Real Presence and primitive locational relations

According to relationalism, space is constituted by the network of spatial relations, such as metric distance relations (e.g., being seven meters apart). If these relations are primitive, then there is a very easy way for God to ensure the Real Presence of Christ: he can simply make there be additional spatial relations between Christ and other material entities, spatial relations that are exactly like the relations that the bread and wine stood in to other material entities.

It might seem contradictory for Christ to stand in two distance relations: for instance, being one mile from me (in one church) and three miles from me (in another). But I doubt this is a contradiction. New York and London are both 5600 km and 34500 km apart, depending on which direction you go.

According to substantivalism, on the other hand, points or regions are real, and objects are in a location by standing in a relation to a point or region. If relations are primitive, again there should be no problem about God instituting additional such relations to make it be that Christ is present where the bread and wine were.

In other words, if location is constituted by a primitive relation—whether to other objects or to space—there is apt to be no difficulty in accounting for the Real Presence. The reason is that we expect, barring strong reason to the contrary, primitive relations to be arbitrarily recombinable.

If location, however, is constituted by a non-primitive relation, there might be more difficulties. For instance, as a toy theory, consider the variant of relationalism on which spatial relations are constituted by gravitational force relations (two objects have distance r if and only if they have masses m₁ and m₂ and there is a gravitational force Gm₁m₂/r² between them). In that case, for God to make Christ present in Waco would require God to make Christ stand in gravitational force relations of the sort that I stand in by virtue of being in Waco. For instance, the earth’s gravitational force on Christ would have to point from Waco to the center of the earth—but since the Eucharist is also in Rome, it would have to point from Rome to the center of the earth as well. And that might be thought impossible. But perhaps there could be two terrestrial gravitational forces on Christ: one along the Waco-geocenter vector and the other along the Rome-geocenter vector. This would require some sort of a realism about component forces, but that’s probably necessary for the gravitational toy theory. And then God would have to miraculously ensure that despite the forces, Christ is not affected in the way he would normally be by these forces. All this may be possible, but it’s less clear than if we have primitive relations.

Python script for Nanson and Black voting

I made a handy little python script for tabulating group votes with more than two candidates using either Black’s Procedure or Nanson’s Method. Both algorithms are Condorcet compliant. The algorithms require as input a text file with the ballots and control information. Once the algorithm finds a winner (or a bunch of tied winners), it deletes them from the ballots and repeats.

For instance suppose sample.txt contains:

method Black's
require 3
ballot Mickey Donald Sonic
ballot Mickey Sonic Donald
ballot Sonic Mickey Donald

Then:

$ python3 vote.py sample.txt
Options: {'method': "Black's", 'require': 3}
Ballots: 3
Valid ballots: 3
Candidates: ['Donald', 'Mickey', 'Sonic']
Method: Black's
position 1 (Condorcet): Mickey
position 2 (Condorcet): Sonic
position 3: Donald

Black’s Procedure works as follows at each stage: see if there is a Condorcet winner; if not, look for a Borda winner (the first person on each ballot gets a (reversed) Borda score of 0 points; the second gets 1 point; and so on; persons not on a ballot get n points where there are n on the ballot; the winner is determined by the lowest sum of points; note that by default the ballots are modified at subsequent stages by deleting candidates who were already selected, which means the Borda scores change from stage to stage). Nanson’s Method deletes everyone with poorer-than-average Borda score, re-ranks, and repeats until there is a winner or a tie. This is also guaranteed to return a Condorcet winner if there is one.

The code marks winners that were Condorcet winners. That may be helpful to group deliberation as it shows that the decision is bit more robust in that case. (Though a Condorcet winner in kth place, with some non-Condorcet winners before, may not mean much if the earlier winners are dubious.)

The require line specifies how many entries a ballot must contain to be valid (by default, all ballots are valid, even ones with varying numbers of votes; any unranked candidates count as tied after the last ranked candidate). The ballot lines contain the candidates that someone voted for, in order from best to least good. Candidate names are case sensitive and cannot contain spaces. To save typing, one can also introduce abbreviations with the key entry. For instance:

method Black's
require 3
key m Mickey
key d Donald
key s Sonic
ballot m d s
ballot m s d
ballot s m d

Previsions for inconsistent credences and arguments for probabilism

Fix a sample space Ω and an algebra F events on Ω. A gamble is an F-measurable real-valued function on Ω. A credence function is a function from a F to the reals. A prevision or price function on a set of set G of gambles is just a function from G to the real numbers. A previsory method E on a set of gambles G and a set of credence functions C assigns to each credence function P ∈ C a prevision E_P on G.

A previsory method on G and C has the weak domination property provided that if f and g are two gambles such as that f ≤ g everywhere on Ω, then E_P(f)≤E_P(g) for every f and g in G and P in C. It has the strong domination property provided that it has the weak domination property and if f < g everywhere on Ω, then E_P(f)<E_P(g). It has the zero property provided that E_P(0)=0.

Mathematical expectation is a previsory method on the set of all bounded gambles and all probability functions. It has the zero and strong domination properties.

The level set integral is a previsory method on the set of all bounded gambles and all monotonic credence functions (P is monotonic iff P(⌀)=0, P(Ω)=1 and P(A)≤P(B) whenever A ⊆ B). It has the zero and weak domination properties.

The level set integral has the strong domination property on the set of weakly countably additive monotonic credence functions, where P is weakly countably additive provided that Ω cannot be written as a countable union of sets each of credence 0. If F (or Ω) is finite, we get weak countable additivity for free from monotonicity.

A previsory method E requires (permits) a gamble f given a credence P provided that E_P(f)>0 (E_P(f)≥0); it requires (permits) it over some set S of gambles provided that E_P(f)>E_P(g) (E_P(f)≥E_p(g)) for every g in S.

A previsory method with the zero and weak domination properties cannot be strongly Dutch-Booked in a single wager: i.e., there is no gamble U such that U < 0 everywhere that the method requires. If it also has the strong domination property, it cannot be weakly Dutch-Booked in a single wager: there is no U such that U < 0 everywhere that the method permits.

Suppose we combine a previsory method with the following method of choosing which gambles to adopt in a sequence of offered gambles: you are required (permitted) to accept gamble g provided that E_P(g₁ + ... + g_n + g)>E_P(g₁ + ... + g_n) (≥, respectively) where g₁ + ... + g_n are the gambles already accepted. Then given the zero and weak domination properties, we cannot be strongly Dutch-Booked by a sequence of wagers, and given additionally the strong domination property, we cannot be weakly Dutch-Booked, either.

Given that level set integrals provide a non-trivial and mathematically natural previsory method with the zero and strong domination properties on a set of credence functions strictly larger than the consistent ones, Dutch-Book arguments for consistency fail.

What about epistemic utility, i.e., scoring-rule, arguments? I think these also fail. A scoring-rule assigns a number s(p, q) to a credence function p and a truth function q (i.e., a probability function whose values are always 0 or 1). Let T be truth, i.e., a function from Ω to truth functions such that T(ω)(A) if and only if ω ∈ A. Thus, T(ω) is the truth function that says “we are at ω” and we can think of s(p, T) as a gamble that measures how far p is from truth.

If E is previsory method on a set of gambles G and a set of credence functions C, then we say that s is an E-proper scoring rule provided that s(p, T) is in G for every p in C and E_ps(p, T)≤E_ps(q, T) for every p and q in C. We say that it is strictly proper if additionally we have strict inequality whenever p and q are different.

If E is mathematical expectation, then E-propriety and strict E-propriety are just propriety and strict propriety.

It is thought (Joyce and others) that one can make use of the concept of strictly propriety to argue for that credence functions should be consistent. This uses a domination theorem that says that if s is a strictly proper additive scoring rule, then for any inconsistent credence function p there is a consistent function q such that s(p, T(ω)) < s(q, T(ω)) for all ω. (Roughly, an additive scoring rule adds up scores point-by-point over Ω.)

However, I think the requirement of additivity is one that someone sceptical of the consistency requirement can reasonably reject. There are mathematical natural previsory methods E that apply to some inconsistent credences, such as the monotonic ones, and these can be used to define (at least under some conditions) strictly E-proper scoring rules. And the domination theory won’t apply to these rules because they won’t be additive. Indeed, that is one of the things the domination theorem shows: if C includes an inconsistent credence function and E has the strong domination property, then no strictly E-proper scoring rule is additive.

So, really, how helpful the domination theorem is for arguing for consistency depends on whether additivity is a reasonable condition to require of a scoring rule. It seems that someone who thinks that it is OK to reason with a broader set of credences than the consistent ones, and who has a natural previsory method E with the strong domination property for these credences, will just say: I think the relevant notion isn’t propriety but E-propriety, and there are no strongly E-proper scoring rules that are additive. So, additiveness is not a reasonable condition.

