Monday, December 31, 2012
Grammatical possibility
Saturday, December 29, 2012
Qualitative probabilities, regularity and nonmeasurable sets
Normally, regularity is formulated as saying that P(A)>0 for every non-empty A. But suppose that instead of working with numerical probability assignments, we work with qualitative probabilities, i.e., probability comparisons. Thus, instead of saying B is at least as likely as A provided that P(B)≥P(A), we might take the relation of being at least as likely as to be primitive, and then give axioms.
Given a theory of qualitative probabilities, it will be possible to define an equiprobability relation ~ such that we can say A~B if and only if A and B are equiprobable. (The typical way would be to say that A~B provided that B is at least as likely as A and A is at least as likely as B.) This relation ~ will satisfy some axioms, but we actually won't need them for the argument. We shall suppose that ~ is defined on some collection of subsets of a sample space, which we will call the measurable sets. Our setup generalizes classical probabilities, as well as hyperreal probabilities, since if we have probability-values, we can say that A~B if and only if P(A)=P(B).
We can plausibly formulate regularity in terms of an equiprobability relation:
- An equiprobability relation ~ is regular if and only if whenever A and B are measurable sets such that A is a proper subset of B, then we do not have A~B.
- An equiprobability relation ~ is rotation-invariant if and only if whenever A and B are measurable sets such that B is a rotation of A, then A~B.
Now, we know that given the Axiom of Choice, and given classical probabilities, there is no way of defining probabilities for all subsets of our circle in a rotation-invariant way. Surprisingly, but very simply, if we assume regularity, we need neither classical probability—any equiprobability relation will do—nor the Axiom of Choice. In fact, we will have a countable nonmeasurable set, so when we add regularity to the mix, we have to sacrifice the measurability of sets that are unproblematically measurable using classical measures.
Theorem: There is no equiprobability relation ~ such that (a) all countable subsets of the circle are measurable; (b) the relation is regular; and (c) the relation is rotation-invariant.
Proof: Let u be any irrational number. Let B be the set of all points on the circle at angles 2πnu (to some fixed axis, say the x-axis), for positive integers n. Let A be a rotation of B by the angle 2πu. Then A is a proper subset of B (A contains all the points on the circle at angles 2πnu for n an integer greater than one, and by the irrationality of u that will not include the point at angle 2πu). So if we had regularity, we couldn't have A~B. But if we had rotation-invariance, we would have to have A~B. ∎
The above proof is based on the counterintuitive fact that there is a subset of the circle, i.e., B, that can be rotated to form a proper subset of itself, i.e., A. (This reminds me of the Sierpinski-Mazurkiewicz paradox and other cases of paradoxical decomposition, though it's much more trivial.)
This is, of course, a trivial modification of the Bernstein and Wattenberg inspired argument here.
Friday, December 28, 2012
Could God have become incarnate as a non-person?
The Logos became incarnate as a human being, to save us from our sins. There would have been no similar point to his becoming incarnate as a cat, an oak or a photon? But could he have done so, if he had a purpose to?
Suppose we say "no" to at least one of the three options (cat, oak or photon). Why would we? Assuming we accept that the Logos could have become incarnate as a human being, we would have to suppose some relevant difference between humans and cats, oaks or photons. What the difference is will depend where one draws the possibility-of-incarnation line. If one thinks that God could have become a cat but not an oak, that's presumably because one thinks that sentience is crucial to the possibility of incarnation. And one will presumably then deny that God could have become a photon. If one thinks God could have become a human but not a cat, then presumably one thinks sapience (and I won't worry about the details of what the consists in, say agency or abstract thought) is crucial.
But we human beings don't always exhibit sentience, much less sapience. We don't exhibit sentience for the first weeks of life after conception, and we don't exhibit sapience until at least around one year after birth. Moreover, when unconscious we do not exhibit sentience and need not exhibit sapience (though, maybe, sapience doesn't always require consciousness). In becoming one of us, the Logos would have become a being that wasn't always sentient or sapient. So if one thinks that sentience or sapience is crucial for incarnation, and yet one accepts that the Logos could become a being like we who does not always have sentience or sapience, one has to say that it is something like the potential for sentience or sapience (depending on which view we are considering) that is a necessary precondition for incarnation or that it is sometimes having sentience or sapience that is necessary.
