Another version of the first-sinner argument against Calvinism

1. (Premise) Any circumstances that are sufficient to determine a person with a character free of moral failings to do wrong are exculpatory for such a person.
2. (Premise) There was a first wrongdoing and it was not done in exculpatory circumstances.
3. So, either the first wrongdoer was not determined by circumstances and character, or the first wrongdoer had antecedent moral failings. (1 and 2)
4. (Premise) the first sinner did not have antecedent moral failings.
5. So, the first wrongdoer was not determined by circumstances and character. (1 and 4)

Premise (2) seems to be a part of the standard Christian picture. Nobody thinks Satan first sinned in circumstances that are exculpatory. Premise (4) follows from the fact that moral failings are evils, and evils came from sin (in the full sense of a wrongdoing the agent is responsible for—"formal" sin in Catholic terminology).

That leaves (1). But consider this line of thought. Suppose I was tortured and under torture I turned in my friends. Am I responsible or has the torture taken away my responsibility? Here is a test. I imagine whether a person free of moral failings would have been determined to do the same under torture of this intensity. If so, then the torture is of sufficient intensity to be exculpatory for me, and presumably likewise for her. (It doesn't quite follow that I am exculpated. For I could still be responsible for my sin in exculpatory circumstances, if my action is overdetermined by the exculpatory circumstances and something I am responsible for.)

Special relativity and the A-theory

I've been thinking about (Special) Relativity Theory and the A-theory of time. The A-theory of time requires an absolute simultaneity. Relativity theory seems to deny an absolute simultaneity.

But need it? Why not instead say this? Relativity theory never even talks about simultaneity. It talks about a different relation: simultaneity relative to a frame. Likewise, relativity theory never talks about spatial or temporal distances. It talks about spatial or temporal distances relative to a frame. These are different concepts: simultaneity, spatial distance and temporal distance are binary relations. Simultaneity relative to a frame, spatial distance relative to a frame and temporal distnace relative to a frame are ternary relations (the frame is a relatum). All the stuff Relativity says about simultaneity relative to a frame, spatial distance relative to a frame and temporal distnace relative to a frame may very well be true. But the A-theorist need not worry at all about this, because that's not what she is talking about.

But wouldn't it follow, then, that contrary to relativity theory there is a privileged reference frame? There are two questions here: (a) Does it follow that there is a privileged reference frame? (b) Would this be contrary to relativity theory? And neither question has a clearly positive answer.

The standard answer to (a) is this. If we have absolute simultaneity and a Minkowski spacetime, then we can define a privileged frame as the inertial frame whose simultaneity relation matches the absolute simultaneity relation. But nothing that was said so far guarantees that any inertial frame has a simultaneity relation that matches our absolute simultaneity relation. Suppose, for instance, that there are two points, a and b, that are absolutely simultaneous but where b is in the forward light-cone of a. Then not only will there be no inertial frame according to which a and b are simultaneous, but there won't be any foliation by spacelike hypersurfaces (which I guess could be our General Relativistic version of a reference frame) according to which they are simultaneous. Granted, that would be weird. But so far nothing has ruled this out. To get a positive answer to (a) we would need a posit that bridges between metaphysics and physics, namely that the absolute simultaneity relation agrees with the simultaneity relation of some frame.

What about (b)? Is it contrary to Relativity for there to be a privileged reference frame? It surely depends on the way in which the frame is privileged. Suppose, for instance, that earth is the only inhabited planet in the universe. Then the reference frame of the earth is privileged as the reference frame of the reference frame of the only inhabited planet in the universe. While earth may not be the only inhabited planet in the universe, it is no business of Relativity Theory to decide that question. So it had better not be contrary to Relativity for one reference frame to be privileged in this way.

Now, it is essential to Einstein's project that no reference frame be privileged with respect to the fundamental laws of nature. But I don't see that the A-theorist need hold anything that implies some reference frame is privileged in this way. The fundamental laws of motion, perhaps, talk only about what moves relative to reference frames, and say nothing about what moves simpliciter.

If the answer to (a) is positive, then presumably our A-theorist should hold that there is no metaphysically privileged reference frame. But Einstein's Principle of Relativity appears to be a (second-order) law of physics rather than a law of metaphysics. Why should we read it as denying a metaphysically privileged frame? Granted, Einstein so read it. But did he have reason to assert such a strengthened Principle?