Are there any strongly E-proper scoring rules in such cases?

[The rest of the post is based on the mistake that E-propriety is additive and should be dismissed. See my discussion with Ian in the comments.]

Sometimes, yes.

Suppose E is previsory method with the weak domination condition on the set of all bounded gambles on Ω. Suppose that E has the scaling property that E_p(cf)=cE_p(f) for any real constant c. (Level Set Integrals have scaling.) Further, assume the separability property that there is a countable set of B of bounded gambles such that for any two distinct credences p and q, there is a bounded gamble f in B such that E_pf ≠ E_qf. (Level Set Integrals on a finite Ω—or on a finite field of events—have separability: just let B be all functions whose values are either 0 or 1, and note that E_p1_A = p(A) where 1_A is the function that is 1 on A and 0 outside it.) Finally, suppose normalization, namely that E_p1_Ω = 1. (Level Set Integrals clearly have that.)

Note that given separability, scaling and normalization, there is a countable set H of bounded gambles such that if p and q are distinct, there exist f and g in H such that E_p requires f over g (i.e., E_pf > E_pg) and E_q does not or vice versa. To see this, let H consist of B together with all constant rational-valued functions, and note that if E_pf < E_qf, then we can choose a rational number r such that r lies between E_pf and E_qf, and then E_p and E_q will disagree on whether f is required over r ⋅ 1_Ω.

Let H be the countable set in the above remark. By scaling, we may assume that all the gambles in H are bounded by 1. Let (f₁, g₁),(f₂, g₂),... be an enumeration of all pairs of members of H. Define s_n(p, T(ω)) for a credence function p in C as follows: if E_p requires f_n over g_n then s_n(p, T(ω)) = −f_n(ω), and otherwise s_n(p, T(ω)) = −g_n(ω).

Note that s_n is an E-proper scoring rule. For suppose that q is a different credence function from p and E_ps_n(p, T)>E_ps_n(q, T). Now there are four possibilities depending on whether E_p and E_q require f_n over g_n and it is easy to see that each possibility leads to a contradiction. So, we have E-propriety.

Now, let s(p, T) be Σ_n = 1^∞ 2⁻ⁿs_n(p, T). The sum of E-proper scoring rules is E-proper, so this is an E-proper scoring rule.

What about strict propriety? Suppose that p and q are credence functions in C and E_ps(p, T)≤E_ps(q, T). By the E-propriety of each of the s_n, we must have E_ps_n(p, T)=E_ps_n(q, T) for all n. Thus, for all pairs of members of H, the requirements of E_p and E_q must agree, and by choice of H, p and q cannot be different.

Forgiveness of sins

It is very plausible that God can forgive wrongs we do to him. But a very difficult question which is rarely discussed by philosophers of religion is how God can forgive wrongs done to beings other than God.

This puzle seems to me to be related to the mystery of the line: “Against you [God], you alone, have I sinned” in Psalm 51:4, a line that seems on its face to contradict the obvious fact that the sins in question (David’s adultery with Bathsheba and murder of Uriah) seem to be primarily against human beings. Perhaps also related is Jesus’s puzzling statement: “No one is good but God alone” (Mark 10:18).

I think the answer to all of these questions may lie in a metaphysics and axiology of participation on which all the value of creatures is value had by participation in God, so that only God is good in the primary sense and only God is sinned against in the primary sense, which in turn gives God the normative power to forgive all wrongs, including wrongs directly against God as such as well as wrongs against God’s goodness as participated in by creatures.

Joint powers

Suppose neither Alice nor Bob has the power to budge the sofa, but together they can lift it. Causal powers belong to substances, and Alice and Bob do not compose a substance, so it seems the causal power to lift the sofa does not belong to the pair as a pair. Rather, the power must belong to the pair in virtue of the substances composing it. But how can that work?

Here is what I used to think. Causal powers come with actuation conditions. So we can say:

1. Alice has the power to lift the sofa when Bob is helping.

2. Bob has the power to lift the sofa when Alice is helping.

But now suppose both are working together. Then both causal powers’ actuation conditions are met. But when each of two causal powers for an effect E is actuated, then E is overdetermined. Thus, the sofa’s upward movement is overdetermined. But that is clearly false. So something is wrong.

Maybe we just need a better account of overdetermination? Or maybe there need to be irreducibly joint powers?

More on fake assertions

In my previous post I argued that if Bob writes and posts a letter of recommendation for himself purporting to be from Alice, and saying all sorts of false stuff like that Bob is very honest, then the contents of the letter are not asserted by Bob, and hence while they are deceptions—and, obviously, immoral—they are not lies.

Here are some more cases that I think support this. In all of the stories, I assume Alice is honest and well-informed.

1. Bob has deceived Alice into thinking that he is actually very honest. She writes him a letter of recommendation asserting this, and Bob reads the letter (e.g., by steaming open the envelope) and mails it to the potential employer.

2. Bob breaks into Alice’s office and finds a letter of recommendation for another guy—a really honest guy—with the same name as Bob. He sends the letter in support of his job application.

3. After an accident, Alice has been engaging in handwriting exercises by writing joke letters of recommendation. One of these joke letters is a letter of recommendation for Stalin as a kindergarten teacher, praising his compassion, and another is a letter for Bob as a bank teller, praising his honesty. Bob breaks into Alice’s office, finds the letter for him. He knows full well it is a joke, since he knows what Alice actually thinks of him, but he posts the letter in support of his job application.

4. Bob has a bunch of monkeys employed randomly typing on typewriters. One day, a monkey produces a letter praising Bob’s honesty and purporting be from Alice. He sends the letter as part of his job application.

5. Bob obtains a letter of recommendation from Alice where one line ends with “Bob is utterly dis-” and the next line begins with “honest.” He carefully erases the “dis-” and posts the letter in support of his job application.

6. The original case where Bob fakes the entire letter.

Case 1: There is no lie in the letter, and nothing in the letter is asserted by Bob. Bob is still being deceitful by knowingly mailing a letter containing false information about him, which false information comes from his deceit of Alice. But there is no lie in the letter.

Case 2: Alice asserts truths in the letter. Bob manipulates the reader into thinking that the letter is about him, which it is not. The reader misunderstands the letter as about Bob. But the one person doing any asserting in the letter is Alice, who cannot be said to be asserting falsehoods.

Case 3: Alice neither asserts that Stalin is compassionate nor that Bob is honest. She is just joking and jokes aren’t assertions. Bob manipulates the reader into misunderstanding the jokes as assertions. No one does any asserting in the letter, certainly not Alice, but also not Bob.

Case 4: This one is a little bit trickier, but in the end it’s hard to see a difference between case 3 and case 4. In both cases, the writer of the letter made no assertions. And Bob just posted it.

Case 5: Here things are, I think, even a little bit murkier. But imagine a version of case 5 where Bob sees his pet monkey playing with an eraser and erasing the “dis-”, and then he posts the letter. In that case, this is just like case 4 with respect to Bob’s authorship, and hence Bob is not lying in the letter. But I also don’t think it matters whether Bob physically does the erasing himself or the monkey does it with Bob’s knowledge. Bob isn’t lying in the letter.

I could imagine someone caviling at my judgment in case 5, so let’s go back to case 4 some more.

Imagine that Bob has all the time in the world on his hands, and he has hired a bunch of monkeys as secretarial staff. Whenever he wants to write a letter, he composes it is in his mind, and then waits for the monkeys to type exactly it at random. When they do so, he posts the letter. This is just an inefficient way of writing letters: the letter is just as fully from Bob as it would be if he typed it himself. If the letter is signed “Bob” and contains claims that Bob knows to be false, Bob is lying in the letter. But note that if the letter is signed “Alice”, this is just case 4, and in case 4, Bob isn’t lying in the letter. So, it looks like whether Bob is or is not lying in the letter depends on whether it purports to be from him, and hence in cases 5 and 6, Bob isn’t lying in the letter either.

Let me push a bit further. Go back to case 1, which was perhaps the clearest case of Bob’s not lying in the letter. Imagine that Bob has the following inefficient technique for avoiding doing any typing himself. When he wants to write a particular letter purporting to be from himself, he finds another person with the same name as his own, and he manipulates them into believing the content of the letter, and then puts them in circumstances where the other person has a reason to honestly write such a letter. He then steals the letter and posts it as if it were his own. This seems, once again, to be a case of an inefficient letter composition procedure, and Bob is the author of the letter, just as much as he would be if he waited for a monkey to type it at random or if he trained a monkey to write it. Yet the main difference between this and case 1 is that in case 1, Bob isn’t purporting the letter to be from himself, but from Alice, which it in fact is. So, if we grant, as I think we have to, that Bob isn’t lying in the letter in case 1, but that he would be if he used the inefficient secretarial technique of manipulating namesakes into writing letters just like the ones he wants, then we have to say that what makes the difference as to whether Bob is lying in a letter that he fully approves of despite knowing it contains falsehoods is whether the letter purports to be from Bob.

