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.

The epistemic force of beauty in laws of nature does not reduce to simplicity

Some people think that simplicity of laws of nature is a guide to truth, and some think beauty of laws of nature is. One might ask: Is the beauty of laws of nature a guide that goes beyond simplicity? Are there times when one could make epistemic decisions about the laws of nature on the basis of beauty where simplicity wouldn’t do the job?

I think so. Here is one case. Suppose we live in a Newtonian universe, and we are discovering fundamental forces. The first one has an inverse cube law. The second has an inverse cube law. These two laws account for most phenomena, but a few don’t fit. Scientists think there is a third fundamental force. For the third force law, we have two proposals that fit the data: an inverse square law and a slightly more complicated inverse cube law. It is, I think, quite reasonable to go for an inverse cube law by induction over the laws.

There is something indeed beautiful about the idea that the same power law applies to all the forces of nature. But if we just go with simplicity, we will go for an inverse square law. However, going for the inverse cube law seems clearly reasonable, and it is what beauty suggests—but not simplicity.

Here is another thought. Sometimes a fundamental law has some particularly lovely mathematical implications. For instance, a conservative force law is connected in a lovely way with a potential. But it need not be the case that a conservative force law is simpler than a non-conservative alternative. (It is true that a conservative force is the gradient of a potential. If the potential can be particularly simply expressed, this makes it easier to express the conservative force law. But we can have a case where the potential is harder to express than the force itself.)

Defining comparative probabilities in terms of conditional probabilities

Suppose we have a full conditional probability P(A∣B) defined for all pairs of events (stipulating that P(A∣⌀) = 1 if we wish). I've proposed two methods for defining a probability comparison using conditional probabilities:

1. A ⪅ B iff P(A∣A∪B) ≤ P(B∣A∪B).

2. A ⪅ B iff P(A−B∣AΔB) ≤ P(B−A∣AΔB), where AΔB = (A−B) ∪ (B−A) is the symmetric difference.

In a footnote in a paper, I wrote about the second ordering, which I incorrectly attributed to de Finetti: “This ordering has the advantage that if A is a proper subset of B, then A < B, but it is somewhat harder to prove transitivity”.

Well, that was an understatement! It’s not just harder to prove transitivity: it’s impossible.

Define:

• Ω: all integers

• E₀: non-negative even integers

• E: positive even integers

• D: positive odd integers.

Let P be a full conditional probability such that:

3. P(E₀∣E₀∪D) = 1/2 = P(D∣E₀∪D)

4. P(D∣D∪E) = 1/2 = P(E|D∪E).

It is easy to see from (3) and (4) that because E₀ and D are disjoint, and so are D and E, then by definition (2) we have E₀ ⪅ D and D ⪅ E. (For disjoint A and B, the definition (2) of A ⪅ B is equivalent to thedefinition (1).) However, E₀ − E = {0}, E − E₀ = ⌀, and E₀ΔE = {0}, so P(E−E₀∣E₀ΔE) = 0 while P(E₀−E∣E₀ΔE) = 1, and thus we cannot have E₀ ⪅ E.

The only question is whether there actually is a full conditional probability satisfying (3) and (4). If there is, then (2) is not transitive in our case.

There is such a full conditional probability. Let Q_n(A) =  ∣ A ∩ [−n,n] ∣ /(2n+1), where $B$ is the cardinality of a set B. Then Q_n is a probability. Let Q be a limit of the Q_n along an ultrafilter. This is a finitely additive hyperreal probability which is non-zero for all non-empty sets. Define P(A∣B) as the standard part of Q(A∩B)/Q(B) for B non-empty. This is a full conditional probability. Moreover, P(A∣B) = lim Q_n(A∩B)/Q_n(B) whenever the latter limit is defined. That limit is defined in the cases of the events involved in (3) and (4), and it is easy to evaluate the limits and see that (3) and (4) are true.

I don’t know what I was thinking when I wrote that footnote. My guess is that I had in my mind a proof sketch that doesn’t work (I have some idea what that might have been).

Whew! I noticed this afternoon that Theorem 1 of this paper of mine was incompatible with the transitivity of comparison (2). This made me really worried that my Theorem 1 was false. But since the comparison (2) isn’t transitive, I can relax.

This raises an interesting potential research problem. The Pruss definition of ⪅ does not satisfy the additivity axiom for comparative probabilities, namely that if C is disjoint from A ∪ B, then A ≲ B if and only if A ∪ C ≲ B ∪ C (it only preserves the left-to-right implication). Definition (2) does satisfy the additivity axiom, which is what I liked about it.

I suspect there is no good definition of comparative probabilities in terms of full conditional probabilities that satisfies the additivity axiom. (One reason for this intuition has to do with the fact that in Figure 1 here there are entries with a Yes in column 3 and a No in column 5.)

So, I now wonder: Is there some good combination of a definition of comparative probabilities in terms of full conditional probabilities with some weakened version of the additivity axiom?