All that said, while there is no incoherence between the A-theory and Relativity Theory, I think there is a plausibilistic argument from Relativity against the A-theory. On any sensible A-theory, the events that appear commonsensically simultaneous are at least approximately simultaneous, and the events that appear commonsensically far from simultaneous are not simultaneous. Moreover, our normal commonsensical measurements of distance (with meter sticks) and time (with clocks) had better approximately match up with the correct absolute distance and absolute time measurements. Now, these ordinary life judgments and measurements match up with those provided by our Relativistic science, relative to some reasonable reference frame. Such a match would be a surprising and vast coincidence if there were no deeper connection between absolute simultaneity, distance and time and simultaneity, distance and time according to a frame.

So, it seems that if we are to have a sensible A-theory, we will want to say that there is some frame whose simultaneity relation at least approximately matches absolute simultaneity, and likewise for temporal and spatial difference. But this sort of global coincidence is precisely the sort of thing for laws of nature to explain. So our sensible A-theorist should have fundamental laws of nature that entail that there is a frame whose simultaneity and distance relations approximately match the absolute ones. But these laws will violate Einstein's strictures against the laws not privileging a frame. For such laws privilege those frames whose relativized relations approximately match the absolute ones over those frames where this does not happen. And so such laws are improbable by Einstein's inductive argument for the Principle of Relativity.

There is also a perhaps more direct argument. On the supposition that there is an absolute simultaneity relation, it is surprising that frames whose simultaneity relation do and do not match are treated equally in the laws of motion. On the supposition that there is no such simultaneity relation, there is no surprise there. So, the laws of motion favor there not being an absolute simultaneity relation in a familiar Bayesian way.

Best-systems accounts of laws and second-order laws

On best-systems accounts of laws, p is a law provided that p is a theorem of the best system, where the best system is the one that optimizes informativeness and simplicity (and maybe fit, if we want probabilistic laws). But we also seem to have second-order laws, such as symmetry principles like Lorentz-invariance. Can we fit such second-order laws into a best-systems account?

Say that a symmetry S is a permutation of the collection of worlds. Given a proposition p, let p^S be a proposition that is true at a world w if and only if p is true at S(w). Say that a propositon is invariant under a class U of symmetries provided that p is true at w if and only if p^S true at w for all S in U. A nice way to formulate invariance principles is then to say that the conjunction of the laws is invariant under U, for some appropriate class of symmetries U.

On a best systems analysis, then, U-invariance holds in a world w if and only if the conjunction of the axioms of w's best system is U-invariant. But that's not enough for us to have a second-order invariance law. To have a second-order invariance law, we need that it be a theorem of the best system that the conjunction of the axioms of the best system is U-invariant.

Suppose w is some world with deterministic first-order Newtonian laws, an absolute distinction between rest and motion, and a second-order laws that says that the laws are Galileian-invariant. Is this likely to work out on a best-systems analysis? Let the first-order Newtonian axioms be N₁,...,N_n. These will be all part of the best-system. Now, N₁,...,N_n do not entail that the laws are Galileian-invariant. For there will be worlds where N₁,...,N_n are laws, but on a best-systems analysis there is a further law, L₀, where L₀ is not Galilean-invariant (perhaps all of the particles initially are at rest, and it that they are all at rest might then make it in as an axiom of the best system). Since the second-order claim that the laws are Galileian-invariant does not follow from N₁,...,N_n, it follows that for the second-order claim to be a law, the best system needs something more than N₁,...,N_n. But it seems possible that (a) it is a second-order law that the laws are Galileian-invariant and yet (b) the best system is just N₁,...,N_n as adding stuff that entails the second-order law to the axioms does not add enough predictive value beyond that N₁,...,N_n already have to justify the loss of simplicity.

Symmetry principles

Here is an interesting inductive argument about the restricted Principle of Relativity in Einstein's little popular book on relativity:

But that a principle of such broad generality should hold with such exactness in one domain of phenomena [mechanics], and yet should be invalid for another [electrodynamics], is a priori not very probable.

