The unavoidability of sin

Aquinas says that without grace we can avoid each individual mortal sin but not all mortal sin, at least not for a significant length of time.

On its face, this seems contradictory. After all, I can avoid the first mortal sin. If I avoid it, then I should be able to avoid the next one. And so on. And hence I should be able to avoid all mortal sin.

But this argument mistakenly agglomerates what one can do. For instance, suppose that there is a mine field with a thousand mines. I know how to defuse a mine, but I have an independent probability of 10% of slipping and detonating the mine. It's correct to say that each mine can be defused by me—being able to do this with 90% reliability is sufficient for this—but it is incorrect to say that I can defuse the whole minefield. It is appropriate to look at the minefield that one is to defuse and think: "This is an impossible task without help."

The person without grace is unable to presently control her future actions in favor of the good. Each individual action is in her power, but she cannot control them all at once. Hence she can rightly look with trepidation at her future moral life and say: "This is an impossible task without help."

But a virtuous person can control future actions in favor of the good en masse, by growing in virtue and resolving to do good.

A weakness of eliminative materialism

I was telling my kids about eliminative materialism, the view that there are only material objects, and that there are no minds, persons, beliefs, perceptions, etc. My kids are used to hearing about nutty philosophical views, such as those of Zeno, but they noticed that the standard tool in defending wacky philosophical views is unavailable here. For while Zeno can say that motion is an illusion, eliminative materialists can't say that thought is an illusion. For illusions are among the things the eliminative materialist eliminates. I hadn't noticed this.

Natural roles

I've been thinking about roles such as being married, being a parent, being a sibling, being a second cousin, being a firefighter, being a president or being a monarch. These roles come along with moral duties, permissions and rights. Each role has entry conditions, such that when one satisfies the entry condition, one falls under the role, and some of the roles have exit conditions. In some roles the entry conditions involve one's agreeing to enter the role—marriage, firefighting, presidency and monarchy are like that—but in some the entry conditions may have nothing to do with one. On the other hand, one need not do anything to become a sibling.

Here's what I am inclined to think about such roles. Our human nature specifies what one might call natural roles. Some of our natural roles directly give us roles. Thus, I think being married and being a parent are natural roles. I am not sure about being a sibling, and I quite doubt that being a second cousin, being a firefighter and being a president are natural roles.

Roles can have sub-roles which have more specific duties, permissions and rights. These sub-roles derive all their normative force from the full role and other conditions. For instance, the role of parent has sub-roles such as: parent of a small child and parent of an adult child. The moral duties, permissions and rights associated with the sub-role derive from the moral duties, permissions and rights associated with the full role together with the fact of the satisfaction of the condition.

My big conjecture is that all roles that are not natural are sub-roles of natural roles. Thus, there is no natural role of second cousin, but there is a natural role of distant relative, perhaps. The duties, permissions and rights of being a distant relative depend on multiple conditions, including the closeness of the relationship and the operative social conventions. So there may be such a thing as the role of second cousin in mid-twentieth century rural Pennsylvania. All the moral normative oomph in this sub-role comes from the moral normative oomph of the natural role of distant relative. It does this by means of the natural role having requirements that are conditional, in this case on closeness of relationship and the social conventions in place. For instance, the natural role of distant cousin may say: "Fulfill those non-immoral conventional duties associated with your degree d of relationship whose fulfillment does not exceed degree of onerousness c(d)", where c(d) is some function setting a cap for how onerous a conventional duty one is required to fulfill in the case of a degree d of relationship (presumably the closer d is, the higher the cap). In this way, conventional duties (which on my view are no more duties in themselves than rubber ducks are ducks) become moral duties when the natural role makes them so. Thus, a conventional duty to come to second cousins' weddings yields a moral rule that one should come to second cousins' weddings, when the onerousness does not exceed c(d), where d is the degree of relationship involved in being a second cousin.

We could have cases of non-natural roles whose moral normative oomph derives from two or more natural roles. In this case, we can say that the non-natural role is a sub-role of an aggregate of natural roles (we may or may not want to say that it is a sub-role of each natural role in the aggregate). Thus, while monarchy may simply be a sub-role of public authority, which is a natural role, being a fire-fighter and being a president may be sub-roles of public authority and being an employee.

There may also be merely conventional roles. Merely conventional roles are of no interest to me here. They do not role-ishly result in moral duties, permissions and rights, and non-moral duties, permissions and rights are of little more interest to me than rubber ducks are to ornithologists. :-)

Why do sets have their members essentially?

