Butterflies

The following argument is valid:

1. Many butterflies are works of art.
2. If God does not exist, then no butterfly is a work of art.
3. So, God exists.

Note that with the progress of genetic engineering, we may have to modify (2) to "If God does not exist, then no butterfly that isn't genetically engineered is a work of art" and then strengthen (1) to say that many non-engineered butterflies are works of art.

The thought behind (1) is not just that butterflies are beautiful, but that the best way to appreciate their aesthetic qualities is to see them as works of art.

A being that knows much

Here is an argument inspired by the Gale-Pruss cosmological argument.

1. (Premise) Every first-order truth is knowable.
2. (Premise) The conjunction of all basic first-order truths exists and is a first-order truth.
3. (Premise) If all the basic first-order truths of a world w₁ hold at a world w₂, then w₂=w₁.
4. Let p be the conjunction of all basic first-order truths. Let w be a world where there is a being, x, that knows p. (By 1 and 2)
5. (Premise) Necessarily, if someone knows p, then p is true.
6. p is true at w. (4 and 5)
7. w is the actual world. (3, 4 and 6)
8. There actually is a being that knows the conjunction of all basic first-order truths. (4 and 7)

Note that the restriction to first-order truths makes this not be just the standard knowability paradox.

I am not giving a definition of basicness. The crucial constraint is that the notion be such that both (2) and (3) hold. If one says that every first-order truth counts as basic, then (3) is very plausible, but (2) is less clear because one worries about a conjunction that has itself as a conjunct. Or maybe we could say that the basic first-order truths are truths expressed by literals, namely truths of the form of the proposition that P(a₁,...,a_n) for some predicate P or its negation. This requires a free logic, an existence predicate, and propositions about all possible non-existent entities. Alternately, we might take "basic" to just mean "fundamental".

Note added later: I forgot the word "basic" in (3) in the original argument.

A reminder about co-authorship

If you're a philosopher or philosophy grad student (especially one at Baylor!), and like one of the arguments on my blog (including old ones in the archives), and would like to co-author with me a paper filling out and expanding on the argument, do email me--I am very open to such activity, as I do not have the time to do this by myself to every argument that I think might have promise. While some of the arguments are only worthy of being blogged--and some on reflection not even that--I think that scattered in the archives, as well as given more recently, there are a number of arguments that could be at the center of a paper.

An ontological argument

Start with:

1. (Premise) Possibly, an unlimited being exists.
2. (Premise) Necessarily, for every proposition q that is possibly true, there is a state of affairs p(q) such that p(q) grounds the possibility of q.
3. (Premise) Necessarily, if s grounds the possibility of x not existing or the possibility of x being limited, then s limits x.
4. (Premise) Necessarily, nothing limits an unlimited being.
5. (S5) If something is possibly necessary, then it is necessary.

It follows that there is an unlimited being. For suppose w is a world that contains an unlimited being u. Suppose it is true at w that u possibly does not exist. Then by 2 and 3, something at w limits u, which violates 4. So it is true at w that u necessarily exists. Suppose it is true at w that u is possibly limited. Then by 2 and 3, something at w limits u, which again violates 4. So, at w it is true that u exists necessarily and is necessarily unlimited. So, by S5, u exists necessarily and is necessarily unlimited.

The most controversial premises, I think, will be 1 and 2.

Analogy and properties

Spot is a cat and Tob is a table. Spot has four legs and Tob has four legs. But "has four legs" in these two claims is analogical. The two claims are ultimately grounded in similar sorts of facts, but not in exactly similar sorts of facts. What makes Spot have four legs is that he has certain organs of locomotion, four in number. What makes Tob have four legs is that it has certain supporting parts, four in number.

Suppose we start talking about legs with people and cats, and only later move on to tables. Then we are analogically extending the concept of quadruple leggedness to pick out that feature that in tables is similar to what quadruple leggedness is or would be like in people and cats.

We could (in the sense corresponding to metaphysical possibility) use a very well trained cat that has particularly rigid ears and tail as a three legged table, by standing it up on on the ears and tail and putting a dish on the tummy. That cat/table would have four legs in the cat sense and three legs in the table sense. We would then say that this object has four legs qua cat and three legs qua table. On my adverbial ontology, the clear way to express this is to say that the object is a cat four-leggedly and a table three-leggedly.

But if this is right, then we should say similar things about Spot and Tob. Spot is a cat four-leggedly while Tob is a table four-leggedly. And these adverbs do different, albeit analogous, things to being a cat and being a table.

The predicate "has four legs" is adverbial. Something satisfies it in virtue of being something or other in a four-legged manner.

There had better be some non-adverbial predicates. Otherwise, we will have a vicious regress or circularity.

I conjecture that all accidental predication ("accidental" in the medieval, not the modal, sense) is adverbial and analogical in this way. One needs to, as it were, fill in an underlying predicate, the "G" in "is a G Fly", in order to make sense of "is F".

The line of thought is inductive: all the accidental predicates that I can think of are analogical in this way.

