Monday, February 19, 2018

Leibniz on PSR and necessary truths

I just came across a quote from Leibniz that I must have read before but it never impressed itself on my mind: “no reason can be given for the ration of 2 to 4 being the same as that of 4 to 8, not even in the divine will” (letter to Wedderkopf, 1671).

This makes me feel better for defending only a Principle of Sufficient Reason restricted to contingent truths. :-)

Life, thought and artificial intelligence

I have an empirical hypothesis that one of the main reasons why a lot of ordinary people think a machine can’t be conscious is that they think life is a necessary condition for consciousness and machines can’t be alive.

The thesis that life is a necessary condition for consciousness generalizes to the thesis that life is a necessary condition for mental activity. And while the latter thesis is logically stronger, it seems to have exactly the same plausibility.

Now, the claim that life is a necessary condition for mental activity (I keep on wanting to say that life is a necessary condition for mental life, but that introduces the confusing false appearance of tautology!) can be understood in two ways:

  1. Life is a prerequisite for mental activity.

  2. Mental activity is in itself a form of life.

On 1, I think we have an argument that computers can’t have mental activity. For imagine that we’re setting up a computer that has mental activity, but we stop short of making it engage in the computations that would make it engage in mental activity. I think it’s very plausible that the resulting computer doesn’t have any life. The only thing that would make us think that a computer has life is the computational activity that underlies supposed mental activity. But that would be a case of 2, rather than 1: life wouldn’t be a prerequisite for mental activity, but mental activity would constitute life.

All that said, while I find the thesis that life is a necessary condition for mental activity, I am more drawn to 2 than to 1. It seems intuitively correct to say that angels are alive, but it is not clear that we need anything more than mental activity on the part of angels to make them be alive. And from 2, it is much harder to argue that computers can’t think.

Thursday, February 15, 2018

Physicalism and ethical significance

I find the following line of thought to have a lot of intuitive pull:

  1. Some mental states have great non-instrumental ethical significance.

  2. No physical brain states have that kind of non-instrumental ethical significance.

  3. So, some mental states are not physical brain states.

When I think about (2), I think in terms similar to Leibniz’s mill. Leibniz basically says that if physical systems could think, so could a mill with giant gears (remember that Leibniz invented a mechanical calculator running on gears), but we wouldn’t find consciousness anywhere in such a mill. Similarly, it is plausible that the giant gears of a mill could accomplish something important (grind wheat and save people from starvation or simulate protein folding leading to a cure for cancer), and hence their state could have great instrumental ethical significance, but their state isn’t going to have the kind of non-instrumental ethical significance that mental states do.

I worry, though, whether the intuitive evidence for (2) doesn’t rely on one’s already accepting the conclusion of the argument.

Beyond binary mereological relations

Weak supplementation says that if x is a proper part of y, then y has a proper part that doesn’t overlap x.

Suppose that we are impressed by standard counterexamples to weak supplementation like the following. Tibbles the cat loses everything but its head, which is put in a vat. Then Head is a part of Tibbles, but obviously Head is not the same thing as Tibbles by Leibniz’s Law (since Tibbles used to have a tail as a part, but Head did not use to have a tail as a part), so Head is a proper part of Tibbles—yet, Head does not seem to be weakly supplemented.

But suppose also that we don’t believe in unrestricted fusions, because we have a common-sense notion of what things have a fusion and what parts a thing has. Thus, while we are willing to admit that Tibbles, prior to its injury, has parts like a head, lungs, heart and legs, we deny that there is any such thing as Tibbles’ front half minus the left lung—i.e., the fusion of all the molecules in Tibbles that are in the front half but not in its left lung.

Imagine, then, that there is a finite collection of parts of Tibbles, the Ts, such that there is no fusion of the Ts. Suppose next that due to an accident Tibbles is reduced to the Ts. Observe a curious thing. By all the standard definitions of a fusion (see SEP, with obvious extensions to a larger number of parts), after the accident Tibbles is a fusion of the Ts.