We can make similar points about some of the other cases. For instance, suppose we agree that there is no lie in the joke letter in case 3. But we can imagine Bob having an inefficient secretarial technique where letters from him are written by getting lots of people to do handwriting exercises until one of them writes something signed “Bob” that has the exact content he wants it to have. In that case, Bob is lying in the letter, if the letter has falsehoods.

If this is right, then lies are tightly connected to a personal endorsement of a claim. If instead of personally endorsing a claim one fakes an endorsement by someone else, one is engaging in deceit but one isn’t lying.

Fake assertions

Suppose Bob faked a letter of recommendation from his dissertation director Alice, in which letter lots of stuff was said which Bob knew to be false, and then posted the letter to Carl.

Bob clearly deceived Carl, or tried to. But did he lie to Carl? Let’s consider three representative example sentences from the letter:

1. I am Bob’s dissertation director.

2. I think the world of Bob.

3. Bob is impeccably honest.

I will also take that Bob knows that Alice is his dissertation director, that Alice thinks poorly of him (which is why he faked the letter) and that he’s dishonest, and I will also assume that Bob thinks the world of himself.

If Bob lied, which of these sentences did he lie in?

One important question is who “I” refers to in the letter. If it refers to Bob, then (1) and (3) are false and (2) is true. If it refers to Alice, then (2) and (3) are false and (1) is true. Basically, we need to decide which of (1) and (2) is true.

It seems clear that by (2), Bob intended to communicate that Alice thinks the world of him, and he had no intention at all to communicate that Bob thinks the world of himself (indeed, perhaps another sentence in the letter is “I have never met a humbler person”). So it seems that “I” refers to Alice, and hence (2) and (3) are false, but (1) is true.

On this reading, Bob has knowingly written two false things: (2) and (3), and one truth: (1). Has Bob lied in the false things he wrote? I have some doubts. The reason is this. What makes lying be lying is that one is betraying a trust that one has solicited in speaking. But Bob has not solicited Carl’s trust in Bob: rather, he is relying on Carl’s trust in Alice. But one can only betray trust in oneself. So Bob cannot betray Carl’s trust in Alice, and hence Bob is not lying when Alice is the object of Carl’s trust. Here’s another way to think about this: To lie is to stand behind a falsehood. But Bob isn’t standing behind the falsehood—he is, instead, putting Alice in front of it, as is clear from the fact that “I” refers to Alice.

In asserting something one implicates that one believes it. But Bob isn’t implicating that he believes it, only that Alice does. And it’s not, it seems, that Bob has canceled the implicature of belief (as one sometimes can, pace Moore). I think Bob not only isn’t lying, but he isn’t asserting anything.

This seems paradoxical. But consider this. Suppose Drew, who is dishonest but not a racist, fakes an open letter from Adolf Hitler, hoping to sell it off to the Holocaust Museum.. The letter contains all sorts of false statements, such as that various minority groups are subhuman. Drew is clearly committing fraud. But is he making racist statements? I don’t think so. Rather, he is faking racist statements by Hitler. Similarly, the falsehoods in the letter are not lies by Drew, for if Drew were lying in the letter, he would be making racist statements. But he is faking, not making, racist statements.

I think the same may be true of Bob: he is faking, not making, various assertions in the letter. There is a difference between Bob and Drew, of course. Drew is not trying to get the audience to believe the fake assertions, but only to believe that they were made. Bob is trying to get the audience to believe the fake assertions. But this difference aside, I still suspect that Bob is deceiving, not lying.

Of course, this difference doesn’t let Bob or Drew off the hook. They have engaged in a massive failure of integrity, indeed in fraud.

But the difference between deceiving and lying could still be relevant. I think a challenge for those of us who think lying is always wrong is to articulate some sort of a theory of clandestine military and police operations that allows for non-lying deceit. If lies require that the liar be taken to be the author, then this opens up the way for various things like Operation Mincemeat being deceit but not lies.

I fear, however, that at this point I am engaging in the kind of casuistry that gives casuistry a bad name. Here is one way of highlighting this. Surely one can’t just write a letter with falsehoods putatively from oneself and claim that one faked one’s own letter, and hence one didn’t lie in it. But now imagine that Alice and Bob conspire to each write a fake letter purporting to be from the other. Surely that shouldn’t escape the moral prohibitions against lying. Maybe, though, it depends on the details of the conspiracy. If Bob is just writing in the letter putatively from Alice things that Alice asked him to write, then the letter is no fake, and Bob is just Alice’s secretary.

Fake counting

When someone’s walking speed is two miles per hour, there are not two things, “one mile per hour walkings”, that are present.

When we say that a sculpture has three dimensions, we are not saying there are exactly three things—dimensions?—that are present in it. But are there not width, height and depth? In a way. But rotate the sculpture by 45 degrees, and “width”, “height” and “depth” refer to measurement along three other axes. There are, it seems, infinitely many axes along which the sculpture can be non-trivially measured.

These are examples of what one might call “fake counting”. We speak as if there were n of something, but the following argument is invalid:

1. There are n Fs.

2. n ≥ 1.

3. So, there are some Fs.

And, similarly, this is invalid:

4. There are exactly two Fs.

5. So, ∃x∃y(F(x)&F(y)&∀z(F(z)→(z = x ∨ z = y))).

In fake counting of Fs, there is counting involved, but it is not counting of Fs. For instance, when we say that the sculpture has three dimensions, we mean something like this:

• there are three mutually perpendicular axes such that the sculpture has non-zero extent along each of them, but there are no four such axes.

So, there is a counting of axes, but it is not a counting of dimensions. If we were counting dimensions, we would have to have say what the first one is, what the second one is and what the third one is, and as the rotation thought experiment shows, that doesn’t work. And the counting of axes doesn’t involve counting axes overall, but rather axes in a particular set of them.

We need to beware of fake counting when making metaphysical arguments for the existence of entities of some sort. For instance, topologists have ways of “counting holes”. But topological properties are invariant under deformations. Now, imagine a pancake with, as we would say, “one hole in the middle”. Well, however we distort the pancake, it has one topological hole. But if we ask where that hole is, there is no topological answer to it (in the animation below, is the hole outlined in red or in blue?). So, topological hole counting is fake counting.

Silencing and epistemic harm

Suppose that Arthur is about to give a lecture on trope theory but the lecture is canceled due to Platonist protests.

It is intuitive to say that unjust epistemic harms have been perpetrated. But on whom?

The primary epistemic harms from Arthur’s silencing are to the audience who is prevented from hearing his arguments, and hence is in a poorer epistemic state to adjudicate the truth of the matter about tropes. Arthur potentially receives some secondary epistemic harms, in that he is deprived of the benefits of challenging questions or of the potential growth of understanding of a subject that a speaker gains by speaking. But notice that these are accidental to his silencing. The loss of the benefit of challenging questions is due to the silencing of the audience, not due to Arthur’s silencing, and the benefit of thinking through one’s position in presenting it could be had by presenting the position to an empty room. However, the audience’s epistemic loss is primary.

Now, Arthur receives all sorts of potentially serious non-epistemic harms. He has been insulted. A promise to him has been reneged on. He has lost the value of being the intentional cause of the audience’s epistemic benefits. His CV is shorter by one item. Perhaps he has lost an honorarium, or at least he has lost the opportunity to have pre-scheduled something else for that day. He may come to be in fear for his personal safety. Some of these harms may result in epistemic harms down the road: for instance, he may abandon a promising line of research as a result of these insults or get a job at a department with poorer research opportunities. But these epistemic harms are secondary to the non-epistemic harms.

I suppose there could be the following epistemic harm to Arthur: if public proclamation of p carries negative consequences for one, then one will be tempted to cease to believe p. If Arthur was in fact right in his views, and he abandons those views due to the opposition, then he will suffer epistemic harm. But while people do change their views because of social consequences, I suspect this is much more common when the social consequences are subtle than when they are highly overt. I suspect that highly overt cases, such as lecture cancelation, are more likely to entrench one in one’s beliefs. Of course, such entrenchment could itself be an epistemic harm, especiallyif if the beliefs are false.

So, Arthur’s being silenced results in:

• primary epistemic harms to the audience

• primary non-epistemic harms to Arthur

• secondary epistemic harms to Arthur.

So, it is correct to say that unjust harm has been done to Arthur, but that harm is not primarily epistemic. The people to whom unjust epistemic harm has been done are the people who would have been in Arthur’s audience.

The same is true if the form the injustice takes is one’s prejudiced refusal to take seriously another’s testimony or arguments. In this case, one is doing injustice to the speaker, but the speaker does not suffer epistemic harms by one’s refusal to take their testimony or arguments seriously. The speaker is insulted—whether they know it or not (the speaker may not know that one is refusing to take them seriously)—but the epistemic harm is to oneself.

