Monday, October 31, 2011

Rational and irrational desires

Odysseus is told by Athena that he is very unlikely to reach Ithaca, unless he suppresses his desire to reach Ithaca. If he does suppress it, he will quickly by accident find his way to Ithaca, and as soon as he is within ten stadia of it, his desire will return. Athena points to a pear from a tree on the banks of Lethe, and tells him that this pear will suppress his desire to reach Ithaca. But Odysseus longs for Ithaca too much to be willing to let go of his desire to return there. He tosses the pear by the wayside and wanders the world for many years.

Odysseus was irrational to hold on to his desire to return to Ithaca. It was thereafter irrational for him to have the desire. Yet the desire to return to his home was a perfectly rational one.

Irene never desired to experience friendship. Finally, one day, Matthew gave her a fallacious argument whose conclusion was that friendship is worth having. Irene didn't see the fallacy, and concluded that friendship is worth having. She then hired Dr. Mesmer to hypnotize her into desiring friendship. A couple of months later, while having an intense desire for friendship, she found the fallacy in Matthew's argument. But while she believes that friendship is not worth having or desiring, she irrationally refuses to hire Dr. Memser to hypnotize the desire for friendship away.

Irene acquired her desire for friendship irrationally and is irrational in holding on to the desire. But the desire for friendship is perfect rational.

Patrick comes to be convinced by Irene, whom he has excellent reason to think to be an epistemic authority even whe she says things that seem absurd (she has said many seemingly absurd things to him, and turned out to be right), that he ought to desire to be the heaviest man on earth. By constantly dwelling on the excellent reasons he has for trusting Irene, and on Irene's advice, he comes to desire to be the heaviest man on earth, and starts to eat.

Patrick acquired his desire to be the heaviest man on earth quite rationally, and is rational in holding on to the desire. But the desire is irrational.

Collectively, the cases force a distinction between (ir)rationally having or acquiring a desire, and a desire being itself (ir)rational.

But now what does the irrationality or rationality of a desire in itself consist in if it does not consist in the irrationality or rationality of the agent who has it in respect of the having of the desire? I suspect that a good answer will have to advert to the human good, to human flourishing, but even so, I don't know how to answer.

Saturday, October 29, 2011

Another objection to a hypothetical desire-satisfaction theory of well-being

According to the simple desire-satisfaction theory of well-being, something would contribute to your well-being, would be good for you, precisely to the extent that it would satisfy your desires. The simple theory is clearly mistaken, because one's desires could be based on false beliefs or mental illness, and so it is easy to come up with examples of desires the satisfaction of which does not make one be well off. Imagine, for instance, that one desires that Patrick flourish, because one believes that Sally is one's long-lost brother, but in fact Patrick is one's long-lost brother's murderer; Patrick's flourishing is not a part of one's well-being.

The standard move is to hypotheticalize the theory by defining well-being in terms of the desires one would have after being informed of all the relevant non-normative facts and being given ideal psychotherapy. There are serious problems with this suggestion (for instance, the order in which one is informed of the non-normative facts can clearly make a difference as to what desires one comes to have.[note 1]

Here I want to focus on one particular difficulty that has struck me. Suppose I have no genuine friends and no prospects for friendship, but I desperately want friendship. Surely, friendship would contribute to my well-being, and I am badly off for not having a friendship. But the following is also conceivable. After ideal psychotherapy, and after being informed of the non-normative fact that there are no prospects for friendship for me, I might stoically suppress the desire for friendship. Indeed, unless one thinks that friendship is an essential aspect of human well-being, it might be quite rational, even rationally required, to suppress the desire under the circumstances. But now it would be absurd to say to someone who desparately wants friendship that she is not badly off for not having one because after ideal psychotherapy she would stoically and rationally suppress that desire.

That's a dreary example. Here's a positive one. Suppose I have no interest in collecting matchboxes. Getting a matchbox would not contribute to my well-being, surely. But it might be that after ideal psychotherapy and being informed of the details of the hobby, I would come to realize that collecting matchboxes is a perfect hobby for me, and come to desire matchboxes. But I don't go for the ideal psychotherapy, and I don't come to desire matchboxes, and so I don't desire matchboxes. How does getting a matchbox contribute to my well-being? (Well, on an objective theory, it might contribute somewhat despite my lack of desire, because the matchbox is intrinsically valuable. But the desire-satisfaction theorist can't say that.)

This post is inspired by William Lauinger's work on well-being. I would not be surprised if some of my examples parallel his.

Friday, October 28, 2011

Representationalism in philosophy of mind and cognitivism in aesthetics and ethics

The contemporary naturalist's best bet for an account of conscious states seems to be representationalism: reducing conscious states to representational states. For independent reasons, I am very friendly to this reduction. Let us assume representationalism.

Representational states represent reality (including the epistemic agent) as being a certain way. But now consider different kinds of aesthetic awareness, say aesthetic awe or aesthetic repugnance or beauty-appreciation. If representationalism is true, each of these states represents reality as being a certain way. But aesthetic statements like "This is sublime", "This is hideous" and "This is beautiful" are connected to kinds of aesthetic awareness. For instance it seems that aesthetic awe gives rise to statements (or exclamations) of "This is sublime", aesthetic repugnance gives rise to "This is hideous" and beauty-appreciation gives rise to "This is beautiful." But once it is granted that a state of aesthetic awareness have representational content, surely the aesthetic statement that it is naturally connected with expresses that representational content. When I am visually aware of a red cube, I say "That's a red cube" and what I say expresses at least a part of the representational content of my awareness.

Thus, once we accept representationalism in the philosophy of mind, we should accept cognitivism in aesthetics. The representational content of aesthetic awe is surely something like the sublime, the representational content of aesthetic repugnance is surely something like the hideous and the representational content of beauty-appreciation is the beautiful.

Granted, the above argument is compatible with the sublime, the hideous and the beautiful being indexical or mind-dependent. That's a matter for further investigation. But the argument does make it difficult to be a non-cognitivist if one is a representationalism.

And the same argument applies in the moral sphere. If representationalism is true, moral admiration and moral repugnance have representational content, and sure if they have representational content, that content is something's being morally admirable or repugnant, respectively.

Objection: Although the aesthetic or ethical feelings have representational content, that content is an inner state of the individual, and not the sort of thing that could be the content of aesthetic or ethical claims. Imagine the content of aesthetic awe is the fluttering of the heart (no doubt it's something more subtle). Then representationalism is satisfied. But plainly the fluttering of the heart is not the object of statements of (say) sublimity. We have two constraints on what could be the object of a statement of sublimity: (1) it has to be appropriately connected to the right sorts of aesthetic consciousness, and (2) it has to fit with enough other things we say. Fluttering of the heart fits with (1)--it is on this toy theory the representational content of aesthetic awe--but not with (2).

Reply: It's certainly true that the fluttering of the heart is not at all a plausible content for aesthetic statements. But likewise it is not a plausible content for aesthetic awareness. Suppose I am having a state of aesthetic awe at a performance of King Lear. The representational content of that state is not the fluttering of the heart. For then there would be no difference in representational content between aesthetic awe at one performance of King Lear and aesthetic awe at another performance of the same play. But there is, since it is a part of the awe, qua conscious state, that it is awe at this performance, and so when the performance is different, the representational state is different. In other words, the fluttering of the heart does not capture the directedness of the awe.

I think there are two routes for the non-cognitivist now. The first is to say that the object is something like the performance's causing fluttering of one's heart. But once we do this, it is hard to resist saying that the sublime just is whatever causes one's heart to flutter. I don't say that the latter is a plausible theory--but is just as plausible as saying that the representational content of aesthetic awe at the performance is the performance's causing of the fluttering of one's heart. (Interestingly, on this view, we seem to have perception of causal features of the world, pace Hume.)