Aquinas on why the incarnation is fitting

I find it striking and interesting that in the article where Aquinas officially addresses whether the Incarnation is fitting, his argument in favor of the fittingness of the Incarnation makes no mention of salvation. In other words, it's clear that the Incarnation would be fitting whether or not we sinned, though Aquinas is inclined to think that had we not sinned, the Incarnation would not have occurred. The focus on epistemic benefits is particularly interesting:

On the contrary, It would seem most fitting that by visible things the invisible things of God should be made known; for to this end was the whole world made, as is clear from the word of the Apostle (Romans 1:20): "For the invisible things of God . . . are clearly seen, being understood by the things that are made." But, as Damascene says (De Fide Orth. iii, 1), by the mystery of Incarnation are made known at once the goodness, the wisdom, the justice, and the power or might of God--"His goodness, for He did not despise the weakness of His own handiwork; His justice, since, on man's defeat, He caused the tyrant to be overcome by none other than man, and yet He did not snatch men forcibly from death; His wisdom, for He found a suitable discharge for a most heavy debt; His power, or infinite might, for there is nothing greater than for God to become incarnate . . ."
I answer that, To each things, that is befitting which belongs to it by reason of its very nature; thus, to reason befits man, since this belongs to him because he is of a rational nature. But the very nature of God is goodness, as is clear from Dionysius (Div. Nom. i). Hence, what belongs to the essence of goodness befits God. But it belongs to the essence of goodness to communicate itself to others, as is plain from Dionysius (Div. Nom. iv). Hence it belongs to the essence of the highest good to communicate itself in the highest manner to the creature, and this is brought about chiefly by "His so joining created nature to Himself that one Person is made up of these three--the Word, a soul and flesh," as Augustine says (De Trin. xiii). Hence it is manifest that it was fitting that God should become incarnate.

Of course, in the next article, Aquinas does talk of the need for the Incarnation for our redemption. It's not an absolute need, but it is necessary for our redemption to be worked in a better and more fitting way (Aquinas compares it to the necessity of a horse for a journey--presumably, one could always walk, but a horse is better). In that article, Aquinas gives ten benefits for the sake of which the Incarnation was fitting in respect of our redemption. It is interesting that penal satisfaction is only one of the ten.

And then Aquinas adds: "And there are very many other advantages which accrued, above man's apprehension." How does he know? Is it just a confidence that God does many, many good things for us? Could one argue that the Incarnation of an infinite being must somehow bring infinitely many benefits, of which only finitely many will be understandable to us?

How large a boost should the priors of simpler laws receive?

Simpler laws should have higher prior probabilities. Otherwise, the curve-fitting problem will kill scientific theorizing, since any set of data can be fitted with infinitely many curves. If, however, we give higher probabilities to simpler laws, then the curves describable have a hope of winning out, as they should.

So simpler formulae need to get a boost? How much of a boost? Sometimes a really big one.

Here's a case. Let's grant that Newton was justified in accepting that the force of gravitation was F=Gmm'/r². But now consider the uncountably many force laws of the form F=Gmm'/r^a, where a is a real number. Now in order for Newton to come to be justified on Bayesian grounds in thinking that the right value is a=2, he would have to have a non-zero prior probability for a=2. For the only way you're going to get out of a zero prior probability of a hypothesis would be if you had evidence with zero prior probability. And it doesn't seem that Newton did.

So Newton needed a positive prior for the hypothesis that a=2. But a finitely additive probability function can only assign a positive probability to countably many incompatible hypotheses. Thus, if Newton obeyed the probability axioms, he could only have positive priors for countably many values of a. Thus for the vast majority of the uncountably many possible positive values of a, Newton had to assign a zero probability.

Thus, Newton's prior for a=2 had to be infinitely greater than his prior for most of the other values of a. So the simplicity boost can be quite large.

Presumably, the way this is going to work is that Newton will have to have non-zero priors for all the "neat" values of a, like 2, 1, π, 1/13, etc. Maybe even for all the ones that can be described in finite terms. And then zero for all the "messy" values.

Moreover, Newton needs to assign a significantly larger prior to a=2 than to the disjunction of all the uncountably many other values of a in the narrow interval between 2−10⁻¹⁰⁰ and 2+10⁻¹⁰⁰. For every value in that interval generates exactly the same experimental predictions within the measurement precision available to Newton. So, all the other "neat" values in that narrow interval will need to receive much smaller priors, so much small that when they're all summed up, the sum will still be significantly smaller than the prior for a=2.

One interesting question here is what justifies such an assignment of priors. The theist at least can cite God's love of order, which makes "neater" laws much more likely.

The vertical harmony of nature?

One kind of harmony of nature is widely noted: laws of nature that hold at one place and time tend to hold everywhere else. This is a kind of horizontal harmony of nature.

But maybe there is also a vertical harmony of nature. Nature has multiple levels. There is the fundamental physics, the chemistry, the biology, the psychology and the sociology of the world; and also, along a parallel hierarchy starting with the chemistry of the world, the geology and astronomy of the world. The unity I am interested in between these levels is subtler: it is that essentially the same scientific methods yield truth at all these levels. Granted, there are modifications. But at all the levels, the same inductive techniques are used, and relatively simple mathematical models are made to fit reality.

