Against group intentional action

Alice, Bob and Carl are triumvirate that unanimously votes for some legislation, for the following reasons:

1. Alice thinks that hard work and religion are intrinsically bad while entertainment is intrinsically good, and believes the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.

2. Bob thinks that hard work and entertainment are intrinsically bad while religion is intrinsically good, and believes the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.

3. Carl thinks that religion and entertainment are intrinsically bad while hard work is intrinsically good, and believes the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

If groups engage in intentional actions, it seems that passing legislation is a paradigm of such intentional action. But what is the intention behind the action here?

When I first thought about cases like this, I thought they were a strong argument against group intentional action. But then I became less sure. For we can imagine an intrapersonal version. Suppose Debbie the dictator was given a card by a trustworthy expert that she was informed contains a truth, with the expert departing at that point. Before she could read it, however, she accidentally dropped the card in a garbage can. Reaching into the garbage can, she found three cards in the expert’s handwriting, two of them being mere handwriting exercises and one being the advice card:

1. Hard work and religion are intrinsically bad while entertainment is intrinsically good, and the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.
2. Hard work and entertainment are intrinsically bad while religion is intrinsically good, and the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.
3. Religion and entertainment are intrinsically bad while hard work is intrinsically good, and the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

Oddly, Debbie’s own prior views are so undecided that she just sets her credence to 1/3 for each of these propositions, and enacts the legislation. What is her intention?

But now I think there is a plausible answer: Debbie’s intention is to increase whichever one of the trio of entertainment, religion and hard work is good and decrease whichever two of them are bad.

Could we thus say that that is what the triumvirate intends? I am not sure. Nobody on the triumvirate has such an abstract intention.

So perhaps we still have an argument against group intentional action, of the form:

1. If there is group intentional action, the triumvirate acts intentionally.

2. Something only acts intentionally if it has an intention.

3. The triumvirate has no intention.

4. So, there is no group intentional action.

Reasons of identity

In paradigm instances of parental action, my reason for action is the objective fact that I am a parent, not because of the subjective fact that I think I'm a parent or identify with being a parent. There are times when it makes sense to act on the subjective fact. If I'm asked by someone (say, a counselor) whether I identify with being a parent, my answer needs to be based on the subjective fact that I so identify. But those are atypical cases. 

I suspect this is generally true: cases when one acts on what one is are primary and cases when one acts on what one identifies as are secondary. It is, thus, problematic to define any feature that is significantly rationally relevant to ordinary action in terms of what one identifies with. 

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.

No fix for Goodman and Quine's counting

In yesterday’s post, I noted that Goodman and Quine’s nominalist mereological definition of what it is to say that there are more cats than dogs fails if there are cats that are conjoint twins. This raises the question whether there is some other way of using the same ontological resources to generate a definition of “more” that works for overlapping objects as well.

I think the answer is negative. First, note that GQ’s project is explicitly meant to be compatible with there being a finite number of individuals. In particular, thus, it needs to be compatible with the existence of mereological atoms, individuals with no proper parts, which every individual is a fusion of. (Otherwise, there would have to be no individuals or infinitely many. For every individual has an atom as a part, since otherwise it has an infinite regress of parts. Furthermore, every individual must be a fusion of the atoms it has as parts, otherwise the supplementation axiom will be violated.) Second, GQ’s avail themselves of one non-mereological tool: size comparisons (which I think must be something like volumes). And then it is surely a condition of adequacy on their theory that it be compatible with the logical possibility that there are finitely many individuals, every individual is a fusion of its atoms and the atoms are all the same size. I will call worlds like that “admissible”.