In my initial example, Arthur’s lecture was on trope theory, a highly theoretical topic. But nothing changes when the topic becomes more personal. Suppose Arthur is silenced and kept from speaking out about the injustices that he has received over his lifetime because of his disability. The primary epistemic harms are, again, to the audience. But Arthur is harmed by being insulted, and prevented from convincing people to stop perpetrating injustice on him. These, however, are not, primarily, epistemic harms.

When I was initially writing this post, I was thinking this was going to be an argument that we shouldn’t think of epistemic violence or epistemic injustice as something that is done to a person who is silenced, but as something that is done to the audience.

But I then realized that “epistemic” in “epistemic violence” and “epistemic injustice” can be understood as qualifying either the types of harm imposed or the means by which the harms are imposed. If we understand it as qualifying the types of harm imposed, then I think my original thesis is quite correct: epistemic violence and injustice are done not to those who are silenced but to those who are prevented from hearing them (and that could, I think, be the silencers themselves). But it seems more faithful to the intent of those who have been writing on these topics to take “epistemic” to qualify the means. And this fits with our usage in some other cases: Bob perpetrates “gun violence” just in case he perpetrates violence with a gun, rather than when he harms someone’s gun collection. When Arthur is silenced, epistemic violence/injustice is done to him because he is unjustly harmed by epistemic means, namely he is harmed by others’ epistemic malpractice (whether in the narrow sense, as when a prejudiced audience refuses to listen, or in a broad sense, when protesters make it impossible for others to listen). But he is not epistemically harmed.

Shapes of holes

The ordinary notion of a hole is kind of dubious. Consider the hole in the thin wavy sheet of rubber on the right. What is the shape of that hole? How thick is it? Is it exactly as thick as the rubber sheet? But the rubber sheet varies in thickness, actually. How does it stretch from its wavy edges to the middle? Does it have a sinewave bump in the middle, to correspond to where there are sinewave bumps in the sheet elsewhere? Or does that depend on the history of its formation (e.g., maybe if the sheet used to have a bump there but then a hole was made--that's how my code generating this picture works--then the hole has a bump, but if the sheet was pre-made with a hole, then the hole is flatter)? I think there really are no good answers to these questions, and hence holes don't exist.

Holes and substantivalism

Suppose substantivalism about space is correct. Imagine now that the following happens to a slice of swiss cheese: the space where the holes were suddenly disappears. I don’t mean that the holes close up. I mean that the space disappears: all the points and regions that used to be in the hole are no longer there (and any air that used to be there is annihilated). The surfaces of the cheese that faced the hole now are at an edge of space itself.

The puzzle now is that in this story we have an inconsistent triad:

1. There is no intrinsic change in the cheese.
2. The slice of cheese no longer has holes.
3. Changing with respect to whether you have holes is intrinsic.

Here are my arguments for the three claims. There is no intrinsic change in the slice of cheese as something outside the cheese has changed—space has been annihilated. The slice of cheese no longer has holes, as it makes sense to talk of the size or shape or volume of a hole, but there is no size or shape or volume where there is no space. And changing with respect to whether you have holes is change of shape, and changes of shape are intrinsic.
It seems that the above story forces you to reject one of the following:

4. Substantivalism about space
5. Intrinsicness of shape.

But there is another way out. Deny (3). Whether you have holes is not intrinsic. What is intrinsic is your topological genus with respect to your internal space and similar topological properties.

Note, also, a lesson relevant to the famous Lewis and Lewis paper on holes: the counting of holes should not involve the counting of regions, but the computation of a numerical invariant, namely the genus.

Characterizing actions by the reasons against them

It is plausible that the reasons for which one chooses an action help determine the kind of action it is. Plausibly, an action is a murder if one performs it because it will kill an innocent.

But it is also interesting that the reasons against which one chooses an action also help determine the character of an action. This is true both in good and bad actions. Some actions are acts of courage in part because they are done contrary to reasons of one’s own safety. And some actions are acts of gross negligence in part because they are done contrary to reasons of the safety of another. If, on the other hand, the reasons of safety did not enter into deliberation at all, the act, in both cases, may well be a case of recklessness.

Intending the end without intending the known means

It is said that:

• he who intends the end intends the means.

But the person who doesn’t know how a computer keyboard works does intend to close circuits when writing an email, even though the closing of circuits is the means to writing emails.

Perhaps, though, when they learn how computer keyboards work, that might change their intention, so that now they intend to close circuits whenever they intentionally type a character? But that is psychologically implausible: the activity of the practical intellect involved in typing is normally unchanged by learning how a keyboard works. (There are, of course, special circumstances where it may change. For instance, if one knows that one is near some very delicate electrical equipment whose functioning could be affected by the closing of these circuits, then one’s deliberation might change.) The knowledge of what happens in typing remains merely non-occurrent knowledge, not affecting the activity of the practical intellect or the will.

One might think, though, that if one occurrently knows that the means to typing an email is the closing of circuits, one is intending to close circuits. But even this need not be true. For instance, a person who is writing a technical article on how keyboards work may well be occurently knowing that their movements are transformed into data in computer memory by means of closing electrical circuits, but this occurrent knowledge may very well still not affect either their practical intellect or their will. (Indeed, when I wrote the opening paragraph of this post, I no doubt occurrently knew how keyboards work, but I don’t think this affected my intentions.)

For one’s knowledge to affect one’s intentions it needs to enter into the deliberation. For that, it needs to be occurrently and practically taken by the agent as practically relevant. For most people under most circumstances, that computer keyboards work by closing circuits is not practically relevant. But if it is Sabbath and one is an Orthodox Jew who believes that closing circuits is forbidden on the Sabbath, then the knowledge is apt to be taken as practically relevant: if one still types, that is apt to become an act of rebellion or of akrasia, and if one refrains from typing, that act is apt to be done as a mitzvah. However, one could imagine the sad case of such an Orthodox Jew who types on the Sabbath anyway, and eventually becomes so calloused that the fact that circuits are being closed stops entering into deliberation, though the fact is still known by the theoretical intellect. Such a person’s intentions may eventually drift to those of the typical gentile.

So, what is one to say about the principle that he who intends the end intends the means? There is of course a trivial version:

• he who intends the end intends the intended means.

Maybe we can do a little better:

• in intending the end one intends the means insofar as they enter into deliberation.

I am not sure this is right, but it’s the best I can do right now.

Note an interesting thing. If this last version is right, then the means may enter into deliberation on the opposite side, against the action. For instance, if one thinks it’s forbidden to close electrical circuits on the Sabbath, but one chooses to do so, that the means involve the closing of electrical circuits is apt to enter into deliberation on the con side of typing, not on the pro side (unless one is positively rebellious).

Internal reference frame

Suppose a long snake is stretched out and its front half is annihilated instantaneously. This presumably instantly destroys the snake's form or soul. So the tip of the snake's tail instantly ceases to be informed by the snake form. But then there will be a reference frame according to which the front half is annihilated before the tail loses its form. In that frame, the tip of the tail still has a snake form at a time at which the snake's front half doesn't exist. That seems wrong. So it seems there should be a privilege to reference frames where the front half is destroyed simultaneously with the tail losing its form. But a global privileged frame is unattractive. Maybe, however, we should suppose that particular substances carry along privileged frames of their own, frames internal to them. Then there will be a privilege frame for each substance, but these frames need not cohere into a global privileged frame.

Foresight and intention

The Principle of Double Effect controversially teaches that when an effect is bad, it is typically worse to intend it than to merely foresee it. I think it's interesting and somewhat refreshing to reflect on reverse cases, where we distinguish between intending and foreseeing a good effect. Obviously it's better if a legislator intends rather than merely foresees that the legislation furthers the public good. Here, it's clear that here a foresight-intention distinction captures something important.

The argument from apparently gratuitous evil

I think the following two claims are plausible:

1. If God exists and there is a lot of evil, then we would expect that some of the evil is such that we cannot see its point.

2. If God doesn’t exist and there is a lot of evil, then we would expect that some of the evil is such that we cannot see its point.

Premise 2 is pretty plausible: without God, and given a lot of evil, we’d expect evils to be pretty much random, some of them connected to goods that give them a point and others not. Now, if God exists and allows for a lot of evil, then there will be a point to all the evil allowed. And it would be intrinsically good for us to see the point of any particular evil, since knowledge is intrinsically good. But given the assumption that God has allowed a lot of evil, it would be surprising if all of this evil was such that its point (a) could be understood by us and (b) it would be on balance good for us to understand its point. In regard to (a), we can cite our cognitive limitations. In regard to (b), we can cite the fact that it is likely that some of the justifications for permissions of evil would involve soul-building, whereas it is very plausible that some soul-building would require techniques that are hidden from its beneficiaries.