Suppose that all the higher levels reduce to the fundamental physics. I think it is still surprising that the methods that work for the reducing level continue to work for the reduced level. And if there is no reduction, then the vertical unity is even more surprising.

There may be a teleological argument here. But I am worried about three flies in the ointment. The first is that perhaps I am exaggerating the unity of methods of investigation between the different levels. In school, we learn about "the scientific method". But in fact the methods of investigation in the different sciences are perhaps rather less similar than talk of "the scientific method" suggests.

The second is that the unity between the levels may simply be an artifact of the method. In other words, we have a certain method of mathematically and inductively modeling reality. And the levels that I am talking about are nothing but areas where the method works fairly well. And there is nothing that surprising that given an orderly fundamental level, among the infinitely many other "levels" (not all in a single hierarchy; just as above we had two separate hierarchies, one going up to sociology and another to astronomy) of description of reality, there will be some that can be modeled using the same methods, and those are the levels we give names like "chemistry" and "geology". In other words, we have a selection bias when we set out the case for vertical order.

If this worry is right, then we should only be surprised by the order we find at the fundamental level. But, my, how surprised we should be by that!

And, further, if we see things in this way, we will see no reason to privilege scientific approaches epistemologically. For there is nothing that special about the sciences. There may be infinitely many levels of description of reality which can be better known using other methods.

The final fly in the ointment is that while there are a number of levels that can be known using the same methods, there seem to be areas where we have genuine knowledge, but the scientific methods do not work: ethics is a particularly important case.

So in the end, I do not know really what to make of the vertical harmony thesis. It bears more thought, I guess.

Choice and certainty

Suppose I am certain that I will choose A? (Maybe an oracle I completely trust has told me, or maybe I have in the past always chosen A; I am not saying anything about the certainty being justified; cf. this paper by David Hunt.) Can I deliberate between A and B, and choose A?

Here is an argument for a positive answer. Suppose I am 99% sure I will choose A. Clearly, I can deliberate between A and B, and freely choose either one (assuming none of the other reasons why I might be unfree apply). The same is true if I am 99.9% sure. And 99.99%. And so on. Moreover, while in such cases it may (though I am not sure even of that) become psychologically harder and harder to choose B, except in exceptional cases (see next paragraph), it should not become psychologically any harder to choose A over B. But if it does not become any harder to choose A over B, why can't I still deliberate and choose A in the limiting case where the certainty is complete?

There will be special cases where this limiting case argument fails. These will be cases where either I suffer from some contingent psychological condition that precludes a choice in the case of certainty or where in the limiting case I lose the reason I had for doing A. For instance, if I am in weird circumstances, there may be actions where the reasons I have for choosing the action depend on my not being certain that I will choose it—maybe you offer me money to choose something that I am not certain I will choose. But apart from such special cases, the probability that I will choose A is irrelevant to my deliberation. And hence it does not enter into my deliberation if I am being rational.

What enters into my deliberation whether to choose A or B are the reasons for and against choosing the options. The probabilities or even certainties of my making one or the other choice normally do not enter into deliberation.

But what if I know that it is impossible to do B? Isn't that relevant? Normally, yes, but that's because it's a straightforward reason against choosing B: one has good reason not to choose to do things which one can't succeed in doing, since in choosing such things, one will be trying and failing, and that is typically an unhappy situation.

But what if I am certain that it is impossible to choose B? Isn't that a reason against choosing B? I am not sure. After all, it might be a reason in favor of choosing B—wouldn't it be really cool to do something impossible? Maybe, though, what the case of impossibility of choice does is it removes all the reasons in favor of choosing B, since reasons involve estimates of expected outcomes, and one cannot estimate the expected outcome of an impossibility, perhaps.

Aggregating data from agents with the same evidence

Consider a case where we have two or more rational agents who have in some sense the same evidence, but who evaluate the force of the evidence differently and who have different priors, and who assign different credences to p. Suppose for simplicity that you are a completely undecided agent, with no evidence of your own, rather than one of the people with the evidence (this brackets one of the questions that the disagreement literature is concerned with—whether if you are one of these agents, you should stand pat or not). What credence should we assign after aggregating the agents' different credences?

An obvious suggestion is that we average the credences. That suggestion is incorrect, I believe.

The intuition I have is that averaging is the right move to make when aggregating estimates that are likely to suffer from normally distributed errors. But credences do not suffer from normally distributed errors. Suppose the correct credence, given the evidence, is 0.9. The rational agent's credences is not normally distributed around 0.9, since it cannot exceed 1 or fall below 0.

However, once we replace the credences with logarithms of odds, as we have learned to do from Turing, where the log-odds corresponding to a credence p is log (p/(1−p)), then we are dealing with the sorts of additive quantities where we can expect normally distributed error. When we are dealing with log-odds, Bayes' theorem becomes additive:

• posterior-log-odds = prior-log-odds + log-likelihood-ratio.