So we get one surprising conclusion from the above thoughts: whether the Ts have a fusion depends on extrinsic features of them, namely on whether they are embedded in a larger cat (in which case they don’t have a fusion) or whether they are standalone (in which case their fusion is the cat). This may seem counterintuitive, but artefactual examples should make us more comfortable with that. Imagine that on the floor of a store there are hundreds of chess pieces spilled and a dozen chess boards. By picking out—perhaps only through pointing—a particular 32 pieces and a particular board, and paying for them, one will have bought a chess set. But perhaps that particular chess set did not exist before, at least on a common-sense notion of what things have a fusion. So, one will have brought it into existence by paying for it. The pieces and the board now seem to have a fusion—the newly purchased chess set—while previously they did not.

Back to Tibbles, then. I think the story I have just told shows that if we deny weak supplementation and unrestricted fusions also suggests something else that’s really interesting: that the standard mereological relations—whether parthood or overlap—do not capture all the mereological facts about a thing. Here’s why. When Tibbles is reduced to a head, we want to be able to say that Tibbles is more than its head. And we can say that. We say that by saying that Head is a proper part of Tibbles (albeit one that is not weakly supplemented). But if Tibbles is more than his head even after being reduced to a head, then by the same token Tibbles is more than the sum of the Ts even after being reduced to the Ts. But we have no way of saying this in mereological vocabulary. Tibbles is the fusion or sum of the Ts when that fusion is understood in the standard ways. Moreover, we have no way of using the binary parthood or overlap relations to distinguish the how Tibbles is related to the Ts from relationships that are “a mere sum” relationship.

Here is perhaps a more vivid, but even more controversial, way of seeing the above point. Suppose that we have a tree-like object whose mereological facts are like this. Any branch is a part. But there are no “shorn trunks” in the ontology, i.e., no trunk-minus-branches objects (unless the trunk in fact has no branches sticking out from it). This corresponds to the intuition that while I have arms and legs as parts, there is no part of me that is constituted by my head, neck and trunk. And (this is the really controversial bit) there are no other parts—there are no atoms, in particular. In this story, suppose Oaky is a tree with two branches, Lefty and Righty. Then Lefty and Righty are Oaky’s only two proper parts. Moreover, by the standard mereological definitions of sums, Oaky is the sum of Lefty and Righty. But it’s obvious that Oaky is more than the sum of Lefty and Righty!

And there is no way to distinguish Oaky using overlap and/or parthood from a more ordinary case where an object, say Blob, is constituted from two simple halves, say, Front and Back.

What should we do? I don’t know. My best thought right now is that we need a generalization of proper parthood to a relation between a plurality and an object: the As are jointly properly parts of B. We then define proper parthood as a special case of this when there is only one A. Using this generalization, we can say:

  • Head is a proper part of Tibbles before and after the first described accident.

  • The Ts are jointly properly parts of Tibbles before and after the second described accident.

  • Lefty and Righty are jointly properly parts of Oaky.

  • It is not the case that Front and Back are jointly properly parts of Blob.

Wednesday, February 14, 2018

Mereology and constituent ontology

I’ve just realized that one can motivate belief in bare particulars as follows:

  1. Constituent ontology of attribution: A thing has a quality if and only if that quality is a part of it.

  2. Universalism: Every plurality has a fusion.

  3. Weak supplementation: If x is a proper part of y, then y has a part that does not overlap x.

  4. Anti-bundleism: A substance (or at least a non-divine substance) is not the fusion of its qualities.

For, let S be a substance. If S has no qualities, it’s a bare particular, and the argument is done.

So, suppose S has qualities. By universalism, let Q be the fusion of the qualities that are parts of S. This is a part of S by uncontroversial mereology. By anti-bundleism, Q is a proper part of S. By weak supplementation, S has a part P that does not overlap Q. That part has no qualities as a part of it, since if it had any quality as a part of it, it would overlap Q. Hence, P is a bare particular. (And if we want a beefier bare particular, just form the fusion of all such Ps.)

