Advent chains

My friend Abigail Tardiff sent me this great Advent paper craft [PDF], which I constructed with the kids today, and hung from a folded-up projection screen.

Attached is an easy Advent activity, suitable for kids or grown-ups. Cut on the lines to make strips, and use a stapler to form the strips into paper chains, which represent the chains of sin and death. Then each day of Advent, starting this Sunday, remove one strip and read it. Except for December 19, which is from the Canticle of Zechariah, they are all Old Testament prophecies of the coming of the Messiah.
Each strip also has a Jesse Tree ornament (not related to the prophecy) on it that you may color and cut out, and hang on a branch. The Jesse Tree tells the history of Salvation, beginning with Creation and ending with Emmanuel, God With Us. During Advent, we tell our children these stories, because they tell why we need a Savior, and how God prepared the world for His coming.
The last seven ornaments are the "O Antiphons" taken from the evening prayers of those days. You can read about them here: http://en.wikipedia.org/wiki/O_antiphons. The hymn "O Come, O Come Emmanuel" is based on the O Antiphons.
Feel free to pass these around. I chose the prophecies, but I kiped uncopyrighted images from various places on the net.
Happy Advent from the Tardiff family.

Some problem sentences for expressivist views in ethics and aesthetics

According to expressivist views in ethics and aesthetics, sentences like "Murder is wrong" or "The Mona Lisa is beautiful" do not express a proposition, but express some kind of an attitude. I think many propositional attitudes are problematic for expressivists. Here is one that is perhaps not: "Sally thinks murder is wrong." This, the expressivist can say, is a sentence expressing the proposition that Sally has the attitude A, where A is the attitude that she would express by saying "Murder is wrong." But some are much harder:

1. Sam knows that cheating on exams is wrong.
2. Alex is unsure whether Picasso's cubist paintings are any good.
3. The younger Augustine feared that fornication might be wrong.
4. Dr. Jones hopes that her proposed experiment is morally acceptable.
5. Mark hopes that the sculpture that he is working on will be beautiful.

Shifty propositions

Say that a proposition p is shifty provided that there are worlds w₁ and w₂ at which p holds, and a proposition q such that q grounds p at w₁ but q does not ground p at w₂.

A proposition p is non-shifty provided that either it cannot have a ground or there is some proposition q such that p entails that q grounds p.

The proposition that Obama or McCain is president is shifty: in some worlds it is grounded in Obama being president and in others it is instead grounded in McCain being president.

Propositions of the form of the proposition that N exists are non-shifty, if "N" directly rigidly refers to a substance.

Propositions expressed by typical simple subject-predicate sentences are shifty if trope theory is true. For instance, that I am sitting is grounded by the proposition that I have s₁, where s₁ is the trope of sitting that I actually have, but could have been equally well grounded by the proposition that I have s₂, where s₂ is some other trope of sitting. On the other hand, if Platonism is true, then sentences whose subject term directly refers and whose predicate expresses a property are going to be non-shifty, as in every world where they are true, they are grounded by some proposition of the form of the proposition that N exemplifies P.

Amusing probability case

Write down a decimal point. Then choose a digit at random, with equal probability 1/10 of each possible digit. Repeat ad infinitum, with all the digits chosen independently. Let X be the number you've written down the infinite decimal expansion of.

Suppose you find out that X is going to be either 1/4 or 1/3. Which of the two is more likely? Answer: 1/4. For there are two ways of getting 1/4: 0.250000... and 0.249999.... But there is only one way of getting 1/3: 0.333333..., and each infinite sequence is equally likely. Thus, intuitively P(X=1/4 | X=1/3 or X=1/4)=2/3. Surprised?

Another interesting fact here. In the technical probability-theory sense, X is uniformly distributed on the interval [0,1]. But in the intuitive sense, it's not. So the technical probability-theory sense does not capture the notion of uniform distribution.

Similarly, the technical probability-theory sense of independence does not capture the intuitive notion of independence. Suppose that a random process uniformly picks out a number Y in the interval [0,1], and suppose you get a dollar if and only if the number is 1/2. Let A be the event that the number picked out is 1/2 and let B be the event that you get a dollar. Then P(A&B)=P(A)=0=P(A)P(B), and hence in the probability-theoretic sense A and B are independent. But intuitively they are far from independent: B is entirely determined by A.

Maybe a better definition of independence for philosophical (though maybe not mathematical) purposes is that both P(A|B)=P(A) and P(B|A)=P(B). And then conditional probabilities should not be defined by ratios of unconditional probabilities.