We can think of the rational agents as having normally distributed errors for their prior log-odds and for their estimate of the evidence's log-likelihood-ratio. (Maybe more can be said in defense of those assumptions.) We idealize, then, by supposing errors to be independent. And in cases where we are dealing with independent normally distributed errors, the best aggregation of the estimates is arithmetic averaging (cf. this post on voting).

If this line of thought works, what we should do is calculate the log-odds corresponding to the agents' credences, average these (somehow weighting by competence, I suppose, if there is competence data), and then calculate the credence corresponding to that average.

This method handles symmetry cases just as ordinary averaging does. If one agent says 0.9 and another says 0.1, then we get 0.5, as we should.

But this method of aggregation yields significantly different results when some of the credences are close to 0 or 1. Suppose we have two agents with credences 0.1 and 0.99. The arithmetic average would be 0.55. But this method recommends 0.77. Suppose we have three agents with credences 0.1, 0.1 and 0.99. The arithmetic average would be 0.40. But our aggregation method yields 0.52. On the other hand, if we have credences 0.02 and 0.8, we get 0.22. All this is correct, under the normal distribution in log-odds error assumption.

If you want to play with this, I made a simple credence aggregation calculator.

This method, thus, accords greater weight to those who are more certain, in either direction. Therefore, the method suffers from the same manipulation problem that the corresponding voting method does. The method will produce terrible results when applied to agents who significantly overestimate probabilities close to 1 or underestimate probabilities close to 0—or when they lie about their credences. That's why I am only advertising this method in the case of rational agents. How useful this is in real life is hard to say. It could be that one just needs to adapt the method by throwing out fairly extreme credences, just as one throws out outliers in science, by taking them to be evidence of credences not formed on the basis of evidence (this need not be pejorative—I am not an evidentialist).

There is, I think, an interesting lesson here that parallels a lesson I drew out in the voting case. In aggregating credences, just as in aggregating votings, we have two desiderata: (1) extract as much useful information as we can from the individual agent data, and (2) not allow individual non-rational or non-team-player agents to manipulate the outcome unduly. These two desiderata are at odds with each other. How far we can trust other agents not to be manipulative affects social epistemology just as it does voting.

But here is a happy thought for those of us who (like me) have high credences in various propositions that are dear to us and where those credences are, we think, evidence-based. For then we get to outvote, in the court of our own minds (for our friends may dismiss us as outliers), more sceptically oriented friends. Let's say my credence that it's objectively wrong to torture those known to be innocent is 0.99999999, but I have two colleagues who incline to irrealism, and hence assign 0.1 to this claim. Even if I accord no greater weight to my own opinions, I still end up with an aggregate credence of 0.99.

NYT's why it's ethical to eat meat contest

The New York Times is having an essay contest (600 words, April 8 due date) on why it's ethical to eat meat.

I think the challenge is somewhat poorly defined. The challenge seems to be to argue for the thesis that it is morally permissible to eat meat when this is not necessary to human survival.

But what does the thesis mean? If it means that in all cases where human survival is not at stake it's permissible to eat meat, the answer is uncontroversially negative. For instance, when you've made a promise to a vegetarian friend not to eat meat today, it won't be permissible to eat meat on that day.

But if it means that in some cases where human survival is not at stake it's permissible to eat meat, that thesis is not particularly controversial. For instance, a typical philosophical vegetarian is unlikely to dispute that it can be permissible to eat meat from non-conscious species (worms?), accidental roadkill when doing so does not encourage further killing, or meat from a predator which one killed in the defense of one's (at least equally cognitively sophisticated) pets.

If, on the other hand, the thesis is that it's permissible to eat meat under typical circumstances obtaining in our culture, then we have to get into a messy discussion of the particular methods of meat farming in our culture, and that's the sort of discussion the contest wants us to avoid.

More on interpersonal data consolidation

This expands on the discussion here.

You query three distinguished astrobiologists each of whom is currently doing Mars research how likely they think there was once life on Mars. They give you probabilities of 0.85, 0.90 and 0.95 respectively for there being life on Mars, and explain the data on which they base their decisions. You find that each of them assigns the correct probability given their data set (hence they each have at least somewhat different data they are working with), given the same reasonable prior. Moreover, you have no other data about whether there was life on Mars.

What probability should you assign to L, the hypothesis that there was once life on Mars? The intuitive answer is: 0.90. But it turns out that what I told you in the preceding paragraph underdetermines the answer. What I said in the preceding paragraph is compatible with any probability other than zero and one, depending on what sort of dependencies there are between the data sets on the basis of which the scientists have formed their respective judgments.

For instance, it could be that they have in common a single very strong piece of evidence E⁻₀ against L but that they each have an even stronger piece of evidence in favor of L. Moreover, their respective pieces of evidence E⁺₁, E⁺₂ and E⁺₃ in favor of L are conditionally independent of each other and of E⁻₀ (on L and on not-L). In such a case, when you consolidate their data, you get E⁻₀,E⁺₁,E⁺₂,E⁺₃. Since each of the E⁺_i is sufficient to significantly undo the anti-L effect of E⁻₀, it follows that when you consolidate all four pieces of data (starting with the same prior), you get a very high probability of L, indeed much higher than 0.90. In a case where the evidence-against is shared but the evidence-for is not, the consolidated probability is much higher than the individual probabilities.