The other route is to entrench and say that the conscious component of aesthetic awe just is an experience of the fluttering of the heart (say), but that we mistakenly describe this by saying it is awe at the performance. Rather, it is awe caused by the performance. I think this is not true to the phenomenology, however.

Thursday, October 27, 2011

First- but not second-order vagueness

Consider a view on which there is first-order vagueness but no higher-order vagueness. Thus, it can be vague that Smith is bald, but it can't be vague that it's vague that Smith is bald. Smith is always definitely definitely bald, or definitely vaguely bald, or definitely definitely non-bald. In respect of higher-order vagueness, we just go epistemicist. Call this the "intermediate view".

Why would one do that? Well, there are two main alternatives. One is vagueness all the way up. The other is sharpness all the way down.

Sharpness all the way down—i.e., epistemicism—has trouble with ordinary language intuitions such as that no one becomes bald by the loss of one normal[note 1] hair. The intermediate view says that here we distinguish: you can become vaguely bald by the loss of one normal hair, and you can become definitely bald by the loss of one normal hair, but you can't move from definitely non-bald to definitely bald by the loss of one normal hair. This may not be exactly what the ordinary language intuition holds, but it arguably does it all the justice that it needs to have done. Moreover, it is sometimes just obvious that someone is neither definitely bald nor definitely non-bald. Sharpness all the way down also has difficulties with incompletely introduced terms, such as when I tell you that every square is a squibble and only quadrilaterals are squibbles, but say and think nothing else. Is a non-square rectangle a squibble? Surely the answer is that the sentence, given how "squibble" was introduced, lacks the precision for the answer. But if we have sharpness all the way down, there should be an answer.

Vagueness all the way up also does justice to the intuition that you can't become bald by losing one normal hair, and does so in exactly the same way as the intermediate view does. It also has the advantage of doing justice to the further intuition that one doesn't become vaguely bald by the loss of one hair. But the further intuition is an intuition about much less common language, quite possibly about technical language (while "vague" is an ordinary word, it's not clear that philosophers use it in the technical sense), and the cost of giving it up is much less. The major advantage of the intermediate view over the vagueness all the way up view is that the intermediate view allows an analysis of vague language in a logically classical metalanguage. This is a way of holding on to the intuition that there is something importantly right about classical logic. The intermediate view opens up theoretical possibilities closed by the vagueness all the way up view—it seems to me to be not uncommon to observe about some view of vagueness that "it has trouble with higher-order vagueness". But if there is no higher-order vagueness, then that's not a problem!

The intermediate view allows one to say that all vague predicates are basically like "is a squibble". If we take the intermediate view in this direction, vagueness is simply due to the fact that there is a gap between the precise cases in which the predicate has been specified to hold and the precise cases in which the predicate has been specified not to hold.

Moreover, the intermediate view allows me to make sense of the intuition which I have—but which many do not share—that the world is fundamentally non-vague, that God creates and knows the world with perfect precision. For any vague predicate P can be replaced with a trio of non-vague predicates that carve up the world more precisely—definitely P, vaguely P and definitely not P—and which of the three predicates applies to x fully determines what we should say in regard to x and P. Moreover, I can further suppose that these three predicates are more fundamental than P whenever P is subject to vagueness.

There is a theistic argument for the view here.

Finally, the intermediate view allows a refinement to my theistic story about vagueness. On that theistic story, all predicates are fully sharp, because God has given us language, either by giving us the predicates directly or by giving certain predicate-production rules that result in sharp predicates. There is nothing particularly mysterious on this view about the source of sharpness or about the appearance of vagueness—one can inherit sharp terminology from someone else and the appearance of vagueness is explained by the fact that God didn't tell us what all the sharp boundaries are (why should he? it would be a massive waste of our time to keep track of them).

But that view has two difficulties. The first is that it doesn't at all do justice to the intuition that one doesn't become bald by the loss of one normal hair. The second is that it seems that we can introduce underdetermined terms like "squibble", and it is odd to suppose that God steps in, or has stepped in when setting up the general predicate-production rules, to fill in the gaps in our linguistic stipulations, even when we intended them not to be filled in, as in the squibble case.

The intermediate view helps with the one-hair problem, as already noted, and solves the second. For instead of supposing that the general predicate-production rules that God has enacted for us (in whatever way that happens, whether by enacting the conventions that underlie them or by making the rules implicit in the teleology of our nature—I like the latter version) always yield sharp predicates, we may suppose that they always sharply yield predicates, some of which are vague, but vague in the sharp way that the intermediate view recognizes.

There is, however, another problem with the intermediate view. It seems that by stipulation we can raise the level of vagueness. For instance, suppose I stipulate a predicate P by stipulating what it definitely applies to and what it definitely does not apply to. And suppose that I make use of vague terms in stipulating these. For instance, let's say I stipulate "acceptable employee" as follows. Someone is definitely an acceptable employee if and only if she does her tasks well enough and does not harm the company. Someone is definitely not an acceptable employee if and only if she harms the company. If I can stipulate "squibble" as I did, I should be able to stipulate "acceptable employee" as I did. But my stipulation used vague terms. Suppose Patrick definitely does not harm the company but only vaguely does his tasks well enough. Then Patrick is vaguely definitely an acceptable employee.

But this problem is easily fixed by replacing the intermediate view with a "bounded vagueness" view on which there is a number n such that every predicate has no vagueness above the nth level. The number n may be very large for ordinary languages like English. And since ordinary languages keep on evolving and adding predicates, the number may be continually increasing. But the point is that it's always finite. And as long as it's always finite, we maintain most of the advantages of the intermediate view, while avoiding the above disadvantage. In particular, we can work with a classical metalanguage. And, as a bonus, we can now do some justice to the intuition that one doesn't become definitely bald by the loss of a hair, though there will be some iterated intuition this won't work for—but that iterated intuition will be much less plausible, I think.

We do lose this argument, but I think we can get around the problem there in another way (e.g., by saying vagueness stays at the level of sentences, not the level of propositions, or by saying that the problem is solved by God having the precisified beliefs—perhaps involving a long iterated list of "definitely" and "vaguely" operators—that ground the vague stuff).

Wednesday, October 26, 2011

Goodman and Quine's nominalism and infinity

The argument in this post is highly compressed. It's even more a note to self than other posts are.

Goodman and Quine have a very clever nominalist metalanguage that lets them handle first-order logic. There seems, however, to be a serious problem in their system. As it stands, the system will be inconsistent in certain infinite worlds, if it allows excluded middle (which it does, being classical). And they do not have the resources for specifying that the worlds they're using are finite.

The problem stems from the fact that Goodman and Quine nowhere specify that the sentences of their target language are finite. Because they fail to specify that, they cannot rule out infinite sentences corresponding to something like "~~~......~~~p". Think of the front part of it as two infinite sequences of smaller and smaller tildes, with the infinite tails touching. (Goodman and Quine work with Sheffer strokes, and it's a touch harder to explain how such an infinite sentence is done with Sheffer strokes, but I think it can be done, too. I will work with ordinary FOL.)

Now, why would you want to rule out such sentences? Well, write N for the doubly infinite sequence of negations, as above. Then by excluded middle, we have Np or ~Np. But ~Np is the same sentence as Np. Hence, we have Np or Np. Hence we have Np. But again ~Np is the same sentence, so we have both Np and ~Np. And that's pretty bad, because everything now follows by explosion, again because we have a classical logic.