Consider first the "sometimes" view. This presumably requires that the incarnation cannot precede the developmental attainment of sentience or sapience (for the incarnation does so precede, we could imagine it terminating, with the destruction of the finite nature, before that attainment). If sapience is the relevant condition, then we get the view that barring miraculous precociousness, God cannot be incarnate as a newborn, which at least to us Christians will be absurd. If sentience is necessary, then we get the view that, again barring miraculous precociousness, the incarnation couldn't have happened simultaneously with conception. (Interestingly, Aquinas actually goes for miraculous precociousness here—his view that we don't come into existence at conception but a significant amount of time thereafter forced him into holding that Jesus came into existence fully formed in Mary's womb.)
Still, the "sometimes" view just seems implausible. Why would the incarnation require initial exercise of sentience or sapience without the need for exercise of sentience or sapience thereafter?
Now consider the potentiality view. This, too, does not seem all that plausible to me. Presumably the pull of saying that God couldn't become a cat or an oak or a photon is that these beings are so very unlike God. But potentiality is very much unlike God's perfect actuality, too. In the end, I think that once one reflects on the fact that human beings often exhibit neither sentience nor sapience, the pull to thinking the Logos couldn't have become a cat or an oak weakens.
How about a photon? There the relevant difference would be something like life. But again it seems hard to see why life is a necessary condition for an incarnation. There are, plausibly, infinitely many attributes as significant as life that God has and that human beings lack. The gaps between the photon and the oak, the oak and the cat, and the cat and the human are infinitely less than the gap between humans and God, a gap that God can bridge, we have assumed.
The above arguments are not very strong. But I think they do give one a presumption in favor of the view that if God can become incarnate as a human, he can become incarnate as any kind of being.
Thursday, December 27, 2012
Is the Internet "the same" as face-to-face social interaction?
I used to claim to think that social interaction by email is not significantly different qua social interaction from face-to-face interaction. But the fact that I typically strongly preferred email interaction to face-to-face interactions is evidence that it's significantly different, and given my introvertive tendencies, it is evidence that it is less social. People like I need significant time "without social interaction" to avoid exhaustion. But writing this post qualifies, even though it is obviously a social activity.
Or maybe it's not correct to characterize introverts as tired out by social interaction. Rather, they are tired out by particular modalities of social interaction. So perhaps there is a response possible to the argument of the preceding paragraph.
"Figuratively"
Everyone I've talked to agrees that statements like the one on the T-shirt are an example of literal language. The wearer is claiming to literally be made figuratively insane.
But here is an oddity. If you say: "Misuse of 'literally' makes me insane", I can say: "Figuratively speaking, that is." My use of "figuratively" attributes figurativeness to your sentence, which sentence is figurative. But in the slogan on the T-shirt, what does "figuratively" attribute figurativeness to? Presumably, the word "insane"? So does the sentence, thus, contain figurative language after all? But the sentence seemed like a piece of literal language. The "insane" is only there in the scope of "figuratively". So does the "figuratively", perhaps, implicitly attribute figurativeness to a different sentence that hasn't actually been uttered, namely the sentence "Misuse of 'literally' makes me insane". Or, more precisely, maybe it attributes figurativeness to the word "insane" as found in that unsaid sentential context? If so, then analyzing actual sentence tokens requires thinking about sentence types or nonactual sentence tokens.
Saturday, December 22, 2012
First and second order desires
My fear of dogs brings involves a paradigmatic first-order desire: a desire to avoid the proximity of unsecured dogs. But a desire to avoid the proximity of unsecured dogs also motivates me to avoid activities that have a sufficiently high (which does not need to be high at all!) probability of leading to being in the proximity of unsecured dogs, activities such as walking to work. This, too, is a paradigmatic feature of this first-order desire.
