Tuesday, April 30, 2013

Vagueness and grounding

  1. If p is a precification of q and p is true, then p grounds q.
  2. If q is vaguely true, then q has a true precification.
  3. So, every proposition that is vaguely true is grounded.
If we could add the thesis that every grounded proposition is non-fundamental (which in another post I argued against), we could conclude that all vaguely true propositions are non-fundamental.  But even without that thesis, groundedness is evidence of non-fundamentality.  So vagueness is evidence of non-fundamentality.

Monday, April 29, 2013

A cardinality argument against five-dimensional universalism

Five-dimensional universalism (hereby stipulated) holds that if f is a partially defined mapping f from worlds to regions such that (a) if f(w) is defined, then f(w) is a nonempty region of w's spacetime and (b) f(w) is nonempty for some w, there is an object Of that exists in every world w for which f(w) is defined and occupies precisely f(w) at w. We will call a function f with the above properties a "modal profile", indeed the modal profile of Of.

I think that to do justice to the vast flexibility of our language about artifacts, if we want to be realists about artifacts, we will need to be five-dimensional universalists. Mere four-dimensionalism mereological universalism is insufficient, because there can be always coincident artifacts with different modal properties.


  1. There is a set of all actual concrete objects.
  2. There is no set of all modal profiles.
  3. If there is no set of all modal profiles and five-dimensional universalism is true, there is no set of all actual concrete objects.
  4. So, five-dimensional universalism is not true.

The argument for (2) is that there are way too many possible worlds with spacetimes to make up a set[note 1], and for each such world w there is a different modal profile[note 2], so there is no set of all modal profiles.

Eternalism and presentism

Here is an argument against eternalism:

  1. If eternalism is true, times are like places.
  2. Times are not like places.
  3. So, eternalism is false.
There are a number of arguments for (2). Many, though not all, of them have something to do with the directionality of time, given that space lacks such directionality. Now consider this parallel argument against presentism, and hence for eternalism:
  1. If presentism is true, times are like worlds.
  2. Times are not like worlds.
  3. So, presentism is false.
There are a number of arguments for (5). Here's a fun one. If I am happy now and miserable at all other times, I'm really unfortunate. If I am happy in the actual world and miserable at all other worlds, I'm really lucky one. In general, misery at other times matters in a way in which misery at other worlds does not.

So how to break this impasse? One way would be to opt for a theory other than eternalism and presentism, say growing block. Another way is to keep on adding disanalogies between times and places or between times and worlds until one of the disanalogies ends up being much stronger. Yet another way, and I think the most promising, is to embrace both (2) and (5), but explain the disanalogy in a way that is compatible with presentism or eternalism (whichever is one's preference).

One should also note that arguments from analogy tend not to be the strongest.

Saturday, April 27, 2013

A just-so story for Genesis 1-3

Consider this argument:

  1. If Christianity is right, every assertion of rightly interpreted Scripture is true.
  2. Genesis 1-3 is rightly interpreted literalistically.
  3. The approximate truth of our best relevant science contradicts the assertions of Genesis 1-3 when these texts are interpreted literalistically.
  4. Our best relevant science is approximately true.
  5. So, Christianity is not right.
Liberal Christians reject (1), and often (2) as well. Young Earth Creationists either engage in revisionary science and deny (3), or they simply deny (4).

The right way out of the argument is, of course, to reject (2). But in this post I want to undercut the argument in a very different way. Basically, I will argue against (3) by offering a just-so story that is compatible with both our best science and a literalistic reading of Genesis 1-3, without scientific revisionism, scientific irrealism, or invocations of divine or demonic deception.

I am not claiming the story is true. In fact, I think it's false. It is in tension with the Thomistic view of the soul which I hold (but I think it may be logically compatible with it—but that's a longish story). As I said, the right way out is to deny (2).

The story is simple. First, everything happens exactly as it is described in Genesis 1-3 interpreted literalistically. Everything, including a light-studded dome ("firmament"), with waters above and below, creation in six days, vegetation without any sun or moon. Eve is literally taken from Adam's side, and so on. Then Adam and Eve sin, exactly as described in Genesis 3. All this happens in a universe ("Paradise") where all of this is possible by the laws of nature.

God then kicks them out of this universe. In the process, he destroys their bodies and puts their souls in stasis. But in Paradise, there was a law of nature that when the forbidden fruit is eaten, a Big Bang will occur (or this could be a miracle), initiating a 14 billion year process leading to some pretty clever apes in a universe better suited to sinners like Adam and Eve. God then takes the matter of two of these clever apes (if animals have souls, he de-souls them first, or perhaps he simply miraculously ensures that these two don't get souls) and instills Adam and Eve's souls in this matter.

And so all the science as to what has happened in the material universe since the Big Bang is right. Of course, science doesn't talk about souls.

A materialist Christian could also run a variant of this story of Adam and Eve being asleep for fourteen billion years, but it would involve some miracles in the physical world and maybe disagreement with science at one point. (Maybe Adam and Eve's brains are put in the bodies of some apes. Or maybe God is capable of so guiding indeterministic processes that there develop two apes that are just like Adam and Eve, and God can replace them with Adam and Eve.)

Of course, I don't believe these stories. But they do show that premise (3) of the anti-Christianity argument is false.

Friday, April 26, 2013

Naturalism and injustice

  1. (Premise) All instances of severe suffering of small children are unjust.
  2. (Premise) Only things agents are responsible for are unjust.
  3. So, all instances of severe suffering of small children are things that agents are responsible for.
  4. If (3), then naturalism is false.

A quick argument for (1): all unfair things are unjust, and all such instances are unfair. The naturalist will, I think, in the end want to deny (1) if she is to remain a naturalist. However I do think a lot of people have a strong intuition that such suffering is not just really bad, but that it is unjust.
Premise (2) is very plausible.

