The Grotius view of lying

Grotius had a weird view: it is never permissible to lie, but “for purposes of natural law”, only assertions to people who had a right to the truth were lies. Nazis at the door, he would have said, have no right to the truth, so one isn’t lying when one asserts known falsehoods to them. This view has always seemed clearly wrong.

But I just realized that there is actually an interesting argument for a very similar view. Start with these three principles:

1. Every lie is an assertion.

2. A defining feature of an assertion is that it is the sort of speech act that the sincerity norm (e.g., “Don’t say what you think is false!”) applies to.

3. No norm applies in contravention of unequivocal moral norms.

Premise (1) is clearly true. Premise (2) is part and parcel of normative accounts of assertion (there is room for variance on what the sincerity norm exactly is, but that variance will not affect our main argument).

Premise (3) is highly controversial. It is a generalization of Aquinas’ principle that immoral “laws” are not really laws. The general idea is that morality not only overrides other norms that contradict it, but as it were sucks all the power out of them. When one knows that ϕing is morally forbidden, responses like “But the law of the land requires it” or “I’d be breaking the rules of the game if I ϕed” make no sense. For there is no normative force against morality. Here are two reasons to accept premise (3). The first is the controversial claim that all norms of action are a species of moral norms. (Here is a theistic argument for this: Norms are appropriately action-guiding; the only thing that can appropriately guide our action is what the love of God requires (we are to love God with all our heart); but to be guided by the love of God and to be guided by morality is the same thing.) The second is that if there are norms other than moral norms, they are created by our normative powers, but it is not plausible that we have the normative power to create norms that stand against the norms of morality (that is, for instance, why immoral promises are null and void).

Then:

4. If the sincerity norm for a speech act ϕ contravenes unequivocal moral norms, the speech act is not an assertion. (By 2 and 3)

5. If the sincerity norm for a speech act ϕ contravenes unequivocal moral norms, the speech act is not a lie. (By 1 and 4)

Now here is one way to fill out the rest of the argument:

6. In Nazi at the door cases, we are morally required to say what we disbelieve (i.e., go against what the sincerity norm would require).

7. So, in Nazi at the door cases, saying what we disbelieve is not a lie. (By 5 an 6)

And that gives us a version of the Grotius view.

My own view is to flip the last two steps of the argument, replacing 6 and 7 with:

8. In Nazi at the door cases, saying what we disbelieve is a lie.

9. So, in Nazi at the door cases it is still false that we are morally required to say what we disbelieve. (By 5 and 8)

10. In Nazi at the door cases, if it is morally permissible to say what we disbelieve, it is morally required.

11. So, in Nazi at the door cases, it is not morally permissible to say what we disbelieve. (By 9 and 10)

But a lot of people balk at 9. And they then have reason to accept the Grotius-like thesis 7.

So, all in all, if one accepts the normative view of assertion and one accepts the contravention principle 3, one has a choice between Kantian absolutism about lying and a Grotius-like view.

An odd argument for an omniscient being

Here’s a funny logically valid argument:

1. The analytic/synthetic distinction between truths is the same as the a priori / a posteriori distinction.

2. The analytic/synthetic distinction between truths makes sense.

3. If 1 and 2, then every truth is knowable.

4. So, every truth is knowable. (1–3)

5. If every truth is knowable, then every truth is known.

6. So, every truth is known. (4–5)

7. If every truth is known, there is an omniscient being.

8. So, there is an omniscient being. (6–7)

I won’t argue for 1 and 2: those are big-picture substantive philosophical questions. I am sceptical of both claims.

The argument for 3 is this. If the analytic/synthetic distinction makes sense, then the two concepts are exclusive and exhaustive among truths: a truth is synthetic just in case it’s not analytic. So, every truth is analytic or synthetic. But if 1 is true and the analytic/synthetic distinction makes sense, then it follows that every truth is a priori or a posteriori. But these phrases are short for a priori knowable and a posteriori knowable. Thus, if 1 is true and the analytic/synthetic distinction makes sense, then every truth is knowable.

The argument for 5 is the famous knowability paradox: If p is an unknown truth, then that p is an unknown truth is a truth that cannot be known (for if someone know that p is an unknown truth, then they would thereby know that p is a truth, and then it wouldn’t be an unknown truth, and no one can’t know what isn’t so).