But now one of the activities that would have a sufficiently high probability of leading to being in the proximity of unsecured dogs would be getting rid of my fear and hence desire for avoidance. If I didn't fear dogs, I wouldn't avoid the proximity of unsecured dogs. Thus, the desire to avoid the proximity of dogs motivates me to avoid getting rid of this very desire. But such motivation is paradigmatically the work of a second-order desire. Yet it comes about through exactly the same means-end reasoning by which the desire to avoid the proximity of unsecured dogs motivates me to avoid walking to work.
This isn't an exceptional case. Normally, the possession of a desire for A helps promote getting A. There could be exceptions: a desire to have many friends might not make one a good friend and joy might be the sort of thing that comes most when not pursued. But normally desires help promote what they are desires for—indeed, that's presumably at least a part of why we have desires. But then, when one reflects on this, the desire for A will motivate one to maintain a desire for A.
Fortunately, however, often the motivation to maintain a desire for A will not be as strong as the motivation for more direct means to A. This contingent fact makes it easier to rid ourselves of desires that we should not have: for even if the desire is very strong indeed, its motivational force for self-maintenance may not be all that strong, and hence we may be able to induce, through reflection on the perniciousness of that desire, a sufficiently strong motivation not to have that screwed-up desire. Notice, though, that at least sometimes that motivation-to-remove-desire will itself be simply a means-to-end motivation in light of a first-order goal. One doesn't want to die of lung cancer—so one works to remove the remove the desire to smoke.
Friday, December 21, 2012
Goedelian ontological argument
Wednesday, December 19, 2012
One Body: released
I got my copies yesterday. They look nice. Here's the blurb from Amazon:
This important philosophical reflection on love and sexuality from a broadly Christian perspective is aimed at philosophers, theologians, and educated Christian readers. Alexander R. Pruss focuses on foundational questions on the nature of romantic love and on controversial questions in sexual ethics on the basis of the fundamental idea that romantic love pursues union of two persons as one body.
One Body begins with an account, inspired by St. Thomas Aquinas, of the general nature of love as constituted by components of goodwill, appreciation, and unitiveness. Different forms of love, such as parental, collegial, filial, friendly, fraternal, or romantic, Pruss argues, differ primarily not in terms of goodwill or appreciation but in terms of the kind of union that is sought. Pruss examines romantic love as distinguished from other kinds of love by a focus on a particular kind of union, a deep union as one body achieved through the joint biological striving of the sort involved in reproduction. Taking the account of the union that romantic love seeks as a foundation, the book considers the nature of marriage and applies its account to controversial ethical questions, such as the connection between love, sex, and commitment and the moral issues involving contraception, same-sex activity, and reproductive technology. With philosophical rigor and sophistication, Pruss provides carefully argued answers to controversial questions in Christian sexual ethics.
"This is a terrific—really quite extraordinary—work of scholarship. It is quite simply the best work on Christian sexual ethics that I have seen. It will become the text that anyone who ventures into the field will have to grapple with—a kind of touchstone. Moreover, it is filled with arguments with which even secular writers on sexual morality will have to engage and come to terms." —Robert P. George, Princeton University
"One Body is an excellent piece of philosophical-theological reflection on the nature of sexuality and marriage. This book has the potential to become a standard go-to text for professors and students working on sex ethics issues, whether in philosophy or theology, both for the richness of its arguments, and the scope of its coverage of cases. " —Christopher Tollefsen, University of South Carolina
"Alexander Pruss here develops sound and humane answers to the whole range of main questions about human sexual and reproductive choices. His principal argument for the key answers is very different from the one I have articulated over the past fifteen years. But his argumentation is at every point attractively direct, careful, energetic in framing and responding to objections, and admirably attentive to realities and the human goods at stake." —John Finnis, University of Oxford
An electronic version (PDF) can be purchased directly from the press.
Tuesday, December 18, 2012
There is such a thing as supererogation
Supererogatory actions are admirable but not obligatory. A sufficient, and perhaps necessary, condition for an action A to be supererogatory is that (a) A is permissible and (b) there is an alternative B to A such that (i) it is permissible to do B instead and (ii) A is more morally praiseworthy than B.