For another case, it could be that each expert started off with a credence of 1/2 in L, and then each expert had a completely different data set that moved them to their respective probabilities. In this case, when you consolidate, you will once again get a probability significantly higher than any of their individual probabilities, since their data will add up.

On the other hand, if they each have in common a single extremely strong peice of evidence in favor of L but also each have a different strong piece of evidence against L, and we've got the right independence hypotheses, then the result of consolidating their data will be a small probability.

Both scenarios I've described are compatible with the setup. And if one assigns numbers appropriately, one can use these two scenarios to generate any consolidated probability strictly between 0 and 1.

The lesson here is that the result of consolidating expert opinions is not just a function of the expert's credences, even if these credences are all exactly right given the evidence the experts have. Consolidation needs to get below the hood on the experts' credences, to see just how much overlap and dependence there is in the evidence that the experts are basing their views on.

We can, however, give one general rule. If the experts are basing their views on entirely independent (given the hypothesis and given its negation) evidence, and are starting with a prior credence of 1/2, then the consolidated odds are equal to the product of the odds, where the odds corresponding to a probability p are p/(1−p). (It's a lot easier to Bayesian stuff in terms of odds or their logarithms.)

Time and sacrifice

Suppose that I am now undergoing suffering S on account of a greater good G. If there was no way of gaining G or something comparable without S or something comparable, and if G obtains, then I would rationally say: "It was worth it."

Notice that for the "It was worth it" judgments, it does not matter whether G is past, present or future. All that matters is that G be actual. You may wonder briefly how one can undergo suffering on account of a greater good. Time travel is the exotic case—I can get a tetanus shot in order to avoid getting tetanus in the Cretaceous. But the humdrum case is where S is a cost of the good G: perhaps I worked really hard to gain G yesterday, and today I am suffering exhaustion.

Suppose, on the other hand, I now undergo suffering S in account of a greater good G that occurs in some other possible world. For instance, I endure penury because I have spent my money building a robot that digs in my backyard looking for diamonds. I fully know that there are no diamonds in Texas, but there is a possible, though I am quite sure non-actual, situation where tomorrow someone will bury a treasure trove of diamonds in my backyard. In that possible situation, I will get very rich. But it is silly to endure actual penury for the sake of merely possible riches.

So for a good G to make a sacrifice S worthwhile, it matters a great deal that the good occur in the actual world. But it does not matter whether G occurs in the past, present or future.

This is unsurprising to eternalists. But it should be puzzling to presentists and growing blockers who think that present goods really exist while future ones do not.

Two presentist ways of seeing worlds

If presentism is true, then right now, call it t₂, the proposition B that Bucephalus exists is false, but it once was true, namely at t₁. Now, at every time a token of the following sentence expresses a truth:

0. For all p, a proposition p is true if and only if it is true at the actual world.

Now, let's imagine ourselves at t₁. Then Bucephalus exists. Thus, B is true. Moreover, (0) expresses a truth, and so B is true at the actual world. So at t₁ the sentence

1. B is true at the actual world

expresses a truth. But now let's return to our time. B is false. But (0) expresses a truth, and so the sentence

2. B is not true at the actual world

does expresses a truth. Thus, (1) expresses a truth when said at t₁ but expresses a falsehood when said at t₂. This shows that either:

3. "The actual world" refers to different worlds at different times

or

4. The proposition that p is true at w can change in truth value, even if "p" and "w" refer rigidly to a proposition and a world, respectively.

Thus, the presentist has two ways of understanding possible worlds. Either possible worlds are tensed, so that at every time we inhabit a different possible world (that's option (3)) or else the "true at" relation is tensed, so that we inhabit the same world at different times, or when we say at t that p is true at w, we say something true if and only if p is true at t at w.

I think there is a problem for (4). Let p be the proposition that horses do or do not exist. Let t be the actual present time. Then p is true at every world, since it's a necessary truth. Now consider a world w where the time sequence does not include t. There are several options for this. Maybe in w, time comes to an end in 2011. Maybe time is discrete in w while in our world it is continuous, and so w either includes no times from our world or else w "skips over" t. Or maybe for some other reason the time sequence in w is radically different from our world's time sequence. Then p is true at w. But on (4), when we say that p is true at w, that is true if and only if p is true at t at w. But nothing is true at t at w, since t isn't a time at w.

Here's a slightly different way to see the point. When p is true at w, it is true either because there are or because there are not horses at w (this is an uncontroversial case of disjunctive grounding). Suppose it's true because there are not horses at w. But at which time are the horses not there at w? After all, w could have horses at some but not other times. Presumably, the relevant time is the present time. On proposal (3), every world comes along with its own present time, and this is fine. But on proposal (4), a world's relevant present time is our present time, and w doesn't have our world's present time.

One could try to solve this with counterpart theory for times. But one can suppose w won't have a counterpart to our time.