It is often said that sets have their members essentially. Why? Normally the finger gets pointed at the axiom of extensionality which says that two sets that have exactly the same members are equal. The axiom of extensionality does not seem to imply by itself that sets have their members essentially. Consider a set theory without ur-elements (i.e., every member of a set is itself a set). Suppose that in the actual world there are two sets, Bob and Jane, where Bob is has no members and Jane's only member is Bob. Then imagine that in every other worlds all the sets are exactly like in the actual world, except that Jane has no members and Bob has Jane as its only member. I do not see—though I could be missing something—how this violates any of the axioms of set theory. If the axiom of extensionality were extended to a transworld context (if A in w₁ has the same members as B does in w₂, then A=B), we would get a contradiction, but the axiom of extensionality is an intraworld axiom.

But nonetheless I have reason to think that sets have their members essentially if the axiom of extensionality is correct, namely that what seem to me to be the two most plausible intuitive kinds of reasons to think sets need not have their members essentially conflict with the axiom of extensionality. We can argue for this by considering paradigmatic cases of the two reasons.

First kind of reasons: Losing members. Let S be the set { Quine, Wittgenstein, Socrates }. Then, the intuition goes, in a world w* where Quine never comes into existence, S only has two members, namely Wittgenstein and Socrates.

But in the actual world there is also the set T={ Wittgenstein, Socrates }. And we have every reason to think that T will exist in w* and have the same two members there that it actually has. Thus, in w*, the sets S and T have the same members, namely Wittgenstein and Socrates, and so in w* we have S=T. Since identity is necessary, it follows that actually S=T. But that's absurd since s has three members and T has two. So we need to reject

Second kind of reasons: Sets defined by descriptions. Let M be the set of all mathematicians. Intuitively, in a world where Hilbert becomes a biologist instead of a mathematician, M still exists, but Hilbert isn't a member of M. The intuition here is that in every world, or at least every world with mathematicians, M has as its members all and only the mathematicians.

But by the same token, if B is the set of all biologists, then in every world where there are biologist B will have as its members all and only the biologists. But there is a world where all and only biologists are mathematicians. In that world, then M=B. But since identity is necessary, it follows that actually M=B, and hence that Darwin was a mathematician and Hilbert a biologist, which is absurd.

Maybe you have some other reasons to doubt the essentiality of set membership?

Marriage and natural kinds

On a New York Times blog, Wedgwood has offered an interesting argument for same-sex marriage. The argument is focused on what he calls "the 'social meaning' of marriage", which "consists of the understandings and expectations regarding marriage that almost all members of society share." He notes that this meaning "cannot include any controversial doctrines", but rather must include only uncontroversial assumptions about the nature of marriage. He then argues that same-sex couples have the same interest in having access to an institution that yields the same understandings and expectations as heterosexual couples do. Therefore, the social meaning needs to be shifted in such a way as to give access to marriage to same-sex couples. (This quick summary doesn't do full justice to Wedgwood's rich piece.)

But now compare the social meaning of "marriage" to the social meaning of "water", defined by the understandings and expectations regarding water that almost all members of society share.

There is indeed such a social meaning of water. And this social meaning of water may well help fix the meaning of the word "water". But it does not fix the meaning on its own, as we see from Twin Earth cases. On Twin Earth, suppose, there are beings like us, with a language like ours, and they have a substance that behaves just as water does, and hence obeys the same ordinary understandings and expectations that water does, but its chemical structure is instead XYZ.[note 1] The Twin Earthers' word "water" does not refer to H₂O, but to XYZ, even though their word "water" has the same social meaning as our word "water" does. Thus, the meaning of the word "water" is not exhausted by the social meaning of water.

If we focus on social meaning as central to meaning, we're going to say something like: "The word 'water' refers to that natural kind in the vicinity of the word's users that best fits with the users' 'understandings and expectations.'" In other words, the meaning of "water" isn't fixed by the social meaning alone: it is fixed by social meaning plus facts about what natural kinds are really exemplified in the vicinity of the speakers. Moreover, water will have non-obvious essential properties, such as that its molecules are a union of two atoms with atomic number 1 with one atom with atomic number 8. We aren't going to find these properties by examining the social meaning of water.

Now go back to marriage. If "marriage" is analogous to "water", the social meaning of marriage does not by itself fix the meaning of the word "marriage". Rather, "marriage" refers to that natural kind of relationships in the vicinity of the speakers that in fact best fits with the language-users' understandings and expectations. But then we would expect marriage to have all sorts of essential properties that go beyond the social meaning. And it could turn out that the only natural kind of relationships in the vicinity that fits with our understandings and expectations also has the essential property of being a union of a man and a woman.