Over the years, I've met people--including myself--who have been troubled by the idea of supererogatory actions and tempted to deny that there is such a thing as supererogation. But here is a pretty conclusive argument that there can be supererogatory actions. You and your friend, both innocent people, are captured by a tyrant. The tyrant sentences your friend to 24 hours of torture. Then the tyrant offers you the option of reducing your friend's torture by any amount of time less than 12 hours. And of course, she notes, any torture taken away from your friend will be given to you, by Public Law Number One: the Preservation of Torment.
Now, many people will say that any taking on of your friend's torture is automatically supererogatory. But I think the sort of people who doubt that there are supererogatory actions won't be impressed--they tend to have a view that morality does indeed sometimes call us to very great sacrifices (and they are right about that, even if they might be wrong about this case).
However, the following is surely true: There is an amount T_{1}<12 such that it is permissible to reduce the friend's torture by T_{1} hours. Indeed, surely, T_{1}=11.99 qualifies. (Argument: reducing one's friend's torture by 11.99 hours, given the cost that one will suffer that torture oneself, is plainly praiseworthy simpliciter, but only permissible actions are praiseworthy simpliciter.) Let B be the action of reducing one's friend's torture by T_{1} hours. Let T_{2} be a number such that T_{1}<T_{2}<12. Let A be the action of reducing one's friend's torture by T_{2} hours and let B be the action of reducing one's friend's torture by T_{1} hours. Then, barring further factors not given in the story: (a) A is permissible (it would be odd if it were permissible to reduce one's friend's torture by, say, 11.99 hours but not by 11.999 hours); (b)(i) B is permissible and (b)(ii) A is more morally praiseworthy than B. Thus, A is supererogatory.
If you think time is discrete, the above example still can be made to work. Suppose for simplicity 11.99 hours is the longest time interval short of 12 hours. Then if you think there is no supererogation, you might think that you're obligated to request that your friend be relieved of 11.99 hours of torture. But as long as the agent in the story doesn't know that 11.99 hours is the longest time interval short of 12 hours there is, she can do something more praiseworthy than requesting the 11.99 hour reduction: she can request 11.999 hours, and as long as she is not certain that 11.99 hours is the most she can get, she thereby risks getting more than 11.99 hours of torture as the cost of trying to relieve more than 11.99 hours, and that's more praiseworthy than just going for 11.99.
Friday, December 14, 2012
Almost necessary beings and the ontological argument
The familiar S5 ontological argument for a necessary being goes:
- Possibly, there is a necessary being.
- So, there is a necessary being. (By S5)
- Possibly, there is an almost necessary being.
- Something exists.
- So, there is an almost necessary being. (By S5. If an almost necessary being exists at one world, it exists at all worlds at which something exists; but something actually exists.)
Could one support (3) without that also giving an equally good argument for (1)? Maybe.
- It quasi-perceptually seems to some mystic that love grounds all being.
- What quasi-perceptually seems to someone is probably possible (or at least conceivable in the two-dimensionalist sense, but that's all we actually need).
- Necessarily, if x grounds all being, then x grounds all being in all worlds in which something exists.
- Necessarily, if love grounds all being, then there is a lover who grounds all being.
- So, probably it's possible that love grounds all being. (6 and 7)
- So, probably it's possible that there is an almost necessary being. (8, 9)
- So, probably there is an almost necessary being.
Thursday, December 13, 2012
Another start on the problem of evil
According to Socrates the greatest goods and evils are moral ones. Call this the "Socratic thesis". On the Socratic thesis, the worst thing that can befall one is to act culpably wrongly. Now, we may divide up the evils of the world into three classes:
- Culpable wrongdoings.
- Harms other than culpable wrongdoings resulting from culpable wrongdoings.
- Harms neither identical with culpable wrongdoings nor resulting from them.
