Friday, June 16, 2017

Optimalism about necessity

There are many set-theoretic claims that are undecidable from the basic axioms of set theory. Plausibly, the truths of set theory hold of necessity. But it seems to be arbitrary which undecidable set-theoretic claims are true. And if we say that the claims are contingent, then it will be arbitrary which claims are contingent. We don’t want there to be any of the “arbitrary” in the realm of necessity. Or so I say. But can we find a working theory of necessity that eliminates the arbitrary?

Here are two that have a hope. The first is a variant on Leslie-Rescher optimalism. While Leslie and Rescher think that the best (narrowly logically) scenario must obtain, and hence endorse an optimalism about truth, we could instead affirm an optimalism about necessity:

  1. Among the collections of proportions, that collection of propositions that would make for the best collection of the necessary truths is in fact the collection of the necessary truths.

And just as it arguably follows from Leslie-Rescher optimalism that there is a God, since it is best that there be one, it arguably follows from this optimalism about necessity that there necessarily is a God, since it is best that there necessarily be a God. (By the way, when I once talked with Rescher about free will, he speculatively offered me something that might be close to optimalism about necessity.)

Would that solve the problem? Maybe: maybe the best possible—both practically and aesthetically—set theory is the one that holds of necessary truth.

I am not proposing this theory as a theory of what necessity is, but only of what is in fact necessary. Though, I suppose, one could take the theory to be a theory of what necessity is, too.

Alternately, we could have an optimalist theory about necessity that is theistic from the beginning:

  1. A maximally great being is the ground of all necessity.

And among the great-making properties of a maximally great being there are properties like “grounding a beautiful set theory”.

I suspect that (1) and (2) are equivalent.

Brute necessities and supervenience

There is something very unappealing about unexplained, i.e., brute, metaphysical necessities that are “arbitrary”. For instance, suppose that someone said that some constant in a law of nature had the precise value it does by metaphysical necessity. If that contant were 1 or π or something like that, we could maybe buy that. But if the constant couldn’t be put in any neat way, could not be derived from deeper metaphysical necessities, but just happened necessarily to be exactly 1.847192019... (in a natural unit system) for some infinite string of digits? Nah! It would be much more satisfactory to posit a theory on which that constant has that value contingently. “Arbitrariness” of this sort is evidence of contingency, though it is a hard question exactly why.

Here is an application of this epistemic principle. It seems very likely that any view on which mental properties supervene of metaphysical necessity on physical ones will involve brute metaphysical necessities that are “arbitrary”.

For instance, consider a continuum of physical arrangements, starting with a paradigmatic healthy adult human and ending with a rock of the same mass. The adult human has conscious mental properties. The rock does not. Given metaphysically necessary supervenience, there must be a necessary truth as to where on the continuum the transition from consciousness to lack of consciousness occurs or, if there is vagueness in the transition, then there must be a necessary truth as to how the physical continuum maps to a vagueness profile. But it is very likely that any such transition point will be “arbitrary” rather than “natural”.

Or consider this. The best naturalist views make mental properties depend on computational function. But now consider how to define the computational function of something, say of a device that has two numerical inputs and one numerical output. We might say that if 99.999% of the time when given two numbers the device produces the sum of the numbers, and there is no simple formula that gives a higher degree of fit, then the computational function of the device is addition. But just how often does the device need to produce the sum of the numbers to count as an adder? Will 99.99% suffice? What about 99.9%? The reliability cut-off in defining computational function seems entirely arbitrary.

It may be that there is some supervenience theory that doesn’t involve arbitrary maps, arbitrary cut-offs, etc. But I suspect we have no idea how such a theory would go. It’s just pie in the sky.

If supervenience theories appear to require “arbitrary” stuff, then it is reasonable to infer that any supervenience is metaphysically contingent—perhaps it is only nomic supervenience.

This line of argument is plausible, but to make it strong one would need to say more about the notion of the “arbitrary” that it involves.