This is induction across laws: from the laws of mechanics having a property, we infer that the laws of electrodynamics have it as well. If induction is within a natural kind, this induction would require there to be the natural kind law.

One can take the Principle of Relativity to be a second-order law (cf. Earman). That's not the only way to think about it. We could, for instance, imagine the following metaphysics: We are realists about ordinary laws, like those of mechanics and thermodynamics, and we think that God has produced these laws. But in choosing which laws to produce, God followed certain "artistic principles". One of these artistic principles is the Principle of Relativity (and similar symmetry principles). I don't know what advantages of this over a hierarchical view on which there are higher order laws that constrain lower order ones. But it's worth thinking about.

Might spacetime be countably dense?

The standard view of spacetime is that it's a manifold based on the real numbers. The view that spacetime is discrete is also sometimes considered. Much more rarely, the idea that spacetime is a "manifold" based on a field like the hyperreals that extends the reals gets considered. But I have never heard anybody wonder whether spacetime might not have a dense but countable structure, like the structure of the rational or algebraic numbers.

I see no philosophical benefits to thinking spacetime might have such a structure. I wonder, though, whether we have any philosophical or empirical reason to think it doesn't. Perhaps there is a consistent first-order axiomatization of the mathematics that physicists actually use. If so, then the downward Loewenheim-Skolem theorem will model it within a countable structure. The algebraic numbers are unlikely to be rich enough, though.

Theistic intentional explanation

It is frequently objected that explanations in terms of the divine will are useless because they can "explain" everything.

One might equally object to quantum mechanics that it can explain any coherent macroscopic state, since all macroscopic outcomes have non-zero probability. This would be a poor objection. For while it could be that any macroscopic state can be given a statistical explanation, these explanations are not all equally good. The statistical explanation of why the cream spreads throughout the coffee is much better than the statistical explanation, invoking flukish probabilities, of why the cream coagulates into a regular nonagon.

Similarly, one might object to ordinary agential explanations. After all, just about anything within the power of humans can be given an agential explanation simply by positing some odd set of motivations. But some of these agential explanations will be better than others.

So it seems to be in the case of divine will explanations. Some are much more plausible than others. The explanation that there is life because God was so impressed with the value of life that he willed there to be life is much better than the explanation that there are platupuses in order to make us laugh. Why is the former a better explanation? One reason is that life is a greater value than laughter. Another is that while there being life is the only way to get the value of life, there being platypuses is not the only, and not the best (giving P.G. Wodehouse his sense of humor and writing ability is an even better way), way to get the value of laughter.

Moreover, notice that just about anything that occurs in a book could be explained by positing some motives or other on the part of the writer. But the explanation is better when these motives cohere well with the motives apparently exhibited elsewhere in the book. Nature throughout seems to exhibit a motive to give reality a mathematical structure and predictability. Explanations in terms of that motive are thus much better than explanations in terms of a one-off motive. In this way, good science, by discovering such structure, will actually help provide very good theistic explanations.

That said, the less good explanations are still explanations. In a typical case where something realizes a value (there may be some exceptions, say if there is some deontic prohibition in the vicinity), this realization of the value will give God a reason to make the case come about, and God will not ignore that value. He will act at least in part on it. So it is true that the platypus exists in part to make us laugh. But a much better explanation will be given by attending to the evolutionary processes that produced it.

Lying and being sincere at the same time

Sam is a politician speaking to a large multilingual audience, and is planning on offering them slogans that uniquely appeal to each language group. By coincidence, there is something, s, that he can say which is such that in Elbonian it means that he loves to hunt while in Baratarian s means that he is an avid cyclist. Sam actually loves to hunt but hates cycling, but knows that saying that he loves to hunt will tend to appeal to Elbonian speakers, whom he also tends to respect and does not wish to deceive, and that saying that he is an avid cyclist will tend to appeal to Baratarian speakers. So Sam utters s.

In so doing, Sam is sincerely asserting to Elbonian speakers that he loves to hunt and lying to Baratarian speakers that he is an avid cyclist. But there is Jane in the audience who is a completely bilingual Elbonian and Baratarian speaker. Did Sam lie to Jane?

