The unthinkable and the ineffable

Suppose that Alice right now thinks about some fact F and no other fact. Then we can stipulate that “Xyzzies” is a sentence whose content is that very fact which Alice is thinking. Thus:

1. If a linguistically identifiable person can think about some fact F to the exclusion of other facts at a linguistically identifiable time, then F can be expressed in a language.

It does not, however, follow that every fact can be expressed in a language. For it’s epistemically possible that there is a fact F such that a person can only think about F if the person is simultaneously thinking about G and H as well, and there may be no way for us to distinguish F from G and H in such a way as to stipulate a term for it.

This may seem like a pretty remote possibility, but I think it’s pretty plausible. There could be some fact F that only God can think. But presumably any fact has infinitely many logical consequences. But since God is inerrant and necessarily thinks all facts, necessarily if God thinks F, he thinks all the infinitely many logical consequences of F as well. And it could well be that we have no way of distinguishing F from some of its logical consequences in such a way that we could delineate F.

So it is possible to accept (1) while holding that some thinkable facts are ineffable.

However, plausibly any fact thinkable by a human can be thought by the human in a specifiably delineated way (the primary fact thought about at t₁, etc.). Thus our thought cannot exceed the possibilities of our language, since for anything we can think we could stipulate that “Xyzzies” means that. (Though, of course, our thought can (and sometimes does) exceed the actualities of our language.) Thus:

2. The humanly ineffable is humanly unthinkable.

Nonetheless, we might make a distinction between two ways of extending human language. A weak extension is one that can be introduced solely in terms of current human language. Stipulations in mathematics are like that: we explain what “continuous” is using prior vocabulary like “limit”. A strong extension is one that requires something extralinguistic, such as ostension to a non-linguistic reality.

3. There are things that are humanly thinkable that are only expressible using a strong extension of human language.

Beyond us

A being that does not represent the world has no conception of what representation might be like, since the being has no conceptions.

A being that lacks consciousness has no conception of what consciousness might be like. The being might have intentionality (our unconscious thoughts, after all, have intentionality), and so might have the contentful thought that there can be beings that have some crucial mental quality that goes beyond the unconscious being’s mentality.

A being that lacks will presumably has no consciousness of what rational will or responsibility might be like. Again, the being might have the concept of beings with “something more” in causation of activity by means of thought.

The distinctions between non-representing and representing, unconscious and conscious, and involuntary and voluntary involve immense qualitative and value gaps. In each of the three cases, we humans exemplify the higher of the two options. At the same time, we are not alone in all these on earth. We share representation with all living things, I suspect. We share consciousness with many animals. But responsibility, I suspect, is ours alone.

I find it implausible to think that we are at the qualitative apex of the space of valuable possibilities. It seems quite likely to me that there could be beings that differ from us in further fundamental valuable qualities in such a way that we are on the lower end, and if we were to meet these beings, we would be unable to grasp what they have which we lack, though we might on testimony, or maybe even empirical observation of behavior, conclude that there is such a thing.

In fact, I suspect there are infinitely many such distinctions, and that God is beyond the higher side of all of them.

In heaven, might we be raised to have the further higher levels? Maybe, but maybe not. However, the mere epistemic possibility of us being gradually raised to acquire infinitely many further such irreducible values is enough to undercut any “argument from boredom” against eternal heavenly life.

Assuming there are infinitely many more such non-V and V pairs, I wonder what this infinity is. Does it have a cardinality?

Open-mindedness and epistemic thresholds

Fix a proposition p, and let T(r) and F(r) be the utilities of assigning credence r to p when p is true and false, respectively. The utilities here might be epistemic or of some other sort, like prudential, overall human, etc. We can call the pair T and F the score for p.

Say that the score T and F is open-minded provided that expected utility calculations based on T and F can never require you to ignore evidence, assuming that evidence is updated on in a Bayesian way. Assuming the technical condition that there is another logically independent event (else it doesn’t make sense to talk about updating on evidence), this turns out to be equivalent to saying that the function G(r) = rT(r) + (1−r)F(r) is convex. The function G(r) represents your expected value for your utility when your credence is r.