I think (4) is plausible, as well. For while some cases of severe suffering of children are things agents are responsible for even if naturalism is true—say, suffering directly imposed by agents—there will be many cases which are not like that. Say, a couple lovingly procreates in order to share their good life with a child, and the child has a congenital disease that causes severe suffering. There is no naturalistically-acceptable agential explanation.

What sort of non-naturalistic agential explanation could be given of these injustices? Here are the three most obvious options:

  • An evil deity.
  • A devil.
  • The Fall.
Moreover, there is a special non-naturalistic story that could be given as to (1) is false: one could hold to reincarnation and say that all instances of severel suffering of small children are fair punishments for a life of wickedness.

Which is the right story? Well, it's not an evil deity and it's not reincarnation.

Thursday, April 25, 2013

The possibility of unicorns

Kripke argued that it is not possible for there to be unicorns. For "unicorn" is a natural kind term. But there are many possible natural kinds of single-horned equines that match our unicorn stories, and there is no possible natural kind that has significantly more right to count as the kind unicorn. So none of them count, and no possible world contains unicorns.

But there is another approach, through vagueness and supervaluationism. Let's say that the term "unicorn" is vague. It can be precisified as a full description of any one of the possible natural kinds of single-horned equines that match our unicorn stories.

Now consider the sentence that Kripke is unwilling to assert but which seems intuitively correct:

  1. Possibly there is a unicorn.
We get to say (1) if we can correctly affirm:
  1. Definitely, possibly there is a unicorn.
And we certainly do get to say this. For (2) holds if and only if:
  1. For every precification U of "unicorn", possibly there is a U.
And assuming that all the precifications of "unicorn" are natural kinds that are possibly instantiated (we don't allow as a precification of "unicorn" something whose eyes are square circles, etc.), (3) is true. And so even if there is no world at which definitely there is a unicorn (though there might be—maybe that world is very rich and contains animals that fall under all possible precifications of "unicorn"; Blake McAllister suggests a multiverse), it is definite that there is a world at which there is a unicorn.

Wednesday, April 24, 2013

An interesting preference structure

Sam invites me to a home-sewn costume party. While I'd love to come, I would much rather not spend the time to sew costume. Sam offers to do it for me. I know that it would take many hours for him to do it, and I would feel bad having him put this effort in when I could do it myself.

This generates a circular preference structure if we restrict to pairwise comparisons, assuming in each case that the third option is not available:

  • Not coming to party beats sewing a costume.
  • Sam's sewing a costume for me beats not coming to the party.
  • My sewing a costume for me beats Sam's sewing a costume for me.

But if all three options are available, then I think I am stuck sewing a costume for me. For I just can't let Sam do the work for me simply because it's a lot of trouble for me, assuming I can do the work myself. Initially my choices were between sewing for myself and missing the party, and I preferred missing the party. But Sam's offering of a third option forced me to switch.

This kind of thing is a way for Sam to manipulate my behavior if I am a nice guy who doesn't want to put Sam to the trouble. In the case at hand, this means that Sam probably should not make me the offer to sew the costume, since by offering, he brings it about that I will go to the trouble myself. In cases where it is important that I go to the party, this manipulation may be perfectly fine—I've used it in an important case several years ago.

Tuesday, April 23, 2013

Length and other predicates

The length of a pencil is measured in a straight line from tip to end. This is equal to the length of the region of space occupied by the pencil. The length of a rope is measured along the rope, so that the length does not change much when the rope is coiled or uncoiled, and so unless the rope is straightened out, the length of the rope is not equal to any dimension of the region of space occupied by the rope. The length of a bow is (typically) the length of the string plus three inches. On the other hand, the length of a computer program is something quite different, not measured in length-dimensions, but in units like lines or lines-of-code or characters.

Similar points apply to almost all other predicates. These are a matter of decision rather than discovery. When we extend our language to start talking about pencils, ropes, bows and programs, we also need to decide how all the many predicates that could apply to them are to be extended. Quantifier pluralism requires predicate pluralism.

Friday, April 19, 2013

Belonging and members

In Ephesians 4:25, Paul talks of us Christians being all members of one another. Karol Wojtyla thought of romantic love as directed at mutual possession (I am grateful to John Crosby for pointing this out). Plausibly, this possession is not like ownership, but more like the body parts are related to the person they belong to.

A sensible reading of these and other texts requires a non-authoritarian view of the relations between ourselves and our body parts, a view on which our authority over our body parts is limited. This has serious repercussions for things like sterilization, gender-reassignment surgery and more radical forms of elective cosmetic surgery.


Suppose I am a plumber, and I take a section of pipe, insert a blowgun dart, and blow.  I just shot a dart out of a blowgun.  When did the pipe turn into a blowgun, though?

Did it happen when I formed the intention to use the pipe as a blowgun?  No: I do not have the power to make new material objects come into existence just by thinking about it.

When I picked up the pipe?  There are at least there is contact.  But surely it's not the right kind of contact.  It would be magic if I could make a new material object come into existence by just picking up a material object with a certain thought in mind.

When I inserted the dart?  Presumably, not any insertion will do, but one with a plan to blow.  For I could just be doing plumbing, using the outer diameter of the dart to measure the inner diameter of the pipe, and that shouldn't turn the pipe into a dart.  Again, we have some magic here--thinking about the pipe in one way while inserting the dart creates a blowgun while thinking about it another way leaves it a boring pipe.  Moreover, putting the dart into the pipe seems to be an instance of loading a blowgun rather than making a blowgun.

When I fired the dart?  Surely, that's too late.  As I lift up the pipe and point it at the target, I am surely pointing a blowgun!

Further complication: I now put the blowgun down among the pipes on the back of my truck, and next day install it in Mr. Smith's sprinkler system.  Does Mr. Smith now come to be a blowgun owner, with the rights, liabilities and responsibilities attendant on having such a weapon (blowguns are illegal in California or Canada, after all)?  Moreover, did I violate my contract with Mr. Smith, for I agreed to install pipes, whereas I installed a blowgun instead?  The last question is perhaps not so pressing--for perhaps the tubularly arranged matter I installed in his garden constitutes both a pipe and a blowgun.