So, here are GQ’s theoretical resources for admissible worlds. There are individuals, made of atoms, and there is a size comparison. The size comparison between two individuals is equivalent to comparing the cardinalities of the sets of atoms the individuals are made of, since all the atoms are the same size. In terms of expressive power, their theory, in the case of admissible worlds, is essentially that of monadic second order logic with counting, MSO(#), restricted to finite models. (I am grateful to Allan Hazen for putting me on to the correspondence between GQ and MSO.) The atoms in GQ correspond to objects in MSO(#) and the individuals correspond to (extensions of) monadic predicates. The differences are that MSO(#) will have empty predicates and will distinguish objects from monadic predicates that have exactly one object in their extension, while in GQ the atoms are just a special (and definable) kind of individual.

Suppose now that GQ have some way of using their resources to define “more”, i.e., find a way of saying “There are more individuals satisfying F than those satisfying G.” This will be equivalent to MSO(#) defining a second-order counting predicate, one that essentially says “The set of sets of satisfiers of F is bigger than the set of sets of satisfiers of G”, for second-order predicates F and G.

But it is known that the definitional power of MSO(#) over finite models is precisely such as to define semi-linear sets of numbers. However, if we had a second-order counting predicate in MSO(#), it would be easy to define binary exponentiation. For the number of objects satisfying predicate F is equal to two raised to the power of the number of objects satisfying G just in case the number of singleton subsets of F is equal to the number of subsets of G. (Compare in the GQ context: the number of atoms of type F is equal to two the power of the number of atoms of type G provided that the number of atoms of type F is one plus the number of individuals made of the atoms of type G.) And of course equinumerosity can be defined (over finite models) in terms of “more”, while the set of pairs (n,2ⁿ) is clearly not semi-linear.

One now wants to ask a more general question. Could GQ define counting of individuals using some other predicates on individuals besides size comparison? I don’t know. My guess would be no, but my confidence level is not that high, because this deals in logic stuff I know little about.

Goodman and Quine and shared bits

Goodman and Quine have a clever way of saying that there are more cats than dogs without invoking sets, numbers or other abstracta. The trick is to say that x is a bit of y if x is a part of y and x is the same size as the smallest of the dogs and cats. Then you’re supposed to say:

1. Every object that has a bit of every cat is bigger than some object that has a bit of every dog.

This doesn’t work if there is overlap between cats. Imagine there are three cats, one of them a tiny embryonic cat independent of the other two cats, and the other two are full-grown twins sharing a chunk larger than the embryonic cat, while there are two full-grown dogs that are not conjoined. Then a bit is a part the size of the embryonic cat. But (assuming mereological universalism along with Goodman and Quine) there is an object that has a bit of every cat that is no bigger than any object has a bit of every dog. For imagine an object that is made out of the embryonic cat together with a bit that the other two cats have in common. This object is no bigger than any object that has a bit of each of the dogs.

It’s easy to fix this:

2. Every object that has an unshared bit of every cat is bigger than some object that has an unshared bit of every dog,

where an unshared bit is a bit x not shared between distinct cats or distinct dogs.

But this fix doesn’t work in general. Suppose the following atomistic thesis is true: all material objects are made of equally-sized individisible particles. And suppose I have two cubes on my desk, A and B, with B having double the number of particles as A. Consider this fact:

4. There are more pairs of particles in A than particles in B.

(Again, Goodman and Quine have to allow for objects that are pairs of particles by their mereological universalism.) But how do we make sense of this? The trick behind (1) and (2) was to divide up our objects into equally-sized pieces, and compare the sizes. But any object made of the parts of all the particles in B will be the same size as B, since it will be made of the same particles as B, and hence will be bigger than any object made of parts of A.

Trope theory and merely numerical differences in pleasures

Suppose I eat a chocolate bar and this causes me to have a trope of pleasure. Given assentiality of origins, if I had eaten a numerically different chocolate bar that caused the same pleasure, I would have had had a numerically different trope of pleasure.

Now, imagine that I eat a chocolate bar in my right hand and it causes me to have a trope of pleasure R, and immediately as I have finished eating that one chocolate bar, I switch to eating the chocolate bar in my left hand, which gives me an exactly similar trope of pleasure, L, with no temporal gap. Nonetheless, by essentiality of origins, trope L is numerically distinct from trope R.