Tuesday, June 13, 2017

Death, dignity and eternal life

One way to look at the difference between the deaths of humans and brute animals is to say that the death of a human typically deprives the human of goods of rational life that the brute animal is not deprived of. While it is indeed an important part of the evil of typical cases of death in humans that they are deprived of such goods, however, focusing on this leads to a difficulty seeing what is distinctively bad about the death of humans who are not deprived of such goods by death, say elderly humans who have already lost the distinctive goods of rational life.

Sure, one can say that the death of a human is the death of a being that normally has the goods of rational life. But it is unclear why the death of a being that normally has the goods of rational life but actually lacks them is worse than the death of a being that actually and normally lacks the goods of rational life.

(Of course, not everybody shares the normative view that there is something distinctively bad about the death of a human being even when the goods of rational life have already been lost. A significant number of people think that euthanasia in such cases is morally licit. But even among those who think that euthanasia in such cases is morally licit, I think many will still think that there is something particularly morally bad about killing such human beings against their clear prior wishes, and those may find something plausible about what I say below.)

How, then, do we explain the distinctive bad in the death of human beings, even ones that lack the distinctive goods of rational life? In the end, I think I would like to invoke human dignity here, but to a significant degree that’s just giving a name to the problem. Instead of invoking and trying to explain human dignity, I want to explore a different option, one that I think in the end will not succeed, but perhaps there is something in the vicinity that can.

Here is a hypothesis:

  • It is the nature of human beings to live forever and never die, but the nature of brute animals is to have a finite life.

If this is true, then death always constitutes a mutilation of the human being. It is what directly deprives the human being of the normative diachronic shape of its life. And killing a human mutilates the human being.

Objection 1: If a murderer didn’t kill her victim, the victim would still have died at some later point.

Response: The murderer is still the proximate cause of the victim’s not living forever. And such proximate causation matters. Suppose that my brother murdered Sally’s brother, and to avenge her brother, in true Hammurabic fashion, Sally seeks to kill me. When she finally comes upon me, I am already falling off a cliff. A moment before I would have hit the ground, Sally shoots and kills me. Sally has murdered me, a grave evil. She is the proximate cause of my death. And that matters, even though it would make little difference to my life if Sally hadn’t killed me.

Objection 2: Even if it is the nature of brute animals to have a finite life, it is not the nature of brute animals to die young. But it is not wrong to kill a brute animal when it is young, even though doing so mutilates the brute animal in much the same way that killing a human mutilates the human by causing her life to be finite if the hypothesis is true.

Response: Agreed: it does mutilate the brute animal to kill it when it is young. But to foreshorten the life of a human being from infinity to a finite amount is much worse—in a sense, infinitely worse—than to foreshorten the life of a brute animal from a longer finite length to a shorter finite length.

Objection 3: Christian faith holds that humans will be resurrected. Thus, killing a human being does not succeed in causing the human being to lose infinite life.

Response: Yes, but according to the hypothesis it is not only the nature of human beings to have an infinite future life but it is also the nature of human beings to have a death-free infinite future life.

Objection 4: Imagine an otherwise unremarkable shrub which has a very special nature: it is supposed to live forever, undying. Destroying this shrub would feel distinctively bad as compared to destroying an ordinary shrub, but still not bad in the same way that killing a human being is. Hence, reference to the normativeness of an infinite future life is not enough to explain the distinctive badness of killing humans.

Response: I think that this objection is decisive. Mere invocation of the normativeness of an infinite deathless life is not enough to solve the problem of the distinctive badness of human death. One still needs something like a story about the special dignity of human beings. But it might be that the hypothesis still helps: it multiplies the synchronic dignity of the human being by something like infinity. So less needs to be accomplished by the dignity part of the account.

Questions that interest me on norm institution and grounding

For any norm Nk that we institute, there is a prior norm Nk − 1 that specifies that when the acts of institution of Nk are performed, then Nk has such-and-such force.