If G is a convex function, then it is continuous on the open interval (0,1). This implies that if one of the functions T or F has a discontinuity somewhere in (0,1), then the other function has a discontinuity at the same location. In particular, the points I made in yesterday’s post about the value of knowledge and anti-knowledge carry through for open-minded and not just proper scoring rules, assuming our technical condition.

Moreover, we can quantify this discontinuity. Given open-mindedness and our technical condiiton, if T has a jump of size δ at credence r (e.g., in the sense that the one-sided limits exist and differ by y), then F has a jump of size rδ/(1−r) at the same point. In particular, if r > 1/2, then if T has a jump of a given size at r, F has a larger jump at r.

I think this gives one some reason to deny that there are epistemically important thresholds strictly between 1/2 and 1, such as the threshold between non-belief and belief, or between non-knowledge and knowledge, even if the location of the thresholds depends on the proposition in question. For if there are such thresholds, then now imagine cases of propositions p with the property that it is very important to reach a threshold if p is true while one’s credence matters very little if p is false. In such a case, T will have a larger jump at the threshold than F, and so we will have a violation of open-mindedness.

Here are three examples of such propositions:

• There are objective norms

• God exists

• I am not a Boltzmann brain.

There are two directions to move from here. The first is to conclude that because open-mindedness is so plausible, we should deny that there are epistemically important thresholds. The second is to say that in the case of such special propositions, open-mindedness is not a requirement.

I wondered initially whether a similar argument doesn’t apply in the absence of discontinuities. Could one have T and F be openminded even though T continuously increases a lot faster than F decreases? The answer is positive. For instance the pair T(r) = e^10r and F(r) =  − r is open-minded (though not proper), even though T increases a lot faster than F decreases. (Of course, there are other things to be said against this pair. If that pair is your utility, and you find yourself with credence 1/2, you will increase your expected utility by switching your credence to 1 without any evidence.)

Knowledge and anti-knowledge

Suppose knowledge has a non-infinitesimal value. Now imagine that you continuously gain evidence for some true proposition p, until your evidence is sufficient for knowledge. If you’re rational, your credence will rise continuously with the evidence. But if knowledge has a non-infinitesimal value, your epistemic utility with respect to p will have a discontinuous jump precisely when you attain knowledge. Further, I will assume that the transition to knowledge happens at a credence strictly bigger than 1/2 (that’s obvious) and strictly less than 1 (Descartes will dispute this).

But this leads to an interesting and slightly implausible consequence. Let T(r) be the epistemic utility of assigning evidence-based credence r to p when p is true, and let F(r) be the epistemic utility of assigning evidence-based credence r to p when p is false. Plausibly, T is a strictly increasing function (being more confident in a truth is good) and F is a strictly decreasing function (being more confident in a falsehood is bad). Furthermore, the pair T and F plausibly yields a proper scoring rule: whatever one’s credence, one doesn’t have an expectation that some other credence would be epistemically better.

It is not difficult to see that these constraints imply that if T has a discontinuity at some point 1/2 < r_K < 1, so does F. The discontinuity in F implies that as we become more and more confident in the falsehood p, suddenly we have a discontinuous downward jump in utility. That jump occurs precisely at r_K, namely when we gain what we might call “anti-knowledge”: when one’s evidence for a falsehood becomes so strong that it would constitute knowledge if the proposition were true.

Now, there potentially are some points where we might plausibly think that epistemic utility of having a credence in a falsehood takes a discontinuous downward jump. These points are:

• 1, where we become certain of the falsehood

• r_B, the threshold of belief, where the credence becomes so high that we count as believing the falsehood

• 1/2, where we start to become more confident in the falsehood p than the truth not-p

• 1 − r_B, where we stop believing not-p, and

• 0, where the falsehood p becomes an epistemic possibility.

But presumably r_K is strictly between r_B and 1, and hence r_K is no one of these points. Is it plausible to think that there is a discontinuous downward jump in epistemic utility when we achieve anti-knowledge by crossing the threshold r_K in a falsehood.