One might think some of the difficulties could be removed by saying that throughout I was dealing with one object, a pipe, which acquired an extrinsic property, being a blowgun.  There need be no magic when a material object acquires an extrinsic property as a result of my thinking.  When I think about your car, your car acquires the property of being thought about by me.  But this is mistaken.  Suppose I add sights to the blowgun.  The sights come to be a part of the blowgun, but they do not come to be a part of the pipe--they are, rather, attached to the pipe.  So the blowgun seems to be a material object distinct from the pipe.

The solution to all this is to deny that there are pipes and blowguns.  There is just matter (or fields) arranged pipewise and blowgunwise.  And for convenience we adopt ways of speaking that make it sound like such objects are among the furniture of the universe.

Wednesday, April 17, 2013

Necessity is not provability

A plausible account of necessity is that p is necessary provided that p can be proved in the correct logical system K and p is possible provided that its negation cannot be proved. Assuming K is axiomatizable and proves enough of the axioms of arithmetic, this account can be shown to be incorrect.

Fix any sentence s in K. It follows from Goedel's Second Incompleteness theorem that there is no K-proof of s's being K-unprovable (for if there were such a proof, then it would follow that there is a K-proof of K's consistency, since if K is inconsistent, then every sentence, including s, can be proved in K). But on the account of modality under consideration, this means that it is possible that s is K-provable, i.e., it is possible that s is necessary.

In other words, this account of modality implies that every sentence is possibly necessary. But it is absurd to think that 0=1 is possibly necessary!

I think much the same reasoning can be used to disprove Swinburne's account of necessity, since where we are not dealing with directly referential rigid designators, Swinburne's account agrees with the provability account.

I am skirting over distinctions between s and its Goedel number, but I think that's a mere technicality to work out in greater precision.

This makes for a nice way to see a relationship between the two incompleteness theorems. The first one tells us that not everything true is provable. From the second we learn that not even everything necessary is provable.

Tuesday, April 16, 2013

Asserting falsehoods

Because of various difficulties in accounts of lying and related phenomena, I've been drawn to the simple theory on which it's always morally wrong to assert falsehoods, though of course if one is justified in believing the falsehood, one isn't culpable (and it's not lying then). But Heath White's argument which I reference in this comment suggests a pretty decisive argument to the contrary. Suppose that my daughter justifiably asserted to her brother: "Daddy won't play Monopoly with us tonight." It is within my power to make this false by playing Monopoly with the kids. But if I were to make it false, then on the view in question it would be the case that my daughter did something morally wrong. And surely I have a significant moral reason to avoid bringing it about that my child did something wrong, even if it would only be inculpably wrong and my bringing it about would not be intentional.

What if Intelligent Design is irrefutable?

Consider this valid argument:

  1. (Premise) Intelligent Design is irrefutable.
  2. (Premise) Intelligent Design is incompatible with evolution.
  3. (Premise) If p is incompatible with q, and p is established, then q can be refuted.
  4. So, evolution is not established.

Premise 3 will be taken to be false by many if "refute" and "establish" are understood in the knowledge sense or even in the sense of knowledge-type justification, since knowledge and knowledge-type justification is not closed under entailment. Some will say that although it is established that I have two hands, and that I am a brain in a vat is incompatible with that, that I am a brain in a vat cannot be refuted. I think this response is mistaken—I know I am not a brain in a vat—but I won't insist on it.

On the other hand, if "refute" and "establish" are taken in the sense of assigning low and high rational probability, respectively, then 3 is surely true: if p and q are incompatible, and one has high probability, the other has low probability. Thus, it seems, that those who claim that evolution is established should either hold that Intelligent Design is compatible with evolution or stop arguing that it is irrefutable.

But perhaps what I just said isn't right. Maybe to refute q is both to get to assigning a low probability to q and to obtain significant incremental disconfirmation. Then 3 is false. For suppose that q has very low rational priors. Then we can establish p without getting any incremental disconfirmation for q—we get incremental confirmation for q, and q ends up with high probability, but p's probability is basically unchanged from the priors. Maybe it is empirically established that I have two hands, but the brain in a vat scenario continues to be ruled out by low priors.

If this response is right, then the evolutionary theorist who wants to claim that evolution is established while yet accepting 1 and 2 needs to hold that the rational priors of intelligent design are low. But are they?

Benevolence towards God

The best kind of love involves benevolence. But how then can a love for God be of the best kind? Aquinas answers this problem basically by saying that we can do good to God by doing good to those God loves--namely, our neighbor. This is a good answer, but I also got another Thomistic answer from my friend Richard Sisca on Saturday: we can rejoice that God enjoys perfect beatitude. For Aquinas there are two ways of having the other's good in one's will: (a) by willing that good things should happen to the beloved; and (b) by mourning the evils and rejoicing in the goods that the beloved suffers or enjoys.

Monday, April 15, 2013

Mellor's argument against circular causation

Mellor's argument against circular causation in Real Time II seems to me to basically come down to the following observation. If we have probabilistic causation of B by A and of A by B, then the four conditional chances involved in the causation, namely P(B|A), P(B|~A), P(A|B) and P(A|~B), fully determine all the unconditional probabilities of A, B and their negations. Namely, only one assignment to P(A) and P(B) does not generate a violation of the laws of probability. (For instance, if P(B|A)=P(A|B)=1/2 and P(B|~A)=P(A|~B)=1/4, then we have to set P(A)=P(B)=1/3, or we will violate the laws of probability.) But Mellor seems to think that in a causal system, we should be able to keep fixed the conditional chances and yet set some unconditional probability howsoever we wish.

Certainly, in ordinary non-looping causation, we can do that. If we have a events A1,A2,... occurring or not occurring in a sequence with no memory (i.e., we have a Markov chain with a binary state space), then no matter what the transition probabilities P(An|An−1) and P(An|~An−1) are, we can arrive at a coherent probability assignment to the system as a whole no matter what we let P(A1) be.