It follows that every substance has a bare particular as a part.

[Bibliographic notes: Sider thinks that something like this argument means that the debate between constituent metaphysicians overlap bare particulars is merely verbal. Not all bare particularists find themselves motivated in this way (e.g., Smith denies 1).]

To me, universalism is the most clearly false claim. And someone who accepts constituent ontology of attribution can’t accept universalism: by universalism, there is fusion of Mt. Everest and my wedding ring, and given constituent ontology, the montaineity that is a part of Everest and the goldenness of my ring will both be qualities of EverestRing, so that EverestRing will be a golden mountain, which is absurd.

But universalism is not, I think, crucial to the argument. We use universalism only once in the argument, to generate the fusion of the qualities of S. But it seems plausible that even if universalism in general is false, there can be a substance S such that there is a fusion Q of its qualities. For instance, imagine a substance that has only one quality, or a substance that has a quality Q1 such that all its other qualities are parts of Q1. Applying the rest of the argument to that substance shows that it has a bare particular as a part of it. And if some substances have bare particular parts, plausibly so do all substances (or at least all non-divine substances, say).

If this is right, then we have an argument that:

  1. You shouldn’t accept all of: constituent ontology, weak supplementation, anti-bundleism and anti-bare-particularism.

I am an anti-bundleist and an anti-bare-particularist, but constituent ontology seems to have some plausibility to me. So I want to deny weak supplementation. And indeed I think it is plausible to say that the case of a substance that has only one quality is a pretty good counterexample to weak supplementation: that one quality lacks even a weak supplement.

Tuesday, February 13, 2018

Theistic multiverse, omniscience and contingency

A number of people have been puzzled by the somewhat obscure arguments in my “Divine Creative Freedom” against a theistic modal realism on which (a) God creates infinitely many worlds and (b) a proposition is possible if and only if it is true at one of them.

So, here’s a simplified version of the main line of thought. Start with this:

  1. For all propositions p, necessarily: God believes p if and only if p is true.

  2. There is a proposition p such that it is contingent that p is true.

  3. So, there is a proposition p such that it is contingent that God believes p. (1 and 2)

  4. Contingent propositions are true at some but not all worlds that God creates. (Theistic modal realism)

  5. So, there is a proposition p such that whether God believes p varies between the worlds that God creates. (3 and 4)

Now, a human being’s beliefs might vary between locations. Perhaps I am standing on the Texas-Oklahoma border, with my left brain hemisphere in Texas and my right one in Oklahoma, and with my left hemisphere I believe that I am in Texas while with my right one I don’t. Then in Texas I believe I am in Texas while in Oklahoma I don’t believe that. But God’s mind is not split spatially in the same way. God’s beliefs cannot vary from one place to another, and by the same token cannot vary between the worlds that God creates.

An objection I often hear is something like this: a God who creates a multiverse can believe that in world 1, p is true while in world 2, p is false. That's certainly correct. But those are necessary propositions that God believes, then--it is necessary that in world 1, p is true and that in world 2, p is false, say. And God has to believe all truths, not just the necessary ones. Hence, at world 1, he has to believe p, and at world 2, he has to believe not p.

Proper classes as merely possible sets

This probably won’t work out, but I’ve been thinking about the Cantor and Russell Paradoxes and proper classes and had this curious idea: Maybe proper classes are non-existent possible sets. Thus, there is actually no collection of all the sets in our world, but there is another possible world which contains a set S whose members are all the sets of our world. When we talk about proper classes, then, we are talking about merely possible sets.

Here is the story about the Russell Paradox. There can be a set R whose members are all the actual world’s non-self-membered sets. (In fact, since by the Axiom of Foundation, none of the actual world’s sets are self-membered, R is a set whose members are all the actual world’s sets.) But R is not itself one of the actual world’s sets, but a set in another possible world.

The story about Cantor’s Paradox that this yields is that there can be a cardinality greater than all the cardinalities in our world, but there actually isn’t. And in world w2 where such a cardinality exists, it isn’t the largest cardinality, because its powerset is even larger. But there is a third world which has a cardinality larger than any in w2.