Each of these three classes of evils is very large. I think we can say that if we confine ourselves to evils happening to humans (bracketing the problems of animal suffering and angelic fall): Class 1 is roughly as large as the Class 2 (granted, some culpable wrongdoings result in many harms; but many culpable wrongdoings stay at the level of an evil thought that leads to no harmful action) and also roughly at least as large as Class 3. So roughly, about a third of the evils of the world are in Class 1.
Next notice that we have a theodicy for Class 1 evils: the free will theodicy, in its different versions (straight free will theodicy, soul-building, autonomy, need for love to be a free response, etc.) By the Socratic thesis, we thus have a theodicy for the greatest evils that occur, and these evils are roughly a third of all the evils that occur to humans. This provides us with some inductive reason to think that there is a theodicy for the rest of the evils: if a theodicy can be found for the greatest evils, and indeed for about a third of the evils happening to humans, then the existence of a theodicy for the rest seems more plausible.
Moreover, some versions of the theodicy for Class 1 evils extend to theodicies for many Class 2 evils. First, our free will would be a bit of a sham if it wasn't effective—if evil choices never resulted in in the chosen state of affairs. (This is less plausible for the worst Class 2 evils.) Second, while terribly harms do befall undeserving people, most of the evils that befall are, I suspect, quite deserved. Those evils, then have a justice theodicy, given a freedom theodicy for the actions that deserved them. (This might shift our count of some evils from Class 3 to Class 2, though we might also say that there are evils in Class 3 that do not result from our culpable wrongdoings, but that on account of our culpable wrongdoings weren't prevented by God.)
Wednesday, December 12, 2012
Supererogation
Supererogation is a difficult concept for me. But there has to be such a thing. If Jones has suffered two hundred weeks of torture to save the lives of two hundred strangers, and then declines the 201st week of torture to save the life of the 201st stranger, Jones does not do wrong. And if he were to accept the torture, he would be acting superegatorily (barring special circumstances).
I doubt the following account is in the end right, but I think it is surprisingly defensible (modulo perhaps some minor tweaks):
- An action is supererogatory if and only if it is permissible and less convenient than some available alternative permissible action.
The account above seems to be subject to simple counterexamples. Let's say it's permissible for me to go to the kitchen, and suppose there are two paths—an easier and a harder one. Then surely both paths are permissible, and the harder one is less convenient, but that doesn't make the less convenient one supererogatory!
To respond I note that it is wrong to pointlessly impose burdens on any person—including oneself. (Argument 1: We are to love all of the people that God loves, and love prohibits pointless imposition of burdens. But I am one of the people God loves. So I am not permitted to impose pointless burdens on myself. Argument 2: What is vicious is impermissible. But pointless imposition of burdens on myself is contrary to the virtues of prudence and hence vicious.) Thus if there is no benefit to anybody from taking the harder path, the harder path is not permissible, and hence is not supererogatory. But suppose that there is a benefit to someone from the harder path: maybe I become physically or morally stronger, or maybe someone else benefits in some way. Then as long as the harder path is permissible (if the benefit is too trivial as compared to the burden, it might not be), it does seem to be supererogatory.
I do suspect that this account of supererogation only stands a chance if we have duties to self, but that's not a weakness of it.
Some people doubt that there are any supererogatory actions. On the above account, it is quite plausible that there are. First, we need to note that surely there are cases where we choose between multiple permissible actions. And second we note that it is very likely that among such choices there are going to be cases where the permissible options are not all equally convenient. And then the less convenient ones will be supererogatory.
Note that if convenience is what is given up in self-sacrifice, then every supererogatory action involves self-sacrifice. Now, self-sacrifice is relative to some alternative that does not involve such a sacrifice. We might then rephrase our definition of supererogation as:
- An action is supererogatory if and only if it is permissible and it is a self-sacrifice relative to some permissible alternative.
Go back to my initial case of Jones. If Jones did undergo the 201st week of torture, he would be doing something permissible, but it would also be permissible for him not to undergo that torture. However, undergoing the torture is less convenient. Again, this sounds like an absurd understatement, but in our technical sense of "convenient", it's not. It sounds a lot better in the language of self-sacrifice: Jones' undergoing the torture is permissible and is a self-sacrifice relative to the alternative of not undergoing it.