One might say: It depends on how Jane understood him. But that's not right. To lie to someone does not require the interlocutor to understand one at all. If I write to you in a letter of recommendation that says that Jim is the worst student I have ever had, while in fact I know he is the best student I have ever had, and you misread the "worst" as "best" in the letter (after all, typically, in a letter of recommendation a superlative is positive, so you're primed to read it as "best"), nonetheless I lied to you, but unsuccessfully. Jane might have understood s in Elbonian, or in Baratarian, or just been confused by s. But nonetheless it seems that Sam both lied and spoke sincerely to Jane. He lied to Jane qua Baratarian speaker, since he asserted to all Baratarian speakers that he was an avid cyclist, and he spoke sincerely to all Elbonian speakers, including Jane, that he loved to hunt.

But this is odd. And it also means that it is difficult to make the token the unit of meaning. And that's a problem for nominalists of the Goodman and Quine variety.

Functionalism, causal theory of content and introspection

We should read functionalism as denying that mental properties like being in pain are neural properties. Rather, there are neural properties that realize mental properties. Thus, the property P of being in pain is the property of being an x that has a property N such that F_P(x,N), where F_P(x,N) is a predicate that says that N is a property exemplified by x that plays the pain role in x. The property N is a realizer of the pain property P in x. The materialist functionalist then says that the realizers of our mental properties are all neural properties.

The causal theory of content, in a functionalist context, identifies the content of a fundamental perception as the relevant cause of the realizer of that perception. Thus, perceptions whose realizers are typically caused by horses have horses as their intentional content.

Now among our perceptions, there are introspective perceptions of our own mental states. When I have pain, I often perceive that I have pain, and sometimes when I am puzzled, I perceive that I am puzzled. But now we have a problem. For the most plausible story compatible with the above about how we form introspective perceptions is that they are caused by the states that realize the mental states that the perceptions are of. Thus, if N is the neural state that realizes my pain, my perception of my being in pain is caused by N. But by the causal theory of content, that perception's content, then, is the neural state N. In other words, instead of perceiving that I am in pain, I perceive that I am in such-and-such a neural state. Yet although the neural state realizes the pain, it is distinct from the pain. The pain is a second-order property, while the neural state is a first-order property.

So, it seems that the causal theory of content plus functionalism predicts that our introspective awareness is neural states rather than of the mental states the the neural states realize. But surely our introspective awareness is of mental states.

But perhaps we can extend the causal theory of content to an explanatory theory of content. And while the second-order state of being in pain perhaps doesn't cause the realizer of the introspective awareness of the pain, nonetheless the state of being in pain explains either the realizer of the introspective awareness or the introspective awareness (a functional state) itself.

But I think this is implausible. For consider extra-mental functional stuff. Thus, green rectangles realize money. As children, we first see the green rectangles as such. Later we see these green rectangles as caught up in a great functional systems, and we come to see not just green rectangles but the money they realize. It is the physical stuff that is the first object of perception, and the perception of functional stuff is built on that. By analogy, then, we might predict that if functionalism and the causal theory of content are correct, then before children introspect to their pain, they introspect to their neural states. But that is deeply implausible. On the contrary, it is the introspection of the mental, not the neural, states that seems to come first, both chronologically and phenomenologically.

Open future and logic

Typical human languages (maybe all natural human language) have a "Was" operator that applies to a sentence s and yields a sentence Was_t(s) that backdates s to t. Sometimes that operator works by shifting context, but in English it normally works by shifting tense and inserting an "at" (though sometimes a more complex formulation is needed). Thus, perhaps, "Was_t(George is talking about Saddam Hussein)" is the sentence: "George was talking about Saddam Hussein at t".

Now, here are two plausible facts about logic and the Was_t(s) operator:

1. If the proposition that s entails the proposition that u, then the proposition that Was_t(s) entails the proposition that Was_t(u).
2. For every proposition p, p entails the proposition that p is true.

Moreover:

3. That Hitler's actions are starting a world war entails that there is or will be a world war.

Let t be September 1, 1939. Then:

4. Hitler's actions were starting a world war at t. (Historical fact)
5. Was_t(Hitler's actions are starting a world war). (Paraphrase of 4)
6. Was_t(There is or will be a world war). (By 1, 3 and 5)
7. Was_t(The proposition that there is or will be a world war is true). (By 1, 2 and 6)
8. The proposition that there is or will be a world war was true on September 1, 1939. (Paraphrase of 7)

But if (8) is true, then open future views are false. For on September 1, 1939, whether there would be a world war depended at least in part on the free choices of Poland's whether or not to honor their promises to Poland, free choices that British and French leaders made over the next two days.