Could Goodman and Quine cleverly exclude such infinite sentences? It seems that they can't do it using their present metalanguage primitives and axioms, nor by means of any straightforward extension of them. For their metalanguage is also classical and first-order, and hence unable to express sentences that "logically imply" that there are finitely many Fs (say, letters in s)—i.e., sentences that are true in all and only all interpretations on which finitely many things satisfy F (this is easy to prove by compactness, and I think does not even require the Axiom of Choice).

That looks pretty much fatal. Except that Goodman and Quine might be able to help themselves if they could use some heavy duty metaphysics to establish that our world either has only finitely many objects or is a single continuous plenum. For given that the world is a single continuous plenum, we might be able to express the idea that a sentence is finite by saying that for every mereological sum of curvy arrows in the world (arrows being certain arrow-shaped parts of the plenum—we need mereological universalism for the system to work) such that every letter of the sentence is at the tail end of exactly one arrow, and every arrow points to a letter of the sentence, and no two arrows point to the same one, every letter of the sentence is pointed to. But this only works given a continuous plenum where there is enough stuff to make enough arrows to ensure this isn't spuriously satisfied. And I doubt there is a good way to express the fact that we have a continuous plenum in the Goodman and Quine system. So the system can only be made to work on a quasi-empirical assumption that the system, apparently, cannot state. And it is bad that whether a logical system is consistent depends on how matter is arranged in the world—if it is arranged in a plenum or there are only finitely many objects, it's consistent, otherwise possibly not.

Another move would be to require that all the letters be exact copies of each other and that they be all in a straight line. There is no way to form "~~~......~~~p" in an Archimedean universe where all the letters are in a straight line. But, again, their logical system will depend for its consistency on the assumption that our world is Archimedean. And that's weird.

Goodman and Quine mention something related to the finiteness issue in footnote 14, in the context of the alternative framed ingredients method. I think the alternative framed ingredients method also requires an assumption of finitude.

Tuesday, October 25, 2011

More about functionalism about location

Functionalism about location holds that any sufficiently natural relation, say between objects and points in a topological space, that has the right formal properties (and, maybe, interacts the right way with causation) is a location relation.

Here is an argument against functionalism. Functionalism is false for other fundamental physical determinables: it is false for mass, charge, charm, etc. There is a possible world where some force other than electromagnetic is based on a determinable other than charge, but where the force and determinable follow structurally the same laws. By induction, functionalism is probably false for location.

Some will reject this argument precisely because they accept something like functionalism for the other physical determinables, and hence deny the thought experiment about the non-electromagnetic force--they will say that if the laws are structurally the same, the properties are literally the same.

I think there is a way to counter the above argument by pointing out a disanalogy between location and other fundamental physical determinables (this disanalogy goes against the spirit of this post, alas). Let's say we live in an Einsteinian world. A Newtonian world still might have been actual. But, plausibly, the Newtonian world's "mass" is a different determinable from our world's mass. Here's why. In our world, mass is the very same determinable as energy (one could deny this by making it a nomic coextensiveness, but I like the way of identity here). In the Newtonian world "mass" is a different determinable from "energy". Therefore either (a) Newtonian "mass" is a different determinable from mass, or (b) Newtonian "energy" is a different determinable from energy, or (c) both (a) and (b). Of these, the symmetry of (c) is pleasing. More generally, it is very plausible that fundamental physical determinables like mass-energy, charge, charm or wavefunction are all law bound: you change the relevant laws (namely, those that make reference to these determinables) significantly, and you don't have instances of these determinables.

But location does not appear to be law bound. "Location" in a Newtonian spacetime and a relativistic spacetime are used univocally. You can have a set of really weird laws, with a really weird 2.478-dimensional space (for fractional dimensions, see, e.g., here), and yet still have location. Maybe there are some formal constraints on the laws needed for locations to be instantiated, but intuitively these are lax.

Plausibly, natural (in the David Lewis sense of not being gerrymandered) physical determinables that are not law bound are functional. If location is a natural physical determinable, which is very plausible on an absolutist view of spacetime, then it is, plausibly, functional. I think an analogous argument can be run on relationism, except that the fundamentality constraint is a bit less plausible there.

One might question the claim that natural physical determinables that are not law bound are functional. After all, if the claim is plausible with the "physical", isn't it equally plausible without "physical"? But the dualist denies the claim that natural determinables that are not law bound are functional. For instance, awareness seems to be a natural determinable (whose determinates are of a form like being aware of/that ..., and nothing else), but the dualist is apt to deny that it's functional.

In any case, one interesting result transpires from the above. It is an important question whether location is law bound. If we could resolve that, we would be some ways towards a good account of spacetime (if it is law bound, proposals like this one might have some hope, if based on a better physics). The account I give above of law boundedness is rather provisory, and a better account is also needed.

Monday, October 24, 2011

A Platonic theory of determinables

In an earlier post, I explored, without endorsing, a Platonic theory of spacetime, on which spacetime is an abstract Platonic entity, and objects are located by virtue of standing in a relation to abstract points of that entity.

This could extend to other determinables.  Consider, for instance, mass, and simplify by supposing presentism or lack of time variation.  An object o could have mass x, where x is some real number (we need a natural unit system for that), precisely in virtue of o's being M-related to the real number x, a Platonic entity, where M is a natural "mass relation".  This works even for much more complex determinables like wavefunctions.  Thus, an object o could have a wavefunction in virtue of being W-related to some abstract function from R3 to C, again assuming presentism or lack of time variation.  To get time variation into the picture, we could suppose that the mass relation relates objects to functions from a time sequence (an internal time sequence?) to reals.
This would help with regard to the epistemology of abstracta even if (contrary to fact, I am inclined to say) abstracta are causally inert.  For even if the number x is causally inert, the event of o being M-related to x is not causally inert (it causes gravitational influences, for instance).

One intuitive difficulty for this theory is that it is now looking logically possible for an object to have two masses or two wavefunctions at any given time.  I do not think this consequence absurd myself.  If the second person of the Trinity became incarnate as two different humans at the same time, which Aquinas thinks is possible (a possibility that we may care about if it turns out that there are fallen non-human rational beings), he might have two different masses at a given time.  Alternately, one can just say that there are brutely necessary restrictions here.

Notice an interesting consequence of this theory.  If a naturalist were to adopt this theory, it might make it easier to get her to accept a non-reductionist theory of mind on which for us to believe a proposition just is to stand in an irreducible belief relation to a proposition.  After all, it is no more philosophically puzzling how one can stand in an irreducible mass relation to a number or function than it is how one can stand in an irreducible belief relation to a proposition.  And it is no more philosophically puzzling how one's standing in a belief relation to a proposition could causally affect one's behavior than how one's standing in a mass relation could.

What bothers me about this theory, as well as the earlier theory of spacetime, is that abstracta are divine ideas.  But it seems wrong to say that mass and location facts are constituted by a relation to God.  That sounds too panentheistic.  But here's one interesting philosophical/theological question.  Aquinas insists that things are the way they are by participation in God.  Thus, Socrates is wise by (natural) participation in God (and Paul is wise by supernatural participation in God).  Does this mean that (a) Socrates' accidental form of wisdom is identical with a participating in God or does it mean that (b) Socrates' accidental form of wisdom is something distinct from but dependent on Socrates' participating in God?  If the former, then the Platonic theory I offered will be no more problematic than Aquinas' view (but of course I'll want to say something like what Aquinas says about one-sided relations, so that the mass relation is a relation to God but there is no corresponding relation of God to the object--maybe the suggestion in this post helps), and in fact Aquinas' view might just be a variant of the Platonic theory.  If the latter, then the Platonic theory is more panentheistic than Aquinas', and insofar as Aquinas seems to me to be as close as one can orthodoxly come to panentheism, I would then reject the Platonic theory.