I think the weakness of the account is it does not make clear why supererogation is particularly praiseworthy. Moreover, even if the account happens to be extensionally correct, I don't think it captures what it is that grounds supererogation.
Tuesday, December 11, 2012
Uniform measure and nonmeasurable sets, without the Axiom of Choice
Given the Axiom of Choice, there is no translation invariant probability measure on the interval [0,1) (the relevant translation is translation modulo 1). But this fact really does need something in the way of the Axiom of Choice. Moreover, the fact only obtains for countably additive measures. Interestingly, however, if we add the assumption that our measure assigns non-zero (presumably infinitesimal) weight to each point of [0,1), then the non-existence of a translation invariant finitely additive measure follows without the Axiom of Choice. I got the proof of this from Paul Pedersen who thinks he got it from the classic Bernstein and Wattenberg piece (I don't have their paper at hand). I am generalizing trivially.
Theorem: Let P be any finitely additive measure taking values in a partially ordered group G and defined on a collection of subsets of [0,1) such that every countable subset has a measure in G. Suppose P({x})>0 for some x in G. Then P is not translation invariant (modulo 1).
Proof: To obtain a contradiction, suppose P is translation invariant. Then P({x})>0 for every x in [0,1). Let r be any irrational number in (0,1), and let R be the set of numbers of the form nr modulo 1, as n ranges over the positive integers. Let R' be the set of numbers of the form nr modulo 1, as n ranges over the integers greater than 1. Then R' is a translation of R by r, modulo 1. Observe that r is not a member of R' since there is no natural number n greater than 1 such that r=nr modulo 1, since if there were, we would have (n−1)r=0 modulo 1, and hence r would be a rational number with denominator n−1. Thus by finite additivity P(R)=P(R')+P({r})>P(R'). Hence, R is a counterexample to translation invariance, contradicting our assumption.
Note 1: On the assumption that the half-open intervals are all measurable and the measurable sets form an algebra (the standard case), translation invariance modulo 1 follows from ordinary translation invariance within the interval, namely the condition that P(A)=P(A+x) whenever both A and A+x={y+x:y in A} are subsets of [0,1).
Note 2: The proof above shows that if P({x})>0 for every x in [0,1), then the set of all positive integral multiples of any fixed irrational number (modulo 1) is nonmeasurable. It is interesting to note that this nonmeasurable set is actually measurable using standard Lebesgue measure. Thus, by enforcing regularity using infinitesimals, one is making some previously measurable sets nonmeasurable if one insists on translation invariance.
Note 3: Bernstein and Wattenberg construct a hyperreal valued measure that is almost translation invariant: the difference between the measure of a set and of a translation of the set is infinitesimal.
Saturday, December 8, 2012
Another argument about simplicity
In an earlier post, I defended the idea (which Trent Dougherty also came up with independently and earlier) that only theory-unexplained entities, or kinds of entities, count against the simplicity of a theory. Here is another argument for this. Start with these two principles:
- If theories T_{1} and T_{2} are otherwise equally evidenced and explanatorily powerful, but T_{1} is simpler, then T_{1} is more epistemically likely to be true than T_{2}.
- The Principal Principle: Epistemic probabilities should (except in exceptional cases) be set to equal objective chances when the latter are available.
- T_{H}: T_{0} is true, no U particles ever get produced except perhaps in a moment by this apparatus, heads will come up, and a U particle will be produced by the apparatus.
- T_{T}: T_{0} is true, no U particles ever get produced except perhaps in a moment by this apparatus, tails will come up, and no a U particle will be produced by the apparatus.
By a very plausible application of the Principal Principle, since the chances of heads and tails are equal as the coin is fair:
- P(T_{T})=P(T_{H}).
But if the number of explained kinds of objects counts against simplicity, then T_{T} is simpler than T_{H}, since according to T_{T} reality includes an extra kind of particle, the U particle. (If one doesn't think reality includes the future, run this thought experiment retrospectively after the explosion.) So by (2), then, P(T_{T})>P(T_{H}). But this contradicts (3). Thus, by modus tollens, the number of explained kinds of objects does not count against simplicity.