One argument for 7 is an Ockham’s Razor argument: it is more plausible to think there is one being that knows all things than that the knowledge is scattered among many. A sketch of a deductive argument for 5 that skirts over some important technical issues is this: if you know a conjunction, you know all the conjuncts; let p be the conjunction of all truths; if every truth is known, then p is known; and someone who knows p knows all.

Pain and water

One way for physicalists to handle the apparent differences between mental and physical properties is to liken the difference to that between water and H₂O. It is a surprising a posteriori fact that water is H₂O. Similarly, it is a surprising a posteriori fact that pain is physical state ϕ₁₃₅ (say).

Now, a posteriori facts are facts that are knowable by observation. But it is not clear that the proposition that pain is physical state ϕ₁₃₅ is knowable by observation.

Here is why. There are two main candidates for what kind of a state ϕ₁₃₅ could be: a brain state or a functional state. The choice between these two candidates depends on how strongly one feels about multiple realizability of mental states. If one is willing to say that only beings with brains like ours—say, complex vertebrates—feel pain, one might identify ϕ₁₃₅ with a brain state. If one has a strong intuition that beings with other computational systems anatomically different from those of complex vertebrates—cephalopods, aliens, and robots—could have consciousness, one will opt for identifying ϕ₁₃₅ as a functional state.

But in fact, assuming pain is a physical state, there is a broad spectrum of physical state candidates for identifying pain with, depending on how far we abstract from the actual physical realizers of our pains while keeping fixed the broad outlines of functionality (signaling damage and leading to aversive behavior). If we abstract very little, only brain states found in humans—and perhaps not all humans—will be pain. If we abstract a bit more, but still insist on anatomical correspondence, then brain states found in other complex vertebrates will be pain. If we drop the insistence on anatomical correspondence but do not depart too far, we may include amongst the subjects of pain other DNA-based organisms such as cephalopods. Further abstraction will let in living organisms with other chemical bases, and yet further abstraction will let in robots. And even when talking of the fairly pure functionalism applicable to robots, we will have serious questions about how far to abstract concepts such as “damage” and “aversive behavior”.

The question of where in this spectrum of more and more general physical states we find the state that is identical with pain does not appear to be a question to be settled by observation. By internal observation, we only see our own pain. By external observation, however, we cannot tell where in the spectrum of more and more general (perhaps along multiple dimensions) physical states pain is present, without begging the question (e.g., by assuming from the outset that certain behaviors show
the presence of pain, which basically forces our hand to a functionalism centered on those behavior).

Objection 1: An experimenter could replace the brain structures responsible for pain in her own brain by structures that are further from human ones, and observe whether she can still feel pain. Where the feeling of pain stops, there we have abstracted too far.

Response: There are serious problems with this experimental approach. First, mere replacement of brain pain centers will not allow one to test hypotheses on which what constitutes pain depends on the larger neural context. And replacement of the brain as a whole is unlikely to result in the experimenter surviving. Second, and perhaps more seriously, if replacements of the brain pain centers commit the same data to memory storage as brain pain centers do, after the experiment the agent will think that there was pain, even if there wasn’t any pain there, and if they have the same functional influence on vocal production as brain centers do, the agent will report pain, again even if there wasn’t any pain there.

Objection 2: We could know which physical state pain is identified with if God told us, and being told by God is a form of a posteriori knowledge.

Response: It seems likely that God’s knowledge of which physical states are pains, or of the fact that water is H₂O, would be a priori knowledge. God doesn’t have to do scientific research to know necessary truths.

Objection 3: We can weaken the analogy and say that just as the identity between water and H₂O is not a priori, so too the identity between pain and ϕ₁₃₅ is not a priori, without saying that both are a posteriori.

Response: This is probably the move I’d go for if I were a physicalist. But by weakening this analogy, one weakens the position that it defends. For it is now admitted that there is a disanalogy between water-H₂O and pain-ϕ₁₃₅. There is something rather different about the mental case.

The paradox of the Jolly Givers

Consider the Grim Reaper (GR) paradox. Fred’s alive at midnight. Only a GR can kill him. Each GR has an alarm with a wakeup time. When the alarm goes off, the GR looks to see if Fred’s alive, and if he is, the GR kills him. Otherwise, the GR does nothing. Suppose the alarm times of the GR’s are 12:30 am, 12:15 am, 12:07.5 am, …. Then Fred’s got to be dead, but no GR could have killed him. If, say, the 12:15 GR killed him, that means Fred was alive at 12:07.5, which means the 12:07.5 GR would have killed him.