It is a quite interesting feature of circular causation that the conditional chances fix all the probabilities. But where is the absurdity?

Well, to be fair to him, Mellor does bring in frequencies. Suppose we have a large number of independent causal loops governed by the same chances happening side-by-side. Suppose the chances are all strictly between 0 and 1. Then any distribution of features between the loops should be metaphysically possible. Thus, we could have five million As and ten million non-As, or equal numbers, or ten million As and five million non-As. But now suppose that frequency of As and non-As in the system does not match the frequency which is determined by the unconditional probability P(A) that can be derived from the conditional chances. Then we can calculate the expected number of As and non-As that there should be after going around the A-B-A loop, and we will find that that expected number doesn't match the number we have. (Mellor does this explicitly.) And so what? Well, one problem here is that it is very unlikely that the expected number not match the observed number. Correct! But it should not surprise us that we have an unlikely scenario when we have started with an unlikely assumption. For the expected distribution of As and non-As is the one given by the unconditional probabilities P(A) and P(~A), and ex hypothesi our distribution departed from that. In unlikely circumstances, an unlikely result. What's strange about that? This is basically the same point that Nicholas Smith made here in the case of a somewhat different argument.

Reducing asserting to promising

Suppose I had a language that had only one kind of speech act: promising. Could I communicate information in the way we do in assertion? Yes! For, suppose I want to communicate that it's raining. Then I could say "I promise to immediately exclaim 'just kidding!' unless it's raining" without exclaiming "just kidding!"

We could even imagine that over time this would get abbreviated to: "It's raining."

Thus, one can reduce assertion, or something very much assertion-like, to promising. Moreover, if we say that this really is normatively equivalent to assertion, then we get an account of the wrongness of lying and a reduction of the normativity of assertion to moral normativity.

One cannot, however, reduce all speech acts to promises. For while promises generate reasons for self, requests (including questions) and commands generate reasons for others.

Saturday, April 13, 2013

The truth norm of assertion

According to the truth norm, p is assertible if and only if p is true. The truth norm is one of four main proposals for a norm of assertion:

  • truth
  • belief
  • justification
  • knowledge

One argument for the truth norm is that it is the only one of the four norms that survives, without modification, the counterexamples in this paper.

Another argument is this. Once you have speech acts that are governed by truth norm, you can get for free speech acts governed by all of the other three norms—or even any other imaginable norm. For if, say, some communication calls for a knowledge norm, you can assert "I know that s" instead of just asserting "s". But it is useful to have a variety of speech acts governed by different norms for different contexts. Sometimes, it is useful to have speech acts governed by the belief norm, sometimes by the knowledge norm, sometimes by an inclination norm ("I am inclined to think that s")[note 1]. And what is useful in language is likely to be realized. But it is better, theoretically, if we do not multiply fundamental illocutionary types and fundamental norms. So if we can have one norm from which the others can be derived, that's great.

Note that a similar move cannot be made with the other norms. Suppose that you think the knowledge norm is the right one. Well, you don't get to make a statement that s governed by the belief norm by saying "I believe that s", for the assertibility condition for that will then be that I know that I believe that s, and that's not the same as as the condition that I believe that s. And so on.

We can generalize the point by saying that the truth norm has a universal property with respect to the class of possible norms of assertion-like speech acts. Think of a norm of assertion as corresponding to a predicate N(x,p,C) that gives a condition for x's asserting p in context C being appropriate, say: x knows p in C (on contextualist accounts of knowledge) or just x knows p. Then our universal property is this: Given speech acts with the truth norm, one can engage in speech acts with a norm equivalent to N, simply by modifying the content of what is said.

For, if Assertion(p) is an appropriate speech act if and only if p is true, then Assertion(<N(x,p,C)>) is an appropriate speech act if and only if N(x,p,C).

Here, the notion of assertion-like speech acts is very, very broad. Consider, for instance, weak denial, which is an assertion-like speech act governed by the norm of non-belief. You can weakly deny that s simply by asserting, with the assertion governed by the truth norm, "It is not the case that I believe that s." (Again, you can't do this in terms of the other norms. For you might fail to believe that s and yet not know or believe or be justified in believing that you fail to believe that s.)

The notion of a universal property comes from Category Theory. I am using it analogously here.

Friday, April 12, 2013


I was thinking about structural differences between a marriage between two parties and an arrangement with more than two parties. Here are two structural differences.

1. Once you have three persons, you have politics, and not just a relationship. You can have alliances.

2. If you are married to one person, a conflict solely about the apportionment of goods between you and the other person can be resolved by your sacrificing your own good. But if there are three or more persons in the marriage, your self-sacrifice won't solve all such problems, because some goods-apportionment conflicts between the people in the marriage will concern goods to parties other than yourself, rather than between your goods and that of another.

Thursday, April 11, 2013

Natural teleology

Breathing is intrinsically directed at oxygenation even if we're breathing helium. But if one knows it's helium, it may not be possible to voluntarily intend it for oxygenation. Imagine I'm in a room filled with helium. I'll pass out and die soon. I first hold my breath. But then I realize there is no point to that. So I let myself go and breathe. After I let myself, go my breathing isn't intentionally directed. But the breathing still has oxygenation as a telos.

So the breathing both is and is not directed at oxygenation? Perhaps what we need to do is to distinguish between action and activity. Breathing is always an activity, but it only becomes an action when it is willed. The activity of breathing always has oxygenation as a telos (maybe not the only one). The action of breathing may have oxygenation as a telos, but need not--breathing can have all sorts of voluntary purposes, such as signalling to a confederate, etc. But such a case, one is signalling to a confederate with breathing, an activity that has oxygenation as a purpose, whether or not it achieves that purpose.

What I said about breathing and oxygenation also applies to intercourse and reproduction to a significant degree.


Isn't it curious that we talk of "bow and arrow" as a unit but not "gun and cartridge" or "crossbow and bolt"?

