Kind relative predication

It is a broadly Aristotelian doctrine that many predicates apply to individuals in a kind-relative way. Call such predicates k-predicates. (We can stipulatively say that God is the sole member of a kind membership in which is identical with himself, or something like that.) For a k-predicate F, what exactly it is for an x to be F depends on what kind of an entity x is. If it is the same thing for x to be such that Fx as for y to be such that Fy, then kinds of x and y are either the same or have something in common (e.g., a higher genus).

Examples of k-predicates are easy to find. To determine whether a given individual "has legs", we first have to see what counts as legs for an individual of that kind. Thus, Peter has legs in virtue of a particular pair of limbs. Which limbs count as legs? That is defined in part by his human nature, or maybe more generally his nature as a member of Tetrapoda. What it would be for an amoeba to have legs is a different, and more poorly defined, question. What it is for a table is fairly well defined in terms of the nature of the table (we could imagine a table with four upward projecting horns at the corners; the nature of a table being to stand on solid ground on its legs, if it has any, would prevent these from counting as legs).

Whether an entity is n inches tall is even more clearly kind-relative. It depends on which axis counts as the "vertical" axis—remember that the entity might be lying on its side for much of the day.

Another kind of predicate is an r-predicate. An r-predicate is a predicate that can only be had by members of one particular (perhaps higher level) kind. Thus, "is a mammal" is an r-predicate, since it can only be had by mammals. And "is Socrates" is also an r-predicate, since it can only be had by a human.

We can perhaps form complex predicates that are neither k- nor r-predicates. Thus, "is not Socrates" is not an r-predicate (all horses and chairs, and most humans satisfy it) and may not be a k-predicate either. Though on the other hand, it may a k-predicate: maybe for non-humans, it holds in virtue of kind difference, while for humans, it holds in virtue of numerical difference within a kind.

There are also some non-contentful predicates, like "is a substance" or "is self-identical" that are neither k- nor r-predicates.

Thesis: All simple, contentful predicates are either k- or r-predicates.

I don't know if the thesis is true. There seem to be counterexamples. Having a particular shape does not seem to be a k- or r-predicate. Likewise, having a certain mass does not seem to be such, either. I suspect such apparent counterexamples can be overcome—but that may be matter for another post.

Adequacy of language

What does it mean to say that language M is at least as "adequate" as language L? One option would be to say that any proposition that L can express is a proposition that M can express. This, I think, is too strong a requirement. For sometimes one language cannot express exactly the same proposition as another language does, but in some sense loses nothing thereby in adequacy, or at least the language is at least as good for theology, ethics, science and ordinary life (that's what I mean by "practically"!) For instance, suppose that L is English and M is a restriction of English to those sentences that end with the conjunct "and each thing is identical to itself". Then M cannot express the proposition that there are horses. But the speakers of M do just as well with respect of theology, ethics, science and ordinary life any way: M is just as good as L for theology, ethics, science and ordinary life. Where a speaker of L would say that there are horses, the speaker of M will say that there are horses and each thing is identical to itself.

I do not have a clear notion of "adequacy" here—suggestions are welcome. Here, for what it is worth, are two interesting examples of how the notion might be useful.

1. Detensing: It is well known that tensed sentences like "I am now in pain" cannot be translated into token-reflexive sentences like "My pain is simultaneous with this utterance" (for instance, the latter sentence entails the occurrence of an utterance). But perhaps one can say, more weakly, that a tenseless language that makes use of token-reflexive forms like this is just as adequate as the tensed language. Certainly, it is just as adequate for ethics, science, ordinary life and probably theology. Instead of saying a sentence of ethics, science, ordinary life and theology like "It is now time for me to partially fulfill my duty of thanking God for the nomic orderliness of the universe", we just say: "This utterance is simultaneous with the time for the partial fulfillment of my duty of thanking God for the nomic orderliness of the universe." In saying this, we are saying something different. Different, yes, but in practice just as useful for ethhics, science, ordinary life and theology.