To some (perhaps Armstrong) this will seem absurd. But I think it’s exactly right. In fact, I think it may even an argument for trope theory. For it seems pretty plausible that as I switch chocolate bars, something changes in me: I go from one pleasure to another exactly like it. But on heavy-weight Platonism, there is no change: I instantiated pleasure and now I instantiate pleasure. On non-trope nominalism, likewise there is no change. It’s trope theory that gives us the change here.

Does one's vote make a difference?

Suppose that there is a simple majority election, with two candidates, and there is a large odd number of voters. Suppose polling data makes the election too close to call. How likely is it that you can decide which candidate wins?

I could look up this stuff, but it’s more fun to figure it out.

A quick and dirty model is this. We have N people other than you voting, each choosing between candidates A and B with probabilities p and 1 − p respectively. You don’t know what p and 1 − p are, but polling data tells you that p is between 1/2 − a and 1/2 + b for some positive numbers a and b. Your vote decides the election provided that exactly N/2 people vote for candidate A. This requires that N be even (if N is odd, at best you can decide between a candidate winning and the election being undecided, so you can’t decide which candidate wins), which has probability 1/2. Given that N = 2n is even, the probability that the other votes are exactly balanced is (a+b)⁻¹ C(2n,n)∫_1/2−a^1/2+bpⁿ(1−p)^n − 1dp, where C(m,n) is the binomial coefficient. Assuming n is large as compared to a and b, the integral can be approximated by replacing its bounds by 0 and 1 respectively, and some work with Mathematica shows that for large n the probability is approximately 1/(N(a+b)).

So what? Well, suppose you think that candidate A will on average make a person in the jurisdiction be u units of flourishing better off than candidate B will, and there are K persons, where K ≥ N + 1 (there are at least as many persons as candidates). So, the expected amount of difference that your voting for A will make is at least Ku/(2N(a+b)). This is at least u/(a+b). Thus, if the polling data gives you a range between 0.48 and 0.52 for the probability of a person’s preferring candidate A, and half of the people in the jurisdiction vote, the expected amount of difference that your vote makes is 25u. This is quite a lot if you think that which candidate wins makes a significant difference u per governed person.

Interestingly, some numerical work with Mathematica also shows that as number of people increases, then the expected amount of difference your vote makes also increases asymptotically, up to the limit of Ku/(2N(a+b)). So for larger jurisdictions, even though the probability of your vote making a difference is smaller, the expected difference from your vote is a bit bigger.

My quick and dirty model is not quite right. Of course, people don’t come to the polls and randomly choose whom to vote for. A more likely source of randomness has to do with who actually makes it to the polls (who gets sick, who has something come up, who decides it’s pointless to vote, etc.). A better model might be this. We have M people eligible to vote, of whom pM want to vote for A and (1−p)M want to vote for B. Some random subset of the M people then votes. My probabilist intuitions say that this is not that different from my model if the number of actual voters is, say, half of the eligible voters. If I had an election that I was eligible to vote in coming, I might try to figure our the more complex model, but I don’t.

Theology and source critical analysis

There is reason to think that a number of biblical texts—paradigmatically, the Pentateuch—were redacted from multiple sources that scholars have worked to tease apart and separately analyze. This is very interesting from a scholarly point of view. But I do not know that it is that interesting from the theological point of view.

Vatican II, in Dei Verbum, famously teaches:

since everything asserted by the inspired authors or sacred writers must be held to be asserted by the Holy Spirit, it follows that the books of Scripture must be acknowledged as teaching solidly, faithfully and without error that truth which God wanted put into sacred writings for the sake of salvation. … However, since God speaks in Sacred Scripture through men in human fashion, the interpreter of Sacred Scripture, in order to see clearly what God wanted to communicate to us, should carefully investigate what meaning the sacred writers really intended, and what God wanted to manifest by means of their words.