I am incline to say not. But that forces me to say that there is no discontinuous upward jump in epistemic utility once we gain knowledge.

On the other hand, one might think that the worst kind of ignorance is when you’re wrong but you think you have knowledge, and that’s kind of like the anti-knowledge point.

Aristotle and Aquinas' Third Way

Aristotle seems to have thought that the earth and the species inhabiting it are eternal. This seems extremely implausible for reasons that should have been available to Aristotle.

It is difficult to wipe out a species, but surely not possible: all it takes is to kill each of the finitely many individuals. Given a species s that cannot have more than n members, and given a long enough time, we would expect there to be a very high probability that all the members of s would have died out during some hour due to random events. Given any finite number of species each with a bound on how many members it can have, and given a long enough time, we would expect with very high probability that all the members would die off.

Now there is a finite limit on how many species there are on earth (as Aristotle knew, the earth is finite), and a finite limit on how many members the species can have (again, the earth is finite). So we should have expected all the species that existed some long amount of time ago to have died out.

The above provides an argument that if the world is eternal, new species can arise. For if new species can’t arise and the world is eternal, then by now there should have been no species left.

How could Aristotle have gotten out of this worry without rejecting his thesis about the eternity of the earth?

One way be to suppose a powerful protector of our ecosystem that would make sure that the species-destroying random events never happen. This protector would either itself have to be sufficiently powerful that it would not be subject to the vicissitudes of chance, or there would have to be an infinite (probably uncountably infinite!) number of such protectors.

Another option would be for Aristotle to reject his thesis that there is only one earth (which was based on theory of gravitation as attraction to the center of the universe: if there were more than one earth they would have both collapsed into the center of the universe by now).

If there were infinitely many earths, then it’s perhaps not so crazy to think that some earth would have lucked out and not had its species die out. Of course, this would not only require Aristotle to reject his thesis that there is only one earth, but also the finitist thesis that there cannot be an infinite number of co-actual things. (Interestingly, given the plausibility that any given species has probability one of dying out given infinite time, and given the countable additivity of probabilities, this way out would require not merely infinitely many earths, but an uncountable infinity of earths. Assuming an Archimedean spacetime for our universe, it would require a multiverse.)

In any case, Aristotle’s commitment to new species not coming into existence (or at least new species of interesting critters; he may be OK with worms coming into existence) is in tension with what he says about the earth’s eternity.

Change and matter

Aristotle’s positing matter is driven by trying to respond to the Parmenidean idea that things can’t come from nothing, and hence we must posit something that persists in change, and that is matter.

But there two senses of “x comes from nothing”:

1. x is uncaused

2. x is not made out of pre-existing materials.

If “x comes from nothing” in the argument means (1), the argument for matter fails. All we need is a pre-existing efficient cause, which need not be the matter of x.

Thus, for the argument to work, “x comes from nothing” must mean (2). But now here is a curious thing. From the middle ages to our time, many Aristotelians are theists, and yet still seem to be pulled by Aristotle’s argument for matter. But if “x comes from nothing” means (2), then theism implies that it is quite possible for something to come from nothing: God can create it ex nihilo.

There are at least two possible responses from a theistic Aristotelian who likes the argument for matter. The first response is that only God can make things come from nothing in sense (2), and hence things caused to exist by finite causes (even if with God’s cooperation) cannot come from nothing in sense (2). But there plainly are such things all around us. So there is matter.

Now, at least one theistic Aristotelian, Aquinas, does explicitly argue that only God can create ex nihilo. But the argument is pretty controversial and depends on heavy-duty metaphysics, about finite and infinite causes. It is not just the assertion of a seemingly obvious Parmenidean “nothing comes from nothing” principle. Thus at least on this response, the argument for matter becomes a lot more controversial. (And, to be honest, I am not convinced by it.)