It’s part of the story that there cannot be any collections with non-existent elements. Thus, one cannot form paradoxical cross-world collections, like the collection of all possible sets. The only collections there are on this story are sets. But we can talk of collections that would exist counterfactually.

The challenge to working out the details of this view is to explain why it is that some sets actually exist and others are merely possible. One thought is something like this: The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world. E.g., if the world contains an infinite sequence of coin tosses, it contains the set of the natural numbers corresponding to tosses with heads.

Saturday, February 10, 2018

Counting goods

Suppose I am choosing between receiving two goods, A and B, or one, namely C, where all the goods are equal. Obviously, I should go for the two. But why?

Maybe what we should say is this. Since A is at least as good as C, and B is non-negative, I have at least as good reason to go for the two goods as to go for the one. This uses the plausible assumption that if one adds a good to a good, one gets something at least as good. (It would be plausible to say that one gets something better, but infinitary cases provide a counterexample.) But there is no parallel argument that it is at least as good to go for the one good as to go for the two. Hence, it is false that I have at least as good reason to go for the one as to go for the two. Thus, I have better reason to go for the two.

This line of thought might actually solve the puzzles in these two posts: headaches and future sufferings. And it's very simple and obvious. But I missed it. Or am I missing something now?

Friday, February 9, 2018

Counting infinitely many headaches

If the worries in this post work, then the argument in this one needs improvement.

Suppose there are two groups of people, the As and the Bs, all of whom have headaches. You can relieve the headaches of the As or of the Bs, but not both. You don’t know how many As or Bs there are, or even whether the numbers are finite or finite. But you do know there are more As than Bs.


  1. You should relieve the As’ headaches rather than the Bs’, because there are more As than Bs.

But what does it mean to say that there are more As than Bs? Our best analysis (simplifying and assuming the Axiom of Choice) is something like this:

  1. There is no one-to-one function from the As to the Bs.

So, it seems:

  1. You should relieve the As’ headache rather than the Bs’, because there is no one-to-one function from the As to the Bs.

For you should be able to replace an explanation by its analysis.

But that’s strange. Why should the non-existence of a one-to-one function from one set or plurality to another set or plurality explain the existence of a moral duty to make a particular preferential judgment between them?

If the number of As and Bs is finite, I think we can do better. We can then express the claim that there are more As than Bs by an infinite disjunction of claims of the form:

  1. There exist n As and there do not exist n Bs,

which claims can be written as simple existentially quantified claims, without any mention of functions, sets or pluralities.

Any such claim as (4) does seem to have some intuitive moral force, and so maybe their disjunction does.

But in the infinite case, we can’t find a disjunction of existentially quantified claims that analysis the claim that there are more As than Bs.

Maybe what we should say is that “there are more As than Bs” is primitive, and the claim about there not being a one-to-one function is just a useful mathematical equivalence to it, rather than an analysis?

The thoughts here are also related to this post.

Thursday, February 8, 2018

Ersatz objects and presentism

Let Q be a set of all relevant unary predicates (relative to some set of concerns). Let PQ be the powerset of Q. Let T be the set of abstract ersatz times (e.g., real numbers or maximal tensed propositions). Then an ersatz pre-object is a partial function f from a non-empty subset of T to PQ. Let b be a function from the set of ersatz pre-objects to T such that b(f) is a time in the domain of f (this uses the Axiom of Choice; I think the use of it is probably eliminable, but it simplifies the presentation). For any ersatz pre-object f, let n(f) be the number of objects o that did, do or will exist at b(f) and that are such that:

  1. o did, does or will exist at every time in the domain of f

  2. o did not, does not and will not exist at any time not in the domain of f

  3. for every time t in the domain of f and every predicate F in Q, o did, does or will satisfy F at t if and only if F ∈ f(t).

Then let the set of all ersatz objects relative to Q be:

  • OQ = { (i,f) : i < n(f) },

where i ranges over ordinals and f over ersatz pre-objects. We then say that an ersatz object (i, f) ersatz-satisfies a predicate F at a time t if and only if F ∈ f(t).