2. Indicatives: Maybe

1. "If the Queen visits me today, I will be prepared"

does not just mean

2. "I will be prepared for the Queen's visit or the Queen won't visit me today or both."

Nor does it just mean

3. "P(I will be prepared | the Queen visits me today) is high"

(claim (3) does not give modus ponens). In fact, plausibly, there is no paraphrase of the indicative conditional except in terms of indicative conditionals (including ones involving "unless" and other variants). Fine. But one can still say that all indicative conditionals could be dropped from English, and the resulting language would be just as adequate. I would not be saying the same thing as (1) if I affirmed the conjunction of (2) and (3), but I would lose nothing by doing so.

Two kinds of scientific accounts

Science can provide two kinds of accounts in response to our observations:

1. Sometimes, science explains why it is that the things we observe are as we observe them.
2. At other times, science will explain why it is that we make the observations we do despite these observations not matching reality.

I think there ought to be a strong preference for the first kind of scientific account. Thus, we should prefer a story on which the universe evolves from a Big Bang to a story on which the universe comes into existence full-blown five minutes ago with false memories of a longer past. Likewise, we should prefer a story on which other people are conscious to a solipsist story explaining why we ascribe consciousness to others.

A false principle concerning desire

I am not the first to discover this particular fallacy—in fact, part of this post is based on ideas I got in conversion from somebody who got them from something he read. But the ideas are no less fun for being mainly not mine.

Consider the following argument for psychological hedonism, the doctrine that the only thing we pursue is pleasure:

1. Whenever we pursue something other than pleasure, we pursue it because it gives us pleasure. (Premise)
2. Therefore, what we really pursue is the pleasure it affords to us.

Not only is (1) false, but (2) does not follow from it. The following inference is invalid:

3. We pursue x because it gives us y.
4. Therefore, what we really pursue is y.

In fact, taken literally, (4) contradicts (3), since if "what we really pursue" is y, and y is not x, then we are not pursuing x (to pursue is the same as to really pursue). But even if we weaken (4) to:

5. Therefore, we pursue y

the inference is still invalid. If I am trying to find a woman who offers water to my camels (cf. Genesis 24), it does not follow that what I am really pursuing is water for my camels. Perhaps I just want the kind of woman who gives water to my camels. Likewise, one might want a particular fig tree in the garden because it yields figs, not because one wants the figs, but because yielding figs is largely constitutive of the health of a tree, and one wants only healthy trees in the garden.

Perhaps the inference works better if we replace (1) by:

6. Whenever we pursue something other than pleasure, we pursue it only because it gives us pleasure.

It's easier to see how (2) might be thought to follow, but (6) now begs the question against the non-hedonist. In any case, the inference is still logically invalid.

In fact, it is even incorrect to conclude from the claim that I seek x solely because it yields y that I want y at all. Suppose that, whimsically, I desire a magical hat that yields rabbits. I only want the hat because it yields rabbits—my whim is that I want to have rabbits pop into existence out of a hat. I can want such a hat for such a reason without having any desire for the rabbits. The rabbits themselves are a nuisance, and I would have no interest at all in rabbits that come into existence in any way other than out of a hat.

It might be objected that then I don't want the hat just because it yields rabbits, but I want the hat because it is a hat that yields rabbits, and so this isn't a counterexample to the inference type. But if so, then the non-hedonist need only say that she doesn't want x just because it yields pleasure, but she wants an x because it is an x that yields pleasure.

The fallacy here also occurs in the Lysis in the argument that if I am friends with x because x yields y, then what I am really friends with is y.

Hedonistic utilitarianism

George is 20 years old and Jake is 50. Neither has friends, or is very likely to make a significant contribution to the pleasure of others, in George's case because of anti-social tendencies, and in Jake's because of severe disability. George hates Jake and pushes him overboard. As Jake flies overboard, George loses his balance and falls in, too. Both call to us for help. We can only pull out one. What should be done?