The second and simpler response is to say that it’s just an empirical fact that there are things in the world that don’t come from nothing in sense (2): oak trees, for example. Thus there in fact is matter. This response is pretty plausible, but can be questioned: one might say that we have continuity of causal powers rather than any matter that survives the generation.

Finally, it’s worth noting that I suspect Aristotle misunderstands the Parmenidean argument, which is actually a very simple reductio ad absurdum:

3. x came into existence.
4. If x came into existence, then x did not exist.
5. So, x did not exist.
6. But non-existence is absurd.

The crucial step here is (6): the Parmenidean thinks the very concept of something not existing is absurd (presumably because of the Parmenidean’s acceptance of a strong truthmaker principle). The argument is very simple: becoming presupposes the truth of some past-tensed non-existence statements, while non-existence statements are always false. Aristotle’s positing matter does nothing to refute this Parmenidean argument. Even if we grant that x’s matter pre-existed, it’s still true that x did not exist, and that’s all Parmenides needs. Likewise, Aristotle’s famous actuality/potentiality distinction doesn’t solve the problem. Even if x was pre-existed by a potentiality for existence, it’s still true that x wasn’t pre-existed by x—that would be a contradiction.

To solve Parmenides’ problem, however, we do not need to posit matter or potentiality or anything like that. We just need to reject the idea that negative existential statements are nonsensical. And Aristotle expressly does reject this idea: he says that a statement is true provided it says of what is that it is or of what is not that it is not. Having done that, Aristotle should take himself as done with Parmenides’ problem of change.

More on the centrality of morality

I think we can imagine a species which have moral agency, but moral agency is a minor part of their flourishing. I assume wolves don’t have moral agency. But now imagine a species of canids that live much like wolves, but every couple of months get to make a very minor moral choice whether to inconvenience the pack in the slightest way—the rest is instinct. It seems to me that these canids are moral agents, but morality is a relatively minor part of their flourishing. The bulk of the flourishing of these canids would be the same as that of ordinary wolves.

Aristotle argued that the fact that rationality is how we differ from other species tells us that rationality is what is central to our flourishing. The above thought experiment shows that the argument is implausible. Our imaginary canids could, in fact, be the only rational species in the universe, and their moral agency or rationality (with Aristotle and Kant, I am inclined to equate the two) is the one thing that makes them different from other canids, but yet what is more important to their flourishing is what they have in common with other canids.

At the same time, it would be easy for an Aristotelian theorist to accommodate my canids. One needs to say that the form of a species defines what is central to the flourishing, and in my canids, unlike in humans, morality is not so central. And one can somehow observe this: rationality just is clearly important to the lives of humans in a way in which it’s not so much these canids.

In this way, I think, the Aristotelian may have a significant advantage over a Kantian. For a Kantian may have to prioritize rationality in all possible species.

In any case, we should not take it as a defining feature of morality that it is central to our flourishing.

One might wonder how this works in a theistic context. For humans, moral wrongdoing is also sin, an offense against a loving infinite Creator. As I’ve described the canids, they may have no concept of God and sin, and so moral wrongdoing isn’t seen as sin by them. Could you have a species which does have a concept of God and sin, but where morality (and hence sin) isn’t central to flourishing? Or does bringing God in automatically elevate morality to a higher plane? Anselm thought so. He might have been right. If so, then the discomfort that one is liable to feel at the idea of a species of moral agents where morality is not very important could be an inchoate grasp of the connection between God and morality.

The overridingness of morality and Double Effect

You’ve been imprisoned in a cell with a torture robot. The cell is locked by a combination lock, and your estimate is that you will be able to open it in a week. If the torture robot is left running, it will stimulate your pain center, causing horrible pain but no lasting damage, and not slowing down your escaping at all. An infallible oracle reveals to you that if you disable the robot, through a random confluence of events this will affect your character in such a way that in a year you will be 0.1% less patient for the rest of your life than you would otherwise be.