Thus, once one has already taken into account the fact that there is a lot of evil, observing that there are evils that we cannot see the point of does not yield much evidence for or against the existence of God. It may, of course, yield some evidence if the degrees of expectation in (1) and (2) are different, but not much.

If this is right, then Rowe-style “evidential” arguments from evil don’t accomplish much beyond the “naive” argument that God wouldn’t allow so much evil.

Of course, one might try to argue that it’s not just the existence of pointless evil that is relevant, but how common it is. But then one would need to get into a messy discussion of just how common it is, and how common one would expect it to be on theism and on atheism.

Perdurance

Here’s an interesting thing. Suppose perdurance is true. Then God cannot be in time. For if perdurantism is true and God is in time, then God is composed of infinitely many temporal parts. But:

1. This violates divine simplicity.

2. These parts are concrete and presumably not created by God, so there are concrete things other than God that God didn’t create.

3. God acts in virtue of the temporal parts acting, but then God’s actions are not the fundamental explainers.

4. The temporal parts are all-knowing, so God is not the only all-knowing entity.

This is utterly unacceptable. So, one cannot both accept perdurantism and that God is in time.

Change, time and contradiction

According to Aristote:

1. Time is the measure of change.

2. The law of non-contradiction says that a thing cannot have and lack the same property in the same respect at the same time.

The law of non-contradiction seems to be the fundamental basis of logic. Yet it presupposes the concept of time, which in turn presupposes that of change. Thus, it seems, for Aristotle, the concept of change is more fundamental than logic itself. That doesn’t seem very plausible to me.

But perhaps there is a different way to understand the “at the same time” qualifier in (2). Sometimes, we give a rule with something we call an exception, but it’s not really an exception. For instance, we could say: “It is an offense to lie to an officer of the law, except unintentionally.” Of course, there is no such thing as an unintentional lie, but it is useful to emphasize that unintentional falsehoods are not forbidden by the rule.

Now, Aristotle is, as far as I know, a presentist. On presentism, the only properties a thing has are its present properties, and it lacks precisely those properties it doesn’t presently have. So it’s not really possible for an object to have and lack the same property, since the having and lacking would have to be both present, and hence at the same time. But it is useful to emphasize that having the same property at one time and lacking it another is not forbidden by the law of non-contradiction, and hence the logically unnecessary qualifier “at the same time”. Strictly speaking, I think “in the same respect” isn’t needed, either.

Molinism and sceptical theism

When we think of God’s reasons for permitting evils, we tend to think of fairly “natural” connections between evils and goods. But given Molinism, there could be some really weird connections. For instance, it could be that if Alice hadn’t been cut off by Bob in traffic today, Carl who witnessed this would have joined a terrorist organization. Not because there is any intrinsic connection between seeing someone get cut off in traffic and joining a terrorist organization, but just because that’s how the conditionals of free will worked out.

Indeed, a Molinist should expect there to be cases where the Molinist conditionals work out the opposite way to the “natural” connections. Thus, we can have cases where becoming more cowardly results in one’s behaving more courageously, just as a Molinist God might know that if the coin loaded in favor of heads would show tails in the next ten tosses while the coin loaded in favor of tails would show heads in the text ten tosses.

So there seems to me to be a very nice affinity between Molinism and sceptical theism.

It’s really too bad that Molinism is false.

Exercise and posession of virtue

The exercise of generosity is good to have. So is its possession. How do the two compare?

One might think the possession of generosity only has value as an instrument towards generous activity. But that seems wrong. It is bad for one to be deficient in generosity even if one will never again have the opportunity to practice generosity (say, because there is no afterlife and one has been marooned on a desert island).

But at the same time, it seems to me that the possession of generosity is of fairly low value as compared to the exercise of it. Suppose I am going to be a coma for the rest of my life and there is no life after death, and I have a choice between two actions, one of which will be generous and the other will increase my generosity (e.g., I have a choice whether I should give some money to a hungry person or to spend it on neurosurgery to eliminate something that blocks me from having much of a virtue of generosity). It seems plausible that I should do the generous deed: living (even in a coma) with generosity is better than living without it, but not by much. Similarly, if I am going to be in a coma for the rest of my life, and I have the opportunity to have one last look at a beautiful landscape, that seems worth doing, even if the price of that look is that I will lose my eyes. It is better to have eyes than not, but if the eyes aren’t going to ever get used, the value of merely having them seems small.

Perhaps, though, in a full Christian picture of life that includes the afterlife, there aren’t going to be cases where one is choosing between the exercise and the possession of generosity. If before the coma I don’t do the generous deed, then maybe I am like the guy who buried his talent, and the generosity will be taken away from me in the next life. Or at least it won’t be increased a hundredfold. I am inclined to say that given the full Christian picture, the exercise of generosity (and other virtues) should generally be chosen over the immediate possession of generosity, but will tend to result in greater possession of generosity.

Conscience and the deontic logic of attempts

When people talk of the value of obedience to conscience, it often makes it sound like there is some sort of a relationship to a mysterious faculty with a mysterious authority.

And that may all be true. But there is also a rather simple and deflationary but still, I think, useful way to think of obedience to conscience.

When I obey my conscience I am just trying to do what I ought thing. There is nothing particularly mysterious about what is right about that. If I ought to do A, I ought to try to do A. I ought to honor my parents. So, I also ought to try to honor them. Similarly, I ought to do what I ought, so I ought to try to do what I ought.

And with respect to the duty to try to do what I ought, it doesn’t matter that due to a mistake on my part I will be unable to do what I ought. That I have wrongly written down my mother’s phone number does not excuse me from trying to call her on her birthday. I ought to dial that number, because not dialing that number would be constitute a failure to try to call her, given my belief that it’s her number. Similarly, even if I am mistaken in thinking that I ought to do B, I still ought to do B, because a failure to do B would be constitutive of a failure to try to do what I ought, given my belief that B is what I ought to do.

(This is all a little less trivial when we realize that the duty to do one’s duty is actually a bit controversial. One might think that one only has first order duties, and lacks the second order duty to see to it that one fulfills the first order duties. But that would, I think, be mistaken. If I know that partaking of alcohol would cause me to neglect my first order duties, I thereby have a second order duty to avoid such partaking.)

Alternate timelines

The following sound right:

1. It is always the case that s if and only if s at all times.

2. It is always the case that s if and only if it is, always was and always will be the case that s.

But suppose, as may very well be the case, that we inhabit a multiverse whose universes all have temporally unrelated time sequences. Then (1) and (2) are apt to disagree. For (2) tells us that it is always the case that s just in case s at all past, present and future times. But in our multiverse scenario, there are times that are not past, present or future (from our point of view—which is surely the point of view we are speaking from). Thus, there is apt to be a difference between what happens at all times and what was, is and will be. For what happens at all times includes stuff that happens in other universes, since there are times that are found in other universes than ours. But what happens in the past, present or future only includes only what happens in our universe.

So, should we take (1) or (2) as the correct reading of “always”? I don’t know. (Note: I am assuming that the quantification over times in (1) is unrestricted, and hence not limited to times in our universe.) It’s a bit puzzling.

Moreover, one would have to say that right now, only a time in our time sequence is present. For if two times are present, they are simultaneous, but there are no simultaneity relations between our times and times in other time sequences. In fact, the times in the other time sequences never were and never will be present, since if they were or will be present, then they would not be temporally unrelated to times in our sequence. It sure sounds odd to talk of times that were not present, are not present and will never be present. But the ontology, nonetheless, seems to make perfect sense.

Or at least it makes perfect sense to me, a B-theorist. Could it make sense to a presentist? I am not sure. The presentist needs to make a distinction between “real” events like World War II and the 2020 Olympics, on the one hand, and merely possible events like the arrival of the Vulcans on Earth in 2053. None of these events are present, but obviously World War II and the 2020 Olympics are in some way real, while the arrival of the Vulcans is a mere fiction. The standard way for presentists to distinguish the arrival of the Vulcans from World War II and the 2020 Olympics is to say that World War II occurred and the 2020 Olympics will occur, while the arrival of the Vulcans neither occurred, nor is occurring nor will occur.

But if there are other time sequences, then the events on these time sequences are more like World War II and the 2020 Olympics than they are like first contact with the Vulcans. I do not see, however, how a presentist can possibly express the kind of reality the other time sequences in our (hypothetical) multiverse have.

(Here is a practical good from being able to make the distinction: It is right and proper to pray for all the beings in all the universes. But we shouldn’t pray for Vulcans and other fictional entities.)

In fact, I think the problem comes up even earlier, before considering any events. I don’t think the presentist can make any sense of the hypothesis of universes temporally unrelated to ours.