Here is a hedonist utilitarian answer (it makes a lot of assumptions, but the assumptions are not crazy). We should pull out George, and have him tried and convicted of murder. Then we should publicly sentence him to a lifetime of pain. We then need to hook him to electrodes in a cell, for the rest of his life. But unbeknownst to the public, the electrodes will deliver intense pleasure for the rest of his life. George will never tell anyone this, because he will be enjoying the pleasure too much. We need to tell George about this before he is sentenced to the lifetime of pain, so that he doesn't get too scared of the sentence.

On hedonist utilitarian grounds shouldn't pull out Jake, because (a) Jake is probably not going to agree to being hooked up to the pleasure-machine, and (b) even if he did, he wouldn't have as long left to live, and hence as much pleasure to experience, as George would, since George is younger. To satisfy the public, we might need to lie that we couldn't pull out Jake. George will support us in that lie. Since contortions of pleasure don't look too different from contortions of pain, we can exhibit George to the public, and this will have a significant deterrent effect on murder.

Yes, I know that hedonist utilitarians will cavil at this and at that in the story. But of course the real reason the story is all wrong is that it is surely wrong to save the life of the murderer while letting his victim drown (unless maybe the victim requests that we save the life of the murderer instead—for instance, if the victim is the murderer's parent).

Conditionals, Adams' Thesis and Molinism

The Theorem below is surely known. But the consequence about Molinism is interesting. It is related to arguments by Mike Almeida.

Definition. The claim A→B is a conditional providing A→B entails the material conditional "if A, then B".

Remark: This is of course a very lax definition of a conditional (B counts as a conditional, as does not-A), so the results below will be fairly general.

Definition. A→B is localized provided A&B entails A→B.

Remark: Lewisian and Molinist subjunctives are always localized.

Definition. Adams' Thesis holds for a conditional claim A→B providing P(A→B)=P(B|A).

Definition. The claim B is (probabilistically) independent of A provided P(B|A)=P(B). (If P(A)>0, this is equivalent to P(A&B)=P(A)P(B).)

Theorem 1. Suppose A→B is a localized conditional. Then Adams' Thesis holds for A→B if and only if A→B is independent of A.

Proof. First note that if A→B is a localized conditional, then, necessarily, A&(A→B) holds if and only if A&B holds. Therefore P(A→B|A)=P(A&(A→B)|A)=P(A&B|A)=P(B|A). Now P(A→B|A)=P(A→B) if and only if A→B is independent of A. ■

Remark: It follows that Molinist conditionals do not satisfy Adams' Thesis. For in Molinist cases, God providentially decides what antecedents of conditionals to strongly actualize on the basis of what Molinist conditionals are true, and hence A is in general dependent on A→B (and thus A→B is in general dependent on A).[note 1]

Omega Nebula and Horseshoe Nebula

It's always fun when a philosophy examples comes up in real life. Last night, I first saw the Omega Nebula in one part of the sky. About an hour or so later, I saw the Horseshoe Nebula in another part of the sky. It was only today that I realized that horseshoes and omegas have the same shape, and checked, and found that in fact the two are the same (the sky had rotated during that hour, so the (relative) locations were different). Of course, it is a necessary truth that the Omega Nebula = the Horseshoe Nebula (necessarily x=x). But it was a truth I only discovered on comparing the Messier and NGC catalogs on my PDA. :-)

Astronomy log

I had a really good time last night (i.e., the night of 9/19/2008) observing with a colleague: M25, M18, M8, M22, M28, C16, Omega Nebula, Ring Nebula, Jupiter with Great Red Spot, and maybe barely NGC6717 with averted vision. And a small asterism or open cluster at or near 18 38 11 +40 11 07 not listed in any of the catalogs I have.

[Edited: Initially, I logged the Omega Nebula twice, once as Omega and once as Horseshoe, because I found it twice, using different catalogs, not noticing it was the same object. I also saw the International Space Station, and an unidentified satellite.]

Deep Thoughts XIV

Every tautology is true.

Compositional universalism and monotheism

Compositional universalism says that any collection of non-overlapping beings makes up a whole. (We might restrict this to material objects, but that would be an ad hoc restriction that would make the doctrine not be a true universalism.) Here, "non-overlapping" should, I think, be read not in the spatial sense, but in the mereological sense.