Presumably many other Christian groups hold something similar.

Now, in the case of a text put together from multiple sources, the question is who the “sacred writers” are. I want to suggest that in the case of such a text, the relevant “sacred writers” are the editors who put the texts together, and especially the ones responsible for a final (though this is a somewhat difficult to apply concept) version, and the intentions relevant to figuring out “What God wanted to communicate to us” are the intentions of the final layer of editing. The books in question, such as Genesis, are not anthologies. In an anthology, an editor has some purposes in mind for the anthologized texts, but the texts belong, often in a more or less acknowledged fashion, to the individual authors. The editorial work in putting the Biblical works together from source material is much more creative—it is genuine form of authorship—which is obvious from how much back-and-forth movement there is. Like in an anthology, we should not take the editor’s intentions to align with the intentions of the source material authors, but unlike in an anthology, the final work comes with the editor’s authority, and counts as the assertion of the editor, with the editor’s intentions being the ones that determine the meaning of the work.

If this is right, then I think we can only be fully confident of dealing with inspired teaching in the case of what the editors intend to assert through the final works. Writers typically draw on a multiplicity of sources, and need not be asserting what these sources meant in their original context—think of the ways in which a writer often repurposes a quote from another. Think here of how Homer draws upon a rich variety of fictional and nonfictional source material, but when he adapts them for inclusion in his work, the intentions relevant to “What the Iliad and Odyssey say” are Homer’s intentions.

If what we want to be sure of is “what God wanted to communicate to us”, then we should focus on the redactors’ intentions. In particular, when there is a tension in text between two pieces of source material, exegetically we should focus on what the editor meant to communicate to us by the choice to include material from both sources. (In a text without divine inspiration, we might in the end attribute a tension to editorial carelessness, but in fact scholars rarely make use of “carelessness” as an explanation for phenomena in great works of secular literature.) I think we should be open even to the logical possibility that the editor misunderstood what the source material meant to communicate, but it is the editor’s understanding that is normative for the interpretation of what the text as a whole is saying.

From a scholarly point of view, earlier layers in the composition process are more interesting. But I think that from a theological point of view, it is what the editor wanted to communicate that matters.

I don’t want to be too dogmatic about this, for three reasons. First, it is possible that the source material is an inspired text in its own right. But, I think, we typically don’t know that it is (though in a Christian context, an obvious exception is where the New Testament quotes Jesus’ inspired teaching). Second, it is possible for a writer or editor who has a deep respect for a piece of source material to include the text with the intention that the text be understood in the sense in which the original authors intended it to be understood, in which case the intentions of the authors of the source material may well be relevant. Third, this is not my field—I could be really badly confused.

An impartiality premise

In an argument that David Lewis’s account of possible worlds leads to inductive skepticism, I used this premise:

1. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) does not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

This is less clear to me now than it was then. Self-locating evidence might be a counterexample to this principle. I know that the tallest person in the world is the tallest person in the world. But suppose I now learn that I am the tallest person in the world. It doesn’t seem entirely implausible to think that at this point it becomes reasonable (or at least more reasonable) to infer that the number of people in the world is small. For on the hypothesis that the number of people is small, it seems more likely that I am the tallest than on the hypothesis that the number of people is large. (Compare: That I won some competition is evidence that the number of competitors was small.)

But I think I can fix my argument by using this premise:

2. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) and that a uniformly randomly chosen person (or other occupied location) is x would not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

There are multiple versions of (b) depending on how the random choice works, e.g., whether it is a random choice from among actual persons or from among possible persons (cf. self-sampling vs. self-indication).

It takes a bit of work to convince oneself that the rest of the argument still works.