On pain of a regress incompatible with the empirical facts of humanity’s finite past, any instituted norm must be grounded in an uninstituted norm. What are these uninstituted norms like?

Are they specific to our human nature or do they apply to all rational beings or are some of one sort and some of the other? Thinking about some issues in ethics, language, epistemology and decision theory has made me think that it is likely that at least some of the uninstituted norms are specific to human nature rather than to all rational beings.

Also, what types of norms are the uninstituted norms, and how do they relate to the types of norms that they ground? For instance, are instituted linguistic norms grounded in uninstituted linguistic norms or in some other kinds of norms, say moral ones?

For those of us who love theoretical simplicity, it would be a great joy if it turned out that all the uninstituted norms were of one type. If so, that type would be the moral. For, plausibly, no norm can ground an instituted norm that has greater force than itself, and moral norms have greater force than any others. In any case, either there are multiple types of uninstituted norms, or they are all moral. In the latter case all norms are moral or derive by institution from moral ones.

Note that the uninstituted norms need not be fundamental. There could be grounding relations between uninstituted norms. For instance, neither the moral norm not to torture the innocent nor the moral norm not to torture innocent blue-eyed people is instituted, but the latter (assuming it really counts as a norm, rather than an application or something like that) is clearly grounded in the former. If it turns out that, as I think, some uninstituted norms are specific to our human nature, it could still be the case that all the uninstituted norms that are specific to our human nature are grounded in a norm not specific to human nature—say, the universal norm to act in accordance with one’s nature.

Furthermore, there are norms that govern rational behavior as such and norms that do not govern rational behavior as such, such as the norm that two legs is good for humans and four legs is good for pigs. What grounding relationships are there between these? Are all the uninstituted norms of one sort or the other, or are they of both sorts?

There is material for interesting dissertations exploring questions like this. Of course, such questions have been explored in multiple contexts, but perhaps not quite in the above structure.

Monday, June 12, 2017

National self-defense

I think many of us have the intuition that it is permissible, indeed often morally required, for a decent country to defend itself against invaders when there is a reasonable hope of victory. The “decent” condition needs to be there: it was not permissible for Nazi Germany to defend itself against the Allies—they had the duty of surrendering. The “reasonable hope” condition needs to be there as well: if the consequence of fighting is nuclear attacks on all one’s cities, one should probably surrender.

If the Ruritanians invade Elbonia, a decent country, with the goal of killing all Elbonians, then at least if there is a reasonable chance of repelling the invaders, it is permissible for the Elbonians to defend themselves with lethal force. Only slightly less clearly, if the Ruritanians intend to cause no physical harm to Elbonians if the Elbonians surrender, but will wipe out Elbonian culture—they will forbid the use of the Elbonian language, ban the national pastime of painting intricate landscapes on pigeon feathers, and so on—then lethal self-defense is still likely to be permissible.

But what if the Ruritanians invade Elbonia simply in order to take away Elbonia’s sovereignty, so that if the Elbonians surrender, they lose sovereignty but nothing else? The Ruritanians won’t kill anyone, won’t disposs any individuals or corporations of their property, won’t interfere with any aspects of Elbonian culture, won’t conscript Elbonians into their military (the Ruritanians have an all-vounteer army), will not harm the Elbonian economical, educational and healthcare systems, etc. But they will take over national sovereignty. Moreover, the Elbonians are confident of this because the Ruritanians have a centuries-long record of expanding their empire on such terms, and many neighboring countries have lost their sovereignty but had no other losses. Furthermore, it is the Elbonians alone that are at issue. For geographic reasons, the Ruritanians are unable to expand any further, and so Elbonians in defending themselves cannot say that they doing so to protect other countries. And there are no other countries in the world capable of imperialism.

It is only permissible to wage war for the sake of a good that is proportionate to the great evils of war, after all. The question here is this: Is maintenance of national sovereignty worth the deaths—both Elbonian and Ruritanian—and manifold other harms of war?