Alan Rhoda has already denied (2), calling it "Pruss's mistake". So he won't have any trouble with this argument. But (2) seems rather more plausible than the conjunctions of the premises of the best arguments for open future views.

I should say that I am not sure (2) is true. What I am more sure of is:

9. For every proposition p, p entails the material conditional: if p exists, p is true.

But the argument can be adapted to use (9) in place of (2), since it's very plausible that the proposition that there is or will be a world war existed at t. The reason why (9) may need to replace (2) is issues with de re propositions. For instance, in a world where Socrates never comes into existence the proposition that Socrates never comes into existence might not exist, and hence might not be true.

More on the Principle of Sufficient Reason and A-theory

Update: My comment of Jan. 19, 2013 may contain a satisfactory answer.

In an earlier post, I argued that asking why it's 2013 presently forces the A-theorist to deny the Principle of Sufficient Reason (PSR). Let me expand on that argument. Here's a thought about my main argument.

Suppose the PSR is true. With the typical A-theorist, consider the proposition q that it is 2013 presently to be a contingent proposition that changes in truth value—it was false a couple of weeks ago and will be false again in a year but is now true. Let p be the ultimate explanation of q. For if PSR is true, there are ultimate explanations. Cf. my PSR book. Then either p is a proposition that is true at all times or it is a proposition only true at some times.

But it is incredible that a proposition that is true at all times (in 2012, in 94994 BC, etc.) would explain why it is 2013 presently.

But if p is true at some but not all times, then p can't be an ultimate explanation, because it is unexplained why p is presently true, rather than this being one of those times where p is false.

So, either way, we get a contradiction.

The typical B-theorist, on the other hand, will say that all true propositions are eternally true, so q, if there is such a proposition at all, is eternally true, and there is no difficulty about explaining it with another eternally true proposition.

Objection: But Pruss has argued at length (say, in the PSR book or here) that it is possible to explain a contingent proposition with a necessary one. So why can't one explain a non-eternal truth with an eternal one, explaining q with an eternal truth p?

Response: I just don't see any plausible way to do it, that's all. I wish I had something more rigorous to say here.

The Adams thesis reconsidered

The following is known as the the Adams Thesis for a conditional →:

1. P(A→B)=P(B|A).

This is very plausible. However, Brian Weatherson expresses a widely shared conviction when he says:

As with so many formal theories, accepting this thesis leads to paradox. Lewis (1976) showed that any probability function Pr satisfying [(1)] would be trivial in the sense that the domain of the function could not contain three possible but pairwise incompatible sentences.

And indeed in the literature, the Lewis result gets used to argue that a conditional cannot have a truth value, since if it had one, that value would have to satisfy the Adams Thesis.

But what Lewis actually showed was somewhat weaker. Lewis showed that triviality results from:

2. P(A→B|C)=P(B|AC).

Now, Lewis perhaps correctly concludes from this that the Adams Thesis can't hold for subjective probabilities. For given a probability distribution P satisfying (1) and any C with P(C)>0, we could imagine another rational agent, or the same one at a later point, who has conditionalized her subjective probabilities on C, and when we apply (1) to her newly conditionalized probabilities we get (2).

But suppose that the probabilities we are dealing with are objective chances. Then one might well accept (1) for the objective chance probability function, without insisting on (2) in general. For instance, a reasonable restriction on (2) would match the restriction on the background knowledge in Lewis's Principal Principle, namely that the background knowledge is admissible, i.e., does not contain any information about B or stuff later than B.

Perhaps, though, clever people will find triviality results for (1), much as Lewis did for (2)? I doubt it. My reason for doubt is that I think I can prove the following two results which show that any probability space, no matter how nontrivial, can be extended to an Adams Thesis verifying probability space for all A with P(A)>0.