Wednesday, April 10, 2013

The flow of time

Famously, D. C. Williams ridiculed the deep-seated intuition (at the heart of Bergson's thought, of course) that there is a flow of time, asking how fast time is flowing, if it's flowing. Some wits have tried to respond: "Always at a second per second." But there is a much better and less trivial answer. And interestingly it is an answer that has a home in the B-theory of time.

The Twin Paradox suggests that we should distinguish the internal time of an individual from something like the generally shared external time of the human community. Thinking about time travel suggests a similar distinction, as Lewis has noted. But once we have a distinction between internal and external time, then we can give a non-trivial answer to Williams' question. The flow of time is measured in terms of external units of time per internal units of time. If external time is defined by the shared life of the human community (that's one among a number of options—we should probably understand "external time" in a context-sensitive way), normally time flows at one external second per second. But if I were to engage in travel at relativistic speeds, it could be that in a month of internal time, eight years of external time would elapse. And if I were to engage in gradual backwards time travel, then I would have a negative rate of time's flow: maybe I would be moving at −1 external century per internal second. (In non-gradual time travel, the rate would be undefined.)

The distinction between internal and external time fits best with the B-theory. So the notion of time's flow, surprisingly, seems to have its home in the B-theory.

Adams' ontological argument

Robert Adams' modal ontological argument in his piece on Anselm in The Virtue of Faith seems not to get much attention. Adams' modal ontological argument doesn't need S5: it only needs the Brouwer axiom pLMp, namely that if p is true, it not only is possible, but it is a necessary truth that p is possible. Here is a version of Adams' argument. Let G be the proposition that God exists. Then as God is by definition a necessarily existent and essentially divine being, that God exists entails that God necessarily exists:

  1. L(GLG).
Add that possibly God exists:
  1. MG.
The proof is simple:
  1. MLG. (By 1 and 2 and K)
  2. ~GLM~G. (Brouwer)
  3. MLGG. (Contraposition on 4)
  4. G. (Modus ponens on 3 and 5)
And by an application of 1, 6, axiom M (the necessary is true) and modus ponens we can even conclude LG, that necessarily God exists.

This doesn't use S4. So worries about the transitivity of possibility are irrelevant here.

Griffin attributes the Brouwer-based argument to Leibniz.

Tuesday, April 9, 2013

Grounding and category theory

I've been searching for the right kind of mathematical structure to think about the phenomenon of grounding or partial grounding with. The orthodoxy is that the right structure is a partial ordering. That the axioms of partial ordering are satisfied by partial grounding has been challenged and defended. But even the critics have tended to take partial grounding to be something like[note 1] a single relation, or perhaps a small collection of related relations, between pairs of propositions or between a proposition and a set of propositions. I've offered two suggestions (first and second) on how to model grounding using graphs. But I now think all of these approaches abstract away too much of the structure of grounding and/or are unable to capture all the prima facie possibilities that a theory of grounding should recognize.

For instance, the relational approach loses sight of the structural fact that one can sometimes have two different grounding relations between a pair of propositions. Let W be the proposition that Smith is drinking water and let H be the proposition that Smith is drinking H2O. Let D be the disjunction of W and H: the proposition that Smith is drinking water or drinking H2O. If we think of grounding as a relation, we can certainly say that H grounds D. But we want to be able to say that there are two groundings between H and D: H grounds D directly by being one of its disjuncts and indirectly by grounding W which is another disjunct. And this structure is not captured by the orthodoxy. The graph approach nicely captures this sort of thing, but it does not adequately capture the compositional structure which is that the indirect grounding that H provides for D is a composition between a grounding by H of W and a grounding by W of D. There are ways to make it capture this, say by identifying composition of grounding with sequences of arrows in the directed graph, but this won't work for infinite sequences of arrows, something that we should not rule out in the formalism. I realized this when trying to finish a paper on grounding and fundamentality.

I am now wondering if the right structure isn't that of a category. Maybe the objects are true propositions. The arrows are groundings, i.e., token grounding-like relations. Every arrow is at least a partial weak grounding (weak, because there are identity arrows). Some arrows may be full groundings. There is a nice associative compositional structure.

There will be further structure in the category. For instance, perhaps, every family of true propositions will have a coproduct, which is the conjunction of the propositions in that family. The canonical injections are the partial groundings that conjuncts give to the conjunction, and the universal property of the coproduct basically says that when a proposition is weakly partially grounded by each member of a family of propositions, then there is a coproduct weak partial grounding arrow from the conjunction of that family to the proposition. This is very nice.

We might also consider a move to a category where the objects are all propositions, but that creates the challenge that we need to keep track of which propositions are true and which groundings are actual. For that p partially grounds q is, in general, a contingent matter.[note 2] Truth and actuality of grounding respects the category structure. Actual groundings only obtain between truths, and compositions of actual groundings are actual.

The category structure is going to yield transitivity of weak partial grounding. There are apparent counterexamples to transitivity. But it is my hope that when we keep track of the additional structure, and think of the token groundings as central rather than the relation of there being a token grounding, the result is not going to be problematic.

It is now interesting to investigate what different category theoretic phenomena occur in the grounding category, and how they connect up with metaphysical phenomena. One thing I'd like to see is if there is a neat category theoretic characterization of full, as opposed to partial, grounding.

I have an intuitive worry about the above approaches. Intuition would suggest that if conjunctions are coproducts, then disjunctions would be products. But they're not. For in general there is no grounding from a disjunction to disjuncts. This could be related to another worry, that because categories include identity arrows, I had to take the arrows to be weak groundings—i.e., I had to allow each proposition to count as having a grounding between itself and itself.

I do not know if category theory will in the end provide a good mathematical home to grounding structures, but I am hopeful.

Privileging objects

Consider this criticism of traditional metaphysics: Traditional metaphysics privileges objects. A natural (though really not all that common) move for philosophers of a certain stripe is to move from a focus on objects to a focus on relationships between objects and to capture this by means of Category Theory. But Category Theory privileges relationships between pairs of objects, and that's at least as bad as privileging single objects.