Thus we have an argument against presentism:

3. It is possible to have each of two temporally unrelated time sequences.

4. If presentism is true, it is not possible to have each of two temporally unrelated time sequences.

5. So, presentism is not true.

Omniscience

The following account of the doctrine of propositional omniscience is incomplete:

1. x is omniscient iff x knows every truth and believes no falsehood.

For suppose that x believes no falsehood and knows every truth but is suffering from retrieval problems for the truths that x believes, in such a way that it takes x a minute to recall what is the capital of China. That’s not omniscience—it’s not sufficiently perfect as knowledge. This suggests to me that omniscience requires occurrent knowledge of every truth: a total contemplation of all of reality.

Moreover, suppose x knows every truth but some of these truths x is not sure of. Again, that’s not omniscience. Nor would it be omniscience if x were sure of every truth and believed no falsehood, but there was some falsehood to which x assigned a small degree of belief—say, a credence of 0.2. (For one, such a being would have probabilistically inconsistent credences, as it would assign credence 1 to the negation of that falsehood.)

So, propositional omniscience should at least be:

2. x is omniscient iff x occurrently knows every truth for sure and has no degree of belief in any falsehood.

And, of course, propositional omniscience is unlikely to be all of omniscience.

The Incarnation and timelessness

Consider the standard argument against the Incarnation:

1. Everything that is God is F (omnipotent, omniscient, impassible, etc.).

2. Everything that is human is non-F.

3. Christ is God and human.

4. So, Christ is F and non-F.

5. Contradiction!

But it is only a contradiction to be F and non-F at the same time: we’ve known this since Aristotle.
Thus the kenotic theologian gets out of the argument by holding that Christ was F prior to the Incarnation and wasn’t F after the Incarnation. (A difficult question for the kenoticist: is he now F?) But that’s contrary to the teaching of the Councils.

However, the “at the same time” observation does not need to lead to kenoticism. In fact, the Christian who is a classical theist should deny that Christ is F and non-F at the same time. For it is strictly false to say that Christ is F at t for any divine attribute F and any time time t, since God has the divine attributes timelessly rather than at a time.

This is not kenoticism. Rather, the view is that Christ is F timelessly eternally and non-F at t (for any t after the beginning of the Incarnation). Kenoticism on this view is metaphysically absurd, because God cannot cease to be F: one can only cease to be something that one used to be, and there is no “used to be” where there is no temporality.

But we sometimes say things like:

• While he was suffering on the cross, Christ was upholding the existence of the universe.

I think there are two ways of make sense of such statements. First, maybe, things that happen timelessly count honorifically as holding at all times. (Compare David Lewis’s idea that abstract objects count as existing in all his worlds.) Second, the statement can be understood as follows:

• While he was suffering on the cross, the following proposition was true: Christ is upholding the existence of the universe.

So, orthodox Christians do not actually need to talk of natures to get out of (1)-(5). Of course, if we want to allow—as I think we should—for the logical possibility of multiple simultaneous incarnations, then the temporal qualification way out won’t help. (Nor will the kenotic solution help in that case, either.)

Note, by the way, that once we realize that there can be timelessly eternal existence, we need to modify Aristotle’s temporal qualification to the law of non-contradiction:

• it is impossible to be F and non-F in the same respect at the same time or both eternally.

More complications for Dutch Book results

Think of a wager as a sequence of event-payoff pairs:

• W = ((e₁, u₁),...,(e_n, u_n)).

There are then two different ways to calculate the expected value of the wager. First, directly:

1. E^D(W)=u₁P(e₁)+...+u_nP(e_n).

Second, indirectly by letting U_W be the utility function defined by W, i.e., U_W = u₁ ⋅ 1_e1 + ... + u_n ⋅ 1_en (where 1_e is the function that is 1 if e happens and 0 otherwise) and then calculating the expected utility of the function U_W:

2. E^I(W)=E(U_W).

If the credence function P is additive, then the two ways are equivalent. But without additivity, they come apart. Moreover, there is more than one way of calculating E(U) if the credences are inconsistent, but for now I will assume the standard Lebesgue sum way where, assuming U has only finitely many values, E(U)=∑_yyP(U = y).

The most common de Finetti Dutch Book Theorem, which says that inconsistent probabilities give rise to a Dutch Book, makes use of the direct way of calculating the values of wagers. Specifically, it considers wagers where you pay an amount x for a chance to win amount y if event E eventuates, and it calculates the value of such a wager as yP(E)−x. However, if instead one uses the indirect method of calculation, the value of such a wager becomes (y − x)P(E)−xP(E^c), where E^c is the complement of E.

This actually makes a real difference to Dutch Book theorems. Consider this inconsistent credence for a coin toss:

• P(H)=1/4

• P(T)=1/4

• P(H&T)=0

• P(H ∨ T)=1.

Then for any credence function U, it turns out that E^I(U)>0 if and only if the expected value of U is positive given the standard consistent fair-toss measure. The reason is this. Either U has the same value at heads and tails or it does not. If it has the same value at heads and tails, then E^I(U) has the same value as the expectation using the fair measure, since P agrees with the fair measure regarding H ∨ T. On the other hand, if U has different values at heads and tails, then E^I(U)=(1/4)U(H)+(1/4)U(T) which is exactly half of the fair measure’s expectation for U, and hence, again, E^I(U)>0 if and only if the fair measure says the expectation is positive. It seems to follow that E^I recommends exactly the same wagers as the standard consistent fair-toss measure.

Except that this isn’t quite true, either. For in addition to two ways of calculating expected values, there are two ways of making decisions on their basis in the case where a sequence of wagers is offered:

3. Accept a wager whose individual expected utility is positive.

4. Accept a wager when the expected utility of the already-accepted wagers combined with the currently offered wager exceeds the expected value of the combination of the already-accepted wagers.

Here, the combination of two wagers is concatenation. For instance ((e₁, u₁),(e₂, u₂)) combiness with ((e₃, u₃)) to form the wager ((e₁, u₁),(e₂, u₂),(e₃, u₃)). Given consistent credences, we have, E(W₁ + W₂)=E(W₁)+E(W₂), and (3) and (4) are equivalent. But, again, for inconsistent credences this additivity property can fail, and so a choice needs to be made between (3) and (4).

Note that (4) is itself an oversimplification. For theoretically, what wagers one accepts earlier on may depend on one’s best estimate as to what wagers will be offered later.

All in all, I know of five utility maximization decision procedures for sequences of wagers, generated by the answers to these questions:

• Direct or indirect utility calculation for a wager? (D or I)

• If indirect, Lebesgue sum or level set integral for calculating expectations? (LSum or LSet)

• If indirect, is the presently offered wager combined with previously accepted wagers in calculating expectations? (Indiv or Combo)

For consistent probabilities, these are all equivalent.

Moreover, there are two kinds of Dutch Books. There are Simple Dutch Books, where from the original position the agent accepts a Dutch Book, and Incremental Dutch Books, where after accepting some wagers, the agent goes on to accept a Dutch Book.

What happens with Dutch Books varies between the different procedures, and I am still working out the details. Say that a credence P is monotonic provided that P(∅)=0, P(Ω)=1 and P(A)≤P(B) whenever A ⊆ B. Here is what I have:

• D: Simple Dutch Books whenever probabilities are inconsistent.

• I+LSum+Indiv: I conjecture Incremental Dutch Books for some but not all inconsistent monotonic credences.

• I+LSum+Combo: I conjecture Incremental Dutch Books for all non-additive credences.

• I+LSet+Indiv: I don’t know.

• I+LSet+Combo: No Dutch Books of either sort for any monotonic credences.

The intellect is not higher than the will

1. The perversion of the higher faculty is worse, other things being equal.
2. Moral wrongdoing is worse than error, other things being equal.
3. Moral wrongdoing is the perversion of the will.
4. Error is the perversion of the intellect.
5. So, the intellect is not higher than the will.

Expected utility and inconsistent credences

Suppose that we have a utility function U and an inconsistent credence function P, and for simplicity let’s suppose that our utility function takes on only finitely many values. The standard way of calculating the expected utility of U with respect to P is to look at all the values U can take, multiply each by the credence that it takes that value, and add:

1. E(U)=∑_yyP(U = y).

Call this the Block Way or Lebesgue Sums.

Famously, doing this leads to Dutch Books if the credence function fails additivity. But there is another way to calculate the expected utility:

2. E(U)=∫₀^∞P(U > y)dy − ∫_−∞⁰P(U < y)dy.

Call this the Level Set Way, because sets of points in a space where some function like U is bigger or smaller than some value are known as level sets.

Here is a picture of the two ways:

On the Block Way, we broke up the sample space into chunks where the utility function is constant and calculated the contribution of each chunk using the inconsistent credence function, and then added. On the Level Set Way, we broke it up into narrow strips, and calculated the contribution of each strip, and then added.