Koons on grim reapers and Kalaam arguments

I've been telling people about this paper by Rob Koons, defending a grim-reaper based Kalaam cosmological argument, but I didn't realize it was online.

Spinoza's argument for internalism about truth

Internalism about truth holds that a belief's being true is a function of things internal to the mind of the believer. Coherentism and Spinoza's extreme rationalism are two kinds of internalisms about truth. Spinoza's argument in the Treatise for the Emendation of the Intellect is basically:

1. If internalism is not correct, truth is not worth having.
2. Truth is worth having.
3. Therefore, internalism is correct.

For (1) to be at all plausible, we need "worth having" to mean intrinsically worth having, and that makes (2) less plausible, though I think (2) remains true. But I deny (1), with or without the qualification, because some things can be intrinsically worth having without being internal or intrinsic to the person. Thus, it is worth having one's friends do well, even though my friends' doing well is not internal or intrinsic to me. Of course my friends' doing well tends to affect me. But not always: my friend could be doing well in my absence, without any contact we me, and that directly makes me better off.

One can also run the argument in terms of knowledge instead of truth. (I think for Spinoza the two come to the same thing! Spinoza thinks knowledge is true belief, but he has high standards for what counts as true belief—beliefs not justified up to Cartesian standards need not apply.)

Actuality, Possibility and Worlds, and Principle of Sufficient Reason for Kindle

I just noticed that my Actuality, Possibility and Worlds book is available for Kindle for under $18. I actually didn't even know (or at least remember) it was available for Kindle.

Update: And The Principle of Sufficient Reason is available for about the same price, too.

Self-organization: Another step in the dialectics

Suppose that it turns out that, given laws of nature like ours, all sorts of neat self-organization—like what we see in evolution—will follow from most sets of initial conditions. Does this destroy the design argument for the existence of God? After all, that there is complexity of the sort we observe appears to cease to be surprising.

A standard answer is: No, because we still need an explanation of why the laws of nature are in fact such as to enable this kind of self-organization, and theism provides an excellent such explanation.

But what if it turns out, further, that in some sense most laws, or the most likely laws (maybe simpler laws are more likely than more complex ones), enable self-organization processes. So not only is it unsurprising that we would get initial conditions that are likely to lead to self-organization, it is also not unlikely that we would have laws that lead to self-organization. It seems that this undercuts the modified design argument.

But I think there is a further design argument. The result that most, or the most likely, laws would likely lead to self-organization would have to be a very deep and powerful mathematical truth. What explains why this deep mathematical truth obtains? Maybe it follows from certain axioms. But why is it the case that axioms such as to lead to that truth obtain? Well, we can say that they are necessary, but that isn't a very good explanation: it is not an informative explanation. (If it turned out that modal fatalism is true, we still wouldn't be satisfied with explaining all natural phenomena by invoking their necessity. Spinoza certainly wasn't, and this he was right about, though he was wrong that modal fatalism is true.) Theism provides a family of deeper and more informative answers: mathematics is grounded in the nature of a perfect being, and hence it is unsurprising that mathematics has much that is beautiful and good in it, and in particular it is unsurprising that mathematics includes self-organization theorems, since self-organization theorems are beautiful and good features of mathematical reality.

I said that theism provides a family of answers, since different theistic theories give different accounts of how it is that mathematical truth is grounded in God. Thus, one might think, with St Augustine, that mathematical truth is grounded in God's intellect. On the theory I defend in my Worlds book, necessary truths—and in particular, mathematical truths—are grounded in the power of God.

There is, of course, an obvious argument from the beauty of mathematics to the existence of God along similar lines. But that argument is subject to the rejoinder that the beauty of mathematics is a selection effect: what mathematics mathematicians are interested in is to a large degree a function of how beautiful it is. (Mathematicians are not interested in random facts about what the products of ten-digit numbers are.) However, I think the present argument side-steps the selection effect worry.

Hormonal contraception and informed consent

In a 2000 article in the Archives of Family Medicine, Larimore argued that because the extremely high effectiveness rate of hormonal contraception is much higher than what one would expect on the basis of its often not very high rate of ovulation suppression, there is very good reason to think a significant portion of the high effectiveness rate is due to preventing implantation of the early embryo. But many women believe that the early embryo is a human being, and hence would take this effect to be a morally unacceptable abortion (and I expect there are additional women who do not take the effect to be utterly morally unacceptable, but for whom such an effect is nonetheless a significant reason against the use of the contraceptive method). Since patient autonomy requires that the patient be informed of those aspects of treatment that are salient given the patient's values and moral beliefs, the physician's duty in the case of such women is to inform the women of the risks of prevention of implantation. Because a physician may not know whether a particular woman consider this factor relevant, Larimore suggests that a physician can say something like: "Most of the time, the pill acts by preventing an egg from forming. This prevents pregnancy. However, women on the pill can still sometimes get pregnant. Some doctors think that the pill may cause the loss of some of these pregnancies very early in the pregnancy, before you would even know you were pregnant. Would knowing more about this possibility be important to you in your decision about whether to use the pill?"