Proposition 1: Let <P,F,Ω> be any probability space. Then there is a probability space <P',F',Ω'> that extends <P,F,Ω> in the sense that there is a function e:F→F' that preserves intersections, unions and complements and where P'(e(A))=P(A), and such that for every A in F with P(A)>0 and every B in F, there is an event A→B in F satisfying P'(A→B)=P(B|A).

This result only yields a probability space verifying the Adams thesis for conditionals where neither the antecedent nor the consequent contains a conditional. Since conditionals that have conditionals in the antecedent and consequent can be at least somewhat hairy, this restriction may not be so bad. And one can iterate the Theorem n times to get an extension that allows the antecedent and consequent to have n conditional arrows in them. But if we are willing to allow merely finite additivity, then we have:

Proposition 2: Let <P,F,Ω> be any probability space and assume the Axiom of Choice. Then there is a finitely additive probability space <P',F',Ω'> (in particular, F' is a field, perhaps not a sigma-field) that extends <P,F,Ω> and is such that for any events A and B with P'(A)>0 there is an event A→B such that P'(A→B)=P'(B|A).

To prove Theorem 2 from Theorem 1, let <P_n,F_n,Ω_n> be the probability space resulting from applying Theorem 1 n times. Let N be an infinite hypernatural number. Then there will be a hyperreal-valued *-probability <*P_N,*F_N,*Ω_N>, and when we restrict this appropriately to include only finitely many iterations of arrows in an event and take the standard part of *P_N we should be able to get <P',F',Ω'>.

Can A-theorists accept the Principle of Sufficient Reason?

According to the A-theory (at least on the versions I am interested in here), it's an objective but temporally changing (and hence contingent) fact that the year 2013 is present (i.e., lit up by the spotlight, on the leading edge of the growing block, or the year containing all real events, depending on the version of the A-theory).

But why is 2013, instead of say 2012 or 10 billion BC, present? Note that our argument will work whether or not by "2013" we mean a rigid designator of a particular year or some sufficiently long definite description.

Well, the one answer I can think of leads to a regress: 2013 is present because last year 2012 was present. For then we ask why last year 2012 was present. And the answer is, presumably, that two years ago 2011 was present.

This regress is infinite or finite. If it's infinite, we have something unexplained—namely why all of these years happened when they did vis-a-vis the present, 2012 a year ago, 2011 two years ago, and so on.

If it's finite, we have something unexplained, namely why it was that N years ago it was the year 2013−N, where N is the age of the world.

Can we give a theistic answer? Let's suppose "2013−N" is a definite description like "the first year of time". So: Why was the first year of time N years ago? Answer: Because N years ago, God created time (or the universe). But now we have a new puzzle: Why did God create time N years ago (i.e., before the lit-up time, before the year containing all real events, or before the leading edge of the growing block) rather than, say, N−1 years ago? (Or maybe better: Why is it N years after God created time?) After all, this is a contingent fact. Last year, it wasn't true that God created time N years ago (i.e., before the lit-up time, etc.)—it was instead true that God created time N−1 years ago.

Let's try our regressive explanation again. For we could say that God created time N years ago, because a year ago it was the case that N−1 years ago God created time. But this regress is actuality a circularity. For if we apply this schema N times, we get to the explanation that N years ago it was the case that 0 years ago God created time. But that's just a fancy way of saying that N years ago God created time, which is what was to be explained. Oops.

Perhaps there is some other explanation. Maybe God created time N years ago because N years ago was the first year of time, and God could only create time in the first year of time. But now we have a new question: Why is it that the first year of time was N years ago? And we better not go down the road of saying that a year ago, the first year of time was N−1 years ago, as then we'll end up, in a finite number of steps, with the claim that N years ago, the first year of time was 0 years ago, which is just a fancy way of sayingt hat the first year of time was N years ago.

If I am right, then the A-theorist cannot explain why it's 2013. The B-theorist denies that there is any such objective fact, except for the trivial fact that at t₀ it's 2013 which holds because of necessary truths about t₀ and 2013.

Now perhaps a given presentist hasn't been convinced by the arguments for the Principle of Sufficient Reason. But nonetheless it counts against a theory that it posits contingent unexplained facts that a competitor does not.

Is there a cosmological argument from objective probability?