I take it that Wedgwood is assuming that marriage is unlike water, that the meaning of marriage is exhausted by its social meaning rather than there being a mind-independent natural kind for the word "marriage" to refer to. But whether this is so is the most fundamental question in the debate, and he gives no case his view on it.

A less deep problem with Wedgwood's line of thought is that, as he himself acknowledges, a part of the shared understanding of marriage is that sexual activity is a normal aspect of marriage. But if same-sex sexual activity is not permissible, then same-sex couples do not have the same interest in an institution that comes along with an assumption of normative sexuality, since it is generally not in one's interest that people assume one is doing something impermissible (whether or not one is actually doing it). Thus, Wedgwood's argument requires that same-sex sexual activity is permissible. But his typical philosophical or theological opponent will deny this permissibility.

Sexual ethics and moral epistemology

Let A be the set of all the sexual activities that our society takes to be wrong. For instance, A will include rape, voyeurism, adultery, bestiality, incest, polygamy, sex involving animals, sex betweeen adults and youth, etc. Let B be the set of all the sexual activities that our society takes to be permissible. For instance, B will include marital sex, contracepted marital sex, premarital sex, etc.

Now there is no plausible philosophical comprehensive unified explanatory theory of sexuality that rules all the actions in A wrong and all the actions in B permissible. There are comprehensive unified explanatory theories of sexuality that rule all the actions in A wrong but that also rule some of the actions in B wrong—the various Catholic theories do this—as well as ruling same-sex sexual activity wrong (which isn't in B at this point). There are also comprehensive unified explanatory theories of sexuality that rule all the actions in B permissible but that also rule a number of the actions in A permissible—for instance, theories that treat sex in a basically consequentialist way do this.

If this is right, then unless a completely new kind of theory can be found, it looks like we need to choose between three broad options:

1. Take all the actions in A to be forbidden, and revise social opinion on a number of the actions in B.
2. Take all the actions in B to be permissible, and revise social opinion on a number of the actions in A.
3. Revise social opinion on some but not all of the actions in A and on some but not all of the actions in B.

Moreover, the best theories we have fall into (1) or (2) rather than (3). So given our current philosophical theories, our choice is probably between (1) and (2).

It is an interesting question how one would choose between (1) and (2) if one had to. I would, and do, choose (1) on the grounds that typically we can be more confident of prohibitions than of permissibles, moral progress being more a matter of discovery of new prohibitions (e.g., against slavery, against duelling, against most cases of the death penalty) than of discovery of new permissions.

It is somewhat easier to overlook a prohibition that is there than to imagine a prohibition that isn't there. An action is permissible if and only if there are no decisive moral reasons against the action. Generally speaking, we are more likely to err by not noticing something that is there than by "noticing" something that isn't there: overlooking is more common than hallucination. If this is true in the moral sphere, then we are more likely to overlook a decisive moral reason against an action than to "notice" a decisive moral reason that isn't there. After all, some kinds of moral reason require significant training in virtue for us to come to be cognizant of them.

If this is right, then prima facie we should prefer (1) to (2): we should be more confident of the prohibition of the actions in A than of the permission of the actions in B.

It would be interesting to study empirically how people's opinions fall on the question whether to trust intuitions of impermissibility or to trust intuitions of permissibility. It may be domain-specific. Thus a stereotypical conservative might take intuitions of impermissibility to be more reliable than intuitions of permissibility when it comes to matters of sex, but when it comes to matters of private property may take intuitions of permissibility to be more reliable, and a liberal might have the opposite view. But I think one should in general tend to go with intuitions of impermissibility.

Effortlessness and omnipotence

All of the extant definitions of omnipotence are missing what seems to me to be an important ingredient. A typical definition says something like: "God can do anything that's logically possible." But that's not quite enough. One needs to specify that God can do everything effortlessly. This is an easy emendation, of course, but an important one.

Eclipse

Here in central Texas, the sun set before the eclipse hit maximum.  But we had some lovely views of the moon gobbling up sunspots as the sun was setting over the lake.  Photos are taken by a Canon G7 camera, using a modified version of the sun funnel for projection.  The scope is an 8" F/4.5.  For the first photo it was stopped down to about 3".  The second photo I think used the full aperture.

