Goodman and Quine and transitive closure

In the previous post, I showed that Goodman and Quine’s counting method fails for objects that have too much overlap. I think (though the technical parts here are more difficult) that the same is true for their definition of the ancestral or transitive closure of a relation.

GQ showed how to define ancestors in terms of offspring. We can try to extend this definition to the transitive closure of any relation R over any kind of entities:

1. x stands in the transitive closure of R to y iff for every object u that has y as a part and that has as a part anything that stands in R to a part of u, there is a z such that Rxz and both x and z are parts of R.

This works fine if no relatum of R overlaps any other relatum of R. But if there is overlap, it can fail. For instance, suppose we have three atoms a, b and c, and a relation R that holds between a + b and a + b + c and between a and a + b. Then any object u that has a + b + c as a part has c as a part, and so (1) would imply that c stands in the transitive closure of R to a + b + c, which is false.

Can we find some other definition of transitive closure using the same theoretical resources (namely, mereology) that works for overlapping objects? No. Nor even if we add the “bigger than” predicate of GQ’s attempt to define “more”. We can say that x and y are equinumerous provided that neither is bigger than the other.

Let’s work in models made of an infinite number of mereological atoms. Write u ∧ v for the fusion of the common parts of both u and v (assuming u and v overlap), u ∨ v for the fusion of objects that are parts of one or the other, and u − v for the fusion of all the parts of u that do not overlap v (assuming u is not a part of v). Write |x| for the number of atomic parts of x when x is finite. Now make these definitions:

2. x is finite iff an atom is related to x by the transitive closure (with respect to the kind object) of the relation that relates an object to that object plus one atom.

3. Axyw iff x and y are finite and whenever x′ is equinumerous with x and does not overlap y, then x′ ∨ y is equinumerous with w. (This says |x| + |y| = |w|.)

4. Say that D_yuv iff A(u−y,u−y,v−y) (i.e., |v−y| = 2|u−y|) and either v does not overlap y or and u ∧ y is an atom or v and y overlap and u ∧ y consists of v ∧ y plus one atom. (This treats u and v as basically ordered pairs (u−y,u∧y) and (v−y,v∧y), and it makes sure that from the first pair to the second, the first component is doubled in size and the second component is decreased by one.)

5. Say that Q⁰yx iff y is finite and for some atom z not overlapping y we have y ∧ z related to something not overlapping x by the transitive closure of D_y. (This takes the pair (z,y), and applies the double first component and decrease second component relation described in (4) until the second component goes to zero. Thus, it is guaranteed that |x| = 2^|y|.)

6. Say that Qyx iff y is finite and Q⁰yx′ for some non-overlapping x′ that does not overlap y and that is equinumerous with x.

If I got all the details right, then Qyx basically says that |x| = 2^|y|.

Thus, we can define use transitive closure to define binary powers of finite cardinalities. But the results about the expressive power of monadic second-order logic with cardinality comparison say that we can only define semi-linear relations between finite cardinalities, which doesn’t allow defining binary powers.

Remark: We don’t need equinumerosity to be defined in terms of a primitive “bigger”. We can define equinumerosity for non-overlapping finite sets by using transitive closure (and we only need it for finite sets). First let T_yuv iff v − y exists and consists of u − y minus one atom and v ∧ y exists and consists of v ∧ y minus one atom. Then finite x and y are equinumerous₀ iff they are non-overlapping and x ∨ y has exactly two atoms or is related to an object with exactly two atoms by the transitive closure of T_yuv. We now say that x and y are equinumerous provided that they are finite and either x = y (i.e., they have the same atoms) or both x − y and y − x are defined and equinumerous₀.

Causation through another's free will

Some people think that causal chains cannot go through other people’s exercises of free will. Thus, if I ask you to do something, and you freely do it, I am not the cause of the action. I think this is mistaken.

Start with this. Suppose I want to stamp out an irregular texture in a piece of aluminum foil. I put the aluminum foil on a soft backing on my CNC router’s bed, and I generate a program for the router by randomly choosing an angle, moving an inch in the direction indicated by the angle (stopping at the edges of the foil) pressing a wooden stick down into the foil, lifting it up, and repeating for a thousand presses. At the end, I will have an irregular texture in the foil. And, clearly, I caused the texture, despite there being randomness in the middle of the causal chain. Nor does it matter for the statement that I caused the irregular texture whether this is pseudorandomness or genuine quantum randomness.