Now, sometimes, a small difference in the degree of a virtue could make a big difference. For instance, perhaps, you will one day be in a position where an extremely arduous task will need to be done to save someone’s life, and you just barely have enough patience for it, so that if you were 0.1% less patient, you wouldn’t do it. You ask the oracle whether something like this will happen if you turn off the robot. The oracle replies: “No, it’s just that you will be 0.1% more annoyed whenever you engage in an arduous task, but that’s never going to push you past any significant threshold—you’re not going to blow up in a big way at your child, or neglect a duty, or anything like that.”

It seems obviously reasonable to disable the robot. Thus, enormous short-term hedonic considerations can win out over tiny long-term virtue considerations. It is thus not the case that considerations of virtue always beat hedonic considerations.

What are we to make, then, of the deep insight—perhaps the most important insight in the history of Western philosophy—about the primacy of morality over other considerations?

Two things. First, moral considerations tend to be much more important than non-moral considerations.

Second, we should never do what is morally wrong, no matter what the price for avoiding it, and no matter how small the wrong. But there is a difference between doing what is morally wrong and doing something morally permissible that makes one less virtuous.

Here is a second case. You and an innocent stranger are in the cell. The robot is set to torture the stranger. The oracle now reveals to you that right after the escape, you will forget the last two weeks of your life, and your life will go the same way whether you disabled the robot or not, with exactly one morally relevant exception: if you have chosen to disable the robot, then one day, feeling peckish and having forgotten your wallet, you will culpably steal a candybar from a cornerstore.

It seems obvious that you should disable the robot, despite the fact that doing so leads to your doing a minor moral wrong. The point isn’t that disabling the robot justifies stealing the candybar—at the time that you steal it, you will have forgotten all about the robot, so there is no justification. The point is that even though you should never do wrong that a good might come of it, nonetheless sometimes for the sake of a great good it is permissible to do something that you know will lead to your later doing something impermissible.

Sometimes theologians have incautiously said things like that the smallest sin outweighs the greatest evil that is not a sin. I think this is incorrect. But what is correct is that you shouldn’t commit the smallest sin for the sake of the greatest good. However, the Principle of Double Effect applies to future sins: you can foresee but not intend that if you perform a certain action—turning off the robot, say—you will commit a future sin.

The badness of non-intentional harming

Consider a trolley problem where on both tracks there is exactly one innocent stranger. Alice is driving the trolley. If she does nothing, the trolley will head down the left track. But the right track will get Alice to her destination three minutes sooner. Alice redirects.

It seems that Alice did something wrong. Yet, why? We can say that she intended to save the person on the left track and get to her destination faster, and did not intend to kill the person on the right track. What went wrong?

One option is this. In the proportionality condition on Double Effect, we need that the outcome chosen have a significantly better consequence than the alternative, and three minutes (normally) is not significant.

But that’s probably not right. There are times when it is permissible to redirect a trolley even when the outcome is a bit worse. For instance, suppose that we have a trolley setup with one person on each track, but things are such that if the trolley hits the person on the right track, the death will be a bit more painful. The trolley is controlled by the person on the right track. It seems obvious that the person on the right track is permitted to redirect the trolley to the right even though the outcome is a bit worse.

Maybe the issue is this. Even though it’s not always wrong to become the non-intentional cause of a grave harm to someone, we have moral reason to avoid becoming such a cause. This fits with our intuitions: we feel really bad when we become such a cause. Murray Leinster’s first novel Murder Madness is all about the horror of a drug that makes one involuntarily kill people (I won’t recommend the novel because of a number of pieces of outrageous racism).

This makes sense from an Aristotelian point of view. For a social organism, helping members of the group is a part of flourishing. This is true for animals that are not moral agents. A meerkat sentinel that saves the group by warning of a danger is thereby flourishing. This is even true in the case of non-intentional cooperative activity. A slime mold that, as part of a stalk, enables reproduction by slime molds that are part of the fruiting body is thereby flourishing. It makes sense, thus, to think that for social organisms harming members of the group is contrary to flourishing whether or not one is morally responsible for the harm, and even when the harm is one that one is not intending.

Scientific realism about mass

While I’ve grown up as a scientific realist, and been trained as one as a philosophy graduate student, and I suppose I still identify as one, I’ve been finding it more difficult to say what scientific realism claims.