Here is an interesting problem for theists who embrace compositional universalism: It seems to follow that there are wholes of which God is a proper part. Thus, there would be a whole consisting of God and the Empire State Building as its parts. Call this whole x. We can get to a reductio in more than one way from this.

1. God and only God is infinite in the fullest sense. But anything that has a part that is infinite in the fullest sense is infinite in the fullest sense. Therefore, x is infinite in the fullest sense. But x is not God, since God is a proper part of x. This contradicts the claim that only God is infinite in the fullest sense.

2. God and only God has within him perfect goodness. But so does x. Which is absurd.

3. The whole is at least as great as its part. Thus, x is at least as great as God. Which is absurd.

4. Every being other than God is wholly created by God. Therefore, x is wholly created by God. But if y is wholly created by God, then every part of y is wholly created by God. Therefore, every part of x is wholly created by God. Therefore, God is wholly created by God, which is absurd.

Colocationism and the Incarnation

According to abundant forms of colocationism, where I sit, there sit an infinity of other individuals. One of these individuals is a human-shaped mound of flesh and blood. Another is a philosopher. Yet another is something one might call a "rigid human figure" (which I will explain below). These individuals are all related to me in having the same matter. Their distinctness can be seen from the fact that they have different persistence conditions. Thus, the mound of flesh and blood can survive my death, while the philosopher came into existence significantly after I did. The rigid human figure does not survive changes in the orientation of bodily parts. According to sparser forms of colocationism, we may have a more limited number of individuals present here—only the objects that fulfill some philosophically explanatory role will be posited. Thus, the sparse colocationist will probably admit the human-shaped mound of flesh and blood, but probably will not admit the philosopher or the rigid human figure.

Colocationism, whether abundant or sparse, multiplies individuals where there is a multiplicity of what one might with significant propriety call "natures". This creates a prima facie problem for the Incarnation. For if such things as being a human and being a mound of flesh and blood count as individual-defining natures, surely so will being a human and being divine. But then the colocationism seems to imply that where Jesus is, there are two distinct individuals, one of whom is human and the other of whom is divine. This is Nestorianism. (We are used to formulations of Nestorianism that use the word "person" instead of "individual". But we could also have used "individual" as our gloss on "hupostasis"—the Greek does not have the personal implications of the Latin. In any case, both of the individuals will be persons, so there will be two persons.)

This is a prima facie problem for colocationism (I assume, of course, that the orthodox doctrine of the Incarnation is true). Colocationism's multiplication of individuals seems to multiply Jesus into at least two individuals. One attempted solution to the problem might be to say that there is only one nature there, maybe "Godhumanhood". But that, of course, threatens monophysitism, unless one can argue that the sense of "nature" here is sufficiently different from that of the Council of Chalcedon. Maybe if one makes the right distinctions, one can get out of this problem. But it's going to be hard (every account of the Incarnation is hard, but here there seems to be an additional difficulty).

Abundant colocationism has a further problem. There will be an infinitude of individuals where Jesus is. A human, a teacher, a carpenter, the King of Israel, the Messiah, etc. Even if somehow we manage to collapse the human and the divine individuals into one—if not, we get Nestorianism—what do we say about all these other individuals? Are they, for instance, individuals worthy of our worship? Presumably not all—for only three individuals in existence are worthy of worship (in the sense of latria): the Father, the Son and the Holy Spirit. The carpenter, the teacher, the King of Israel, the Messiah and the mound of flesh and blood all share the matter of the (incarnate) Son, but are distinct from the Son, and a fortiori from the Father and the Holy Spirit. But there is something more than a little odd about saying that the Messiah is not to be worshiped. Moreover, while this particular multiplication of individuals may not violate the letter of the Council of Ephesus, it seems to be very much in the Nestorian spirit. If it was bad to have two individuals, having an infinitude surely is also problematic. We want to be able to say that the individual that Jesus' disciples were taught by was the Son of God. But when Jesus' disciples were taught by the teacher (whom else do we attribute teaching to in the primary sense but the teacher?), then it seems they were taught by someone other than the Son of God on the colocationist view.