I don’t know. A state is a valuable form of human community. The destruction of a state is prima facie a bad thing. But if the goods of culture and ordinary life are maintained, it does not seem to be a great bad. Suppose that there was no invasion, but the Elbonians voluntarily voted to join the Ruritanian Empire. Then while there would be some bad in the loss of the Elbonian state, it need not be a tragedy, and on balance it could even be for the good. It is, of course, gravely wrong for the Ruritanians to bludgeon the Elbonians into joining their Empire. But the good of sovereignty just might not be great enough for the Ruritanians to have a moral justification to resist to the death.

If this is right, then sometimes the mere fact that a war is one of just national self-defense is not enough to justify fighting. Do such perfectly clean cases occur? I doubt it: imperialist countries aren’t likely to be as nice as my hypothetical Ruritanians. However, one might have cases that are slightly less clean, where the expected damage to local culture is likely to be small relative to the expected harms of a protracted war, even if that war can be won by the defenders. Moreover, in real-life cases one needs to consider the value of policies that discourage future such attacks by this and other imperialist countries. If all small countries surrendered as soon as there was a Ruritanian-style invasion, then we could expect Ruritanians and others to mount a lot more invasions, which could indeed be harmful.

So our initial intuition about the permissibility of national self-defense is, I think, roughly right, though only roughly.

Thursday, June 1, 2017


Some properties that a thing has partially or wholly explain other properties the thing has or doesn’t have. For instance, my having a body partially explains my being in Waco and wholly explains my having a body or horns. Some properties that a thing has do not explain, even partially, what other properties the thing has or doesn’t have. Call such properties “explanatorily fundamental”.

So, here’s a theory. The primary essential properties of a thing are the explanatorily fundamental properties of the thing. The primary essential properties are both essential in the medieval explanatory sense and the contemporary modal sense (properties a thing cannot exist without).

What about the case of Christ, who is essentially divine and essentially human, and yet prior (in the order of explanation) to the incarnation was not human? Here’s what we could say: Divinity is the one and only primary essential property of Christ. But humanity is a secondary essential property. A secondary essential property of a thing is the sort of property that (a) is not a primary essential property of that thing, but (b) normally is the primary essential property of its possessor. In the case of Christ, his divinity is explanatorily prior to his humanity, but normally a thing’s humanity does not have any property of that thing explanatorily prior to it.

Tuesday, May 30, 2017

Drawers for small electronic components

With various ongoing projects, I've acquired a lot of small electronic components in little baggies. To store them, I designed a set of customizable plastic drawers that I could run off on my 3D printer. I am stingy, so I tried to reduce the amount of plastic that goes into them by making dividers thin and a grid-like pattern for the walls.

Location, causation and transsubstantiation

Here’s a fun thought experiment. By a miracle (say) I am sitting in my armchair in Waco but my causal interaction with my environment at the boundaries of my body would be as if I were in Paris. There is a region of space in Paris shaped like my body. When a photon hits the boundary of that region, it causally interacts with me as if I were in Paris: I have the causal power to act at a distance to reflect Parisian photons as if I were in that region in Paris. Alternately, that photon might be absorbed by me: I have the causal power to absorb Parisian photons. As a result, it looks to Parisians like I am in Paris, and as I look around, it looks to me like Paris is all around me. The same is true for other interactions. When my vocal cords vibrate, instead of causing pressure changes in Texan air, they cause pressure changes in chilly France. As I walk, the region of space shaped like my body in Paris that is the locus of my interaction with Parisians moves in the usual way that bodies move.

Furthermore, my body does not interact with the environment in Waco at all. Wacoan photons aimed at my body go right through it and so I am invisible. In fact, not just photons do that: you could walk right through my body in Waco without noticing. My body is unaffected by Texan gravity. It is simply suspended over my sofa. As I wave my hand, my hand does in fact wave in Texas, but does not cause any movement of the air in Texas—but in Paris, the region of space in which I interact with the Parisians changes through the wave, and the air moves as a result. When I eat, it is by means of Parisian food particles that come to be incorporated into my Wacoan body.