I am finding myself pulled to the idea that objective probabilities must always go back to objective tendencies. If so, then on pain of vicious regress, we must have a metaphysically necessary being (prima facie, perhaps an aggregate of multiple beings) which has objective tendencies that ultimately ground all objective probability. Such a being cannot be merely abstract as it is causally efficacious.

Moreover, I am inclined to think rational subjective probabilities are our attempt at modeling the relevant objective probabilities in the face of our ignorance. If so, then if one denies such a first cause, one cannot coherently engage in probabilistic reasoning about reality.

A lot of work is needed to work out a cosmological argument along these lines. I don't have it worked out. It's just an intuition based on a lot of thought about probability theory.

Fair infinite lotteries: What can we say?

Let Ω be a countably infinite set, and suppose that we have a fair lottery on Ω: one member is going to be picked "at random", with no biases. For a subset A of Ω, we can ask how likely it is that the random element is a member of A.

Now, the probabilistic question we asked is independent of any kind of structure on Ω. We have no biases in the choice of an element of Ω. Consequently, we would expect our probabilities to be permutation-invariant. There is only one set of permutation-invariant finitely-additive real-valued probabilities on Ω that assigns probabilities to all finite sets, namely the assignment that P(A)=0 if A is finite, P(A)=1 if A is cofinite (i.e., the complement Ω−A is finite) and P(A) is undefined if A is neither finite nor cofinite.

But suppose we are not satisfied with undefined probabilities in the interesting case where A is neither finite nor cofinite, and want probabilities defined for all subsets of Ω. Let's work with qualitative probabilities, i.e., a pair of relations < and ~ on subsets of Ω, representing being less likely than and being equally likely as. As minimal conditions, let us assume:

1. < is transitive and irreflexive (A<A never happens)
2. ~ is an equivalence relation
3. if A~B, B<C and C~D, then A<D
4. if A⊆B, then A~B or A<B
5. ∅<Ω.

Say that A≤B provided A~B or A<B.

Since we want our probabilities to be order independent, we shall assume:

6. if π is a permutation of Ω, then A~πA,

where πA is the set of all π(x) for x in A.

Now, up to permutation, the following are all the types of subsets of Ω:

• finite subset of cardinality n, for n a natural number (0,1,2,...)
• cofinite subset whose complement has cardinality n, for n a natural number
• subset that is neither finite nor cofinite.

I.e., any two finite (or any two cofinite) sets of cardinality n are equivalent under permutation of Ω, and any two sets that are neither finite nor cofinite are likewise equivalent, and by (6) any permutation-equivalent sets have equal probability. Let F_n be any finite subset of cardinality n, let C_n be any cofinite subset whose complement has cardinality n, and let U be any subset that is neither finite nor cofinite. By 1-6, we then have:

7. ∅=F₀≤F₁≤F₂≤...≤U≤...≤C₂≤C₁≤C₀=Ω.

There is no way of satisfying 1-7 with an additive probability ordering, i.e., one satisfying the de Finetti additivity axiom that if A and B, as well as A and C, are disjoint, then A∪B<A∪C if and only if B<C.

But for intuition's sake we can model this probability ordering with interval-valued probabilities as follows. Let a and u be two positive infinitesimals such that u/a is infinite. Then P(F_n)=[na,na], P(C_n)=[1−na,1−na] and P(U)=[u,1−u].

In any case, the resulting measure is counterintuitive. If Ω is the natural numbers, then all sets that are neither finite nor cofinite have equal probability by permutation equivalence. And so the probability of our random number being divisible by 100 is the same as the probability of its being divisible by 2.

Moreover, we get the following problem. Suppose that we have an infinite number of tickets, one per each of an infinite number of ticket holders. One ticket is picked at random, without disclosure of which ticket it is. Then every ticket holder tosses a fair and independent die. Let A be the set of tickets held by ticket holders who tossed 6 and let B be the set of tickets held by ticket holders who didn't toss 6.

Question: Is it likelier that the winning ticket is a member of B than that it is a member of A, i.e., is it likelier that the holder of the winning ticket tossed a non-six than that he or she tossed a six?

Answer 1: Yes. Suppose Fred is the holder of the winning ticket. Then the probability that Fred tossed non-six is 5/6 while the probability that he tossed six is 1/6.