For instance, what does it mean to be a realist about mass in a Newtonian context? A naive thought is that for each physical object, there is a positive real number, the mass of the object, which mathematically enters into the laws of nature such as F = ma and F = Gm₁m₂/r². But that seems to commit one to there being some odd objective facts, such as to which objects have the property that the square of their masses is less than their mass—a property that barely seems to make any sense, since normally in physics, we don’t compare masses with squares of masses, as they are measured in different units.

A more sophisticated thought is that there is a determinable mass, and a family of determinates, with various mathematical relations between them, with the family isomorphic with the positive real numbers with respect to the relations, but without necessarily a single isomorphism being privileged. But this more sophisticated thought is much more philosophy than physics: physicists hypothesize entities like forces and particles and the like, but not such entities like determinables and determinates. Indeed, this approach commits one to the denial of nominalism, and surely realism about mass in a Newtonian context shouldn’t commit one to such a controversial metaphysical thesis.

Is there some alternative? Maybe, but I don’t know.

Hyperreal worlds

In a number of papers, I argued against using hyperreal-valued probabilities to account for zero probability but nonetheless possible events, such as a randomly thrown dart hitting the exact center of the target, by assigning such phenomena non-zero but infinitesimal probability.

But it is possible to accept all my critiques, and nonetheless hold that there is room for hyperreal-valued probabilities.

Typically, physicists model our world’s physics with a calculus centered on real numbers. Masses are real numbers, wavefunctions are functions whose values are pairs of real numbers (or, equivalently, complex numbers), and so on. This naturally fits with real-valued probabilities, for instance via the Born rule in quantum mechanics.

However, even if our world is modeled by the real numbers, perhaps there could be a world with similar laws to ours, but where hyperreal numbers figure in place of our world’s real ones. If so, then in such a world, we would expect to have hyperreal-valued probabilities. We could, then, say that whether chances are rightly modeled with real-valued probabilities or hyperreal-valued probabilities depends on the laws of nature.

This doesn’t solve the problems with zero probability issues. In fact, in such a world we would expect to have the same issues coming up for the hyperreal probabilities. In that world, a dartboard would have a richer space of possible places for the dart to hit—a space with a coordinate system defined by pairs of hyperreal numbers instead of pairs of real numbers—and the probability of hitting a single point could still be zero. And in our world, the probabilities would still be real numbers. And my published critiques of hyperreal probabilities would not apply, because they are meant to be critiques of the application of such probabilities to our world.

There is, however, a potential critique available, on the basis of causal finitism. Plausibly, our world has an infinite number of future days, but a finite past, so on any day, our world’s past has only finitely many days. The set of future days in our world can be modeled with the natural numbers. An analogous hyperreal-based world would have a set of future days that would be modeled with the hypernatural numbers. But because the hypernatural numbers include infinite numbers, that world would have days that were preceded by infinitely (though hyperfinitely) many days. And that seems to violate causal finitism. More generally, any hyperreal world will either have a future that includes a finite number of days or one that includes days that have infinitely many days prior to them.

If causal finitism is correct, then “hyperreal worlds”, ones similar to ours but where hyperreals figure where in our our world we have reals, must have a finite future, unlike our world. This is an interesting result, that for worlds like ours, having real numbers as coordinates is required in order to have both causal finitism true and yet an infinite future.

Causation and contingency

A correspondent yesterday reminded me of a classic objection to the “inductive” approach to the causal principle that all contingent things have causes in the context of cosmological arguments. As I understand the objection, it goes like this:

1. Granted, we have good reason to think that all the contingent things we observe do have causes. However, all these causes are contingent causes, and so we have equally good inductive support to think that all contingent things have contingent causes. Thus, to extend this reasoning to conclude that the cosmos—the sum total of all contingent things—has a cause is illegitimate, since the cosmos cannot have a contingent cause on pain of circularity.