Monday, April 8, 2013

Seeing air

Last night, I took the big kids to the swimming pool.  When I entered the swimming pool area, I saw something that looked wavy and shimmery.  It was water.  Some time later, on my daughter's advice, I submerged my head while wearing goggles and looked upward.  I saw something that looked wavy and shimmery.  By symmetry with my previous seeing, it had to be... air.

So air is visible, even in moderate quantities.  (Of course one could see air in sufficient quantities, because of how it would dim the light.)

Friday, April 5, 2013

Design, evolution and many worlds

The following image graphs the outcome of a simulation of a random process where a particle starts in the middle of the big square and moves by small steps in random directions until it reaches an edge.

It sure looks from the picture like there was a bias in the particle in favor of movement to the right, and that the particle was avoiding the black lines (you can see at a few points where it seems to be approaching the black lines and then jumps back) and searching for the red edge on the right.  If you saw the particle behaving in this way, you might even think that the particle has some rudimentary intelligence or is being guided.  To increase the impression of this, one could imagine this particle doing something like this through a complex labyrinth.

But in fact the picture shows a run that doesn't involve any such bias or intelligence or guidance.  However, it took 23774 runs until I got the one in the picture!  What I did is I had the computer repeatedly simulate random runs of a particle, throwing out any where where the particle hit the black boundary lines before it hit the red edge.  In other words, there is a bias at work  However, it is not a bias in the step-by-step movements of the particle, but a selection bias--to get the picture above, I had to discard 23773 pictures like this:

Sampling multiple cases with a selection bias can produce the illusion of design.  Most cases look like the second diagram, but if I only get to observe cases that meet the criteria of hitting the red edge before hitting any black edge, I get something that looks designed to meet the criteria (it looks like the process is run by biased chances, whereas the bias comes from conditioning on the criteria).

Now, suppose that evolutionary processes occur at a large number of sites--maybe a very large number of planets in a single large universe or maybe in a large number of universes.  Suppose, further, that intelligence is unlikely to evolve at any particle evolutionary site.  Maybe most sites only produce have unicellular critters or vast blankets of vegetation.  But a few produce intelligence.  We then will have a sampling bias in favor of the processes happening at sites where intelligence results.  And at such sites, the evolutionary processes will look like they have a forward-looking bias in favor of the production of intelligence, just as in my first diagram it looks like there is a bias in favor of getting to the red line and avoiding the black lines (think of the diagram as phase space, with the black lines marking where total extinction occurs and the red line marking where there is intelligent life).  

This means that we will have the appearance of design aimed at intelligence.  This forces a caution for both intelligent design theorists and evolutionary theorists if there is reason to think there is a large number of sites with life.  

The intelligent design theorists need to be very cautious about taking apparent end-directedness in the phylogenetic history that led to human beings to be evidence for design.  For given a large number of life sites and the anthropic principle, we expect to see apparent directedness at the production of intelligence in the process, just as in my first picture there is apparent red-directedness and black-avoidance.  This means that intelligent design theorists would do well to focus on apparent design in lineages that do not lead to us, since such design is not going to suffer from the same anthropic selection bias.  The cool stuff that spiders do is better fodder for intelligent design arguments than the mammalian eye, because the mammalian eye "benefited" from anthropic selection.  However, this also weakens the design arguments.  For design (pace some prominent intelligent design theorists) involves offering an explanation of a phenomenon in terms of the reasons a plausible candidate for a designer would likely be responsive to.  If the phenomenon is one that promotes the development of intelligent life, the design explanation could be quite good, for there are excellent reasons for a designer to produce intelligent life--intelligent life is objectively valuable.  But if the phenomenon is a spider's catching of flies, the reasons imputed to the designer become less compelling, and hence the design explanation becomes weakened.  

On the other hand, evolutionary theorists need to be careful in making precise generalizations about things like rates of beneficial mutations that apply equally to our ancestors and to the ancestors that other organisms have not in common with us.  For given a large number of sites where life develops, we would expect differences in such things due to the anthropic sampling bias.

This also suggests that we actually could in principle have evidence that decides between the hypotheses: (a) intelligent design, (b) naturalistic evolution at a small number of sites and (c) naturalistic evolution at a large number of sites.  

Suppose we find that the rate of favorable mutations among our ancestors was significantly higher than  the rate of favorable mutations not among our ancestors.  This offers support for (c), and maybe to a lesser degree for (a), but certainly against (b).  But suppose we find equality in the rates of favorable mutations among our ancestors and among our non-ancestors.  Then that offers evidence against (c) and depending on whether the rates are as we would expect on evolutionary grounds, it is evidence for (b) or for (a).  

I am assuming here that the number of sites is finite or there is some way of handling the issues with probabilities in infinite cases.

A regress of reasons

Suppose I vote to admit Artur to our program because Artur is Polish. A colleague then criticizes me, not for my voting to admit Artur, who is indisputably an excellent candidate, but for my voting to admit Artur for the reason I did. The colleague says that I should have voted to admit Artur because of his qualifications, not his ethnicity. And of course the colleague would be right in the criticism.

It is a familiar phenomenon that when we make a decision, we can be morally criticized (by others or by ourselves) for acting on the reasons we did. Sometimes it is not up to us which reasons we act on—we might, after all, not be free at all, but be compelled to act on the reasons we act on. Such cases could lead to a paralysis: I know what I ought to do, and I know the reason I ought to do it for, but I am unable to do the action for the right reason.

I suspect that when we conscientiously realize that we are in a position where we are moved to do what we know is the correct action, but we know we are moved by reasons we should not be moved by, typically we are not helpless. We can in fact get ourselves to act for the right reasons. It would be too paradoxical if it were typical in such cases that the only way we had to avoid acting for the wrong reasons was to avoid the correct action.