Now, suppose that I replace the random number generator with code that robo-posts trolling comments on people's blogs, reads the responses, and generates random numbers from their hashes. Now, troll-feeders' free actions are an essential part of the causal chain leading to the irregular texture. But surely I have caused the irregular texture just as much as in the previous cases.

Fundamental mereology

It is plausible that genuine relations have to bottom out in fundamental relations. E.g., being a blood relative bottoms out in immediate blood relations, which are parenthood and childhood. It would be very odd indeed to say that a is b’s relative because a is c’s relative and c is b’s relative, and then a is c’s relative because a is d’s relative and d is c’s relative, and so on ad infinitum. Similarly, as I argued in my infinity book, following Rob Koons, causation has to bottom out in immediate causation.

If this is right, then proper parthood has to bottom out in what one might call immediate parthood. And this leads to an interesting question that has, to my knowledge, not been explored much: What is the immediate parthood structure of objects?

For instance, plausibly, the big toe is a part of the body because the big toe is a part of the foot which, in turn, is a part of the body. And the foot is a part of the body because the foot is a part of the leg which, in turn, is a part of the body. But where does it stop? What are the immediate parts of the body? The head, torso and the four limbs? Or perhaps the immediate parts are the skeletal system, the muscular system, the nervous system, the lymphatic system, and so on. If we take the body as a complex whole ontologically seriously, and we think that proper parthood bottoms out in immediate parthood, then there have to be answers to such questions. And similarly, there will then be the question of what the immediate parts of the head or the nervous system are.

There is another, more reductionistic, way of thinking about parthood. The above came from the thought that parthood is generated transitively out of immediate parthood. But maybe there is a more complex grounding structure. Maybe particles are immediately parts of the body and immediately parts of the big toe. And then, say, a big toe is a part of the body not because it is a part of a bigger whole which is more immediately a part of the body, but rather a big toe is a part of the body because its immediate parts are all particles that are immediately parts of the body.

Prescinding from the view that relations need to bottom out somewhere, we should distinguish between fundamental parts and fundamental instances of parthood. One might have one without the other. Thus, one could have a story on which we are composed of immediate parts, which are composed of immediate parts, and so on ad infinitum. Then there would be fundamental instances of the parthoood relation—they obtain between a thing and its immediate parts—but no fundamental parts. Or one could have a view with fundamental parts while denying that there are any fundamental instances of parthood.

In any case, there is clearly a lot of room for research in fundamental mereology here.

An infinite chain can't have two ends

Say that a chain C is a collection of nodes with the following properties:

1. Each node is directly connected to at most two other nodes.

2. If x is directly connected to y then y is directly connected to x (symmetry).

3. C is globally connected in the sense that for any non-empty proper subset S of C, there is a node in S and a node outside of S that are directly connected to each other.

(This is a different sense of “chain” from the one in Zorn’s Lemma.)

Fun fact: Every infinite chain has at most one endpoint, where an endpoint is a node that is directly connected to only one other node.

I.e., one cannot join two nodes with an infinite chain.

Corollary: We cannot join two events by an infinite chain of instances of immediate causation.

I've occasionally wondered if there is a useful generalization of transitive closure to allow for infinite chains, and to my intuition the fact above suggests that there isn't.

Towards a counterexample to Weak Transitivity for subjunctives

Transitivity for a conditional → says that if A→B and B→C, then A→C. For subjunctive conditionals this rule is generally taken to be invalid. If I ate squash (B), I would be miserable eating squash (C). If I liked squash (A), I'd eat squash (B). But it doesn't follow that if I liked squash, I'd be miserable eating squash.

Weak Transitivity says that if A→B, B→A and A→C, then A→C. The squash counterexample fails, for it's false that if I were eating squash (B), I'd like squash (A).

I don't know whether Weak Transitivity is valid. But here's something that at least might be a counterexample. Suppose a heavy painting hangs on two strong nails. But if one nail were to fail, eventually--maybe several days later--the other would fail. The following seem to be all not unreasonable:

1. If the right nail failed (B), the left nail would fail because of the right's failure (C).
2. If the left nail failed (A), the right nail would fail because of the left's failure (D).