The sparse colocationist has less trouble, perhaps. She might only have the flesh and blood and the human being to deal with. To avoid literal Nestorianism, she will say that the human being is the same individual as the Logos. The mound of flesh and blood will still be problematic, though, in light of the fact that we are told by Scripture that the Logos became flesh. We do not want to leave the flesh and blood too far outside the bounds of the Incarnation.

Approximation in mathematics

The New York Times has an interesting article arguing that imprecise approximational intuitions--the ability to quickly and roughly reckon things--are crucial to success at abstract mathematics.

To someone whose mathematical work was in real-number based areas--analysis and probability theory--this is very plausible on introspective grounds. But I wonder how true it is in more algebraic fields. I've never been very good at higher algebra--things like the Sylow theorems were very difficult for me (I still passed the algebra comp, but it came noticeably less naturally to me than the analysis comp), perhaps in part because my approximational intuitions were close to useless.

Anecdotal data suggests to me that there are two distinct kinds of mathematical skills. There are the skills involved in analysis, skills tied to problems that are real-number based (complex numbers are real-number based, of course, since C is just the cross product of R with itself), often visualizable, and where approximation and limiting procedures may be relevant. And then there are the skills involved in more algebraic fields, where (as far as I can) approximation gets you nowhere, and while visualization is helpful, the visualization is much more symbolic (visualizing a path of a brownian particle is pretty straightforward, one visualizes quotient groups either explicitly in symbols like "A/H" or perhaps in some strange and highly abstract diagrams). I don't know where to put the combinatorial--it may somewhat straddle the divide (a lot of visualization is involved), but I think is very algebraic in nature.

It is quite possible for a person to be really good at one of these, without being very good at the other. There are fields of mathematics that call upon both sets of skills. And there is an asymmetry: I think the analysis-type skills may be of very little use to mathematicians working in very algebraic areas, but just about every mathematician working in an analysis-type area needs to be able to do algebraic manipulation (though I have a strong preference for proofs in analysis-type fields where the algebraic manipulation is just a way of making precise what is intuitively obvious).

Confidentiality

You ask me: "Did Owen tell you in confidence that he is looking for another position?" He didn't, and in fact Owen and I have never talked about the question. What can I say? It seems I can truthfully and with a clear conscience answer: "No." After all, Owen never confided in me, so I owe him no duties of confidentiality.

But if I make it a policy to answer such questions honestly in cases where Owen has reposed no relevant confidence in me, then I make myself into a non-intentional betrayer of secrets. For you can then tell whether Owen has shared a relevant confidence in me simply by asking me about it—if I answer, then he has not, and if I do not answer, then he has. Moreover, in typical cases you can also deduce, with some probability, what the confidence was. For it is more likely that Owen would take the trouble to request confidentiality about his looking for a new position than about his being satisfied with his present post.

By answering in the negative when no confidence has been reposed in me, then, I decrease my ability to keep confidences on other occasions. It seems, then, that a good thing to say is: "If he did tell me so, I wouldn't be able to share it with you. And if he did not tell me so, I still shouldn't tell you that, since then you'd be able to tell when confidence has been reposed in me."

But what is kind of tricky is that there are cases where this response does not seem satisfactory from Owen's point of view. Suppose that Owen never committed a certain pecadillo, but I am such a close friend of his, that had he done it, he would have immediately told me about it in confidence. If I am asked whether Owen confessed the pecadillo to me, and he had not, then it seems the very best thing for Owen's reputation is a clear denial from me. But a policy of such denials makes me a poorer keeper of confidences for my friends. So there is a bit of a dilemma here.

Presumably, the thing to do is to say that the duty to remain an effective keeper of confidences when one has not had a secret confided to one is only a prima facie duty. It is, simply, a good thing to be an effective keeper of confidences, but sometimes we need to act in ways that makes us less effective at keeping secrets, just as sometimes we need to act in ways that will make us less good racketball players (a philosopher I know gave up a professional racketball career to go into philosophy). To be an effective keeper of secrets is a genuine good, but there are incommensurable goods that might justify becoming a less effective keeper of secrets. There is nothing surprising here. In fact, examples are easy to find. Learning to keep a poker face, for instance, makes one a more effective keeper of secrets, but increases one's temptations to dishonesty.