There is also going to be some trickiness coordinating the location determinable with the other determinables.  We want to be able to say things like "x is beige on its left side".  Working this out may require me to abandon the heuristic that there is nothing special about location--that it's just another relation.

"I know my Redeemer lives"

It is a conceit of modern secular society that faith is belief in the absence of evidence or knowledge. That is not how Scripture sees faith. The New Testament constantly talks of us knowing God, knowing the grace of Jesus Christ, and knowing all sorts of things that are the content of faith. In Scripture, faith and knowledge are quite compatible. What may not be compatible is faith and vision, or direct apprehension of the truth. In fact, I think the way to right distinction in a Christian context is between knowing naturally and knowing by faith: but both are species of knowledge.

Aristotle in the Rhetoric defines "pistis" ("faith") as a persuasion by means of the character of the speaker. In the New Testament, "faith" has two aspects: there is the aspect of entrusting oneself to Christ and the aspect of believing. The belief aspect fits very well with what Aristotle says: what we believe by faith is that which we believe on the basis of the perfect character of God.

Belief on the basis of another's character can certainly be knowledge. A friend tells me something. She's got the sort of character that I can't imagine her saying it unless she knew it. I believe her. That's "faith", but it's also a species of knowledge.

Sunday, October 23, 2011

The deep question for the philosophy of spacetime

There is more than one way of putting this point, so the assumptions I will make are not at all essential, and I don't even endorse the assumptions. Assume absolutism about spacetime. On one reading of absolutism, there is then a location relation between objects and points or regions of spacetime (on another reading there is an object- or point-valued location determinable). Depending on the version of absolutism, the location relation may correspond to the predicate is wholly located at, is at least partly located at or is exactly located at (I may be leaving out some options).

Now the deep question is this: What is it that makes a relation between objects and points or regions of a topological space be a location relation? (The question can also be put on relationism. Then the question is what is it that makes a family of relations between objects be a family of spatial, or spatiotemporal, relations.)

There are two extreme answers.

Location monism: There is just one location relation. In a Newtonian and in an Einsteinian world and in a 12-dimensional discrete universe, one and the same relation relates objects to points or regions of a topological space, obviously a very different topological space in each case.

Location functionalism: Any natural (sufficiently natural? perfectly natural?) relation between objects and points in a topological space, where the topological space is either concrete and cosntituted as a topological space by natural relations, or abstract as in this post, is a location relation. What the axioms are will depend on which location relation one takes as fundamental as well as on difficult metaphysical issues. Supposing that the relation is being exactly located at, and the spatial relata are regions, then the axioms might be very lax. In fact they might be nothing but:

  • If xLR and x is a part of y, then there is a unique region R' that contains R such that yLR'.
  • If yLR and x is a part of y, then there is a unique region R' that is contained in R such that xLR'.
(If Thomistic part nihilism is true, do this with virtual parts.) On this view, a relation of being exactly located at any phase space with a topology will count as a location relation as long as the relation is in fact natural. One might additionally add some more axioms, such as that no object is exactly located at two distinct regions (though I myself am inclined to deny that as it's incompatible with my best account of transsubstantiation), but the result about phase spaces will remain true. If one wants to rule them, one can either insist that in fact there is no phase space location in which is natural or disallow abstract topological spaces as relata, despite the benefits of allowing them. One might also add an axiom that makes this be a location in spacetime, by using causation. For instance, we might require the topological space to have a partial ordering on its points (we might add something about how the ordering should play nice with the topology), which we will call "at least as late as", and then extend this to a relation between regions: R' is at least as late as R provided that every point y of R' is at least as late as some point x of R and every point x of R has some point y of R' such that y is at least as late as x. Then add:
  • Normally, if event E causes event E', and E and E' are exactly located at R and R' respectively, then R' is at least as late as R.

Monism and functionalism are extreme theories because functionalism classifies as locational as many relations as anybody could possibly reasonably want to do that to and monism classifies as locational as few as anybody who thinks location is real reasonably could.

I incline to functionalism here.

Friday, October 21, 2011

Monads are spatiotemporal

I've noticed that people sometimes talk as if Leibniz's monads were not spatiotemporal objects. But that seems to me to be just like saying that physicalist reductionists don't believe in mental states. The physicalist reductionist believes in mental states—she just thinks that they are nothing but physical states. Similarly, Leibniz believes in spatial relations between monads—he just thinks that they reduce to the clarity and confusion of conscious and, especially, unconscious perceptions. We shouldn't confuse the reductionist with the eliminativist.

Things are less clear with regard to causation, because Leibniz quite often talks about how there is no causation between monads, and yet he gives what seems to be a reduction of causation between monads to explanatory relations between intra-monadic states. One reading of what's going on there is that Leibniz himself recognizes that his reduction is unsatisfactory, and hence that this is only a causal-like relation rather than real causation. It may be important to his view here that the is a meaningful distinction between this quasi-causation and genuine causation, because in his system there is a place for each: genuine causation occurs within a monad and between God and a monad (I'm stipulatively using "monad" in a way that excludes God; I am not clear whether Leibniz would call God a monad; he has one passage where he referred to God as a monad and then crossed it out), while quasi-causation occurs between monads.

Thursday, October 20, 2011

Hyperlinked Summa Theologica for Kindle

A colleague told me that the Summa Theologica files he found for the Kindle weren't great. So here is mine. It's in Mobipocket format. (I already posted the epub one.)

A Platonic substantivalist theory of spacetime

On standard substantivalist theories of spacetime, there is a special contingent concrete entity spacetime and locational properties of objects depend on the objects' relations to the parts or components of spacetime. There is more than one way of spelling this out, depending on which relations one takes to be primitive and which parts or components of spacetime the relations relation the objects to. For instance, you can have a point theory, on which the fundamental relation is something like being at least partly located at, or you can have a region theory, on which the fundamental relation is something like being wholly located within, exactly occupying or spatially overlapping.

Substantivalist theories have at least two costs: (1) they bloat the ontology and do so with a fairly mysterious object or objects; and (2) as Leibniz argues in his correspondence with Clarke, they make it mysterious why the contents of space (or spacetime) are where they are rather than all shifted in some direction.

I will offer, without in any way endorsing (in fact, I am strongly inclined to disendorse the theory due to some theological worries and as I am mildly inclined towards ifthenism in mathematics), a substantivalist story that helps with cost (1), and when combined with theism also helps with (2), and that may even have the bonus of helping with the epistemology of mathematics. I will do this in the context of a point theory where L will be the being at least partly located at relation, but it can be done equally well with a region theory.

The theory is simple. Physicists and mathematicians model spacetime with a mathematical object, say Euclidean four-dimensional space, Minkowski space or some curved Riemannian manifold. Don't take that to be a model: take it to be the reality. In other words, objects are literally, and not merely within a model, located at points of a mathematical object.

Put that way, it just struck me that this sounds mysterious. How could we be "inside" a mathematical object? Aren't mathematical objects just sets? But how can one be inside a set, except metaphorically, unless you yourself are a mathematical object? But really, there need be nothing much to this. To be at least partly located at a point x of a mathematical object is just to be L-related to x, and the relation L may just be a fundamental relation. There is nothing mysterious about being related to mathematical objects. If there are any mathematical objects, you're related to many of them by the thinking about relation. There is nothing particularly mysterious about supposing another relation that material (or enmattered) objects can have to mathematical objects. And why couldn't that be the location relation? Granted, it sounds odd to say we're inside mathematical objects, but I think the weirdness comes from not paying attention to the shift in the meaning of "inside" between ordinary cases where one material object is inside another and the idea of an object being inside a region of spacetime. It would be weird indeed if we were inside a mathematical object in the first sense but it isn't so weird to say we're inside a mathematical object in the second sense. In fact, we might take the second sense of being inside as somewhat metaphorical even on an ordinary substantivalist theory. One might worry that only mathematical objects can be inside a mathematical object, but there is little reason to think that. One might as well think that only regions can be in a region. Besides, a set can have concrete members in a set theory with ur-elements, and that's a way of being "inside".