An initial response is that (1) as it stands appears to rely on a false principle of inductive reasoning:

2. Suppose that all observed Fs are Gs, and that all observed Fs are also Hs. Then we have equally good inductive support for the hypothesis that all Fs are Hs as that all Fs are Gs.

But (2) is false. All observed emeralds are green and all observed emeralds are grue, where an emerald is grue if it is green and observed before 2100 or it is blue and not observed before 2100. It is reasonable to conclude that all emeralds are green but not that they are all grue. Or even more simply, from the facts that all observed electrons are charged and all observed electrons are observed, it is reasonable to conclude that all electrons are charged but not that all electrons are observed.

Nonetheless, this response to (1) does not seem entirely satisfying. The predicate “has a contingent cause” seems to be projectible, i.e., friendly to induction, in a way in which “is grue” or “is observed” are not.

Still, I think there is something more to be said for this response to (1). While “has a contingent cause” is not as obviously non-projectible as “is observed”, it has something in common with it. We are more suspicious of inductive inferences from all observed Fs being Gs to all Fs being Gs when being G includes features that are known prior to these observations to be concommitants of observation. For instance, consider the following variant of the germ theory of disease:

3. All infectious diseases are caused by germs that are at least 500 nm in size.

Until the advent of electron microscopy, all the infectious diseases whose causes were known were indeed caused by germs at least 500 nm in size, as that is the lower limit of what can be seen with visible light. But it would not be very reasonable to have concluded at the time that 500 nm is the lower limit on the size of a disease-causing germ. Now, something similar is happening in the contingent cause case. All observable things are physical. All physical things are contingent. So being contingent is a concommitant of being observed.

Finally, there is another epistemological problem with (1). The fact that some evidence gives as good support for q as for p does not mean that q is as likely to be true as p given the evidence. For the prior probability of q might be lower than that of p. And indeed that is the case in the reasoning in (1). The prior probability that everything contingent has a contingent cause is zero, precisely for the reason stated in (1): it is impossible that everything contingent have a contingent cause! But the prior probability that everything contingent has a cause is not zero.

Two more counterexamples to utilitarianism

It’s an innocent and pleasant pastime to multiply counterexamples to utilitarianism even if they don’t add much to what others have said. Thus, if utilitarianism is true, I have to do so. :-)

Suppose you capture Hitler. Torturing him to death would appal many but, given fallen human nature, likely significantly please hundreds of millions more. This pleasure to hundreds of millions could far outweigh the pain to one. Moreover, even of those appalled by the torture, primarily only Nazis and a handful of moral saints would actually feel significant displeasure at the torture. For being appalled by an immoral action is not always unpleasant except to someone with saintly compassion—indeed there is a kind of pleasure one takes in being appalled. Normally in the case of counterexamples to utilitarianism one worries about making people more callous, the breakdown of law and order, giving a bad example to others, and so on. But the case of Hitler is so exceptional that likely the negative effects from a utilitarian point of view would be minimal if any.

One might think that an even better thing to do from the utilitarian point of view would be to kill Hitler painlessly, and then mark up his body so it looks like he was tortured to death, and publically lie about it.

Yet it is wrong to torture even Hitler, and it is wrong to lie that one has done so (especially if only for public pleasure).

Bailey's Priority Principle

Andrew Bailey formulated and defended the Priority Principle (PP), that we think our thoughts in a primary rather than inherited way. His main argument for PP is a two-thinkers argument: if I think my thoughts in an inherited way, then something else—the thing I inherit the thoughts from—thinks them as well, but there aren’t two thinkers of my thoughts. While this argument is plausible, I think it skirts around the main intuition behind the PP. That intuition is that there is something implausible about us being thinkers in a derivative way. This intuition, however, is quite compatible with there being something that derives its thoughts from us, but not so Bailey’s argument, which (unless I am missing something) equally rules out the hypothesis that we inherit our thoughts and the hypothesis that our thoughts are inherited by something else.