Even bracketing the question whether contraception and abortion are morally permissible, Larimore is right about what is required what the current consensus on patient autonomy and informed consent. I've had a look at the titles and often abstracts of the 55 papers listed as citing Larimore's, and surprisingly none of them appears to be an argument to the contrary (though maybe some contain such an argument in their body). One interesting recent study of women in Western and Eastern Europe found that only 2% can correctly identify all the mechanisms of oral contraceptives and the IUD (for which the postfertilization effect is probably even greater), but that 73% said that their healthcare provider should inform them about effects that occur after fertilization even when these effects are before implantation. So not only is the information salient to many women, it is information that many women want.

It seems to me that pro-choice physicians should be impressed by the need to obtain informed consent for such postfertilization effects insofar as a significant part of the reasoning for the pro-choice position involves considerations of women's autonomy.

Liar and truthteller questions

Here are some fun questions:

1. Is the answer to this question negative?
2. Is the answer to this question positive?
3. What is the answer to this question?

The last one is due to my six-year-old.

Punishing what harms no one else

The following is a plausible Liberal principle:

1. It is only appropriate punish that which harms someone or something else, or is intended or sufficiently likely to do so.

(If one thinks that exposing someone to a sufficient probability of harm is itself a harm, one can simplify this. Note also that what counts as sufficiently likely is relative to the degree of punishment and degree of harm. Doing something that has a one in a million chance of causing me a hangnail is probably not deserving of punishment, except maybe of the most trivial sort, but doing something that has a one in a million chance of blowing up New York may well deserve serious punishment—cf. Parfit on small chances.)

I shall argue against this principle. Recall Mill's very plausible insistence that:

2. Being subject to social opprobrium is a kind of punishment.

(And one often would rather pay a hefty fine than be subject to social opprobrium, so it can be a heavy punishment.) Now observe:

3. Some irrational beliefs are appropriately subject to social opprobrium even though they harm no one else, are not intended to harm anyone else and are not sufficiently likely to do so.

For instance, consider someone's really crazy conspiracy theoretic beliefs which were formed irrationally, out of a desire to be different, rather than out of an honest investigation of the truth, and suppose that this is someone whom no one is likely to believe, and hence someone harmless in these beliefs. Or consider the racist beliefs of someone who is too prudent to ever act on them because she does not wish to risk social disapproval.

Therefore:

4. It can be appropriate to punish something that harms no one else, and is neither intended nor sufficiently likely to do so.

Now, one can get out of this consequence if one makes some sort of a communitarian assumption that no man is an island, that one person's irrationality is a constitutive part of the community's being thus far irrational, and is eo ipso harmful to other members of the community even if they do not themselves follow this irrationality, since now they are made to be participants in a community that exhibits this irrationality. But if one allows such "extended harms", then the principle (1) becomes uninteresting. Likewise, if one brings in "extended harms" to God, where God is said to be harmed in an extended sense provided that one acts against his will.

Could one turn this around and make it an argument for tolerance of irrationality? This would involve insisting on (1) and concluding that harmless irrationality should not be the subject of opprobrium. Yet such opprobrium seems to be an important part of what keeps us rational, and it seems obviously appropriate, especially when the irrationality is a result of the agent's moral failings.

The value of knowledge

A couple of days ago, an interesting thing happened. Our Department secretary emailed me, in my capacity as Graduate Director, to find out if student A was eligible for an MA degree. When I got to checking, I confused student A with student B, and checked that student B is eligible. Then I emailed our secretary and said that it was all fine. Consequently, I assume, she formed a justified belief that student A was eligible for an MA degree. But this justified true belief wasn't knowledge, since it relied on my testimony, and I did not know whether A was eligible for an MA.

Shortly thereafter, I realized my mistake. I then checked whether A was eligible, and found that indeed A was eligible. Next, I wondered what to do. Our secretary did not know that A was eligible. But she did, as a result of my mistake, have a justified and, as it happened, true belief. I could easily turn her justified true belief into knowledge by emailing her about what happened.

If knowledge has a value over and beyond the value of justification and truth, then I had a reason to email her. But it seems like it would be pointless to send a correction email under the circumstance (or at least, giving her knowledge would not be a point). And, if it would be pointless, then it seems that there is no value over and beyond the value of justification and truth.