The presentist can then do with ersatz objects anything that the eternalist can do with non-ersatz objects, as long as we stick to unary predicates. In particular, she can do cross-time counting, being able to say things like: “There were more dogs than cats born in the 18th century.”

Extending this construction to non-unary predicates is a challenging project, however.

Presentism and counting future sufferings

I find it hard to see why on presentism or growing block theory it’s a bad thing that I will suffer, given that the suffering is unreal. Perhaps, though, the presentist or growing blocker can say that is a primitive fact that it is bad for me that a bad thing will happen to me.

But there is now a second problem for the presentist. Suppose I am comparing two states of affairs:

  1. Alice will suffer for an hour in 10 hours.
  2. Bob will suffer for an hour in 5 hours and again for an hour in 15 hours.

Other things being equal, Alice is better off than Bob. But why?

The eternalist can say:

  1. There are more one-hour bouts of suffering for Bob than for Alice.

Maybe the growing blocker can say:

  1. It will be the case in 16 hours that there are more bouts of suffering for Bob than for Alice.

(I feel that this doesn’t quite explain why it’s B is twice as bad, given that the difference between B and A shouldn’t be grounded in what happens in 16 hours, but nevermind that for this post.)

But what about the presentist? Let’s suppose preentism is true. We might now try to explain our comparative judgment by future-tensing (1):

  1. There will be more bouts of suffering for Bob than for Alice.

But what does that mean? Our best account of “There are more Xs than Ys” is that the set of Xs is bigger than the set of Ys. But given presentism, the set of Bob’s future bouts of suffering is no bigger than the set of Alice’s future bouts of suffering, because if presentism is true, then both sets are empty as there are no future bouts of suffering. So (3) cannot just mean that there are more future bouts of suffering for Bob than for Alice. Perhaps it means that:

  1. It will be the case that the set of Bob’s bouts of suffering is larger than the set of Alice’s.

This is true. In 5.5 hours, there will presently be one bout of suffering for Bob and none for Alice, so it will then be the case that the set of Bob’s bouts of suffering is larger than the set of Alice’s. But while it is true, it is similarly true that:

  1. It will be the case that the set of Alice’s bouts of suffering is larger than the set of Bob’s.

For in 10.5 hours, there will presently be one bout for Alice and none for Bob. If we read (3) as (4), then, we have to likewise say that there will be more bouts of suffering for Alice than for Bob, and so we don’t have an explanation of why Alice is better off.

Perhaps, though, instead of counting bouts of suffering, the presentist can count intervals of time during which there is suffering. For instance:

  1. The set of hour-long periods of time during which Bob is suffering is bigger than the set of hour-long periods of time during which Alice is suffering.

Notice that the times here need to be something like abstract ersatz times. For the presentist does not think there are any future real concrete times, and so if the periods were real and concrete, the two sets in (6) would be both empty.

And now we have a puzzle. How can fact (6), which is just a fact about sets of abstract ersatz times, explain the fact about how Bob is (or is going to be) worse off than Alice? I can see how a comparative fact about sets of sufferings might make Bob worse off than Alice. But a comparative fact about sets of abstract times should not. It is true that (6) entails that Bob is worse off than Alice. But (6) isn’t the explanation of why.

Our best explanation of why Bob is worse off than Alice is, thus, (1). But the presentist can’t accept (1). So, presentism is probably false.

Wednesday, February 7, 2018

A really weird place in conceptual space regarding infinity

Here’s a super-weird philosophy of infinity idea. Maybe:

  1. The countable Axiom of Choice is false,

  2. There are sets that are infinite but not Dedekind infinite, and

  3. You cannot have an actual Dedekind infinity of things, but

  4. You can have an actual non-Dedekind infinity of things.

If this were true, you could have actual infinites, but you couldn’t have Hilbert’s Hotel.