To me, to Wacoans and to Parisians it looks in all respects like I am in Paris. But I am in Waco.

Or am I? There is a view on which the causal facts that I’ve described imply that I am in Paris, namely the view that spatial relationships reduce to causal relationships. It is an attractive view to those like me who like reductions.

But this attractive view threatens to be heretical. Christ’s body is here on earth in the Eucharist, as well as in heaven in the more normal way for a body to be. But while the body is surely visible in heaven and interacts with Mary and any other embodied persons in heaven, it does not interact physically with anything on earth. Granted, there is spiritual interaction: Christ’s presence in the Eucharist is a means of grace to recipients. But that probably isn’t the sort of interaction that would ground spatial location.

There is, however, a way to modify the causal reduction of location that handles the case of the Eucharist. Actual causal interactions do not seem to be enough to ground location. The reduction very likely needs needs dispositional causal interactions that typically ground causal counterfactuals like:

  1. If Parisians were to shine a flashlight into that dark alley, they’d see me.

However, dispositions can be masked. For instance, sugar is still soluble even if God has promised to miraculously keep it from dissolving when it is placed in water. In such a case, the sugar still has the disposition to dissolve in water, but fails to ground the counterfactual:

  1. The lump would be dissolved were it placed in water.

We might, thus, suppose that when the Mass is being celebrated in Waco, Christ comes to have the dispositional causal properties that would ordinarily be contitutive of his being present in Waco, such as the disposition to reflect Texan photons, and so on. But by miracle, all these dispositions are masked and do not result in actual causal interaction. The unmasked dispositions are those corresponding to spiritual interaction.

Here’s an interesting lesson. The kind of causal-reductive view of location that I’ve just considered seems to be one of the least transsubstantiation-congenial views of location. But, nonetheless, the transsubstantiation can still be made sense of on that view when the view is refined. This gives us evidence that transsubstantiation makes sense.

And we can now go back to the story of my being in Waco while interacting in Paris. The story was underspecified. I didn’t say whether I have the dispositions that go with being in Waco. If I do, these dispositions are being miraculously masked. But they may be enough to make me count as being in Waco. So on the story as I’ve told it, I might actually be both in Waco and in Paris.

Final question: Can external temporal location be similarly causally grounded? (Cf. this interesting paper.)

Thursday, May 25, 2017

Can destruction be good for something?

It is good for a mouse to occupy a limited region of space: if a mouse were cat-sized, it would be incapable of excellent engagement in many of its characteristic behaviors (scurrying around in narrow passages). If time is relevantly like space, we would expect that there be things for which it is good that they occupy a limited interval of time--i.e., it is good for them to die, or at least good for them to die in a particular way. (It is good for a mouse to be spatially bounded--but only certain kinds of spatial bounds, those delimited by healthy skin and fur, are good for the mouse.)

One category of things whose destruction is a part of their flourishing is things whose purpose is to give rise to something else. For instance, sperm and egg are destroyed in giving rise to a zygote, and that it is their flourishing to be destroyed in this manner. But that's not the only category. It may be a part of the flourishing of a skin cell that it perish in order to make way for a newer skin cell. Both of these categories are subsumed in the category of things directed at the good of something other than themselves.

But I think human beings are not like that.

Tuesday, May 23, 2017

Natural Law decision theory

One of the things I’ve learned from the St Petersburg Paradox and Pascal’s Wager is that we are rationally required to have attitudes to risk that significantly discount tiny chances of benefits, rather than to maximize expected utility. This requirement is rational because failure to have such attitudes to risk makes one subject to two-person diachronic Dutch Books. But it is also clearly irrational to significantly discount large chances of benefits.