A new kind of project

I did something new and fun this fall: I wrote a computer science paper. It's an analysis of the conditions under which a device equipped with a camera and an accelerometer can identify its position relative to two observed landmarks with known positions. Except for a measure zero set of singular cases with infinitely many solutions, there are always at most two solutions for device positions (this was previously known), and I found necessary and sufficient conditions for there to be a single solution. In particular, if the two landmarks are at the same altitude, there is always a single solution, unless the device is at the same altitude as the landmarks.

I implemented the algorithm on a phone (code here). In the screenshot, the markers 1 and 2 are landmarks, identified and outlined in green with OpenCV library code, and then the phone uses their positions and the accelerometer data to predict where the control markers 3 and 4 are on the screen, outlining them in red.

For someone like me who does some philosophy of science, it was an interesting experience to actually do a real experiment and collect data from it.

I am planning at some point to try to implement the algorithm using infrared LEDs under a TV and the accelerometer and infrared camera inside a right Nintendo Switch joycon. To that end, over the last couple of days I've reverse-engineered two of the joycon infrared camera blob identification modes.

Aristotelian sciences

There is an Aristotelian picture of knowledge on which all knowable things are divided exhaustively and exclusively into sciences by subject matter. This picture appears wrong. Suppose, after all, that p is a fact from one science—say, the natural science fact that water is wet—and q is a fact from another science—say, the anthropological fact that people pursue pleasure. Then the conjunction p and q does not belong to either of these science, or any other science.

One might cavil that a conjunction isn’t another fact over and beyond the conjuncts, that to say p and q is to say p and to say q. I am sceptical, but it’s easy to fix. Just replace my counterexample with something that isn’t a conjunction but is logically equivalent to it, say the claim that it’s not the case that either p or q is false.

Actual result utilitarianism implies a version of total depravity

Assume actual result utilitarianism on which there are facts of the matter about what would transpire given any possible action of mine, and an action is right just in case it has the best consequences.

Here is an interesting conclusion. Do something specific, anything. Maybe wiggle your right thumb a certain way. There are many—perhaps even infinitely many—other things you could have done (e.g., you could have wiggled the thumb slightly differently) instead of that action whose known consequences are no different from the known consequences of what you did. We live in a chaotic world where the butterfly principle very likely holds: even minor events have significant consequences down the road. It is very unlikely that of all the minor variants of what you did, all of which have the same known consequences, the variant you chose has the best overall consequences down the road. Quite likely, the variant action you chose is middle of the road among the variants.

So, typically, whatever we do, we do wrong on actual result utilitarianism.

There is no canonical way to define a regular comparative probability in terms of a full conditional probability

I claim that there is no general, straightforward and satisfactory way to define a total comparative probability with the standard axioms using full conditional probabilities. By a “straightforward” way, I mean something like:

1. A ≲ B iff P(A−B|AΔB) ≤ P(B−A|AΔB)

or:

2. A ≲ B iff P(A|A∪B) ≤ P(A|A∪B) (Pruss).

The standard axioms of comparative probability are:

3. Transitivity, reflexivity and totality.

4. Non-negativity: ⌀ ≤ A for all A

5. Additivity: If A ∪ B is disjoint from C, then A ≲ B iff A ∪ C ≲ B ∪ C.

A “straightforward” definition is one where the right-hand-side is some expression involving conditional probabilities of events definable in a boolean way in terms of A and B.

To be “satisfactory”, I mean that it satisfies some plausible assumptions, and the one that I will specifically want is:

6. If P(A|C) < P(B|C) where A ∪ B ⊆ C, then A < B.

Definitions (1) and (2) are straightforward and satisfactory in the above-defined senses, but (1) does not satisfy transitivity while (2) does not satisfy the right-to-left direction of additivity.

Here is a proof of my claim. If the definition is straightforward, then if A ≲ B, and A′ and B′ are events such that there is a boolean algebra isomorphism ψ from the algebra of events generated by A and B to the algebra of events generated by A′ and B′ such that ψ(A) = A′, ψ(B) = B′ and P(C|D) = P(ψ(C)|ψ(D)) for all C and D in the algebra generated by A and B, then A′ ≲ B′.