A Hawthorne answer to the GR paradox is that the GRs together killed Fred, though no one of them did.

Here’s a simple variant that shows this can’t be true. You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world, nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its orange in the stocking. The alarm times are the same as in the previous story.

The analogous Hawthorne answer would have to be that the JGs together put an orange in the stocking. But then one of the JGs would need to be missing his orange. But no one of the JGs is missing his orange, since no one of them took it out of his pocket. So, the orange would have had to come out of nowhere.

And, to paraphrase a very clever recent comment, if it came out of nowhere, why would it be an orange, rather than, say, a pear?

I think the JG paradox also suggests an interesting link between the principle that nothing comes from nothing and the rejection of supertasks.

Explanation and understanding

In the 1960s, it dawned on philosophers of science that:

1. Other things being equal, low-probability explanation confers equally good understanding as high-probability explanation.

If I have a quantum coin that has a probability 0.4 of heads and 0.6 of tails, and it yields heads, I understand why it yielded heads no less well than I would have had it yielded tails—the number is simply different.

On the other hand, the following thesis (which for years I’ve conceded to opponents to low-probability explanations):

2. Other things being equal, low-probability explanations are less good than high-probability ones.

Finally, add this plausible comparative thesis:

3. What makes an explanation good is how much understanding it confers (or at least would confer were it true)

which plausibly fits with the maxim that I’ve often been happy to concede that the job of an explanation is to provide understanding.

But (1)–(3) cannot all be true. Something must go. If (2) goes, then Inference to Best Explanation goes as well (I learned this from Yunus Prasetya’s very recent work on IBE and scientific explanation). I don’t want that (unlike Prasetya). And (1) seems right to me, and it also seems important to defending the Principle of Sufficient Reason in stochastic contexts.

Reluctantly, I conclude that (3) needs to go. And this means that I’ve overestimated the connection between explanation and understanding.

A simpler formulation of the paradox of short pains

On reflection, my paradox of short pains can be simplified. Start with:

1. Whether I have had a pain does not depend on the future.

2. It is impossible for me to have a pain that lasts less than a picosecond.

3. I once started to be in pain for the first time in my life.

Now imagine that the first pain in my life just started half a picosecond ago. Then anybody who has the power to annihilate me in the next quarter (say) picosecond has the power to make it be the case that I had not yet had a pain, since if they so annihilate me, then I won’t have had a pain by (2). But whether someone will annihilate me in the next quarter second is a fact about the future, so whether I have had a pain depends on the future.

To put this in Ockhamist terminology, if we accept (2), then we have to accept that facts about whether one has had a pain can be soft facts.

I like the idea of denying (1), though I think this may make presentists (and others) uncomfortable.

I also like the idea of saying that the argument equivocates between phenomenal and physical time. The duration of pain is a phenomenal duration that does not correspond in a precise way to a physical duration.

Preprint: Conditional, Regular Hyperreal and Regular Qualitative Probabilities Invariant Under Symmetries

Abstract: Classical countably additive real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value---namely, zero---to cases that are impossible as well as to cases that are possible. There are three non-classical approaches to probability that can avoid this drawback: full conditional probabilities, qualitative probabilities and hyperreal probabilities. These approaches have been criticized for failing to preserve intuitive symmetries that can easily be preserved by the classical probability framework, but there has not been a systematic study of the conditions under which these symmetries can and cannot be preserved. This paper fills that gap by giving complete characterizations under which symmetries understood in a certain "strong" way can be preserved by these non-classical probabilities, as well as by offering some results to make it plausible that the strong notion of symmetry here may be the right one. Philosophical implications are briefly discussed, but the main purpose of the paper is to offer technical results to inform more sophisticated further philosophical discussion.

Preprint here.

More on the problem of short pains

Consider these two very plausible theses:

1. Whether I have pain at time t does not depend on any future facts.

2. There is a length of time δt such that you cannot have a pain lasting no more than δt, but you can have a pain lasting 4δt.

Now imagine that I feel a pain that lasts from t₀ to t₂ = t₀ + 4δt. Let t₁ = t₀ + (1/2)δt. Suppose it is now t₁. Then I am in pain. But now I claim that this counterfactual is true:

3. Were it to be the case that I was to be annihilated at t₀ + (3/4)δt, I wouldn’t have felt any pain now.

Why? For if I were so annihilated, my pain could only have lasted (3/4)δt, which is too short for a pain by (2).