But where are the lines to be drawn? Maybe it’s not worth enduring an hour of sitting on an uncomfortable chair for a 1/101000 chance of any finite length of bliss, but enduring an hour of sitting in such a chair for a 45% chance of 1000 years of bliss is worthwhile. As long as we thought the decisions were to be made on the basis of expected utility, we could have said that the lines are to be non-arbitrarily drawn by multiplying probabilities and utilities. But that fails.

It is possible, I suppose, that there is a metaphysically necessary principle of rationality that says where the line of the negligibility of chances is to be drawn. Perhaps an hour in the uncomfortable chair for a 1/101000 chance of a finite benefit cannot possibly be worthwhile, but for a 1/106 chance of a large enough finite benefit it is worth it, and there is a cut-off precisely at π ⋅ 10−9. But the existence of any such a metaphysically necessary cut-off is just as implausible as it is to think that the constants in the laws of nature are metaphysically necessary.

(Vagueness is of no help. For even if the cut-off is vague, the shape—vague or exact—of the vagueness profile of the cut-off will still look metaphysically contingent.)

One could leave it to the individual. Perhaps rationality requires each individual to have a cut-off but where the cut-off lies is up to the individual. But rationality also places constraints on that cut-off: the person who is unwilling to sit in an uncomfortable chair for an hour for a 45% chance of 1000 years of bliss is irrational. (I deliberately made it 45%. The cut-off isn’t at 1/2, which would be satisfyingly non-arbitrary.) And where the constraints on the cut-off lie is itself something to be explained, and again it is implausible that it is metaphysically necessary.

In morals, we also have similar cut-off phenomena. It is morally wrong to put someone in prison for life for stealing an ordinary book, while a week of community service is morally permissible. Whence the cut-off? The problem in both cases comes from two features of the situation:

  1. We have a parameter that seems to have a normative force independent of our minds.

  2. That parameter appears to be contingent.

Utilitarianism provides an elegant answer, but no analog of that answer seems to apply in the rationality/risk case. Kantianism is out of luck. Divine command theory provides an answer, but one whose analogue in the case of rationality is quite implausible: it is irrational to be unwilling to sit in the uncomfortable chair for the 45% chance of the great benefit, rather than forbidden by God.

Natural Law, on the other hand, provides a framework for both the moral and the rational cases by saying that the parameter necessarily comes from our nature. Our nature is independent of our minds, and hence we do justice to (1). But while it is presumably not a contingent fact that we have the nature we do, it is a contingent fact that the persons that inhabit the world have the natures they do. Humans couldn’t have these normative risk or moral parameters other than they do, but there could easily have existed non-humans somewhat similar to us who did. The explanation is parallel to the Kripkean explanation of the seeming arbitrariness of water having two hydrogen atoms. Water couldn’t have had a different number of hydrogen atoms, but something similar to water could have had.

More and more, I think something like Natural Law is a powerful framework in normative areas outside of what is normally construe to be moral theory: in decision theory and epistemology. (I hedge with the “normally construe”, because I happen to think that both decision theory and epistemology are branches of moral theory.)

Wednesday, May 17, 2017

Could God be divinity?

Here's a plausible thesis:

  1. If it is of x's essence to be F, then Fness is prior to x.
This thesis yields a fairly standard argument against the version of divine simplicity which identifies God with the property of divinity. For if God is divinity, then divinity is prior to divinity by (1), which is absurd.

But (1) is false. For, surely:

  1. It is of a property's essence to be a property.
But propertyhood is a property, so it is of propertyhood's essence to be a property, and so propertyhood is prior to propertyhood if (1) is true, which is absurd. So, given (2), we need to reject (1), and this argument against the God=divinity version of divine simplicity fails.

What else might properties do?

Suppose that we think of properties as the things that fulfill some functional roles: they are had in common by things that are alike, they correspond to fundamental predicates, etc. Then there is no reason to think that these functional roles are the only things properties do. It is prima facie compatible with fulfilling such functional roles that a property do many other things: it might occupy space, sparkle, eat or think.