Now consider a full conditional probability P on the interval [0,1] such that P(A|[0,1]) is equal to the Lebesgue measure of A when A is an interval. Let A = (0,1/4) and suppose B is either (1/4,1/2) or (1/4, 1/2]. Then there is an isomorphism ψ from the algebra generated by A and B to the same algebra that swaps A and B around and preserves all conditional probabilities. For the algebra consists of the eight possible unions of sets taken from among A, B and [0,1] − (A∪B), and it is easy to define a natural map between these eight sets that swaps A and B, and this will preserve all conditional probabilities. It follows from my definition of straightforwardness that we have A ≲ B if and only if we have B ≲ B. Since the totality axiom for comparative probabilities implies that either A ≲ B or B ≲ A, so we must have both A ≲ B and B ≲ A. Thus A ∼ B. Since this is true for both choices of B, we have

7. (0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2].

But now note that ⌀ < {1/2} by (3) (just let A = ⌀, B = {1/2} and C = {1/2}). The additivity axiom then implies that (1/4,1/2) < (1/4, 1/2], a contradiction.

I think that if we want to define a probability comparison in terms of conditional probabilities, what we need to do is to weaken the axioms of comparative probabilities. My current best suggestion is to replace Additivity with this pair of axioms:

8. One-Sided Additivity: If A ∪ B is disjoint from C and A ≲ B, then A ∪ C ≲ B ∪ C.

9. Weak Parthood Principle: If A and B are disjoint, then A < A ∪ B or B < A ∪ B.

Definition (2) satisfies the axioms of comparable probabilities with this replacement.

Here is something else going for this. In this paper, I studied the possibility of defining non-classical probabilities (full conditional, hyperreal or comparative) that are invariant under a group G of transformations. Theorem 1 in the paper characterizes when there are full conditional probabilities that are strongly invariant. Interesting, we can now extend Theorem 1 to include this additional clause:

6. There is a transitive, reflexive and total relation ≲ satisfying (4), (8) and (9) as well as the regularity assumption that ⌀ < A whenever A is non-empty and that is invariant under G in the sense that gA ∼ A whenever both A and gA are subsets of Ω.

To see this, note that if there is are strongly invariant full conditional probabilities, then (2) will define ≲ in a way that satisfies (vi). For the converse, suppose (vi) is true. We show that condition (ii) of the original theorem is true, namely that there is no nonempty paradoxical subset. For to obtain a contradiction suppose there is a non-empty paradoxical subset E. Then E can be written as the disjoint union of A₁, ..., A_n, and there are g₁, ..., g_n in G and 1 ≤ m < n such that g₁A₁, ..., g_mA_m and g_m + 1A_m + 1, ..., g_nA_n are each a partition of E.

A standard result for additive comparative probabilities in Krantz et al.’s measurement book is that if B₁, ..., B_n are disjoint, and C₁, ..., C_n are disjoint, with B_i ≲ C_i for all i, then B₁ ∪ ... ∪ B_n ≲ C₁ ∪ ... ∪ C_n. One can check that the proof only uses One-Sided Additivity, so it holds in our case. It follows from G-invariance that A₁ ∪ ... ∪ A_m ∼ E ∼ A_m + 1 ∪ ... ∪ A_n. Since E is the disjoint union of A₁ ∪ ... ∪ A_m with A_m + 1 ∪ ... ∪ A_n, this violates the Weak Parthood Principle.

Restricted composition and laws of nature

Ted Sider famously argues for the universality of composition on the grounds that:

1. If composition is not universal, then one can find a continuous series of cases from a case of no composition to a case of composition.

2. Given such a continuous series, there won’t be any abrupt cut-off in composition.

3. But composition is never vague, so there would have to be an abrupt cut-off.

Consider this argument that every velocity is an escape velocity:

4. If it’s not the case that every velocity is an escape velocity from a spherically symmetric body of some fixed size and mass, then one can find a continuous series of cases from a case of insufficiency to escape to a case of sufficiency to escape.