But this means that we should be able to exercise some power of choice over the reasons for which we choose. To choose an action A for reason R is itself an action, and it is possible to have reasons for and against this action, since choice requires reasons. This threatens a regress of reasons.

One way to arrest the regress is to say that, in practice, our freedom eventually runs out. Eventually, we have no freedom to choose our reasons, but only freedom to choose the action. That may be the case sometimes. But I think a more interesting case would be where we can choose to act on a reason for that very reason, thereby arresting the regress. My love for my children, perhaps, not only provides me with reason to spend time with my children, but reason to do things out of love for my children.

Thursday, April 4, 2013

Transgender realism, abortion, animalism and colocationism

There are two major families of views on our relationship to the biological world. On animalism, we are animals of the species homo sapiens. Animalism comes in two varieties: physicalist animalism says that we are purely physical animals and dualist animalism says that some or all animals, including all of us, have non-physical features such as non-physical mental states or a soul (of a Cartesian or an Aristotelian sort). On colocationism, wherever one of us is present, there is an animal of the species homo sapiens present as well, but we are not identical to such an animal. There are multiple varieties of colocationism. On the constitution view, we are wholly constituted by our associated animals. Typically, such constitution theorists are physicalists—the animals are purely physical and hence so are we. The other main variety of colocationism is further-aspect dualist colocationism on which our associated animals are purely physical, but we are not. This includes a view on which we are souls (which count as located wherever the ensouled bodies are), a view on which we are a composite of an animal and a soul and a view on which we are partly constituted by an animal and partly constituted by a non-physical aspect. The debate on animalism versus colocationism is thus to a significant degree orthogonal to the debate between physicalists and dualists.

If animalism is true, then a normal adult, say Sally, used to be a fetus, and to have killed that fetus would have been to kill Sally, and it would have deprived Sally of even more than killing Sally now would. Thus, animalism strongly suggests that abortion is wrong, though violinist-type arguments could be used to try to resist that conclusion. On the other hand, colocationist views are much more congenial to pro-choice philosophers, and hence appear to be somewhat dominant in the pro-choice moral philosophy scene. For if colocationism is true, then it could be that the human animal existed significantly before Sally came to be colocated with it, and if so, then killing that human animal in abortion would not have been a killing of Sally. Though, a colocationist could also think that colocation started at fertilization and hence a killing of the fetus would also be a killing of the colocated Sally.

So whether animalism or colocationism is the right metaphysics of us is very relevant to the moral status of abortion.

Now I will cautiously wade into waters that are rather unfamiliar to me, and I apologize if I use terminology in non-standard ways. The question of animalism versus colocationism appears to be very relevant to the question of transgender realism. Let Type I Transgender Realism (1TR) be the view that some people literally are men in female bodies or women in male bodies. Let Type II Transgender Realism (2TR) be the claim that some people who had female bodies and felt that they were or should be men are now, after gender reassignment surgery and hormonal treatment, literally men, and some people who had male bodies and felt that they were or should be women are now, after gender reassignment surgery and hormonal treatment, literally women. If 1TR is true, so is 2TR: surely a man in a female body does not cease to be a man after the body is surgically modified to be more male-like. But at the same time, the law in a number of jurisdictions tracks 2TR but not 1TR, requiring surgery for legal classification as male or female.

Now, it seems very plausible that whether a human animal is male or female (or hermaphrodite) depends on biological criteria very much like those by which we ask whether an elephant or a gecko or maybe even a plant is male or female (or hermaphrodite). These criteria do not depend on psychological states but on whether the organism is such that it should produce its own sperm or such that should produce its own eggs (or both). It is also very plausible that men are male (though they may be more or less feminine) and women are female (though they may be more or less masculine). So if we are human animals, then whether we are male or female, and hence whether we are men or women, depends solely on biological criteria, and 1TR is false.

Moreover, if we are human animals, then 2TR is also false, at least given the current surgical methods. If we remove a mouse's female reproductive system and reshape what remains to look like male genitalia, and treat with hormones, what we have is a female mouse that has lost its reproductive system and behaves like a male. It might be more complicated if a functioning male reproductive system is transplanted. But I think it would still be true that the resulting mouse isn't such that it should produce sperm. Moreover, the mouse doesn't produce its own sperm—it produces the donor's sperm. Here's another route to the conclusion that even a functioning male reproductive transplant doesn't turn the female mouse male. After mere removal of a female (respectively, male) mouse's reproductive system, what we have is a female (respectively, male) mouse that is missing a reproductive system. But now imagine two identical twin female mice, A and B. Both have their female reproductive systems removed. But B then has a male reproductive system added, and then removed. If B became male upon addition of the male reproductive system, then B should still be male after removal thereof—a male does not cease to be a male after losing the reproductive system, but becomes a mutilated male. But A and B may be exactly alike at the end of suffering all this cruelty. It would then be odd to say that of two exactly similar mice, one is male and one is female. So we should say that they are both female, and hence B was female all along, even while having the male reproductive system.

Maybe an animalist could get out of this argument by distinguishing between sex and gender, and denying the idea that a man is an adult male human and a woman is an adult female human. Instead, perhaps, a man is an adult masculine human and a woman is an adult feminine human. The appeal to non-human animals in my argument then becomes irrelevant because only human animals can be men and women. On this story, there will be a disnalogy between the triple of terms "human", "woman" and "man" and triples like "chicken", "hen" and "rooster". A hen is a female chicken, but a woman need not be a female human. While this animalist-compatible view would let one preserve 1TR and 2TR, it would not be compatible with the aspiration that "a woman in a man's body" may have to be really female. It is my impression it is more the genderqueer than the transgendered who use phrases like "male woman" or "female man". Besides the idea of literally male women and female men seems problematic.

