplementation. And indeed I think it is plausible to say that the case of a substance that has only one quality is a pretty good counterexample to weak supplementation: that one quality lacks even a weak supplement.

Tuesday, February 13, 2018

Theistic multiverse, omniscience and contingency

A number of people have been puzzled by the somewhat obscure arguments in my “Divine Creative Freedom” against a theistic modal realism on which (a) God creates infinitely many worlds and (b) a proposition is possible if and only if it is true at one of them.

So, here’s a simplified version of the main line of thought. Start with this:

  1. For all propositions p, necessarily: God believes p if and only if p is true.

  2. There is a proposition p such that it is contingent that p is true.

  3. So, there is a proposition p such that it is contingent that God believes p. (1 and 2)

  4. Contingent propositions are true at some but not all worlds that God creates. (Theistic modal realism)

  5. So, there is a proposition p such that whether God believes p varies between the worlds that God creates. (3 and 4)

Now, a human being’s beliefs might vary between locations. Perhaps I am standing on the Texas-Oklahoma border, with my left brain hemisphere in Texas and my right one in Oklahoma, and with my left hemisphere I believe that I am in Texas while with my right one I don’t. Then in Texas I believe I am in Texas while in Oklahoma I don’t believe that. But God’s mind is not split spatially in the same way. God’s beliefs cannot vary from one place to another, and by the same token cannot vary between the worlds that God creates.

An objection I often hear is something like this: a God who creates a multiverse can believe that in world 1, p is true while in world 2, p is false. That's certainly correct. But those are necessary propositions that God believes, then--it is necessary that in world 1, p is true and that in world 2, p is false, say. And God has to believe all truths, not just the necessary ones. Hence, at world 1, he has to believe p, and at world 2, he has to believe not p.

Proper classes as merely possible sets

This probably won’t work out, but I’ve been thinking about the Cantor and Russell Paradoxes and proper classes and had this curious idea: Maybe proper classes are non-existent possible sets. Thus, there is actually no collection of all the sets in our world, but there is another possible world which contains a set S whose members are all the sets of our world. When we talk about proper classes, then, we are talking about merely possible sets.

Here is the story about the Russell Paradox. There can be a set R whose members are all the actual world’s non-self-membered sets. (In fact, since by the Axiom of Foundation, none of the actual world’s sets are self-membered, R is a set whose members are all the actual world’s sets.) But R is not itself one of the actual world’s sets, but a set in another possible world.

The story about Cantor’s Paradox that this yields is that there can be a cardinality greater than all the cardinalities in our world, but there actually isn’t. And in world w2 where such a cardinality exists, it isn’t the largest cardinality, because its powerset is even larger. But there is a third world which has a cardinality larger than any in w2.

It’s part of the story that there cannot be any collections with non-existent elements. Thus, one cannot form paradoxical cross-world collections, like the collection of all possible sets. The only collections there are on this story are sets. But we can talk of collections that would exist counterfactually.

The challenge to working out the details of this view is to explain why it is that some sets actually exist and others are merely possible. One thought is something like this: The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world. E.g., if the world contains an infinite sequence of coin tosses, it contains the set of the natural numbers corresponding to tosses with heads.

Saturday, February 10, 2018

Counting goods

Suppose I am choosing between receiving two goods, A and B, or one, namely C, where all the goods are equal. Obviously, I should go for the two. But why?

Maybe what we should say is this. Since A is at least as good as C, and B is non-negative, I have at least as good reason to go for the two goods as to go for the one. This uses the plausible assumption that if one adds a good to a good, one gets something at least as good. (It would be plausible to say that one gets something better, but infinitary cases provide a counterexample.) But there is no parallel argument that it is at least as good to go for the one good as to go for the two. Hence, it is false that I have at least as good reason to go for the one as to go for the two. Thus, I have better reason to go for the two.

This line of thought might actually solve the puzzles in these two posts: headaches and future sufferings. And it's very simple and obvious. But I missed it. Or am I missing something now?

Friday, February 9, 2018

Counting infinitely many headaches

If the worries in this post work, then the argument in this one needs improvement.

Suppose there are two groups of people, the As and the Bs, all of whom have headaches. You can relieve the headaches of the As or of the Bs, but not both. You don’t know how many As or Bs there are, or even whether the numbers are finite or finite. But you do know there are more As than Bs.

Obviously:

  1. You should relieve the As’ headaches rather than the Bs’, because there are more As than Bs.

But what does it mean to say that there are more As than Bs? Our best analysis (simplifying and assuming the Axiom of Choice) is something like this:

  1. There is no one-to-one function from the As to the Bs.

So, it seems:

  1. You should relieve the As’ headache rather than the Bs’, because there is no one-to-one function from the As to the Bs.

For you should be able to replace an explanation by its analysis.

But that’s strange. Why should the non-existence of a one-to-one function from one set or plurality to another set or plurality explain the existence of a moral duty to make a particular preferential judgment between them?

If the number of As and Bs is finite, I think we can do better. We can then express the claim that there are more As than Bs by an infinite disjunction of claims of the form:

  1. There exist n As and there do not exist n Bs,

which claims can be written as simple existentially quantified claims, without any mention of functions, sets or pluralities.

Any such claim as (4) does seem to have some intuitive moral force, and so maybe their disjunction does.

But in the infinite case, we can’t find a disjunction of existentially quantified claims that analysis the claim that there are more As than Bs.

Maybe what we should say is that “there are more As than Bs” is primitive, and the claim about there not being a one-to-one function is just a useful mathematical equivalence to it, rather than an analysis?

The thoughts here are also related to this post.