What is kind of interesting to me about this case is that it seems one has prima facie duties of confidentiality towards people whose confidences one does not actually possess. I think this is because one has good reason to be ready with the offices of a friend (understood broadly—we should be a friend or neighbor to all), and hence to act in ways that make one a more effective friend. Maybe we should see this reason as grounded in what one owes fellow human beings, or maybe in what one owes oneself, or maybe in what one owes God.

And confidentiality is not the only such case. For instance, one likewise has reason to avoid budgeting one's money and time in such a way that one has no margin to help friends in need.

There is nothing earthshaking or deeply surprising here. I just wanted to think through these issues, and as often, my way of thinking them through is by writing.

Could it contingently be the case that the laws of nature hold necessarily?

Here is a nice cautionary tale about being careful with scope and modality. A friend asked me whether one could argue from the possibility of the laws of nature being necessary to laws of nature being actually necessary. I answered in the affirmative, thinking about S5, and imagining the following argument:

1. Possibly, the laws of nature hold necessarily. (Premise)
2. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
3. Therefore, the laws of nature hold necessarily.

But the argument is badly fallacious. One way to see the fallacy is to note that in (2) the proposition p is referred to rigidly, while in (3) "the laws of nature" are a definite description. Compare the fallacious argument:

4. Possibly, what I will write on the board holds necessarily. (Premise)
5. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
6. Therefore, what I will write on the board holds necessarily.

Claim (4) is true: it is possible that I will write on the board something that holds necessarily (e.g., "5+7=12"). But one cannot conclude to (6), since after all I might also write something that holds merely contingently or not at all.

Another way to see the fallacy is to see it as a scope ambiguity in (4). For (6) to follow, (4) must be read as saying that possibly p holds necessarily, where p is what I will write on the board. But that claim is unjustified. All I am justified in saying is that possibly I will write something that holds necessarily.

And in fact we can see that it is prima facie possible that it a merely contingent fact that the laws of nature hold necessarily. Suppose that the laws of nature come in two classes. Some laws of nature are metaphysically necessary and some are not. For instance, prima facie, it might be a metaphysically necessary law that electrons have electric charge, but a metaphysically contingent law that opposite charges attract. Then it might turn out that there is a world w where there are no metaphysically contingent laws. It would then be true at w that all the laws of nature hold necessarily. But this would be only contingently the case, because there are worlds that have some of the contingent laws as well.

Interestingly, we can make the argument from (1) and (2) to (3) work if we add the following premise:

7. Necessarily: (For all p, either it is a law that p is a law, or it is a law that p is not a law).

For then, by (1), suppose that w₁ is a world where the laws hold necessarily. Suppose now that w₂ is any world and p is any law in w₂. Then, p either is or is not a law in w₁. If p is a law in w₁, then it is a law in w₁ that p is a law. But the laws in w₁ hold necessarily, so it follows that p holds necessarily. And if p is not a law in w₁, then it is a law in w₁ that p is not a law. Since the laws in w₁ hold necessarily, p is not a law in w₂, which is absurd. Hence, every law of w₂ holds necessarily.

Note: The above argument assumed that "p is a law" entails p, and that the claim "law p holds necessarily" entails that p holds necessarily (I did not additionally assume that "law p holds necessarily" entails that necessarily p is a law, though that does follow from (7)).

[I fixed some typos, and more importantly edited (2) and (4) in response to Comment #1 below. The original version of the post had "p" instead of "necessarily p" in the consequents of (2) and (4), which made the arguments not work. This emphasizes again the first sentence of the post.]

Almeida on moral status of fetuses and newborns

Mike Almeida has posted a very interesting argument on prosblogion to the effect that fetuses and newborns have the same moral status as adults. There is lots of good discussion there. He then posted an interesting follow-up argument.