Here's a bolder move to defend (4) against our argument: The accessibility relation between worlds differs between times. The proposition p isn't true at all worlds, but only at all accessible worlds (this may or may not involve a denial of S5—S5 does not say that all worlds are accessible, but only that accessibility is an equivalence relation). And a world is only accessible if it includes the present time (or a counterpart to it?). This has the implausible consequence that what is metaphysically possible changes with time. For instance, if in w the time sequence comes to an end with 2011, then the proposition that w is actual was possible in 2011, but is no longer possible. But it's implausible that what is metaphysically possible changes with time.

If this is right, then the presentist should embrace (3). But is (3) plausible? Do we really live in different worlds at different times?

The presentist's other move is simply to abandon talking about worlds, and instead talk about, say, abstract times (in the Crisp sense).

Presentism and future-based theodicies

Suppose that:

1. There was or is a time t at which God was justified in permitting the then-present horrendous evils of the world only in reference to non-present future goods that God would bring out of them.

For instance, it may be that prior to the Incarnation, God's justification in permitting the evils that then existed involved the Incarnation. Or it may be that many evils are only justified in reference to an afterlife in which they are redeemed and defeated.

For Augustinian reasons according to which evil is but a privation, I am not that sure of (1). But the practice of offering answers to the problem of evil almost always makes significant reference to the future, and this makes me think that (1) is generally taken to be fairly plausible.

Now suppose presentism is true. Then only present goods exist. Imagine that the time t mentioned in (1) is present. Then the present horrendous evils of the world are only justified in reference to future goods, which don't exist. But how can a horrendous evil—say, an instance of truly intense suffering—be justified in reference to something that simply and certainly does not exist? A world that contains horrendous evils and no justifying goods seems to be a pretty bad world, the kind of world that it is hard to see—except on Augustinian grounds that (1) rejects—how God could create it.

Granted, it may be true at t that there will be a justifying good. But is the fact that there will be a good itself a good? Is it a good during the Peloponnesian War that Tolstoy will one day write Anna Karenina? The eternalist can say that during the Peloponnesian War it was true that Tolstoy's writing of Anna Karenina exists. But the presentist cannot say such a thing. Or, more weakly, maybe it is a good thing that a good will eventuate. But the present good of its being the case that a good will eventuate is a shadow of the value of the eventuating good itself. Theodicy is hard enough without one having to use shadows of values.

But perhaps what in general (and not just in the divine case) justifies permitting an evil isn't an actual good, but an likelihood of a good? But the relevant kind of likelihood is epistemic. And if presentism is true, then at t God knows for sure there is no justifying good.

If this line of thought is correct, then a presentist cannot make use of future-directed theodicies. And to the extent that (1) is plausible, a theist should not be a presentist.

Beauty, observation and objectivity

The following fact is typically seen as evidence for the subjectivity of beauty:

1. Very long necks look beautiful to the Padaung but not to contemporary Americans.

But the following fact is not typically seen as evidence for the subjectivity of beauty:

2. Van Gogh's Wild Roses is looks very beautiful by visible light but not so much by x-ray.

Why do we not see (2) as evidence for subjectivity about beauty? I think the answer is simple: Wild Roses is no more meant to be viewed by x-ray light than the Moonlight Sonata is meant to be viewed visually in Fourier transform. Wild Roses and the Moonlight Sonata are intended to be beautiful in those respects that are perceived visually or aurally, respectively, and they succeed admirably at these aims.

We can be a bit more subtle here. A microscopic examination of Wild Roses is not going to reveal the relevant beauty of the work, nor will an auditory examination of the individual notes of the Moonlight Sonata. These works are beautiful in respect of those features that are salient to the appropriately trained "eye" or "ear"—and of course it is not the literal eye or ear that is mainly being trained.

But why not say the same thing about long neck of the Padaung woman, then? She intends her long neck to be beautiful in those respects that are salient to trained Padaung observers. Maybe she is beautiful in respect of her long neck in contemporary North America, too, but we lack the training to make salient to us the features that make her beautiful.

Moreover, it is important to note that the features that make something beautiful may very well be relational features. A part of what can make a work beautiful is precisely the allusions to other works and to the outside world—what makes Anna Karenina a great work of art is in part that the people in it are like real people (which does not mean that every work of literature needs to have people in it that are like real people). Thus it may be that the Padaung woman's long neck is beautiful in part precisely by its relation to particular social practices (and hence when she travels to North America, and is no longer appropriately related to these social practices, she ceases to be beautiful in respect of her neck). Recognizing the aesthetic role of such relations is not a form of subjectivism, relativism or contextualism—it is no more that than recognizing the aesthetic role of the reality of the characters in Anna Karenina makes one a subjectivist, relativist or contextualist.

(Of course, there is also the possibility that the Padaung are simply wrong in their aesthetic judgment. But it is hard to say that without their training. And the possibility of their being wrong is significant evidence for objectivism about beauty.)