A really bad moral dilemma

Here would be a really bad kind of moral dilemma:

• It is certain that unless you murder one innocent person now, you will freely become a mass murderer, but if you do murder that innocent person, you will freely repent of it later and live an exemplary life.

If compatibilism is true, such dilemmas are possible—the world could be so set up that these unfortunate free choices are inevitable. If compatibilism is false, such dilemmas are impossible, absent Molinism.

We might have a strong intuition that such dilemmas are impossible. If so, maybe that gives us another reason to reject compatibilism and Molinism.

Proportionality and deterrence

There are many contexts where a necessary condition of the permissibility of a course of action is a kind of proportionality between the goods and bads resulting from the course of action. (If utilitarianism is true, then given a utilitarian understanding of the proportionality, it’s not only necessary but sufficient for permissibility.) Two examples:

• The Principle of Double Effect says it is permissible to do things that are foreseen to have a basic evil as an effect, if that evil is not intended, and if proportionality between the evil effect and the good effects holds.

• The conditions for entry into a just war typically include both a justice condition and a proportionality condition (sometimes split into two conditions, one about likely consequences of the war and the other about the probability of victory).

But here is an interesting and difficult kind of scenario. Before giving a general formulation, consider the example that made me think about this. Country A has a bellicose neighbor B. However, B’s regime while bellicose is not sufficiently evil that on a straightforward reading of proportionality it would be worthwhile for A to fight back if invaded. Sure, one would lose sovereignty by not fighting back, but B’s track record suggests that the individual citizens of A would maintain the freedoms that matter most (maybe this is what it would be like to be taken over by Alexander the Great or Napoleon—I don’t know enough of history to know), while a war would obviously be very bloody. However, suppose that a policy of not fighting back would likely result in an instant invasion, while a policy of fighting back would have a high probability of resulting in peace for the foreseeable future. We can then imagine that the benefits of likely avoiding even a non-violent takeover by B outweigh the small risk that despite A’s having a policy of armed resistance B would still invade.

The general case is this: We have a policy that is likely to prevent an unhappy situation, but following through on the policy violates a straightforward reading of proportionality if the unhappy situation eventuates.

One solution is to take into account the value of follwing through on the policy with respect to one’s credibility in the future. But in some cases this will be a doubtful justification. Consider a policy of fighting back against an invader—at least initially—even if there is no chance of victory. There are surely many cases of bellicose countries that could successfully take over a neighbor, but judge that the costs of doing so are too high given the expected resistance. But if the neighbor has such a policy, then in case the invasion nonetheless eventuates, whatever is done, sovereignty will be lost, and the policy will be irrelevant in the future. (One might have some speculation about the benefits for other countries of following through on the policy, but that’s very speculative.)

One line of thought on these kinds of cases is that we need to forego such policies, despite their benefits. One can’t permissibly act on them, so one can’t have them, and that’s that. This is unsatisfying, but I think there is a serious chance that this is right.

One might think that the best of both worlds is to make it seem like one has the policy, but not in fact have it. A problem with this is that it might involve lying, and I think lying is wrong. But even aside from that, in some cases this may not be practicable. Imagine training an army to defend one’s country, and then having a secret plan, known only to a very small number of top commanders, that one will surrender at the first moment of an invasion. Can one really count on that surrender? The deterrent policy is more effective the fiercer and more patriotic the army, but those factors are precisely likely to make them fight despite the surrender at the top.

Another move is this. Perhaps proportionality itself takes into account not just the straightforward computation of costs and benefits, but also the value of remaining steadfast in reasonably adopted policies. I find this somewhat attractive, but this approach has to have limits, and I don’t know where to draw them. Suppose one has invented a weapon which will kill every human being in enemy territory. Use of this weapon, with a Double Effect style intention of killing only the enemy soldiers, is clearly unjustified no matter what policies one might have, but a policy to use this weapon might be a nearly perfect protection against invasion. (Obviously this connects with the question of nuclear deterrence.) I suppose what one needs to say is that the importance of steadfastness in policies affects how proportionality evaluation go, but should not be decisive.

I find myself pulled to the strict view that policies we should not have policies acting on which would violate a straightforward reading of proportionality, and the view that we should abandon the straightforward reading of proportionality and take into account—to a degree that is difficult to weigh—the value of following policies.

Another argument on the interpretation of Matthew 5:32 and 19:9

Mark (10:11-12) and Luke (16:18) have rather simple and straightforward statements on divorce and remarriage: if you divorce and remarry, you’re in adultery. A standard interpretation is the Strict View:

• (SV) Divorce does not actually remove the marriage, and so if you remarry, you’re still married to the previous party, and hence are committing adultery.

It’s usual in the Christian tradition to restrict this to consummated Christian marriage, and I will take that for granted.

However, Matthew has a more complex set of prohibitions:

• Matthew 5:32: Anyone who divorces his wife, except on account of porneia, makes her commit adultery, and anyone who marries a divorced woman commits adultery.

• Matthew 19:9: Anyone who divorces his wife, except due to porneia, and marries another commits adultery.