Back to the theory. Well, actually, we're done with the fundamental layer. We just have a mathematical object S and L is a relation between objects and points of S. If we want to define other locational relations, such as being wholly located in, and so on, we need to do more work, but I will leave those details to the reader.

Now, there will be a question of what sort of an object S is and how it is related to its points. I think the right answer on the theory is: We don't know. That goes too far beyond the empirically observable. But we can speculate. For instance, in standard mathematical fashion, we might well take S to be a set with some structure, say a topological or a metric one, and "the points of S" will then be members of that set. For instance, in a Euclidean setting, we could just take S to be R4, the set of all quadruples (x,y,z,t) where x, y, z and t are real numbers.

There is an arbitrariness worry here. Why this set and not another? There is actually a lot of arbitrariness here. For instance, in my Euclidean example, I talked of "real numbers". But what are real numbers? On a fairly standard view, they are objects in the universe of sets, constructed with some construction procedure, like Dedekind cuts or Cauchy sequences. Different construction procedures yield different but isomorphic real fields. Why is our spacetime based on the real field R rather than some isomorphic field R*? That's a good question, but notice that (a) it is not clear that the question is insuperable, and (b) the question is no more problematic than ordinary substantivalism. It is not clear that the question is insuperable because some constructions are more elegant than others, and God could have simply chosen a particularly elegant denizen of the universe of sets to be our spacetime (e.g., I have a non-arbitrary preference for the Cauchy sequence method over the Dedekind cut method because it seems to me to have a greater generality). And it's no more problematic than ordinary substantivalism because presumably there are many possible spacetimes we could have inhabited, and we could have inhabited different portions of them.

Now as to the advantages of this theory. First, bloat. Many philosophers think we need mathematical entities, especially those of set theory, anyway, independently of questions of spacetime. If they're right (and I am not sure of this), then there is no additional ontological bloat. If spacetime is a set, and we're already committed to sets, then we've added no new and mysterious object to the ontology.

Second, the Leibniz problem. The problem is why would the contents of spacetime be where they are rather than shifted over by symmetry of the spacetime. Now, this problem is lesser in curved spacetimes where there may not be any symmetries of the right sort. So perhaps given general relativity we don't need to worry about it so much. But let's worry about it. To make the worry as big as we can, take Leibniz's setting where spacetime can be modeled with the Euclidean space R4. Well, on the view at hand, spacetime is some mathematical object, and let's suppose it just is R4. Then Leibniz's worry arose from the fact that all points in spacetime were exactly alike, and so there would be no reason for an object to be at one point rather than another. But it is false that all points in R4 are exactly alike. For the points in R4 are quadruples of numbers, and it is false that all numbers are exactly alike. On the contrary, 0 and 1 are very different indeed, both arithmetically—anything multiplied by 0 is 0, but anything multiplied by 1 is itself—and on the standard set-theoretic construction of arithmetic, 0 is empty and 1 has a member. So since not all points are exactly alike, Leibniz's argment fails. If God exists, then he can make a non-arbitrary choice of where to locate things. For instance, he might put some theologically important event, like the first object being created, at (0,0,0,0).

Finally, I said that this might help with epistemology of mathematics. The major problem with the epistemology of mathematics is that on standard views mathematical entities do not stand in causal relations, so it is hard to see how we can know about them. But on this view, while mathematical objects may still be causally inert, enmattered objects have causally relevant relations to mathematical entities. Which locations things are at is important to causal explanation in the sciences. We can, arguably, even see where an object is. So relations to those mathematical entities that make up spacetime become causally relevant and perhaps even observable, and that may help us get a foothold in the mathematical realm (interestingly, this would make something like analysis or geometry be the basic part of mathematics). I think that this is a stretch, and there are probably better ways of getting an epistemology of mathematics going (option 1: theism; option 2: observe that while mathematical entities are causally inert they are not explanatorily inert), but it does help somewhat.

Wednesday, October 19, 2011

Plato might have been a "nominalist"

I was reading the SEP entry on nominalism by Rodriguez-Pereyra. Rodriguez-Pereyra sees nominalism as basically the rejection of causally inert non-spatiotemporal entities. If so, then Plato might have been a nominalist. It seems that Plato did not think the Form of the Good was causally inert--it caused the good arrangement of things in the universe. I don't know if Plato generalized from that case, but he might well have--he might have taken all of the Forms to be capable of causing things to be like them. So, for all I know, Plato was a nominalist.

And Leibniz might have been was a nominalist despite going on and on about abstract objects, because he thought of them as ideas guiding God's deliberation, and hence perhaps we should say that on his view they had a causal role in creation.

This isn't a big deal. Rodriguez-Pereyra's account nicely captures a rejection of modern forms of Platonist.

I wonder, too, whether a belief in Newtonian space is compatible with nominalism by this definition. Newtonian space seems to be causally inert (perhaps unlike the Riemannian manifold of General Relativity). And it may be a category mistake to say that space is spatiotemporal. Though maybe it's fine to say that space is spatiotemporal in some trivial sense.

A reduction of spatial relations to an outdated physics

Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time t between distinct particles a and b as (mamb/Fab)1/2, where Fab is the magnitude of the gravitational force of a on b at t, and ma and mb are the masses at t of a and b respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between a and a to be zero. For every t, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let S be a three-dimensional Euclidean space. Let P be the set of all particles. Let T be the real line. For each object a in P, let Ta be the set of times at which a exists, and let ma(t) be the mass of a at t. For any pair of objects a and b and time t in both Ta and Tb, let Fab(t) be the magnitude of the gravitational force of a on b at t. Let Q be the set of all pairs (a,t) such that t is a member of Ta. Say that a function f from Q to S is an admissible position function provided that:

  1. If t is a member of both Ta and Tb, then Fab(t)=ma(t)mb(t)/|f(b,t)−f(a,t)|2.
  2. f''(a,t) is equal to the sum over all particles b distinct from a of (f(b,t)−f(a,t))Fba(t)/(ma(t)|f(b,t)−f(a,t)|).
The laws can then be taken to say that the world is such that there is an admissible position function. We can then relativize talk of location to an admissible position function, which plays the role of a reference frame: the location of a relative to f at t is just f(a,t).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).

Tuesday, October 18, 2011


I'm going to offer three arguments for a conclusion I found quite counterintuitive when I got to it, and which I still find counterintuitive, but I can't get out of the arguments for it.

Argument 1. There is a game being played in my sight. The player chooses some value (e.g., a number, a pair of numbers, etc.) and gets a payoff that is a function of the value she chose and some facts that I have no information whatsoever about. Moreover, the payoff function is the same for each player, and the facts don't change between players. I see Jones playing and choosing some value v. I don't get to see what payoff Jones gets. What value should I choose? I think there is a very good case that I should choose v, just as Jones did. After all, I know that I have no information about the unknown facts, but for all I know, Jones knows something more about them than I do (if that's not true, then I do know something about the unknown facts, namely that Jones doesn't know anything about them).