And for reference, here's the sun in the afternoon, before the eclipse:

There are no morally neutral voluntary actions

1. Every action either is or not a legitimate expression of love for God and/or his creatures.
2. Every legitimate expression of love for God and/or his creatures is morally good.
3. A voluntary action that is not a legitimate expression of love for God and/or his creatures (either because it is not an expression of this love or because it is not a legitimate such expression) is morally wrong.
4. So, every voluntary action is either morally good or morally wrong.

Emphasis

• Fred may be bald, but he's not bald (he's got ten hairs).
• The biologists may know a lot of biological facts, but the mathematicians know a lot of mathematical facts.
• The roasted chicken may be healthy, but it's not healthy (it's dead).

What's the emphasis doing? I think it's some sort of narrowing of the semantic fence around the paradigm cases and the focal meaning.

Propriety and open-mindedness

A scoring rule s(i,r) measures the closeness between one's credence assignment r to a proposition and the truth value, i=0 for false and i=1 for true. I shall assume scoring rules to be continuous. Smaller scores are better.

A scoring rule is proper provided that by your own lights it does not tell you to expect a better (i.e., smaller) score if you just change your credence from r. Given a credence of r in the proposition in question, your expected score from adopting a credence of r' is rs(1,r')+(1−r)s(1,r'). So a proper scoring rule says that this function of r' achieves a minimum at r'=r. A strictly proper scoring rule achieves a minimum only at r'=r.

A scoring rule is open-minded provided that by your own lights it does not tell you to expect worse (i.e., bigger) score if you learn the truth value of some other proposition (we can think of this as the result of a binary experiment). If a scoring rule is not open-minded, then there will be circumstances where score-optimization with respect to one proposition sometimes requires you to shut your ears to other facts. A scoring rule is strictly proper provided that optimizing your score with respect to p requires you to be willing to learn the truth value of any proposition q that by your lights is not independent of p.

As Director of Graduate Studies, I have to attend graduation whenever one of our graduate students gets his Ph.D. On previous occasions, this has been very onerous. But this time I took a notepad and had a lot of fun doing math. In particular, I proved:

• A scoring-rule is proper if and only if it is open-minded.
• A scoring-rule is strictly proper if and only if it is strictly open-minded.

I've since learned from Richard Pettigrew that the implication from propriety to open-mindedness was basically already known. I don't know if the other implication was known as well.

My proof used the standard representation of scoring-rules in terms of convex fucntions, though it turns out that there are simpler proofs at least of the left-to-right implications.

Moreover, the left-to-right implications yield a proof of Good's Theorem. Just use the negative of expected utilities in practical decisions made optimally on the basis of credence r in p, given p and given not p, to define s(1,r) and s(0,r) respectively. It is trivial that this is a proper scoring rule, at least modulo continuity (but the simpler proofs of the left-to-right implications don't use continuity; I think continuity can anyway be proved in this case, but haven't checked details). Hence s is open-minded. But open-mindedness for this rule s is basically what Good's Theorem says.

Agreement and tensed propositions

Suppose I assert a proposition p, and you assert p. Then we agree, don't we?

But suppose that propositions are tensed in the way presentists typically think, so that they change in truth value. Then it could happen that I assert p, you assert p, and yet we disagree. For instance, suppose we are watching a gymnast practice. I say: "He's only now got the hang of it!" A minute later you say: "He's only now got the hang of it!" We plainly disagree. Yet on the tensed proposition view, we asserted the same proposition. On an untensed proposition view, we asserted different propositions, say <He's only got the hang of it by t₁> and <He's only got the hang of it by t₂>, where "t₁" and "t₂" are names of times (this works best with a non-Millian view of names).

A necessary condition for me to be convinced by your assertion is that I come to believe what you asserted. But not so if propositions are tensed. You say: "She's scoring a goal." Reflecting on your reliability for a bit, I accept your assertion. But if propositions are tensed, what I come to believe is not the tensed proposition that she's scoring a goal, but the tensed proposition that she's scored a goal. It's not easy to get a tenable view of untensed propositions that gives the right answer here, but it's probably not impossible.

"Thank goodness it's over!"

Some people think that the sentiment expressed by "Thank goodness it's over!" makes no sense apart from an A-theory of time on which there is an absolute present. But the parallel sentiment expressed by "Thank goodness this isn't happening to me!" surely had better make sense apart from some theory on which there is some "absolute I".