Friday, December 7, 2012
Parental duty
Another excerpt from my forthcoming One Body book, this time from the discussion of gamete donation (challenge to the reader: find the relevance of this to gamete donation):
Now, it is not merely the duty of the parents to bring it about that the children are cared for and appropriately educated morally, religiously and academically. Rather, it is the duty of the parents to care for and educate the child—i.e., to do it themselves. In caring for and educating the child, parents will make use of the help of others, including that of family members, friends, and professionals. How much the parents can rely on the help of others before they have failed in their duty of caring for and educating the child will depend on the circumstances.
There are thus two aspects of the parental duty: (a) caring for and educating, and (b) ensuring that the child is cared for and educated. In other words, there is the aspect of parental activity and the aspect of results. These two aspects need to be balanced prudently, and, moreover, balanced with other duties the parents may have; how they are balanced will depend on particular circumstances. In no cases will it be desirable and rarely will it be possible for the parents directly to care for and educate the child in all respects with the help of no one else. Moral education, for instance, requires contact with virtuous people of a significant variety of different characters, not just the parents. Academic education should typically include education in subjects in which the parents lack competency. The need to work to earn money to provide for the child can force the parents to delegate a significant degree care to a third party.
Here is an observation worth making. In most couples, there will be specialization. Thus, the mother might be working long hours to earn the money needed to diaper, feed, clothe, and house the child, while the father might be changing the diapers, feeding, clothing, and otherwise taking care of the child for most of the day. It might seem that in such cases, each parent will be neglecting an aspect of the parental responsibility to himself or herself care for and educate the child. But we can respond to this by noting that parents should be friends of each other, and bringing in an idea from Aristotle’s Nicomachean Ethics. Aristotle considers what value there in having good friends. He observes that friends share a life, a friend is “another self,” and one can be active through one’s friend’s activity: what the friend does virtuously is something that accrues to oneself.
Thursday, December 6, 2012
Conditional probability and nonmeasurable sets
Let P be Lebesgue measure on the three-dimensional cube [0,1]^{3}. Assume the Axiom of Choice. Then there will be P-nonmeasurable sets (e.g, as there is no finitely additive (much less countably additive, rigid-motion-invariant probability measure on all subsets of [0,1]^{3} by the Banach-Tarski paradox. Now let F be the Lebesgue measurable subsets of [0,1]^{3}. One might then hope that one can define P(X|Y) for all X and Y in F, as long as Y is non-empty.
Turns out that one can't, at least if one expects finite additivity and rigid-motion invariance. The reason for that is that if we let Y be (the surface of) a sphere in [0,1]^{3}, then all subsets of Y will be in F, but there is no finitely additive rotation invariant probability measure on all subsets of a sphere, by the Hausdorff paradox. This problem disappears if we restrict ourselves to the Borel subsets of [0,1]^{3}, but the extension to the Lebesgue measurable ones is epistemologically very plausible—obviously any subset of a null set should be a null set.
But here's a funny thing. While I haven't checked all the details—it's grading time so I can only give this so much thought—it turns out that one can sensibly define P(X|Y) for all X in F and all non-measurable Y. The easiest case is where Y is maximally non-measurable, i.e., all its measurable subsets have null measure and all its measurable supersets have full measure. In that case, one can simply define P(X|Y)=P(X). Moreover, one can naturally extend this measure to all X's in F_{Y}, where F_{Y} is the smallest σ-field generated by F and Y (basically by setting P(X∩Y|Y)=P(X) and P(X∩Y^{c}|Y)=0 for X in F).
This means that the Popper function approach to conditional probability on which P(X|Y) is defined for all X and Y in a single field or σ-field is not general enough, at least if we want P(X|[0,1]^{3}) to be defined for all Lebesgue measurable X's. For in fact it seems we have more freedom as to what Y's we get to plug in and less as to what X's.