Now, suppose that the game is the game of assigning credences (whether these be point values, intervals, fuzzy intervals, etc.) to a proposition p, and that the payoff function is the right epistemic utility function measuring how close one's credence is to the actual truth value of p. If I should maximize epistemic utility, I get the conclusion that if I know nothing about p other than that you assign to it a credence r, then I should assign to it credence r. Note: I will assume throughout this post that the credences we are talking about are neither 0 or nor 1—there are some exceptional edge effects in the case of those extreme credences, such as that Bayesian information won't shift us out of them (we might have special worries about irreversible decisions, which may trump the above argument).

I find this result quite counterintuitive. My own intuition is that when I know nothing about p other than the credence you assign to p, I should assign to p a downgrade of your credence—I should shift your credence closer to 1/2. But contradicts the conclusion I draw from the above argument.

I can get to the more intuitive result if I have reason to think Jones is less risk averse than I am. In the case of many reasonable epistemic utility measures, risk averseness will push one towards 1/2. So perhaps my intuition that you should downgrade the other's credence, that you should not epistemically trust the other as you trust yourself, comes from an intuition that I am more epistemically risk averse than others. But, really, I have little reason to think that I am more epistemically risk averse than others (though I do have reason to think that I am more non-epistemically risk averse than others).

Argument 2: Suppose I have no information about some quantity Q (say, the number of hairs you've got, the gravitational constant, etc.) other than that Jones' best estimate for Q is r. What should my best estimate for Q be? Surely r. But now suppose I have no information about a proposition p, except that Jones' best estimate for how well p is supported by her evidence is r. Then my best estimate for how well p is supported by Jones' evidence is r. And since I have no evidence to add to the pot, and since my credence should match evidential support (barring some additional moral or pragmatic considerations, which I don't have reason to think apply, since I have no additional information about p), I should have credence r. (Again, it doesn't matter if credences are points or intervals vel caetera.)

Let me make a part of my thinking more explicit. If I have no further information on Q, which Jones estimates to be r, it is equally likely that Jones is under-estimating Q as that Jones is over-estimating Q, so even if I don't trust Jones very much, unless I have specific information that Jones is likely to over-estimate or under-estimate, I should take Q as my best estimate. If Q is the degree to which p is supported by Jones' evidence, then the thought is that Jones might over-estimate this (epistemic incautiousness) or Jones might under-estimate it (undue epistemic caution). Here the assumption that we're not working with extreme credences comes in, since, say, if Jones assigns 1, she can't be under-estimating.

Argument 3: This is the argument that got me started on this line of thought. Imagine two scenarios.
Scenario 1: I have partial amnesia—I forget all information relevant to the proposition p, including information as to how reliable I am in judgments of the p sort. And I don't gain any new evidence. But I do find a notebook where I wrote that I assign credence r to p. I am certain the notebook is accurate as to what credence I assigned. What credence should I assign?
Scenario 2: Same as Scenario 1, except that the notebook lists Jones' credence r for p, not my credence. And I have no information on Jones' reliability, etc.

In Scenario 1, I should assign credence r to p. After all, I shouldn't downgrade (I assume upgrading is out of the question) credences that are stored in my memory, or else all my credences will have an implausible downward slide absent new evidence, and it shouldn't matter whether the credence is stored in memory or on paper.

But I should do in Scenario 2 exactly what I would do in Scenario 1. After all, barring information about reliability, why take my past self to be any more reliable than Jones? So, in Scenario 2, I should assign credence r, too. But the partial amnesia is doing no work in Scenario 2 other than ensuring I have no other information about p. So, given no other information about p, I should assign the same credence as Jones.

Final off-the-cuff remark: I am inclined to take this as a way of loving one's neighbor as oneself.[note 1]

Monday, October 17, 2011

Fictional entities

A student tells me: "Patrick Jones stole my laptop. Could I get an extension?" But the student's laptop has never been stolen, and the student knows it. Clearly the student has to be lying. Or does she? Suppose that we accept a realism about fictional entities, and suppose that shortly before coming to class, the student wrote a very short story in which Patrick Jones steals her laptop after she writes her homework. On realist views of fictional characters on which one can correctly say "Odysseus was resourceful", the student has told me the sober truth. (That she features in the story is clearly not a problem—stories can include real entities.) That seems to be a good reason to reject such views. This argument leaves untouched realist views on "Odysseus is a fictional character" is literally true, but "Odysseus was resourceful" should be taken to be elliptical for something like "Odysseus was resourceful in the Odyssey."

Friday, October 14, 2011

Folk psychology and scientific practice

All contemporary science evidentially depends on folk psychology. For instance:

  • it is assumed that other scientists tend to say what they observed, and "observation" is a term of folk psychology; the evidence that they tend to say what they observed is based on evaluation of their motives;
  • it is essential to the modern scientific enterprise that one be able to assume that technicians are doing what they claim, rather than producing equipment that displays fraudulent results; but the evidence that they are doing this is based on theories about the technicians' motives;
  • the concept of reporting depends on folk psychological concepts; if nobody has intentions, nobody reports any results; but the reports of others are central to contemporary scientific practice.
Thus, those who are scientifically motivated to attack folk psychology are cutting the branch on which they sit.

Thursday, October 13, 2011

Trust and faith

Aquinas tells us that human sociality is partly, maybe even largely, exhibited in our common epistemic project. Now, I think trust is absolutely central to this project. Trust is the glue that binds the individual epistemic projects into a joint project, and more generally that binds us in non-epistemic contexts. In particular, it is natural to trust others, and we always have a pro tanto moral reason to trust another. A failure to take another person's assurance of something as a reason to trust in the assured thing (a claim, a commitment, etc.) is a failure to treat the other person as a co-participant in the joint project, and is a partial denial of our common sociality.

So we always have pro tanto moral reasons to trust others. There may, of course, be overriders or defeaters for these moral reasons. Still, a habit of appropriately taking into account the moral reason to trust others because of our common sociality is a virtue. This virtue is balanced between mistrust and credulity, and we can call it "proper trust."

This, I think, makes it intelligible how faith can be a virtue. Faith is a species of deep proper trust in God—a proper trust so deep that it cannot be a work of nature. Still, like other theological virtues it builds on a natural virtue, in this case proper trust.

Interestingly, though, the theological virtue, unlike the natural, may well cease to be a mean. For there is no such thing as trusting God too much, as he is perfectly trustworthy. This is a feature faith shares with charity, which is a supernatural love for God. For while one can idolatrously overestimate the object of love for a creature, one cannot overestimate the object of love for God. So there is a sense in which charity also is not a mean (not an original view). I do not, at this point, know exactly what I should say about hope.

Wednesday, October 12, 2011


Let's say that when you feel that you have a disease but you don't have it, you have the disease of hypochondria. But what if you feel that you have hypochondria and you don't feel that you have any other disease? :-)

Tuesday, October 11, 2011

An infinity argument for incommensurability

Consider these plausible claims:

  1. If worlds w1 and w2 contain the exact same individuals, and each individual in w2 is better off than she is in w1, then w2 is a more valuable world.
  2. A world can contain an infinite number of individuals.
  3. If worlds w1 and w2 differ only in respect of which particular individuals exist in them, and perhaps some further value-insignificant respects, if an identity of indiscernibles principle requires it, then w1 and w2 are equal in value.
  4. If worlds w1 and w2 differ only in respect of w1 lacking one individual that exists in w2 and that has a good life in w2, and perhaps in some value-insignificant way, then w2 is more valuable than w1.
  5. If a and b are equally valuable, then c more (less) valuable than b if and only if c is more (less) valuable than a.
  6. Being more valuable than is transitive.
  7. Nothing is more valuable than itself.
Now, making liberal use of (2), imagine this sequence of worlds:
  • w1: God plus an infinite sequence of spatiotemporally disconnected individuals x1, x2, ... who are almost exactly alike, differing only in that xi enjoys on balance flourishing of value i (plus any other insignificant differences needed to avoid violation of the identity of indiscernibles)
  • w2: just like w1, except that the individuals are all different: y1, y2, ...
  • w3: just like w2, except that yi now enjoys on balance flourishing of value i+1, for all i
  • w4: just like w3, except that in the place of yi we have xi+1, for all i
Now, notice that w4 is basically w1 minus the individual x1, with perhaps some value-insignificant differenecs. By (4), it follows that w1 is more valuable than w4. By (3), it follows that w2 has the same value as w1. By (1), it follows that w3 is more valuable than w2. By (3), it follows that w4 has the same value as w3. By (5) and (6) it follows that w4 is more valuable than w4. Which contradicts (7).