Can we produce arguments that the things that fulfill the functional roles that properties are defined by cannot occupy space, sparkle, eat or think? It is difficult to do so. What is it about properties that rules out such activity?

Here's one candidate: necessity. The functional roles properties satisfy require properties to exist necessarily. But all things that occupy space are contingent. And all things that sparkle or eat also occupy space. So no property occupies space, sparkles or eats. (Yes, this has nothing to say about thinking.) Yeah, but first of all it's controversial that all properties are necessary. Many trope theorists think that typical tropes are both contingent and properties. Moreover, it may be that my thisness is a property and yet as contingent as I am. Second, it is unclear that everything that occupies space has to be contingent. One might argue as follows: surely, for any possible entity x, it could be that all space is vacant of x. But it does not follow that everything that occupies space has to be contingent. For we still have the epistemic possibility of a necessary being contingently occupying a region space. Christians, for instance, believe that the Second Person of the Trinity contingently occupied some space in the Holy Land in the first century--admittedly, did not occupy it qua God, but qua human, yet nonetheless did occupy it--and yet the standard view is that God is a necessary being. (Also, God is said to be omnipresent; but we can say that omnipresence isn't "occupation" of space, or that all-space isn't a region of space.)

So the modal argument isn't satisfactory. We still haven't ruled out a property's occupying space, sparkling or eating, much less thinking. In general, I think it's going to be really hard to find an argument to rule that out.

Here's another candidate: abstractness. Properties are abstract, and abstracta can't occupy space, sparkle, eat or think. But the difficulty is giving an account of abstracta that lets us be confident both that properties are abstract and that abstract things can't engage in such activities. That's hard. We could, for instance, define abstract things as those that do not stand in spatiotemporal relations. That would rule out occupying space, sparkling or eating--but the question whether all properties are abstracta would now be as difficult as the question whether a property can occupy space. Likewise, we could define abstract things as those that do not stand in causal relations, which would rule out sparkling, eating and thinking, but of course anybody who is open to the possibility that properties can do these activities will be open to properties standing in causal relations. Or we could define abstractness by ostension: abstract things are things like properties, propositions, numbers, etc. Now it's clear that properties are abstracta, but we are no further ahead on the occupying space, sparkling, eating or thinking front--unless perhaps we can make some kind of an inductive argument that the other kinds of abstracta can't do these things, so neither can properties. But whether propositions or numbers can do these things is, I think, just as problematic a question as whether properties can.

All in all, here's what I think: If we think of the Xs (properties, propositions, numbers, etc.) as things that fulfill some functional roles, it's going to be super-hard to rule out the possibility that some or all Xs do things other than fulfilling these functional roles.

For more related discussion, see this old contest.

Tuesday, May 16, 2017

Pascal's Wager and the bird-in-the-hand principle

My thinking about the St Petersburg Paradox has forced me to reject this Archimedean axiom (not the one in the famous representation theorem):

  1. For any finite utility U and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.
Roughly speaking, one must reject (1) on pain of being subject to a two-player Dutch Book. But rejecting (1) is equivalent to affirming:
  1. There is a finite utility U and a non-zero probability ϵ > 0, such that no gamble that offers a probability ϵ of getting some finite benefit is better than certainty of U.
With some plausible additional assumptions (namely, transitivity, and that the same non-zero probability of a greater good is better than a non-zero probability of a lesser one), we get this bird-in-the-hand principle:
  1. There is a finite utility U and a non-zero probability ϵ > 0, such that for all finite utilities V, the certainty of U is better than a probability ϵ of V.
Now, Pascal's Wager, as it is frequently presented, says that:
  1. Any finite price is worth paying for any non-zero probability of any infinite payoff.
By itself, this doesn't directly violate the bird-in-the-hand principle, since in (3), I said that V was finite. But (4) is implausible given (3). Consider, for instance, this argument. By (3), there is a finite utility U and a non-zero probability ϵ > 0 such that U is better than an ϵ chance at N days of bliss for every finite N. A plausible limiting case argument suggests that then U is at least as good as an ϵ chance at an infinite number of days of bliss, contrary to (4)--moreover, then U+1 will be better than an ϵ chance at an infinite number of days of bliss. Furthermore, in light of the fact that standard representation theorem approaches to maximizing expected utility don't apply to infinite payoffs, the natural way to argue for (4) is to work with large finite payoffs and apply domination (Pascal hints at that: he gives the example of a gamble where you can gain "three lifetimes" and says that eternal life is better)--but along the way one will violate the bird-in-the-hand principle.