Background: A set is Dedekind-infinite if and only if it is the same cardinality as a proper subset of itself. Given the countable Axiom of Choice, one can prove that every infinite set is Dedekind infinite. But we need some version of the Axiom of Choice for the proof (assuming ZF set theory is consistent). So without the Axiom of Choice, there might be infinite but not Dedekind-infinite sets (call them “non-Dedekind infinite”). Hilbert’s Hotel depends on the fact that its rooms form a Dedekind infinity. But a non-Dedekind infinity would necessarily escape the paradox.

Granted, this is crazy. But for the sake of technical precision, it’s worth noting that the move from the impossibility of Hilbert’s Hotel to the impossibility of an actual infinite depends on further assumptions, such as the countable Axiom of Choice or some assumption about how if actual Dedekind infinities are impossible, non-Dedekind ones are as well. These further assumption are pretty plausible, so this is just a very minor point.

I think the same technical issue affects the arguments in my Infinity, Causation and Paradox book (coming out in August, I just heard). In the book I pretty freely use the countable Axiom of Choice anyway.

Tuesday, February 6, 2018

Another intuition concerning the Brouwer axiom

Suppose the Brouwer axiom is false. Thus, there is some possible world w1 such that at w1 our world w0 is impossible. Here’s another world that’s impossible at w1: the world w at which every proposition is both true and false. Thus at w1, possibility does not distinguish between our lovely world and w. But that seems to me to be a sign that the possibility in question isn’t making the kinds of distinctions we want metaphysical modality to make. So, if we are dealing with metaphysical modality, the Brouwer axiom is true.

You can't run this argument if you run this one.

Generating and destroying

Thanks to emails with a presentist-inclining graduate student, I have come to realize that on eternalism there seems to be a really interesting disanalogy between the generation of an item and its destruction, at least in our natural order.

Insofar as you generate an item, it seems you do two things:

  • Cause the item to have a lower temporal boundary at some time t1.

  • Cause the item to exist simpliciter.

But insofar as you destroy an item, you do only one thing:

  • Cause the item to have an upper temporal boundary at some time t2.

You certainly don’t cause the item not to exist simpliciter, since if an item is posited to exist simpliciter, it exists simpliciter (a tautology, of course). It is a conceptual impossibility to act on an existing item and make it not exist, though of course you can act on an item existing-at-t1 and make it not exist-at-t2 and you can counteract the causal activity of something that would otherwise have caused an item to exist. However, existing-at-t isn’t existing but a species of location, just as existing-in-Paris isn’t existing but a species of location.

I suppose one could imagine a world where generation always involves two separate causes: one causes existence simpliciter and another selects when the object exists. In that world there would be an analogy between the when-cause and the destroyer.

(Maybe our world is like that with respect to substance. Maybe only God causes existence simpliciter, while we only cause the temporal location of the substances that God causes to exist?)

I suppose one could see in all this an instance of a deep asymmetry between good and evil.

Monday, February 5, 2018

A heuristic argument for the Brouwer axiom

Suppose that:

  1. We cannot make sense of impossible worlds, but only of possible ones, so the only worlds there are are possible ones.

  2. Necessarily, a possible worlds semantics for alethic modality is correct.

  3. Worlds are necessary beings, and it is essential to them that they are worlds.

Now, suppose the Brouwer axiom, that if something is true then it’s necessarily possible, is not right. Then the following proposition is true at the actual world but not at all worlds:

  1. Every world is possible.

(For if Brouwer is false at w1, then there is a world w2 such that w2 is possible at w1 but w1 is not possible at w2. Since w1 is still a world at w2, at w2 it is the case that there is an impossible world.)

Say that the “extent of possibility” at a world w is the collection of all the worlds that are possible at w. Thus, given 1-3, if Brouwer fails, the actual world is a world that maximizes the extent of possibility, by making all the worlds be possible at it. But it seems intuitively unlikely that if worlds differ in the extent of possibility, the actual world should be so lucky as to be among the maximizers of the extent of possibility.

So, given 1-3, we have some reason to accept Brouwer.