But by (3), whether I feel pain now depends on whether I will shortly be annihilated, contrary to (1).

Hence, we need to reject one of (1) and (2).

It is hard to reject (2). After all, imagine the sequence of times: a second, a quarter second, a sixteenth of a second, …, 2⁻⁴⁰ seconds. Clearly I can feel a pain that lasts a second. The last of these is less than a picosecond, and clearly I can’t feel a pain that short. So somewhere in that sequence I must reach a δt which is too short for a pain, but where 4δt isn’t too short for a pain.

I think denying (1) isn’t as bad as it may seem.

But perhaps a less counterintuitive move is to deny that phenomenal times and physical times are as closely correlated as they intuitively seem. Here is a possible story. Phenomenal times are discrete points while physical times are continuous (or discrete on a much finer timescale). You can feel a pain that is located at exactly one phenomenal time. The spacing between the phenomenal times corresponds to a fairly large (and non-uniform) spacing between physical times, say of the order of magnitude of a millisecond. So, you can feel a pain that is there one millisecond and gone the next, but you may feel it at exactly one point of phenomenal time.

As far as I can tell, it is not possible to run the annihilation argument while keeping careful track of the continuous physical and discrete phenomenal timelines. I guess this is a way of rejecting (2) by making it not make sense.

Here is a third way out of the argument. Imagine that what it is to have a pain at t is to have had some constitutive physical or spiritual process P have lasted some threshold period of time δt. On this view, before P lasted over a period of δt, there was no pain: pain only starts once P has lasted δt. We might now suppose that δt is something like a millisecond. Then it is possible to have a pain that lasts only a picosecond: for that, all we need is the underlying process P to have lasted δt plus a picosecond—and only the last picosecond of that process would have constituted a pain. But we no longer need to make the implausible claim that we can be aware of picosecond-scale stuff. For in paining, our awareness is the awareness of the underlying process P, and that process always needs to have taken something in the millisecond range for it to constitute a paining.

This way out of the argument also has the consequence that it is not possible to have a pain before completing the first δt of one’s existence. Pain is not a momentary property.

The third way out will not appeal to dualists who think phenomenal states are fundamental.

Promising to try

Here’s a natural thing to say. The sentence

1. “I promise to try to ϕ”

should be analyzed as straightforwardly an instance of the schema:

2. “I promise to ψ”

for an action ψ, in the special case where ψing is the action of trying to ϕ.

I am now fairly convinced that this is wrong: that (1) is not a mere special case of (2).

Here are two cases to make one more friendly to this.

• I promised you to try to call you tomorrow. Come tomorrow, I called you accidentally, i.e., without trying to. If the content of my promise was literally to try to call you, then I haven’t fulfilled that, and I have reason to call you, this time intentionally. But that’s silly.

• I promised to try to email you a paper tonight. Come tonight, I try, but my email software puts up the dialog: “Your email did not go through due to a temporary problem. The problemm is now resolved. Should I send it? Yes/No.” I press “No” on the grounds that my promise has been fulfilled: I already tried. That’s vicious, surely.

There are easy ways out of the two cases that are compatible with the straightforward analysis of promising-to-try. But the cases are nonetheless suggestive of the fact that “I promise to try to ϕ” should not be read too literally.

My promise to you to ψ typically accomplishes three things:

• It creates a reason for me to ψ

• It tells you that I will ψ

• It creates an obligation of apology or compensation if I do not ψ (regardless of whether I am culpable).

In particular, because it tells you that I will ψ, it normally creates an expectation in you that I will ψ, which is apt to lead to your organizing your life around my ψing.

I suggest that a promise to try to ϕ instead accomplishes these three things:

• It creates a reason for me to ϕ

• It tells you that I will make a reasonable effort to ϕ

• It creates an obligation of apology or compensation if I do not make a reasonable effort to ϕ.

In other words, I think that my promising to try to ϕ gives rise to exactly the same reason in me as promising to ϕ would. However, it holds back the assurance that I will ϕ, replacing it with an assurance that I will try. Now whether you expect me to ϕ will depend on your judgment of how likely my attempts are to succeed. And by using the weaker wording, I am typically implicating to you an uncertainty about success which should give you some evidence of my unreliability. Finally, my promising to try to ϕ weakens the duties of apology or compensation to only apply when I didn’t make a reasonable effort.

In other words, to promise and to promise-to-try are two different kinds of speech acts, and it is obviously useful to have both.