There are several puzzles here. First, unlike in Mark and Luke, we have exceptions for porneia, a generic term for sexual immorality. There are two main interpretations of this exception:

1. Except when the wife has committed sexual immorality (most commonly, adultery).

2. Except when the “marriage” constitutes sexual immorality.

Reading (1) supports the Less Strict View:

• (LSV) Except when your spouse has committed adultery, divorce does not actually remove the marriage, and so if you remarry, you’re still married to the previous party, and hence are committing adultery.

Reading (2) is based on the observation that not every legal marriage is genuinely a marriage: the Romans, for instance, might have allowed a couple to marry despite their being too closely related from the Christian point of view. In such a case, their “marriage” is not a real marriage but incest, a form of sexual immorality, and divorce is not only permissible, but a very good idea. Note that on reading (2), we can but need not suppose that Jesus verbally included the exception—the inspired author might have added it for clarification because the issue came up for converts, much as we put things in square brackets within a quote to clarify the author’s meaning (there were no brackets in Greek, of course).

Reading (2) has the advantage that it explains how all three Gospels can be inspired, even though Mark and Luke have unqualified statements of SV, since on reading (2) it is true that divorcing one’s wife and remarrying is never permitted, but it is permissible, of course, to divorce one’s partner in an immoral sexual relationship that non-Christian society may call “marriage”. Note that the Greek for “his wife” can literally just mean “his woman”, which makes the disambiguation especially appropriate.

But I want to turn towards a different and more complex argument for SV. Notice that in Matthew 5:32, instead of us being told that the man who divorces his wife (or woman) commits adultery, we are oddly told that he makes her commit adultery. But being a betrayed spouse does not constitute adultery! What’s going on? Well, the good interpretations that I’ve seen note that the social context is a society where it is very difficult to be a woman without a husband. There will thus be significant social pressure to marry or become a concubine, either of which would constitute adultery against the first husband. The realities of the day were such that very likely she would succumb to the pressure, and the first husband would have caused her to commit adultery, and thereby he would have earned himself something worse than a millstone about the neck (Matthew 18:6). This reading also nicely explains why Matthew 5:32, unlike the three other texts, does not mention the man marrying another. For the woman is going to be exposed to the social pressure to join herself to another man whether or not her (first) husband marries another.

Note that this reading of “makes her commit adultery” prima facie works on both readings of the porneia exception. On the reading where the porneia is the wife’s adultery against her husband, obviously if she is already committing adultery, by divorcing her he isn’t making her commit adultery. On the reading where the porneia is constituted by the immorality of the first “marriage”, because the woman wasn’t really married to the man, if she goes and marries another, she isn’t committing adultery.

Nonetheless, there is a serious problem for this reading of “makes her commit adultery” on the Less Strict View and reading (1). While Matthew 5:32 does not talk of the man marrying another, often the man will marry another. So now imagine this story. There is a valid marriage between Alice and Bob with no adultery, but Bob divorces Alice, and marries Charlene. At this point, Bob is committing adultery against Alice on both SV and LSV. Thus, if LSV is correct, then Alice is entitled to divorce Bob and marry another, say Dave. But if she avails herself of this, she isn’t committing adultery. In other words, if LSV is correct, in many cases the first wife will be able to avoid committing adultery without going against social pressures: she need only wait for her first husband to marry, and then the “except on account of porneia” clause on interpretation (1) frees her (and since he’s already legally divorced her, she doesn’t need to do any legal paperwork). (Of course, there will still be less common cases where she is stuck, namely when the man fails to remarry. But such a case wouldn’t be the rule, and Matthew 5:32 implies that leading the woman to adultery is the rule rather than an exception.)

On SV, the problem for the reading of “makes her commit adultery” entirely disappears. Whether or not the man remarries, there is social pressure for the divorced wife to marry, and in doing so, she would be committing adultery against the man.

Interestingly, there is a historically represented view that avoids the Strict View, allows our interpretation of “makes her commit adultery” and avoids the above interpretative problem, namely the quite awful Asymmetric View:

• (AV) A woman is not permitted to remarry after a divorce, whether or not the first husband committed adultery against her, but a man is permitted to remarry after a divorce if, and only if, the first wife committed adultery against him.

Additionally, AV also explains why neither of the texts in Matthew has an exception for porneia in the “anyone who marries a divorced woman” clause, a minor weak point for LSV. (On SV and reading (2) of porneia, we just note that one need not repeat a parenthetical clarification every time.)

In fact, while there was controversy in the early centuries of Christianity over remarriage and divorce following adultery, I understand that it was mainly a controversy between advocates of SV and AV, not between advocates of SV and LSV. However, AV was rightly lambasted by St. Jerome for being sexist, and I assume almost nobody wants to defend it now.

Thus to sum up my argument for SV:

1. One of SV, AV and LSV is true, as they are the historically plausible Christian views on marriage.

2. The right interpretation of “makes her commit adultery” is the social pressure interpretation.

3. This interpretation is incompatible with LSV.

4. AV is false.

5. Therefore, SV is true.