This doesn't, however, destroy Pascal's Wager. But it does render the situation more messy. If the probability ϵ of the truth of Christianity is too small relative to the utility U lost by becoming a Christian, then the bird-in-the-hand principle will prohibit the Pascalian gamble. But maybe one can argue that little if anything is lost by becoming a Christian even if Christianity is false--the Christian life has great internal rewards--and the evidence for Christianity makes the probability of the truth of Christianity not be so small that the bird-in-the-hand principle would apply. However, people's judgments as to what ϵ and U satisfy (2) will differ.

Pleasantly, too, the bird-in-the-hand principle gives an out from Pascal's Mugger.

Friday, May 12, 2017

More on St Petersburg

I’ve been thinking about what assumptions generate the St Petersburg paradox. As stated, the paradox depends on the assumption that we should maximize expected utility, an assumption that will be rejected by those who think risk aversion is rational.

But one can run the St Petersburg paradox without expected utility maximization, and in a context compatible with risk aversion. Suppose finite utilities can be represented by finite real numbers. Assume also:

  1. Domination: If a betting portfolio B is guaranteed to produce at least as good an outcome as A no matter what, then B is at least as good as A.

  2. Archimedeanism: For any finite utility U and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.

  3. Transitivity: If C is better than B and B is at least as good as A, then C is better than A.

(Note: For theistic reasons, one might worry about Construction when the Vi are very negative, but we can restrict Construction to positive finite utilities if we add the assumption in Archimedeanism that V can always be taken to be positive.)

For, given these assumptions, one can generate a gambling scenario that has only finite utilities but that is better than the certainty of any finite utility. Proceed as follows. For each positive integer n, let Vn be any finite utility such that probability 1/2n of Vn is better than certainty of n units of utility (this uses Archimedeanism; the apparent use of the Axiom of Choice can be eliminated by using the other axioms, I think) and Vn ≥ Vn − 1 if n > 1. Toss a fair coin until you get heads. Let your payoff be Vn if it took n tosses to get to heads.

Fix any finite utility U. Let n be a positive integer such that U < n. Then the gambling scenario offers a probability of 1/2n of getting at least Vn, so by Domination, Transitivity and the choice of Vn, it is better than U.

And the paradoxes in this post apply in this case, too.

If we have expected utility maximization, we can take Vn = 2n and get the classic St Petersburg paradox.

Given the plausibility of Domination and Transitivity, and the paradoxes here, it looks like the thing to reject is Archimedeanism. And that rejection requires holding that there is a probability ϵ so small and finite utility U so large that no finite benefit with that probability can outweigh U.

Wednesday, May 10, 2017

Teleology and the direction of time

It would be depressing to think that one will never swim as fast as one is swimming today. But it would uplifting to think that that one has never swum as fast as one is swimming today.

I used to think the direction of time was defined by the predominant direction of causation. That may be the case, but if one takes humanistic cases like the above as central, one might think that perhaps the predominant direction of teleology is a better way to define the direction of time. Of course, telê are there to be achieved, and so the direction of teleology needs to fit well with the direction of causation, at least in the case of things that concern us. Moreover, there is some reason to think that teleology is behind all causation—causation aims at an effect.