[Edited on March 16, 2011, to remove near-contradiction. -ARP]

An account of laws

According to the Lewisian best system analysis of laws, a proposition p is a fundamental law if and only if it is an axiom in the best system. There is room for variation in the concept of a best system, but a standard version in deterministic settings is that the best system comprises only truths and optimizes the brevity of its axioms and the informativeness of its theorems about the world. The biggest problem for me with the best system account is that the fact that something is an axiom in the best system simply does not make it be explanatory.

I think this is a better account. A proposition p is a fundamental law of nature provided that:

1. p is an axiom in the best system, and
2. God wills p, as such.

(I am not sure if the will in (2) should be taken to be antecedent or consequent. If miracles are counterinstances to laws, it must be antecedent. But a lot of people think that's a bad account of miracles, and that allows it to be consequent.)

The "as such" in (2) rules out a case where God instead of willing p, wills something that entails p.

This account solves the explanatory problem with Lewis's account by making the fundamental laws be explanatory. They are not explanatory directly because they are axioms in the best system, but rather because God wills them.

Interestingly, I think (1) may imply (2) in the actual world, by divine omnirationality. For that p has the kind of simplicity and fecundity that axioms in the best system are going to have gives God a reason to will p. And since p in fact holds, presumably God willed p. The only exception is going to be if p reports the sort of thing that God has reason to distance himself from. Suppose, for instance, that everyone who is tempted a certain way sins. Then that universal generalization might be a best system axiom, but God has reason not to will it. But in fact it does not seem that any axioms in our world's best system are going to be like that—such regularities don't seem to be far-reaching enough. All the candidates we hear about from physics are propositions that God does not seem to have reason to distance his will from.

If this is right, then in the actual world, all the axioms in the best system are fundamental laws, and Lewis is contingently right. Moreover, this line of thought shows that the fact that p is an axiom in the best system makes it likely that God wills it. Consequently, as long as we know that God exists, we get to keep the epistemological benefits of Lewis's system.

Of chocolates and laws

According to David Lewis, the fundamental laws of nature are those propositions that collectively optimize a balance of the twin desiderata of informativeness and simplicity. (You can get maximal informativeness by listing all the facts about the world, at the expense of maximal complexity of description, and you can get maximal simplicity by saying nothing, at the expense of minimal informativeness.) And laws are what follows from the fundamental laws.

But laws are explanatory. While Lewis's laws aren't.

Imagine a finite world w₁ which contains a powerful contingent magical being who creates a very large number N of golden boxes, and places a chocolate in each one, to fulfill some aesthetic goal—chocolate goes well with gold. No other explanation of the chocolate content of the boxes exists at w₁.

We can ask at w₁ why the third golden box contains a chocolate. And the answer is that the magical being put a chocolate in every golden box.

Now imagine a world w₂ just like w₁ but where the magical being doesn't exist and the boxes and chocolates come into existence ex nihilo. Nothing gets added to w₂ that wasn't there in w₁. Plainly at w₂ there just is no explanation of why the third golden box contains a chocolate.

If the number of boxes N is large enough, the proposition that every box contains a chocolate will be informative enough that it'll be included in the system that maximizes informativeness and simplicity (as N increases, the informativeness of the proposition increases but its simplicity stays constant).

But then if Lewis's account of laws is correct, then at w₂ it will be a law that all golden boxes contain chocolates. But laws are explanatory. So at w₂ we'll be able to explain why the third golden box contains a chocolate. But we said that at w₂ there is no explanation of this!

So the Lewisian account of laws is wrong.

Options: (1) Deny that laws are explanatory. (2) Abandon the Lewisian account of laws. (3) Deny that it is possible to have uncaused chocolates.

I think that (3) by itself won't do, because we can run the argument on a counterpossible. And (1) is unattractive. That leaves (2).

In summary, I think "Lewisian laws" aren't explanatory, and hence aren't laws.

Principle of Double Effect as an action-requiring principle

Normally the Principle of Double Effect (PDE) is taken to be a permissive principle: it gives permissibility conditions for actions that are foreseen to have an evil effect. Roughly, the conditions are: the action is not in itself wrong; the evil is not intended as an end or as a means; and the evil is proportionate to an intended good.

Suppose that we adopt the thesis that refraining from an action is itself a kind of action, and that refraining can have effects just as any other action can. Call this the Refraining Parity Thesis (RPT).

Given RPT, it turns out that many of the actions that the PDE is used to justify are actually actions that the PDE actually requires. Suppose, for instance, that dropping a bomb on the enemy headquarters will cause a handful of civilian deaths but the deaths of the military leadership will lead to an early end to the war, saving thousands of civilian lives. Given RPT, consider the action of refraining from bombing. Refraining from bombing (considered maybe as an action of a military commander) has an intended good effect, the saving of the lives of the civilians near the headquarters, and an unintended evil effect, the deaths of thousands of civilians later. It is very plausible that the evil is disproportionate to the good, and hence PDE does not allow one to refrain from bombing.