I think the controversial assumptions are (2) and (3). It's really hard to deny (1), (4), (5), (6) or (7). So, either there can't be an actual infinity or (3) is false. Now, the falsity of (3) would imply a really radical form of incommensurability: situations that are exactly alike except for the particular identities of the individuals involved (and whatever identity of indiscernibles further requires) can differ in value.

I want to hold on to (2). Plainly a world can have an infinite future containing an infinite number of individuals. So I reject (3), and thus accept the radical incommensurability claim above.

And I think the incommensurability claim is independently plausible.

Monday, October 10, 2011

It is more than 2.588 times as important to avoid certainty about a falsehood than to have certainty about a truth

William James discusses two kinds of people: there is the person whose epistemic life is focused on getting to as many truths as possible and there is the person whose epistemic life is focused on avoiding falsehoods. So, there is truth-pursuit and error-avoidance. We don't want to have one without the other. For instance, the person who just desires truth, without hating error, might just believe every proposition (and hence also the negation of every proposition) and thus get every truth—but that's not desirable. And a chair has perfectly achieved the good of not believing any falsehood. So a good life, obviously, needs to include both love of truth and hatred of error. But how much of which? William James suggests there is no right answer: different people will simply have different preferences.

But it turns out that while there may be different preferences one can have, there is a serious constraint. Given some very plausible assumptions on epistemic utilities, one can prove that one needs to set more than 2.588 times (more precisely: at least 1/(log 4 − 1) times) as great a disvalue on being certain of a falsehood as the value one sets on being certain of a truth!

Here are the assumptions. Let V(r) be the value of having credence r in a true proposition, for 1/2≤r≤1. Let D(r) be the disvalue of having credence r in a false proposition, again for 1/2≤r≤1. Then the assumptions are:

  1. V and D are continuous functions on the interval [1/2,1] and are differentiable except perhaps at the endpoints of the interval
  2. V(1/2)=D(1/2)=0
  3. V and D are increasing functions
  4. D is convex
  5. The pair V and D is stable.
Assumption 1 is a pretty plausible continuity assumption. Assumption 2 is also a reasonable way to set a neutral value for the utilities. Assumption 3 is very plausible: it is better to be more and more confident of a truth and worse to be more and more confident of a falsehood. Assumption 4 corresponds to a fairly standard, though controversial, assumption on calibration measures. It is, I think, quite intuitive. Suppose that p is false. Then you gain more by decreasing your credence from 1.00 to 0.99 than by decreasing your credence from 0.99 to 0.98, and you gain more by decreasing your credence from 0.99 to 0.98 than by decreasing your credence from 0.98 to 0.97. You really want to get away from certainty of a falsehood, and the further away you are from that certainty, the less benefit there is in getting away. And the convexity assumption captures this intuition.

Finally, the stability condition (which may have some name in the literature) needs explanation. Suppose you have assigned credence r≥1/2 to a proposition p. Then you should expect epistemic utility rV(r)−(1−r)D(r) from this assignment. But now suppose you consider changing your credence to s, without any further evidence. You would expect to have epistemic utility rV(s)−(1−r)D(s) from that. Stability says that this isn't ever going to be an improvement on what you get with s=r. For if it were sometimes an improvement, you would have reason to change your credence right after you set it evidentially, just to get better epistemic utility, like in this post (in which V(r)=D(r)=2r−1—and that's not stable). Stability is a very plausible constraint on epistemic utilities. (The folks working on epistemic utilities may have some other name for this condition—I'm just making this up.)

Now define the hate-love ratio: HL(r)=D(r)/V(r). This measures how much worse it is assign credence r to a falsehood than it is good to assign r to a truth.

Theorem. Given (1)-(5), HL(r)≥(u−1/2)/(1/2+(log 2)−u+log u).

Corollary. Given (1)-(5), HL(1)≥1/(log 4 − 1)>2.588.

In other words, you should hate being certain of a falsehood more than 2.588 times as much as you love being certain of a truth.

Note 1: One can make V and D depend on the particular proposition p, to take account of how some propositions are more important to get right than others. The hate-love ratio inequality will hold for each proposition, then.

Note 2: There is no non-trivial upper bound on HL(1). It can even be equal to infinity (with a logarithmic measure, if memory serves me).

Here is a graph of the right hand side of the inequality in the Theorem (the x-axis is r).

Let me sketch the proof of the Theorem. Let Ur(s)=rV(s)−(1−r)D(s). Then for any fixed r, stability says that Ur(s) is maximized at s=r. Therefore the derivative Ur'(s) vanishes at s=r. Hence rV'(r)−(1−r)D'(r)=0. Therefore V'(r)=(1−r)D'(r)/r. Thus, V(r) is the integral from 1/2 to r of (1/x−1)D'(x)dx. Moreover, by convexity, we have that D'(r) is an increasing function. One can then prove that the hate-love ratio D(r)/V(r) will be minimal when D' is constant (this is actually the hardest part of the proof of the Theorem, but it's very intuitive), i.e., when D is linear, and an easy calculation then gives the value for the hate-love ratio on the right hand side of the inequality in the Theorem.

Saturday, October 8, 2011

Going beyond the evidence out of love for the truth

I want to have correct beliefs. Consider some proposition, p, that isn't going to (directly or indirectly, say by epistemic connections to other propositions) affect my actions in a significant way, say that there is life outside our galaxy. Suppose that my evidence supports p to degree r with 0<r<1. What credence should I assign to p? The evidentialist will say that I should assign r. But that's not the answer from decision theory on the following model of my desires.

As I said, I want to be right about p. If I assign credence 1/2 to p, I get no utility, regardless of whether p is true or not. If p is true and I assign credence 1 to p, then I get utility +1, and if I assign credence 0 to p then I get utility −1. Between these two extremes, I interpolate linearly: If p is true, the utility of credence s is 2s−1. And this gives the right answer for credence 1/2, namely zero utility. If, on the other hand, p is false, then I get utility +1 if I gave credence 0 to p, and I get utility −1 if I gave credence −1 to p, and linearly interpolating tells me that the utility of credence s is 1−2s.

These utilities are a kind of model of love of truth. I want to have the truth firmly in my grasp, and I want to avoid error, and there is complete symmetry between love of truth and hatred of error. And nothing else matters.

What credence should I, from a self-interested decision-theoretic standpoint, assign to p? Well, if I assign credence s to p, my expected utility will be:

  • U(r,s)=r(2s−1)+(1−r)(1−2s)=(2s−1)(2r−1).
It is easy to see that if r>1/2, then I maximize utility when s=1. In other words, on the above model, if nothing matters but truth, and the evidence favors p to any degree, I do best by snapping my credence to 1. Similarly, if r<1/2, then I do best by snapping my credence to 0. The only time when it's not optimal to snap credences to 0 or 1 is when r=1/2, in which case U(r,s)=0 no matter what the credence s is.