Is there a way to argue for PP in concert with this intuition, namely to argue that whether or not there are two thinkers of my thoughts, I am their primary thinker? Such an argument would also escape the following apparent counterexample. Social organizations can have thoughts, derivative in a complex way from their members’ thoughts. But now suppose I join a club, and everyone else resigns membership. Then the club’s opinion on matters relavant to the club’s subject matter comes to be inherited from me. So now there are two thinkers, the club and me, though I am the primary one. This case (which to be fair I am not completely sure of) is a counterexample to Bailey’s argument but not to its conclusion.

My students came up with two closely related arguments, which we might put something like this. First, among our thoughts are intentions. If these are derivative, we are puppets of the primary intender, contrary to our freedom. Second, some of our thoughts are deliberate. It is a contradiction in terms that we think deliberately and yet our deliberate thought is inherited from a prior deliberate thinker—puppetry is incompatible with deliberativeness.

These arguments do not directly show that we are always primary thinkers, so they immediately imply only a weaker version of the PP (WPP), namely that sometimes we think non-derivatively. WPP is still interesting. For instance, it rules out standard perdurantist theories on which we inherit all our thoughts from our temporal parts. Furthermore, WPP makes PP moderately likely: for it is plausible that if there is any thought-inheritance it always goes in the same direction.

That said, maybe there is some reason to accept WPP without PP. Here is one kind of case. Possessing a concept is, perhaps, a way of thinking. But given some moderate semantic externalism, sometimes we possess a concept—say, of a quark—by inheriting it from an expert. Or suppose that the extended mind thesis is true, so that we count as knowing some things because they recorded on our devices. Maybe electronic devices don’t have knowledge, so this isn’t exactly knowledge inheritance. But imagine that you train a parrot to remember all your credit card numbers (a foolish idea) and you carry the parrot with you always. Now you inherit the knowledge of the numbers (under some description common between you and the parrot, definitely not “credit card number”) from the parrot. I am dubious of the extended mind thesis, but there is no need to stick one’s neck out. WPP does justice to many of our intuitions.

Correction to "Goodman and Quine's nominalism and infinity"

In an old post, I said that Goodman and Quine can’t define the concept of an infinite number of objects using their logical resources. Allen Hazen corrected me in a comment in the specific context of defining infinite sentences. But it turns out that I wasn’t just wrong about the specific context of defining infinite sentences: I was almost entirely wrong.

To see this, let’s restrict ourselves to non-gunky worlds, where all objects are made of simples. Suppose, further, that we have a predicate F(x) that says that an object x is finite. This is nominalistically and physicalistically acceptable by Goodman and Quine’s standards: it states a physical feature of a physical object, namely its size qua made of simples. (If the simples all have some finite amount of energy with some positive minimum, F(x) will be equivalent to saying x has a finite energy.)

Now, this doesn’t solve the problem by itself. To say that an object x is finite is not the same as saying that the number of objects with some property is finite. But I came across a cute little trick to go from one to the other in the proof of Proposition 7 of this paper. The trick transposed to the non-gunky mereological setting is this. Then following two statements are equivalent in non-gunky worlds satisfying appropriate mereological axioms:

1. The number of objects x satisfying G(x) is finite.

2. There is a finite object z such that for any objects x and y with G(x) and G(y), if x ≠ y, then x and y differ inside z (i.e., there is a part of z that is a part of one object but not of the other).

To see the equivalence, suppose (2) is true. Then if z has n simples, and if x is any object satisfying G(x), then all objects y satisfying G(x) differ from x within these n simples, so there are at most 2ⁿ objects satisfying G(x). Conversely, if there are finitely many satisfiers of G, there will be a finite object z that contains a simple of difference between x and y for every pair of satisfiers x and y of G (where a simple of difference is a simple that is a part of one but not the other), and any two distinct satisfiers of G will differ inside z.

I said initially that I was almost entirely wrong. In thoroughly gunky worlds, all objects are infinite in the sense of having infinitely many parts, so a mereologically-based finiteness predicate won’t help. Nor will a volume or energy-based one, because we can suppose a gunky world with finite total volume and finite total energy. So Goodman and Quine had better hope that the world isn’t thoroughly gunky.