5. Given such a continuous series, there won’t be any abrupt cut-off in escape velocity.

6. But escape velocity is never vague, so there would have to be an abrupt cut-off.

It’s obvious that we should deny (5). There is an abrupt cut-off in escape velocity, and there is a precise formula for what it is: (2GM/r)^1/2 where G is the gravitational constant, M is the mass of the spherical body, and r is its radius. As the velocity of a projectile gets closer and closer to the (2GM/r)^1/2, the projectile goes further and further before turning back. When the velocity reaches (2GM/r)^1/2, the projectile goes out forever. There is no paradox here.

Why think that composition is different from escape velocity? Why not think that just as the laws of nature precisely specify when the projectile can escape gravity, they also precisely specify when a bunch of objects compose a whole?

My suspicion is that the reason for thinking the two are different is thinking that composition is something like a “logical” or maybe “metaphysical” matter, while escape is a “causal” matter. Now, universalists like David Lewis do tend to think that the whole is a free lunch, nothing but the “sum of the parts”, in which case it makes sense to think that composition is not something for the laws of nature to specify. But if we are not universalists, then it seems to me that it is very natural to think of composition in a causal way: when a proper plurality of xs are arranged a certain way, they cause the existence of a new entity y that stands in a composed-by relation to the xs, just as when a projectile has a certain velocity, that causes the projectile to escape to infinity.

Some may be bothered by the fact that laws of nature are often taken to be contingent, and so there would be a world with the same parts as ours but different wholes. That would bother one if one thinks that wholes are a free lunch. But if we take wholes seriously, it should no more bother us than a world where particles behave the same way up to time t₁, and then behave differently after t₁ because the laws are different.

Humeans have good reason to reject the above view, though. If the laws of composition are to match our intuitions about composition, they are likely to be extremely complex, and perhaps too complex to be part of the best system defining the laws on a Humean account of laws. But if we are not Humeans about laws, and think the simplicity of laws is merely an epistemic virtue, the explanatory power of laws of composition might make it reasonable to accept very complex such laws.

That said, we all have reject the simple causal version of the above view, where a proper plurality composing a whole causes the whole’s existence. For instance, I am composed by a plurality of parts that includes my hair, but my hair is not a cause of my existence: I would have just as much existed had I never developed hair. So a more complex version of the causal view is needed: initial parts (maybe the DNA in the zygote that I started as) causally contribute to the existence of the whole, but the causal relation runs in a different direction with respect to later parts, like teeth: perhaps I and my teeth together cause the teeth to be parts of me.

(I don’t endorse the more complex causal view either. I prefer, but still do not endorse, an Aristotelian alternative: when y is in a certain condition, it causes the existence of all of the parts. This is much neater because the causation always runs in the same direction.)

More on full conditional probabilities and comparative probabilities

I claim that there is no general, straightforward and satisfactory way to define a total comparative probability with the standard axioms using full conditional probabilities. By a “straightforward” way, I mean something like:

1. A ≲ B iff P(A−B|AΔB) ≤ P(B−A|AΔB)

or:

2. A ≲ B iff P(A|A∪B) ≤ P(B|A∪B).

The standard axioms of comparative probability are:

3. Transitivity, reflexivity and totality.

4. Non-negativity: ⌀ ≤ A for all A

5. Additivity: If A ∪ B is disjoint from C, then A ≲ B iff A ∪ C ≲ B ∪ C.

A “straightforward” definition is one where the right-hand-side is some expression involving conditional probabilities of events definable in a boolean way in terms of A and B.

To be “satisfactory”, I mean that it satisfies some plausible assumptions, and the one that I will specifically want is:

6. If P(A|C) < P(B|C) where A ∪ B ⊆ C, then A < B.

Definitions (1) and (2) are straightforward and satisfactory in the above-defined senses, but (1) does not satisfy transitivity while (2) does not satisfy the right-to-left direction of additivity.