Synchronization and the unity of consciousness

The problem of the unity of consciousness for materialists is what makes activity in different areas of the physical mind come together into a single phenomenally unified state rather than multiple disconnected phenomenal states. If my auditory center active in the perception of a middle C and my visual center is active in the perception of red, what makes it be the case that there is a single entity that both hears a middle C and sees red?

We can imagine a solution to this problem in a computer. Let’s say that one part of the computer has and representation of red in one part (of the right sort for consciousness) and a representation of middle C in another part. We could unify the two by means of a periodic synchronizing clock signal sent to all the parts of the computer. And we could then say that what it is for the computer to perceive red and middle C at the same time is for an electrical signal originating in the same tick of the clock to reach a part that is representing red (in the way needed for consciousness) and to reach a part that is representing middle C.

On this view, there is no separate consciousness of red (say), because the conscious state is constituted not just by the representation of red (say) in the computer’s “visual system”, but by everything that is reached by the signals emanating from the clock tick. And that includes the representation of middle C in the “auditory system”.

The unification of consciousness, then, would be the product of the synchronization system, which of course could be more complex than just a clock signal.

This line of thought shows that in principle the problem of the unity of consciousness is soluble for materialists if the problem of consciousness is (which I doubt). This will, of course, only be a Pyrrhic victory if it turns out that no similar pervasive synchronization system is found in the brain. The neuroscience literature talks of synchronization in the brain. Whether that synchronization is sufficient for solving the unity problem may be an empirical question.

The above line of thought also strongly suggests that if materialism is true, then our internal phenomenal timeline is not the same as objective physical time, but rather is constructed out of the synchronization processes. It need not be the case for this that the representation of red and the representation of middle C happen at the same physical time. A part further from the clock will receive the synchronizing signal later than a part closer to the clock, and so the synchronization process may make two events that are not simultaneous in physical time be simultaneous in computer time. I suspect that a similar divide between mental time and physical time is true even if dualism is (as I think) true, but for other reasons.

Bennett's positive and negative instrumentality

Bennett offers this account of positive vs. negative instrumentality. If the volume of the space of possible bodily movements occupied by doing A is greater than that occupied by doing not-A, then doing A is a negative instrumentality; if it is less, then it is positive. Thus, raising one’s hand is positive: one can, for instance, raise, lower or keep one’s hand level, and raising occupies less volume of movement space than not-raising.

Here’s a curious consequence. Let M be the maximum speed at which I can move. Let A be moving at a velocity of magnitude greater than half of M. Then A occupies more of the space of possible bodily movements than not-A and hence counts as negative by Bennett’s criteria.

Why? Well, velocity is a vector: it has a magnitude and a dimension. The relevant action space (assuming the movement is two-dimensional—we can’t fly) is a disc of radius M. The subset occupied by not-A is the a (closed) disc of radius M/2. The area (the two-dimensional analogue of volume) occupied by not-A is (1/4)πM². The area of the whole movement space is πM². The area occupied by A is thus πM² − (1/4)πM² = (3/4)πM². Thus, A occupies three times the area occupied by not-A. Hence, not-A is a positive action and A is a negative action.

This seems quite wrong.

Postdoc at Baylor

“Baylor University seeks application for a postdoctoral fellowship for up to three years from those working in virtue ethics, moral psychology, or social philosophy. Suitable candidates will have a background in psychology or neuroscience that equips them to participate in data collection, qualitative analysis, study design and intervention development with our team of philosophers, psychologists, and practitioners.”

Details here.

Microphysics and philosophy of mind

Much (but not all) contemporary philosophy of mind is written as if microphysics were fundamental physics. But as far as I know, only on those interpretations of quantum mechanics that disallow indeterminacy as to the number of particles can microphysics be fundamental physics. The most prominent such interpretation is Bohmianism. On most other interpretations, the most we can say about the number of particles is that we are in a superposition between states with different numbers of particles. But reality has to have determinate numbers of fundamental entities. The picture of reality we get from both relativity theory and mainstream interpretations of quantum mechanics other than Bohmianism and its close cousins is that fundamental physical reality consists of global entities such as the spacetime manifold or the wavefunction of the universe rather than microscopic entities like particles. (I am attracted to a non-mainstream interpretation on which the fundamental physical entities may include mid-sized things like dogs and trees.)