It turns out that if the credence function P is at least monotone, so that P(A)≤P(B) if A ⊆ B, a condition strictly weaker than additivity, then an agent who maximizes utilities calculated the Level Set Way will not be Dutch Booked.

Here is another fact about the Level Set Way. Suppose two credence functions U₁ and U₂ are certain to be close to each other: |U₁ − U₂|≤ϵ everywhere. Then on the Block Way, their expected utilities may be quite far apart, even assuming monotonicity. On the other hand, on the Level Set Way, their expected utilities are guaranteed to be within ϵ, too. The difference between the two Ways can be quite radical. Suppose a coin is tossed, and the monotone inconsistent credences are:

• heads: 0.01

• tails: 0.01

• heads-or-tails: 1

• neither: 0

Suppose that U₁ says that you are paid a constant $100 no matter what happens. Both the Block Way and the Level Set Way agree that the expected utility is $100.
But now suppose that U₂ says you get paid $99 on heads and $101 on tails. Then the Block Way yields:

• E(U₂)=0.01 ⋅ 99 + 0.01 ⋅ 101 = 1

while the Level Set Way yields:

• E(U₂)=1 ⋅ 99 + 0.01 ⋅ 2 = 99.02

Thus, the Block Way makes the expected value of U₂ ridiculously small, and far from that of U₁, while the Level Set Way is still wrong—after all, the credences are stupid—but is much closer.

So, it makes sense to think of the Level Set Way as harm reduction for those agents whose credences are inconsistent but still monotone.

That said, many irrational agents will fail monotonicity.

Presentism and the Cross

1. It is important for Christian life that one unite one’s daily sacrifices with Christ’s sufferings on the cross.

2. Uniting one’s sufferings with something non-existent is not important for Christian life.

3. So, Christ’s sufferings on the cross are a part of reality.

4. So, presentism is false.

Velocity and teleportation

Suppose a rock is flying through the air northward, and God miraculously and instantaneously teleports the rock, without changing any of its intrinsic properties other than perhaps position, one meter to the west. Will the rock continue flying northward due to inertia?

If velocity is defined as the rate of change of position, then no. For the rate of change of position is now westward and the magnitude is one meter divided by zero seconds, i.e., infinite. So we cannot expect inertia to propel the rock northward any more. In fact, at this point physics would break down, since the motion of an object with infinite velocity cannot be predicted.

But if velocity (or perhaps momentum) is an intrinsic feature that is logically independent of position, and it is merely a law of physics that the rate of change of position equals the velocity, then even after the miraculous teleportation, the rock will have a northward velocity, and hence by inertia will continue moving northward.

I find the second option to be the more intuitive one. Here is an argument for it. In the ordinary course of physics, the causal impact of physical events at times prior to t₁ on physical events after t₁ is fully mediated by the physical state of things at t₁. Hence whether an object moves after time t₁ must depend on its state at t₁, and only indirectly on its state prior to t₁. But if velocity is the rate of change of position, then whether an object moves via inertia after t₁ would depend on the position of the object prior to t₁ as well as at t₁. So velocity is not the rate of change of position, but rather a quality that it makes sense to attribute to an object just in virtue of how it is at one time.

This would have the very interesting consequence that it is logically possible for an object to have non-zero velocity while not moving: God could just constantly prevent it from moving without changing its velocity.

Guessing and omniscience

Suppose that yesterday you guessed that today I’d freely mow the lawn, and today I did freely mow the lawn. Then, the correctness of your guess is a doxastic good you possessed.

(Note: If the future is open, so that there was no truth yesterday that today I’d mow the lawn, it’s a little tricky to say when you possessed it. For when you guessed, it wasn’t true that you possessed the doxastic good of guessing correctly. Rather, now that it has become the case that this doxastic good is attributable to you.)

Now no one can have a doxastic good that God lacks. Thus, God had to have at least guessed the same thing yesterday. And God has no doxastic bads. So, God never gets anything wrong. But the only plausible way it can be true that

1. God always gets right the things we guess right, and

2. God never gets things wrong

is if God has comprehensive knowledge of the future.

The local five minute hypothesis, the Big Bang and creation

The local five minute hypothesis is that the earth, with everything on it, and the environment five light-minutes out from it, come into existence five minutes ago.

Let’s estimate the probability of getting something like a local five minute hypothesis by placing particles at random in the observable universe. Of course, in a continuous spacetime the probability of getting exactly the arrangement we have is zero or infinitesimal. But we only need to get things right to within a margin of error of a Planck distance for all practical purposes.

The volume of the observable universe is about 10⁸⁰ cubic meters. The Planck volume is about 10⁻¹⁰⁵ cubic meters. So, getting a single particle at random within a Planck volume of where it is has a probability of about 10⁻¹⁸⁵.

But, if we’re doing our back-of-envelope calculation in a non-quantum setting (i.e., with no uncertainty principle), we also need to set the velocity for the particles. Let’s make our margin of error be the equivalent of moving a Planck distance within ten minutes. So our margin of error for velocity in any direction will be about 10⁻³⁵ meters in 600 seconds, or about 10⁻³⁸ meters per second. Speeds range from 0 to the speed of light, or about 10⁸ meters per second, so the probability of getting each of the three components of the velocity right is about 10⁻⁴⁶, and since we have three directions right is something like 10⁻¹³⁸. The probability of getting both the position and velocity of a particle right is then 10^{−(185 + 138)} = 10⁻³²³. Yeah, that’s small. Also, there are about 100 different types of particles, and there are a few other determinables like spin, so let’s multiply that by about 10⁻³ to get 10⁻³²⁶.

The total mass of planetary stuff within around five light minutes of earth—namely, Earth, Mass and Venus—is around 10²⁵ kilograms. There are no more than about 10²⁵ atoms, and hence about 10²⁷ particles, per kilogram. So, we have 10⁵² particles we need to arrange within our volume.

We’re ready to finish the calculation. The probability of arranging these many particles with the right types and within our position and velocity margins of error is:

• (10⁻³²⁶)¹⁰⁵² ≈ 10^{−102.5 × 1052} ≈ 10⁻¹⁰⁵⁵.

Notice, interestingly, that most of the 55 comes from the number of particles we are dealing with. In fact, our calculations show that basically getting 10^N particles in the right configuration has, very roughly, a probability of around 10^{−10N + 3}.

So what? Well, Roger Penrose has estimated the probability of a universe with an initial entropy like ours at 10⁻¹⁰¹²³. So, now we have two hypotheses:

• A universe like ours came into existence with a Big Bang

• The localized five minute hypothesis.

If there is no intelligence behind the universes, and if probabilistic calculations are at all appropriate for things coming into existence ex nihilo, the above probability calculations seem about right, and the localized five minute hypothesis wins by a vast margin: 10⁻¹⁰⁵⁵ to 10⁻¹⁰¹²³ or, roughly, 10¹⁰¹²³ to 1. And if probabilistic calculations are not appropriate, then we cannot compare the hypotheses probabilistically, and lots of scepticism also follows. Hence, if there is no intelligence behind the universe, scepticism about everything more than five minutes ago and more than five light minutes from us follows.

1+1=3 or 2+2=4

On numerical-sameness-without-identity views, two entities that share their matter count as one when we are counting objects.

Here is a curious consequence. Suppose I have a statue of Plato made of bronze with the nose broken off and lost. I make up a batch of playdough, sculpt a nose out of it and stick it on. The statue of Plato survives the restoration, and a new thing has been added, a nose. But now notice that I have three things, counting by sameness:

• The statue of Plato

• The lump of bronze

• The lump of playdough.

Yet I only added one thing, the lump of playdough or the nose that is numerically the same (without being identical) as it. So, it seems, 1+1=3.

Now, it is perfectly normal to have cases where by adding one thing to another I create an extra thing. Thus, I could have a lump of bronze and a lump of playdough and they could come together to form a statue, with neither lump being a statue on its own. A new entity can be created by the conjoining of old entities. But that’s not what happens in the case of the statue of Plato. I haven’t created a new entity. The statue was already there at the outset. And I added one thing.

Maybe, though, what should be said is this: I did create a new thing, a lump of bronze-and-playdough. This thing didn’t exist before. It is now numerically the same as the statue of Plato, which isn’t new, but it is still itself a new thing. I am sceptical, however, whether the lump of bronze-and-playdough deserves a place in our ontology. We have unification qua statue, but qua lump it’s a mere heap.

Suppose we do allow, however, that I created a lump of bronze-and-playdough. Then we get another strange consequence. After the restoration, counting by sameness:

• There are two things that I created: the nose and the lump of bronze-and-playdough

• There are two things that I didn’t create: the statue of Plato and the lump of bronze.

But there are only three things. Which makes it sound like 2+2=3. That’s perhaps not quite fair, but it does seem strange.