On the other hand, if colocationism is true, it is much easier to hold to 1TR and 2TR. Sure, Sally's associated animal (the animal that she is partly or wholly constituted by) may be male, but perhaps maleness and femaleness in a human person is not simply determined by whether the human animal is male or female. Colocationism could allow one to hold to 1TR without revisionary biology and without the oddness of saying that Sally is a male woman. Moreover, colocationism makes it plausible that sexual reassignment surgery could be a valuable thing: it is fitting that a man be associated with a male animal and a woman with a female animal, and while my arguments above suggest that surgery will not change the sex of the associated animal, it could somewhat improve the fit between the person and the associated animal.

Of course, colocationism by itself does not imply 1TR or 2TR: one could still think that a person is a man if and only if the person is associated with an adult male human animal and that a person is a woman if and only if the person is associated with an adult female human animal. But colocationism opens options beyond that.

So the debate between animalism and colocationism is not only highly relevant to the abortion debate but also to the question of transgender realism. Settling the question between the animalists and colocationists would not completely settle the latter two questions, but it would lead to significant progress.

Let me end by saying, without argument, that we are primates and all primates are animals. Hence animalism is true.

Wednesday, April 3, 2013

Metric temporal logics and induction

Prior's metric temporal logic comes with two sentential operators, Pn and Fn, which respectively mean something like "n ago in the past" and "in n in the future", where n is a duration. On Prior's metric logic, omnitemporal universal quantification becomes a triple conjunction. Thus,

  1. Always, all ravens are black
  1. (x)(RxBx)&(n)Pn(x)(RxBx)&(n)Fn(x)(RxBx),
i.e., all ravens are black, all ravens have always been black and all ravens will always be black. But this is unsatisfactory. Suppose my data is that all ravens have always been observed to be black. This data does not significantly support the first and third conjunct of (2), but only the second. To conclude to (2) is like trying to draw inferences about jade, which is in fact a disjunction of jadeite and nephrite, on the basis of observations of jadeite alone. Consequently, if Prior's metric temporal logic captures fundament temporal structure, induction from past to future is dubious indeed.

Suppose now that time is in fact discrete. Then one could use a temporal logic with only one sentential operator @n, where n is an integer, which means "in n moments". Since n could be negative, zero or positive, this can be used to talk about the past, present and future. And (1) becomes:

  1. (n)@n(x)(RxBx),
an elegant and simple hypothesis that it is reasonable to take to be supported by the fact that all pastly observed ravens were black.

But logic shouldn't be tied to how things actually are. So this temporal logic is only plausible if time not only is (which is controversial enough) but must be discrete. And that would be very controversial.

Could one get this temporal logic working if time weren't discrete? One would need some way of measuring temporal distance that would allow distances to be negative (past), zero (present) and positive (future). Fixing a unit system, say seconds, makes it easy to do that. But there are three problems with such an approach. First, logic should not be dependent on a unit system. Second, temporal logic should apply in all worlds with time, while all the units of time (Planck times, seconds, etc.) that we have in our world are dependent on the particular laws of nature (just look at how a Planck time or a second is defined). Finally, a third problem is that if times ranged over, say, the real numbers, then the system wouldn't work in worlds with a non-Archimedean timeline.

One could, however, try the following move. The basic tense logic operator is @(n,u) where n is a real number and u is a duration, and we can read it as: in n us. Then (1) becomes something like:

  1. (n)(u)@(n,u)(x)(RxBx).
This even works in non-Archimedean timelines—u could be an infinitesimal or infinite duration. But I think there is a difficulty. All our observations of past black ravens can be explained by:
  1. (u)@(−1,u)(x)(RxBx),
i.e., the claim that all ravens have been black, which is just as simple as (4), but less committive. Why not go for (5) instead of (4)? Induction is still in danger. Besides, all of this metric time stuff is going to be a failure in worlds where there is no metric for time.

So if presentism is true, our best bet for an induction-friendly metric temporal logic is if time is discrete.

What if eternalism is true? Then our most plausible temporal logic is a non-metric one with an operator like #t, which says that something is true at t, where t is a possible time. Our omnitemporal quantification (1) becomes:

  1. (t)(Time(t)→#t(x)(RxBx)),
where Time(t) says that t is a time of the actual world. Could a presentist endorse (6)? I think so. But I also think there would be a pressure on the presentist to explain Time(t) as something like a triple disjunction: t is a past time or t is the present time or t is a future time. Certainly, that's how it will have to look for Crisp-style ersatz times. And now we have the problem of induction again, because plausibly our observations should be taken instead to justify belief that:
  1. (t)(Past(t)→#t(x)(RxBx)).

But perhaps the presentist could explain Time(t) tenselessly. Maybe: Time(t) if and only if #t(0=0). (This won't work on Crisp-style ersatz times. But it might work for times that are durations from a beginning--assuming there has to be a beginning.) I.e., t is a time of the actual world if and only if something (and what better candidate than a tautology?) is true at t. Without the triple disjunction in the definition of an actual time, we now have some hope.

All that said, I doubt that a non-metric #t operator with a tenseless and changeless definition of Time(t) will be attractive to the presentist. For I think the typical presentist wants truths about what happens at different times to be grounded in tensed truths, something like in Prior's metric temporal logic.

Note: The metric logic renderings also need something that says that a given n isn't out-of-range (before a beginning of time or after an end of time).

Tuesday, April 2, 2013

If consciousness causes collapse and physics is simple, then naturalism is false

  1. If the right interpretation of Quantum Mechanics is that consciousness causes collapse (ccc), then naturalism is false or the fundamental laws of physics are extremely complex.
  2. The fundamental laws of physics are not extremely complex.
  3. So if the right interpretation of Quantum Mechanics is that consciousness causes collapse, then naturalism is false.

Why accept (1)? Well, on the ccc interpretation, the fundamental laws of physics will have to have sufficient complexity to describe conscious states. And if naturalism is true, descriptions of conscious states will be very complex, since mental or even teleological properties will not be fundamental, or even close to fundamental, given our best naturalistic theories.

I find (2) very plausible, and so do a number of physicists I suspect, but I don't know if I have a naturalistically acceptable argument for it.