However, a colleague suggests that perhaps the right interpretation of the situation isn't that there is no value in knowledge over and beyond justification and truth, but that there is so little value that it is outweighed by the disvalue of bothering a very busy person with yet another email.

The fecundity of Spinoza's claims

Say that the fecundity of a claim in a logically interconnected text, like Spinoza's Ethics, is the number of claims that logically depend on it. Using the Tredwell adjacency data, I sorted the claims in Spinoza's Ethics in order of decreasing fecundity. We can measure the fecundity in percentages: the percentage of the claims that depend on the given claim.

The result is here. (The explanations of what the items are are here, from Tredwell.) Fecundity is a measure of how fundamental a given claim is to the system.

The twelve most fecund claims, with their number of dependants, are:

1. 1A04: 300 (77.3%)
2. 1D03: 296 (76.3%)
3. 1D04: 295 (76.0%)
4. 1D05: 292 (75.3%)
5. 1A06: 291 (75.0%)
6. 1A01: 291 (75.0%)
7. 1P01: 290 (74.7%)
8. 1A05: 290 (74.7%)
9. 1P04: 290 (74.7%)
10. 1P02: 289 (74.5%)
11. 1P03: 289 (74.5%)
12. 1P05: 289 (74.5%)

Unsurprisingly, they're all from Part I of the Ethics, and unsurprisingly the first six are all axioms or definitions.

The most fecund is Axiom 4, that the cognitio (understanding?) of the effect depends on the cognitio of the cause, which, through Spinoza's overreading of it (it sounds like a weak claim, and that's why we are tempted to agree, but in fact it is a strong claim), becomes the root of the epistemologically central 2 Proposition 7, which says that the order and connection of things is the order and connection of ideas. In fact, it is largely through this 2P07 that Axiom 4 gets its fecundity: 2P07 has a fecundity of 60%, and assumes nothing other than 1A04.

The most fecund derived claim is 1 Proposition 4, that distinct things must differ in attribute or mode.

Unsurprisingly, the ontological argument is central: the fecundity of 1 Proposition 11, that God exists, is 73%.

The most fecund claim from outside of Part 1 is the aforementioned 2 Proposition 7, whose centrality cannot be denied.

There are 103 propositions that have zero fecundity.

Axiom 2 of Part I has zero fecundity in the database I am using, as do 5A02 and 2A08. Due to the limitations of my method, axioms and definitions with zero fecundity don't appear in the results I linked to, though I may fix that eventually. The case of Axiom 2 of Part I interesting and surprising, since it basically states Spinoza's version of the Principle of Sufficient Reason. My feeling is that it is implicitly used all over the place.

The least fecund axiom that actually gets used is 5A01, about contrary actions, which has only one dependent. The next, somewhat more fecund axiom is 4A01, at 4% fecundity, which says that for any thing, there is a stronger thing which can destroy it.

Surprisingly to me, the least fecund axiom from Part 1 is 1A03, at 8%, which basically affirms that causation is deterministic. This may initially suggest that Spinoza's causal determinism is not as central to his thought as it is normally thought to be. But that might be too quick, because I suspect that much if not all of the deterministic import of 1A03 is found in 1A04, especially as interpreted by 2P07 and with the understanding the the logical connections between ideas are always entailment relations.

Axiom/definition dependencies in Spinoza's ethics

Using the Tredwell adjacency data, I generated a list of axiom/definition dependencies for all the propositions in Spinoza's ethics. See the Tredwell adjacency data site for a key to what the nodes (e.g., "2P07") are.

Spinoza graphs

R. F. Tredwell has the data for generating a logical dependency graph of Spinoza's Ethics. I converted the data into DOT format so it can be used with GraphViz, and used GraphViz's dot to generate visual representations.

Here is a giant zoomable graph of the whole Ethics.  (There a "View original" link there, to download the original jpeg.)

Potentially more interesting are the individual graphs of the five parts of the Ethics.  Each graph includes the items from the part in question, as well as the dependencies from earlier parts.  (Again, there is a "View original" link for each which downloads the jpeg file.)

• Zoomable Part 1 [Printable PDF]
• Zoomable Part 2 [Printable PDF]
• Zoomable Part 3 [Printable PDF]
• Zoomable Part 4 [Printable PDF]
• Zoomable Part 5 [Printable PDF]

Here is some additional material:

• All my DOT files, including a plain text version of Tredwell's adjacency data and a perl script to generate the DOT file, and a bash script to generate the svg and jpeg files.
• SVG versions of the graphs: full Ethics, Part 1, Part 2, Part 3, Part 4 and Part 5.