Sometimes, pretending microphysics is fundamental physics is excusable. For certain discussions, it doesn’t matter what the fundamental physics is—the arguments work equally well for global and local fundamental entities. In other cases, all that matters is relative fundamentality. Thus, facts about chemistry might be held to be more fundamental relative to biology, and facts about microphysics might be fundamental relative to chemistry, even if the microphysics facts themselves are not fundamental simpliciter, being reducible, say, to facts about global fields.

But even when the arguments do not formally rely on fundamental physics being microphysics, it is risky in a field so reliant on intuition to let one’s intuitions be guided by acting as if fundamental physics were microphysics. And doing this is likely to mis-focus one’s critical attention, say focusing one more on the puzzle of why the functioning of various neurons produces a unified consciousness than on the puzzle of how the functioning of a handful of global entities results in the existence of billions of minded persons.

Bostock v. Clayton County

In Bostock, the Supreme Court ruled that hiring discrimination against a gay person is discrimination on the basis of sex, and hence forbidden, because one wouldn’t refuse to, say, hire a woman who is attracted to men, and hence to refuse to hire a man who has the same “trait”, namely being attracted to men, is discrimination on the grounds of his sex.

Here is a clear counterexample to this line of reasoning. Consider an employer who refuses to hire a man who claims in a job application to be a woman on the grounds that this man is a liar. (Suppose this is a man in every socially accepted sense of the word: he is biologically male, he socially identifies as a man in every context other than this interview, etc.) Such an employer would not refuse to hire a woman with the same “trait”, namely claiming to be a woman. Hence by the Bostock reasoning, the employer discriminates on the basis of sex. But this is absurd: the basis for the discrimination is not the sex of the prospective employee, but lying about one’s sex. Similarly, discrimination against a white person who claims to be African American on the grounds of a mismatch between their claims and reality is not discrimination against white people.

In other words, the basis for the discrimination is not the sex of the candidate but the relationships between the candidate’s actual sex and the candidate’s claimed sex.

And logically speaking, this is all very much like the gay case, where the basis for the discrimination is not the sex of the candidate but the relationship between the candidate’s sex and the sex of the persons the candidate is attracted to.

I am not claiming that it is morally wrong to be attracted to persons of the same sex in the way in which it is wrong to lie (or in any any other way, for that matter). Nor am I claiming that it is reasonable or legal for an employer to discriminate on the basis of such attraction. All I am claiming is that such discrimination is not discrimination on the basis of the candidate’s sex.

Objection: There is an important difference between the trait of being attracted to men and the trait of claiming to be a man. Being attracted to men is essentially the same trait whether it is found in a man or a woman, while claiming to be a man is radically different when it is found in a man and in a woman, since it is truth-telling in the one case and lying in the other.

Response: This response would require the court to settle the question whether indeed the trait of being attracted to men is basically the same trait when found in men and when found in women, in a way in which the trait of claiming to be a man is not the same trait when found in men and when found in women. That is perhaps the real philosophical question here, and it is presumably precisely what the employer in question would dispute. The court cites the example of how discriminating on the grounds of interracial marriage is racial discrimination. Now, here I would say that the trait of marrying a person of race R is the same trait whether found in a person of race R or not. But clarifying exactly what it means to be basically the same trait is very difficult.

Disclaimer: I am no lawyer or legal scholar, just a philosopher with an eye for counterexamples.

Weak invariance of full conditional probabilities

In two papers (here and here), I explored two different concepts of symmetry for conditional probabilities. The concept of strong invariance says that P(gA|B)=P(A|B) for a symmetry g as long as A and gA are subsets of B. The concept of weak invariance says that P(gA|gB)=P(A|B) for a symmetry g. In some special cases, the weak concept implies the strong concept.

Anyway, here’s an interesting thing: the weak concept does not capture our symmetry intuitions. Take perhaps the simplest case, a lottery on the set of integers Z, and say that the symmetries are shifts. It turns out that there is a weakly shift-invariant full conditional probability P such that:

1. P({m}|{m, n}) = P({n}|{m, n}) (singleton fairness)

2. P(A|A ∪ B)=0 and P(B|A ∪ B)=1 whenever B has infinitely many positive integers and A has finitely many positive integers.

Condition (2) implies that it is more likely that the winning ticket is a power of two than that that is a negative integer. So weak shift invariance is very far from strong invariance.

(And in fact one can have strong invariance for the lottery on Z if one wants. One can even have have strong invariance under shifts and reflections if one wants.)

The proof is a modification of West's proof of a result for qualitative probabilities.