Hausdorff Paradox and conditional probabilities

The Hausdorff Paradox shows that given the Axiom of Choice, there is no finitely additive probability measure defined for all subsets of the surface of a ball that is invariant under rotations—that assigns the same measure to a subset of the surface of the sphere and to a rotation about any axis through the center of that subset. Because of results like that, the standard Lebesgue measure on the surface of a ball is only defined for some subsets, i.e., the measurable ones.

Now, in classical probability, we can only define conditional probability when we condition on an event A with non-zero probability. We then use the formula P(B|A)=P(B&A)/P(A). Some have tried to come up with axiomatizations that allow for conditioning on all non-empty measurable sets, including zero-probability ones.

We should not hold our breath for success. Here's why. Let C be a solid ball of unit volume. Let P be Lebesgue measure on C: the measure of a subset of C is just the volume of the subset. We expect the Lebesgue measure on C to be invariant under rotations about all axes through the center of the ball. We would also expect that the conditional probability to be thus invariant. I.e., if r is a rotation about an axis through the center, we would expect it to be that P(r(B)|r(A))=P(B|A). Unfortunately this cannot be done, at least not if one assumes the axioms that P(−|A) is a finitely additive measure on the P-measurable subsets of C and P(A|A)=1. For let A be the surface of a ball concentric with C but of smaller radius. Then the volume of A is zero: A is a two-dimensional surface, after all. Moreover, Lebesgue measure has the property that every subset of a set with zero measure is also measurable (and has zero measure). So P(−|A) will be a finitely additive probability measure on all subsets of A. If we have our rotation invariance condition, then P(B|A)=P(r(B)|r(A))=P(r(B)|A) since r(A)=A (the sphere A is invariant under rotation through its center). So, P(−|A) will be a finitely additive rotation-invariant probability measure on all subsets of A, which violates the Hausdorff Paradox (assuming the Axiom of Choice).

Put it differently: Any conditional probability assignment that allows conditioning on all non-empty subsets will exhibit an unacceptable rotational bias.

Morality without virtue

Habits induces correlations between choices in similar epistemic circumstances. A person who has behaved courageously in the face of physical danger on ten past occasions is more likely to be a physically courageous person, and therefore is more likely to behave courageously now when again facing physical danger, even when we control for the considerations on the basis of which we are deciding, unlike a fair coin which is not more likely to land hands just because on the ten past occasions it has done so. Our choices, moreover, modify our habits thereby even further increasing these correlations.

Now imagine persons that do not have habits in this sense. They make their choices based on the considerations present in each case, on a case by case basis. The fact that they have braved physical danger ten times does not make it more likely that they will brave it now, as long as one controls for the considerations present in the cases. Moreover, I will suppose that the motivational strength of the considerations they are deciding on the basis of does not change over time. They always find duty to have a certain pull and they always find convenience to have a certain pull, and the degree of pull does not change.

Such persons would not have character in the way we understand character. They would have neither virtues nor vices. They would have much less control over the shape of their lives. For we can shape our futures by inducing in ourselves a certain character. They could, however, influence the shape of their lives through rational means, by gaining new beliefs or by creating reasons to act.

It would be difficult for such beings to live in community. But we could imagine that one of their very clever engineers builds a mechanical sovereign who enforces basic rules for harmonious living through harsh punishment. For although there are no correlations between choices made in similar circumstances, one could change the circumstances to increase the weight of considerations in favor of actions that conduce to harmonious living. Or a prophet could convince the people that great rewards for virtue and harsh punishments for vice follow in an afterlife, and that, too, would conduce to harmony. But that still wouldn't be virtue.

The lives of such beings would be less storied. They would not exhibit the good of making a certain kind of person out of oneself. There are, indeed, many goods that they would lack.

But without any virtues or vices, these beings could still could have morally significant freedom. They could freely choose, on a case-by-case basis, whether to follow duty or some other consideration. Many of the familiar moral norms that bind us could apply to them. It would be no more permissible for them to kill the innocent or build palaces on the backs of the suffering poor than it is for us. They would have one fewer reason in favor of doing the right thing, though. If I build a palace on the backs of the suffering poor, I become a more vicious person. That wouldn't happen to them. But they would still have the simple reason that it's wrong to do this, together with extrinsic considerations coming from hope of reward or fear of punishment. And that I will become a more vicious person is only an extrinsic consideration against committing an action, anyway.

Virtue ethicists will probably disagree with me that such beings couldn't have morality. So much worse for virtue ethics. Virtues are an important component of human moral life, but I think they are not a component of moral life in general as such, just as physical interaction is an important component of human moral life, but isn't a component of moral life in general (angels have a moral life but as far as we know they have no physical interaction).