Wednesday, December 5, 2012
Limiting God to solve the problem of evil
Long ago, I remember reading with great curiosity Rabbi Kushner's Why Bad Things Happen to Good People? How disappointing that Kushner's intellectual answer seemed to be that God isn't omnipotent. (His practical answer not to worry about the question but just to do good is much better.) The idea of limiting divine attributes in part to answer the problem of evil has recently had some defense (e.g., here and in the work of open theists), so I guess it's time to blog the objection to Kushner—which applies to the others as well—that I had when I read him, with some elaboration.
Basically, the objection is that as long as God remains pretty good, pretty smart (he was smart enough to create us!) and powerful enough to communicate with us (Kushner at least accepts this), then serious cases of the Problem of Evil remain. Moreover, these cases do not seem significantly easier to solve than the cases of the Problem of Evil that were removed. Consequently, the intellectual benefit with regard to the Problem of Evil is small. And the intellectual loss with regard to the simplicity of the theory is great—the theory that God has all perfections is far simpler.
Start by considering a deity whose goodness is unlimited but whose knowledge and power are fairly limited.
Consider, first, the problem of polio. This is certainly a horrendous evil. And the limited deity could have alleviated a significant portion of the problem hundreds of years earlier simply by whispering into some people's ears how to make a vaccine—surely any deity smart enough to create this world would be smart enough to figure out how to make vaccines. Maybe the limited deity couldn't have prevented all cases, in the way that an unlimited God could. But given that neither did the wholesale prevention happen nor did the partial prevention by vaccines happen as early as it could have.
Consider, second, the many cases where innocent people suffered horrendously at the hands of attackers, where the attack could have been prevented if the people had been warned. Even a deity of limited power and knowledge should be able to see, for instance, that the Gestapo are talking about heading for such-and-such a house, and could then warn the occupants. (I am not saying that such warnings were never given—for all I know, they were in a number of cases. But I am saying that there are many cases where apparently they were not.)
Moreover, even if one limits the goodness of the deity, and only claims that he is pretty good, the problem remains. For unless the deity had a very serious reason not to tell people about vaccines and not to warn the innocent victims of horrendous attacks, it seems plausible that the deity did something quite bad in refraining from helping, so bad as to be incompatible with being pretty good. (If the deity had a reason that fell a little short of justifying the refraining, then that might be compatible with being pretty good; but a reason would have to be pretty serious for it to fall only a little short of justifying the refraining when the evils are so horrendous.) So even if one thinks that the deity has limited power and knowledge and is only pretty good, the problem of finding very serious reasons for the deity's non-interference remains.
Granted, the problem is diminished, especially if one has decreased the belief in divine goodness. But notice that the decrease in belief in divine goodness is the most religiously troubling aspect of a limited God doctrine. And even that does not make the problem go away.
Moreover, the sorts of things one can then plausibly say about the remaining problems of evil are things that, I suspect, the traditional theist can say as well about this and many other cases. Perhaps God does not prevent all attacks on innocent people (for all we know, he prevents many) because he wants humans to have effective freedom of will. Perhaps he wants to give victims opportunities for forgiveness of their aggressors in an afterlife. Perhaps God does not prevent disease because he wants us to help our neighbor and to develop medical science to this purpose. Or to give us an opportunity to join him on the cross in redeeming humankind. Or perhaps God prevents many evils, but his purposes do not allow him to prevent all, and some arbitrary line-drawing is needed. I am not saying that these answers are sufficient (though I think some contain a kernel of something right), but only that they can be equally used in the case of a limited and unlimited God, and in the case of an unlimited God such answers may well have rather general applicability.
Tuesday, December 4, 2012
Reducing sets
I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.
Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.
Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.
Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.
Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)
We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)
There are many other ways of packaging pluralities.
Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.
Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.
Sunday, December 2, 2012
Feeling cold? Apply to our graduate program!
Of course, the really great thing is the warmth of the graduate student community.
We've extended our deadline for the Baylor Philosophy PhD program this year until January 2.
The anole photo is from today, from the path by the river on campus (or just off campus?). The butterflies are from Thanksgiving, though I saw a number today, too.