There will, however, be cases where PDE allows but does not require an action. Suppose that I could save your life by jumping on a grenade. The PDE allows me to jump. The intended effect is saving your life by the absorption of kinetic energy, the foreseen evil is my death, and my death is not disproportionate to saving your life. But refraining from jumping is also permissible. The intended effect is saving my life, the foreseen evil is your death, and your death is not disproportionate to saving my life.

I don't know if RPT is true. I am inclined to think that refraining is not on par with positive action.

Paul Symington on Sophie's choice

Paul Symington has an intriguing post on Sophie's choice, where he defends the thesis that if virtue ethics is right she should have refused to cooperate. I disagree, but I think the post is really worth thinking about.

Animal consciousness

Sometimes I come up with an argument such that I can't tell for sure if it's more a joke or a really interesting argument. The following is a case in point:

1. (Premise) If some non-human earthly animals are conscious, all normal mammals are conscious.
2. (Premise) There have ever been several orders of magnitude more non-human mammals than humans.
3. (Premise, plausibly a consequence of 2) If all normal mammals are conscious, I should very strongly expect not to experience reality as a human.
4. (Premise) I experience reality as a human.
5. So, probably, not all normal mammals are conscious. (By 3 and 4)
6. So, probably, no non-human earthly animals are conscious.

As for (1), if some non-human earthly animals are conscious, a line must be drawn as to where consciousness is found. There are two main plausible places to draw such a line: (a) humans versus other animals, and (b) animals with sophisticated brains versus other animals. If we draw the line in the second place, all normal mammals will be conscious. As for (2), I don't have data as to how many mammals there are on earth. I saw an unreferenced "400 million" online, and a referenced somewhat smaller estimate for the number of birds (and I could run the argument with birds, too, I think). There are apparently roughly as many rats and mice in the world as humans. And there have been non-human animals for millions of years before there were humans.

I think the difficult philosophical question is whether (3) is true and what sense can be made of it.

I am more inclined to see this argument as a joke, or maybe as a challenge to figure out how anthropic arguments work.

Inductive scepticism and multiverses, theistic and secular

You have a minor ailment. You wonder to yourself if it's going to clear up by itself within a week. Suddenly you receive utterly conclusive evidence that an omniscient and perfectly honest being is speaking with you. Now consider the following case:

1. The being informs you that there is a twin-earth, where everything has hitherto been exactly like on earth. In particular, twin-earth contains a duplicate of you, whose life has hitherto been exactly like yours. In particular, your twin has the same minor ailment you do. Moreover, the being informs you that exactly one of you and your twin will have their ailment clear up within a week.

The following is clear: at the end of this, you should not be at all confident that your ailment will clear up within a week. I claim that this is still true in each of the following variant case:

2. Just like (1), except that the being also gives you some additional information afterwards. He tells you that on both earth and twin-earth, superb medical studies of the ailment have been done, and they have concluded that in the vast majority of cases like yours, the ailment clears up within a week.

The additional information about the medical studies is trumped by the information that of you and your twin exactly one will have the ailment clear up. You still should not be confident that your ailment will clear up in a week. Next suppose:

3. Just like (2), except that instead of the being telling you about the studies, you knew ahead of the communication from the being about these studies in your world, and after the being told you about the twin, you inferred that there must be such studies on twin-earth, too, since twin-earth has up to now been just like earth.

If in case (2) you shouldn't be confident that your ailment will clear up in a week, likewise you shouldn't be confident in case (3). Next consider:

4. Just like (3), except now the being informs you that there is an infinite number of such twin-earths, and if we let E be the set whose elements are earth and these twin-earths, then on infinitely many members of E, you or your twin (as the case might be) recovers within a week, and on an equal infinity of members of E, you or your twin does not recover.

Multiplying the twin-earths infinitely in this way should still not allow you to be confident that you'll be one of the lucky ones who recovers within a week. Finally, consider:

5. Just like (4), except that the twin-earths have not been exactly alike, but extremely similar, with any differences being far several orders of magnitude below the ability of our scientific experiments to discriminate between.

This should not make a difference. You still should not be confident of recovering within a week.

Now, if we are justifiably sure that there are infinitely many universes, then we should expect that there are universes that contain near twins like in (5), and we should expect that infinitely many of them your near twin recovers and on an infinity of them your near twin does not recover, and you should not be confident that you'll recover in a week. And if you aren't sure about there being infinitely many universes, but you simply assign a high probability to that hypothesis, then you should significantly lower your confidence that you will recover, below the confidence given by our best medical studies.

This applies to David Lewis's plurality of worlds. It seems to apply to scientific multiverse theories. And it applies to theistic versions of Lewis's theory, like those by Donald Turner or Klaas Kraay, on which only universes worthy of being created exist, since there plausibly is a just as big infinity of worlds worthy of creation where you don't recover in a week as ones where you do. (This is clear if our universe is far enough above the cut-off line that your failing to recover in a week will not push it below the cut-off line. If our universe is too close to the cut-off, then to ensure worthiness, some kind of a compensating good would also have to be present.)

Thus, infinite multiverse theories should sap our confidence in scientific predictions. This is particularly problematic for scientific multiverse theories.

I think the controversial move will be the transition from (3) to (4) actually.