So, love of truth, on the above model, requires me to go beyond the evidence: I should assign the extreme credence on the side that the evidence favors, howsoever slightly the evidence favors it.

Now, if I can't set the credence directly, then I may still be able to ask someone to brainwash me into the right credence, or I might try to instill a habit of snapping credence to 0 or 1 in the case of propositions that don't affect my actions in a significant way.

The conclusion seems wrong. So what's gone wrong? Is it decision theory being fundamentally flawed? Neglect of the existence of epistemic duties that trump even truth-based utilities? The assumption that all that matters here is truth (maybe rational connections matter, too)?

Objection: The rational thing to do is not to snap the credence to 0 or 1, but to investigate p further, which is likely to result in a credence closer to 0 or to 1 than r, as Bayesian convergence sets in.

Response 1: If you do that, you miss out on the higher expected utility during the time you're investigating.

Response 2: In some cases, you may have good reason to think that you're not going to get much more evidence than you already have. For instance, suppose that currently you assign credence 0.55 to p, and you have good reason to think you'll never get closer to 0 or 1 than a credence of 0.60 or 0.40. It turns out that you can do an eventual expected utility calculation comparing two plans. Plan 1 is to just snap your credence to 1. Then your eventual expected utility is U(0.55,1)=0.1. Plan 2 is the evidentialist plan of seeking more evidence and proportioning your belief to it. Then, plausibly, your eventual expected utility is no bigger than aU(0.60,0.60)+(1−a)U(0.40,0.40), where a is the probability that you'll end up with a subjectively probability bigger than 1/2 (that you'll end up exactly at 1/2 has zero chance). But U(0.60,0.60)=U(0.40,0.40)=0.04. So you'll do better going with Plan 1. Your eventual utility (and here you need to look at what weight should be assigned to considerations of how much more valuable it is to have the truth earlier), however, will be even better if you try to have the best of both worlds. Plan 3: investigate until you it looks like you are unlikely to get anything significantly more definite, and then snap your credence to 0 or 1. You might expect, say, to get credence 0.60 or 0.40 after such investigation, and then your utility will be aU(0.60,1)+(1−a)U(0.40,1)=0.2. This plan combines an evidentialist element together with a non-evidentialist snapping of credences way past the evidence.

Tuesday, October 4, 2011

Does belief distribute over conjunction?

The distribution thesis is that:

  • Necessarily, if x believes that p and q, then x believes that p and x believes that q.
I'm going to give three arguments against the distribution thesis.

Argument 1:

  1. It is not possible to believe that p while believing that not-p.
  2. It is possible to believe that p&q while believing that not-p.
  3. So, it is possible to believe that p&q while not believing that p.
I am inclined to think (1) is false, but some people do accept it. As for (2), imagine a case where you believe not-p, and it's one of your slow days, and someone you trust tells you that p&q. So, you believe that p&q. But since it's one of your slow days, it takes an extra moment or two before you see that this conflicts with p&q, and until then you keep on believing both p&q as well as p.

Argument 2:

  1. If it is possible to assign a higher probability to the proposition that p&q than to the proposition that p, then it is possible to believe that p&q while not believing that p.
  2. It is possible to assign a higher probability to p&q than to p.
  3. So, it is possible to believe that p&q without believing that p.
Claim (4) is pretty plausible. If it's possible to assign a higher probability to the conjunction, it should be possible to have a case where the conjunction's probability is just above the cut-off line for belief and the first conjunct's probability is just below it. And if that's possible, it should be possible to believe the conjunction without believing the first conjunct. Claim (5) follows from Tversky and Kahneman's work on the conjunction fallacy.

Argument 3: My mathematics dissertation director gave me this advice on how to write mathematics papers: you can skip an obvious step, but you cannot skip two obvious steps in a row. The point is that two obvious steps in a row may be quite unobvious. Say that a belief "transfers over an inference" if and only if it is not possible to believe the premises of the inference without believing the conclusion of the inference. For instance, the distribution thesis says preicsely that belief transfers over conjunction elimination.

  1. Belief does not transfer over inferences that can be unobvious.
  2. The inference from the claim that p&(q&r) to the claim that q can be unobvious, since it takes two applications of conjunction elimination.
  3. If the distribution thesis is correct, then belief transfers over the double conjunction elimination inference from p&(q&r) to q.
  4. So, the distribution thesis is not correct.

Monday, October 3, 2011

Leibniz on metaphysics

[I]n fact, metaphysics is natural theology, and the same God who is the source of all goods is also the principle of all knowledge. (Letter to Countess Elizabeth) 

On a relativism about beauty

Consider a naive relativist theory on which, necessarily,
  • y is beautiful to x if and only if x takes y to be beautiful.
This cannot be a complete theory about beauty.  After all, exactly the same theory can be given for ugliness:
  • y is ugly to x if and only if x takes y to be ugly.
Since nothing has been said that distinguishes beauty from ugliness, the theory cannot be complete.  

Moreover, there is a further oddness about the theory as I've given it.  According to the theory, the fundamental concepts are relational: being beautiful (or ugly) to.  But on the right hand side of the biconditionals we have the monadic beautiful (or ugly).  If someone fully accepts the theory, she won't take anything to be beautiful simpliciter, but only beautiful to her.  So, perhaps the relativist should say:
  • y is beautiful to x if and only if x takes y to be beautiful to her.
One serious problem with this is that then nothing is beautiful to the self-conscious objectivist, since the self-conscious objectivist takes nothing to be beautiful to her--she does not have any relational "is beautiful to" predicate.

And consider another problem.  Suppose I am essentially logically omniscient, so that if p and q are logically equivalent, then it is an essential property of me that I believe p if and only if I believe q.  Applying this to the biconditional, I get:
  • It is an essential property of me that: I believe that y is beautiful to me if and only if I believe that I take y to be beautiful to me.
But to take something to be beautiful to me is just to believe it is beautiful to me.  So:
  • It is an essential property of me that: I believe that y is beautiful to me if and only if I believe that I believe that y is beautiful to me.
But that is surely wrong: logical omniscience should not imply omniscience about my internal states.

Maybe, though, I am being too cognitivist about "takes y to be beautiful to her".  Maybe to take y to be beautiful isn't to believe anything about y but to have a certain appreciative attitude to y.  That takes care of the problem of the objectivist and the logically omniscient individuals.  

But we still have another problem.  Imagine that I love Mozart.  I go to a Mozart violin concert, and then during the intermission I get a message about a family emergency and I need to go home.  The first part of the concert was beautiful to me.  Tomorrow I hear that the second half of the concert was even better in the respects I appreciate.  I conclude that I missed some beautiful performances.  But on the appreciative attitude version of "takes", that's false.  For the second half of the concert wasn't beautiful to me, since I didn't take it to be beautiful in the appreciative sense.  

A familiar response is to amend the right-hand-side of the biconditional to replace "I take y to be beautiful (to me)" with "I would take y to be beautiful (to me) if I experienced y."  But probably not.  After all, had I stayed for the second half of the concert, worries about the family emergency and guilt that I am enjoying someone's fiddling while my home burns (literally or figuratively) would have spoiled my enjoyment of the concert.  Of course this is just a special case of the problems that go under the head of "the conditional fallacy."  

So we would have to idealize: I would take y to be beautiful if I experienced y in ideal observing circumstances.  But as the above example shows, the ideal observing circumstances need to include being in the right mental state.  We better not define the right mental state as the one in which one's appreciation is correct, since then our theory isn't subjectivist any more.  And given that what one appreciates can be so heavily dependent on one's emotional state, it seems that at this point matters are hopeless.  

And all the same goes for similar theories about morality.