So, by Weakening (if P→Q and Q entails R, then P→R):

3. If the left nail failed (A), the right would fail (B).
4. If the right nail failed (B), the left would fail (A).

If Weak Transitivity holds, then:

5. If the left nail failed (A), the left nail would fail because of the right's failure (C).

But surely (2) and (5) aren't true together.

As I said, I am not sure if Weak Transitivity is valid. If it is, then there is something wrong with (1)-(4), probably with (1) and (2). Maybe there is. But the example should at least give one reason not to be very confident about Weak Transitivity. (There is another reason: Weak Transitivity is incompatible with the non-triviality of the Adams Thesis for subjunctives.)

Paradoxes of comparison

There are three sets, A, B and C, each consisting of the same number of people, whose lives are endangered by the same sort of danger, and whose future prospects as far as you know are on par. There are also three hungry kids, x, y and z, who will survive if you don't give them breakfast, but who would benefit from your giving them breakfast once (you have no opportunity to do anything more for them). Suppose you have a choice between two actions:

1. Save all the people in A and feed nobody.
2. Save all the people in B and feed x.

Now, it sure seems like 2 is the better action than 1. One might even formulate a general principle:

• (*) If an action saves the same number of lives and feeds more hungry children, and all else is on par, then it's better.

But actually this is false. For suppose that B is a proper subset of A, and there are 100 people in A who are not in B. Since we said that A and B have the same number of people, this can only be the case when A and B are infinite sets. In this scenario, if one goes for 2, there will be 100 people whom one won't be saving. So we should modify (*), perhaps to:

• (**) If an action saves the same number of lives and feeds more hungry children, and the sets of lives saved and children fed are disjoint between the two actions, and all else is on par, then the action is better.

But paradox ensues when we specify that A and B have no people in common, but C is a subset of A missing 100 people, and then add the option:

3. Save all the people in C and feed y and z.

For by one application of (**), 2 is better than 1, and by another application of (**), 3 is better than 2. But 3 is not better than 1, because the 100 people in A who aren't in C die if you go for 3, and that's not balanced out by the two children fed.

(There is a literature on infinite utilities, and I am not claiming any originality for this case.)

One could take this as yet another argument against the transitivity of "better than". But that doesn't get us out of paradox, since denying that transitivity is itself paradoxical. Moreover, there is already something paradoxical in having to deny (*)—that principle sure seemed plausible.

We could conclude that one can't have infinite sets of people, and make this be one of the family of arguments against actual infinites. Maybe.

But I want to do something else here. I think this, like a number of other paradoxes (which need not all involve infinity; I have a hunch that White's puzzle, as per Joyce's reply discussed in the link, is in this family), is due to us having two ways of comparing. We have an uncontroversial and unproblematic inclusion or domination comparison. It is uncontroversial that all other things being equal, if you can save all the people in A or all the people in B, and the people in A are a proper subset of those in B, then you should save the people in B. It is uncontroversial that if p entails q, then q is at least as likely as p. And so on.

But we also insist on comparing apples to oranges, comparing where there is no inclusion or domination relation. Typically, five oranges are more valuable than one apple, and five apples are more valuable than one orange. To make such comparisons we often assign numbers—say, cardinalities, utilities, prices or probabilities—to the things we are comparing, but we can also just make ordinal comparisons without assigning numbers (I didn't assign any utilities when I gave the ethical story).

I think a lot of paradoxes have the consequence that comparisons without domination are fishy. They need not satisfy transitivity. They might suffer from some arbitrariness. In the ethical sphere, this can be manifested in incommensurability of options. In probability theory, this surfaces in difficulties surrounding infinite sample spaces or nonmeasurable sets (as in White's puzzle, since nonmeasurable sets and non-exact probabilities are of a piece, I think).

Yet we need comparisons-without-domination.

So what should we do? In the ethical sphere, perhaps what we need is basically what Aquinas says about the order of charity. Aquinas thinks that when choosing between an equal benefit to one's parent or to a stranger, one should bestow the benefit on one's parent. But what if the benefit to the stranger is greater? If only slightly greater, we should still benefit our parent. But if much greater, we should benefit the stranger. But where is the line drawn? Aquinas refuses to answer. There is no rule, it seems. Rather, this is just somehting for the wise and virtuous agent to know. And maybe there is an analogue to this answer in the case of the non-ethical paradoxes.