Here is a proof of my claim. If the definition is straightforward, then if A ≲ B, and A′ and B′ are events such that there is a boolean algebra isomorphism ψ from the algebra of events generated by A and B to the algebra of events generated by A′ and B′ such that ψ(A) = A′, ψ(B) = B′ and P(C|D) = P(ψ(C)|ψ(D)) for all C and D in the algebra generated by A and B, then A′ ≲ B′.

Now consider a full conditional probability P on the interval [0,1] such that P(A|[0,1]) is equal to the Lebesgue measure of A when A is an interval. Let A = (0,1/4) and suppose B is either (1/4,1/2) or (1/4, 1/2]. Then there is an isomorphism ψ from the algebra generated by A and B to the same algebra that swaps A and B around and preserves all conditional probabilities. For the algebra consists of the eight possible unions of sets taken from among A, B and [0,1] − (A∪B), and it is easy to define a natural map between these eight sets that swaps A and B, and this will preserve all conditional probabilities. It follows from my definition of straightforwardness that we have A ≲ B if and only if we have B ≲ B. Since the totality axiom for comparative probabilities implies that either A ≲ B or B ≲ A, so we must have both A ≲ B and B ≲ A. Thus A ∼ B. Since this is true for both choices of B, we have

7. (0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2].

But now note that ⌀ < {1/2} by (3) (just let A = ⌀, B = {1/2} and C = {1/2}). The additivity axiom then implies that (1/4,1/2) < (1/4, 1/2], a contradiction.

I think that if we want to define a probability comparison in terms of conditional probabilities, what we need to do is to weaken the axioms of comparative probabilities. My current best suggestion is to replace Additivity with this pair of axioms:

8. One-Sided Additivity: If A ∪ B is disjoint from C and A ≲ B, then A ∪ C ≲ B ∪ C.

9. Weak Parthood Principle: If A and B are disjoint, then A < A ∪ B or B < A ∪ B.

Definition (2) satisfies the axioms of comparable probabilities with this replacement.

Here is something else going for this. In this paper, I studied the possibility of defining non-classical probabilities (full conditional, hyperreal or comparative) that are invariant under a group G of transformations. Theorem 1 in the paper characterizes when there are full conditional probabilities that are strongly invariant. Interesting, we can now extend Theorem 1 to include this additional clause:

6. There is a transitive, reflexive and total relation ≲ satisfying (4), (8) and (9) as well as the regularity assumption that ⌀ < A whenever A is non-empty and that is invariant under G in the sense that gA ∼ A whenever both A and gA are subsets of Ω.

To see this, note that if there is are strongly invariant full conditional probabilities, then (2) will define ≲ in a way that satisfies (vi). For the converse, suppose (vi) is true. We show that condition (ii) of the original theorem is true, namely that there is no nonempty paradoxical subset. For to obtain a contradiction suppose there is a non-empty paradoxical subset E. Then E can be written as the disjoint union of A₁, ..., A_n, and there are g₁, ..., g_n in G and 1 ≤ m < n such that g₁A₁, ..., g_mA_m and g_m + 1A_m + 1, ..., g_nA_n are each a partition of E.

A standard result for additive comparative probabilities in Krantz et al.’s measurement book is that if B₁, ..., B_n are disjoint, and C₁, ..., C_n are disjoint, with B_i ≲ C_i for all i, then B₁ ∪ ... ∪ B_n ≲ C₁ ∪ ... ∪ C_n. One can check that the proof only uses One-Sided Additivity, so it holds in our case. It follows from G-invariance that A₁ ∪ ... ∪ A_m ∼ E ∼ A_m + 1 ∪ ... ∪ A_n. Since E is the disjoint union of A₁ ∪ ... ∪ A_m with  A_m + 1 ∪ ... ∪ A_n, this violates the Weak Parthood Principle.