Sameness without identity

Mike Rea’s numerical-sameness-without-identity solution to the problem of material constitution holds that the statue and the lump have numerical sameness but do not have identity. Rea explicitly says that numerical sameness implies sharing of all parts but not identity.

Does Rea here mean: sharing of all parts, proper or improper? It had better not be so. For improper parthood is transitive.

Proposition. If improper parthood is transitive and x and y share all their parts (proper and improper), then x = y.

Proof: But suppose that x and y share all parts. Then since x is a part of x, x is a part of y, and since y is a part of y, y is a part of x. Moreover, if x ≠ y, then x is a proper part of y and y is a proper part of x. Hence by transitivity, x would be a proper part of x, which is absurd, so we cannot have x ≠ y. □

So let’s assume charitably that Rea means the sharing of all proper parts. This is perhaps coherent, but it doesn’t allow Rea to preserve common sense in Tibbles/Tib cases. Suppose Tibbles the cat loses everything below the neck and becomes reduced to a head in a life support unit. Call the head “Head”. Then Head is a proper part of Tibbles. The two are not identical: the modal properties of heads and cats are different. (Cats can have normal tails; heads can’t.) This is precisely the kind of case where Rea’s sameness without identity mechanism should apply, so that Head and Tibbles are numerically the same without identity. But Tibbles has Head as a proper part and Head does not have Head as a proper part. But that means Tibbles and Head do not share all their proper parts.

Here may be what Rea should say: if x and y are numerically the same, then any part of the one is numerically the same as a part of the other. This does, however, have the cost that the sharing-of-parts condition now cannot be understood by someone who doesn’t already understand sameness without identity.

The present king of Ruritania

Suppose I am a quack and I announce:

1. These green pills cured the king of Ruritania of lung cancer.

I am lying, of course. The green pills never cured anyone of lung cancer.

But wait. To lie, I have to assert. To assert, there has to be a proposition that is being expressed. But (1) doesn’t express any proposition, because “Ruritania” is a non-referring name.

Maybe, then, (1) is not a lie, but something that is wrong for the same reason that a lie is wrong. For instance, on Jorge Garcia’s account, lying is wrong as it’s a betrayal of the trust solicited by the very same act. If so, then my pretend assertion of (1) might be wrong for exactly the same reason as a lie.

The point can also be made without relying on non-referring proper names. Suppose Jones has lied, cheated, stolen, plagiarized and defenestrated his friends, but reporting doesn’t make his character black enough for my purposes. So I say:

2. Dr. Jones has lied, cheated, stolen, plagiarized, defenestrated his enemies, and garobulated his friends.

This doesn’t express a proposition. But it’s just as bad as a lie.

Perdurance and particles

A perdurantist who believes that particles are fundamental will typically think that the truly fundamental physical entities are instantaneous particle-slices.

But particles are not spatially localized, unless we interpret quantum mechanics in a Bohmian way. They are fuzzily spread over space. So particle-slices have the weird property that they are precisely temporally located—by definition of a slice—but spatially fuzzily spread out. Of course, it is not too surprising if fundamental reality is strange, but maybe the strangeness here should make one suspicious.

There is a second problem. According to special relativity, there are infinitely many spacelike hyperplanes through spacetime at a given point z of spacetime, corresponding to the infinitely many inertial frames of reference. If particles are spatially localized, this isn’t a problem: all of these hyperplanes slice a particle that is located at z into the same slice-at-z. But if the particles are spatially fuzzy, we have different slices corresponding to different hyperplanes. Any one family of slices seems sufficient to ground the properties of the full particle, but there are many families, so we have grounding overdetermination of a sort that seems to be evidence against the hypothesis that the slices are fundamental. (Compare Schaffer’s tiling requirement on the fundamental objects.)

A perdurantist who thinks the fundamental physical entities are fields has a similar problem.

A supersubstantialist perdurantist, who thinks that the fundamental entities are points of spacetime, doesn’t run into this problem. But that’s a really, really radical view.

An “Aristotelian” perdurantist who thinks that particles (or macroscopic entities) are ontologically prior to their slices also doesn’t have this problem.

Book in Progress: Norms, Natures and God

I have begun work with a working title of Norms, Natures and God, which should be a book on how positing Aristotelian natures solves problems in ethics (normative and meta), epistemology, semantics, metaphysics and mind, but also how, especially after Darwin, to be an intellectually satisfied Aristotelian one must be a theist. The central ideas for this were in my Wilde Lectures.

There is a github repository for the project with a PDF that will slowly grow (as of this post, it only has a table of contents) as I write. I welcome comments: the best way to submit them is to click on "Issues" and just open a bug report. :-)

The repository will disappear once the text is ready for submission to a publisher.

Perdurance and slices

One of the main problems with perdurance is thought to be that it makes intrinsic properties be primarily properties of slices, and only derivatively of the four-dimensional whole.

The most worrisome case of this problem has to do with mental properties. For if our slices have the mental properties primarily, and we only have them derivatively, then that leads to a sceptical problem (how do I know I am a whole and not a slice?) and besides violates the intuition that we have our mental properties primarily.

But someone who accepts a perdurantist ontology and accepts the idea that we are four-dimensional wholes does not have to say that intrinsic properties are primarily had by slices. For a property that involves a relation to one’s parts can still be intrinsic (having one’s parts is surely intrinsic!). Now instead of saying that, say, Bob has temporary property P at time t in virtue of his slice B_t at t having P, we can say that Bob has P in relation to B_t. This is very similar to how relationalist endurantists say that we have our temporary properties in relation to times, except that times are normally thought of as extrinsic to the object, while the slices are parts of the objects.

In fact, this helps save some intuitions of intrinsicness. For instance, it seems to be an intrinsic property of me that my heart is beating. But if t is now and A_t is my slice now, then A_t does not seem to intrinsically have the property of heart-beat. It seems that heart-beat is a dynamical property dependent not just on the state of the object at one time but also at nearby times. Thus, if we want to attribute heart-beat to A_t primarily, then heart-beat will not be intrinsic, as it will depend on A_t as well as slices A_t′ for t′ near t. But if we see my present heart-beat as a property of the four-dimensional worm, a property the worm has in relation to A_t (as well as neighboring times), then heart-beat can be an intrinsic property—and it can be had primarily by me, not my slices.

It is plausible that mental properties are dynamical as well: that one cannot tell just from the intrinsic properties of a three-dimensional slice whether thought is happening. (This is pretty much certain given materialism, but I think is plausible even on dualism.) So, again, mental properties aren’t going to be intrinsic properties of slices. But they can be primarily the intrinsic properties of four-dimensional persons, had in relation to their slices.

Persistence and internal times

Here are some desiderata for a view of the persistence of objects:

1. Ordinary objects can change with respect to intrinsic properties.

2. Ordinary objects are the primary bearers of some of the changeable intrinsic properties.

3. Ordinary objects are literally present at multiple times.

Endurantism is usually allied with some sort of view on which temporary properties are had in relation to times, and hence the temporary properties are relational and not intrinsic. Perdurantism violates 2: it is the stages, not the ordinary objects, that are the primary bearers of the temporary intrinsics. And no primary bearer of a property can change with respect to it. Exdurantism violates 3: ordinary objects only exist at a single time.

Here is a view that yields all three desiderata. Objects have internal times, and these internal times are literally parts of the objects. Changeable intrinsic properties are relational to the internal times: an object is, say, straight at internal time t₁ and bent at internal time t₂.

Let’s go through the desiderata. The internal times are parts of the object, and a property obtaining in virtue of relations between one’s own parts can still be intrinsic. Shape, for instance, might be had in virtue of the spatial relationships between the parts of an object—and yet this does not rule out shape being intrinsic (indeed, for David Lewis it’s paradigmatically intrinsic). Similarly, consciousness properties in a split brain might be had relationally to a brain hemisphere, but are still intrinsic since brain hemispheres are parts of the patient. Thus we can have (1).

Moreover, while parts—namely, internal times—are used to account for change, the parts are not the primary bearers of the changeable intrinsic properties. The changeable intrinsic properties to be relational between the ordinary object and the times, but that does nothing to rule out the possibility that some of these properties are primarily had by the object as a whole.

Ordinary objects can be literally present at multiple times. One can ensure this either in an endurantist way, so that the ordinary objects are multiply temporally located 3D objects, or in a four-dimensionalist way, so that the ordinary objects are 4D. Note that the endurantist version may require the ordinary object to have parts—namely, the internal times—that do not themselves endure but that only exist for an external instant. But there is no problem with an enduring object having a short-lived part.

There is another variant of the view. The internal times could be taken to be abstract objects instead of parts of the ordinary object. Arguably, a property that is had in virtue of a relation to an abstract object is not thereby objectionably extrinsic. If it were, then strong Platonists would all count as denying the existence of intrinsic properties.