Answer 2: No. We know for sure (i.e., with probability 1) that infinitely many ticket holders tossed six and that infinitely many did not. Thus, A is neither finite nor cofinite, and the same goes for B. Thus A~B: the two sets have equal probability.

Ooops! Intuitively, Answer 1 is the right one. But it's hard to escape the reasoning in favor of Answer 2. So what should we say? Maybe that probability breaks down in infinite lotteries. And if so, then we better not be in an infinite multiverse.

Infinite lotteries and nonmeasurable sets

This may be a longshot, but I wonder if the problems of fair countably infinite lotteries and nonmeasurable sets aren't species of the same problem, for the following reason. The standard construction of nonmeasurable sets is the Vitali sets. But the Vitali sets on [0,1) have the following property: There is a countable collection of disjoint Vitali sets whose union is [0,1) and which are such that each of the sets is a translation (modulo 1) of every other set in the collection. Since we want translation-equivalent sets to have the same probability, this collection of Vitali sets implements a fair lottery with countably many tickets--the dart tossed at [0,1) intuitively has the same probability of hitting any particular Vitali set.

Here's another way to put it. Suppose we had a plausible probability for the individual outcomes of a fair countably infinite lottery. Then we could reasonably assign that value to the probability of an infinitely thin dart uniformly aimed at [0,1) hitting any given Vitali set.

(A Vitali set in this context contains exactly one element from each equivalence class of numbers in [0,1) under the equivalence relation x~y if and only if y is a translation of x, modulo 1, by a rational amount; the collection in question consists of any given Vitali set plus all its translations by rational amounts in [0,1).)

Hyperintegers, cardinality and probability

Suppose F is an ordered field extending the reals. Then a subset Z of F closed under addition, subtraction and multiplication can be called a set of hyperintegers provided that for any x in F there is a unique n in Z such that n≤x<n+1. Every real closed field has a set of hyperintegers. Given a set of hyperintegers, we can call the non-negative ones "hypernaturals". Introduce the notation [x] for the set of all the hypernaturals smaller than x, for x in F. If x is itself a hypernatural, then [x] = {0,...,x−1}.

Here's an amusing and perhaps surprising little fact:

• If F has hyperintegers Z, and M is an infinite element in F, then [M] has at least the cardinality of the continuum, and in particular is uncountable.

(An infinite element is bigger in absolute value than every real.)

If F is a field of hyperreals, this follows from the fact that [M] is an internal set and not finite.

The proof is very simple. For x in F, let floor(x) be the unique n in Z such that n≤x<n+1. Let A be the set of numbers of the form floor(xM) for x a real number in [0,1). Observe that A is a subset of [M]. Moreover, A has the same cardinality as the set of real numbers in [0,1), since the function f(x)=floor(xM) from the real numbers in [0,1) onto A is one-to-one. To see that it's one-to-one, observe that if f(x)=f(y) for real x and y, then |xM−yM|<1. Unless x=y, then M<1/|x−y|, and M is finite. So x=y. So A has the cardinality of the continuum as the set of real numbers in [0,1) does. Thus, [M] has at least the cardinality of the continuum, since A is a subset of [M].

This helps improve on and slightly generalize the argument here that infinitesimals are too small to model outcomes of infinite fair countable lotteries. Suppose we say that the probability of getting an outcome n in a lottery with possible outcomes 0,1,2,... is an infinitesimal u in an ordered field F extending the reals that has hyperintegers. But u is of the right size (if it's the reciprocal of a hyperinteger) or just slightly too small (otherwise) for being the probability of getting outcome n in a fair lottery on the set [1/u]. But the set [1/u] is much much bigger than the set {0,1,2,...}, since [1/u] is uncountable while {0,1,2,...} is countable.

An argument that consciousness is reducible

Suppose that over the next hour, you are consciously and constantly perceptually aware of a red wall, while I am consciously aware of nothing red—I am in a room where everything is blue—except that for exactly one nanosecond, half way through the hour, I have induced in me your intrinsic mental state, followed by a restoration of my previous state. Thus, if I was consciosu of something red during that nanosecond, I don't remember it afterwards.

But was I ever conscious of something red? I doubt it. A mental state that short just "doesn't register". Plausibly, given how humans are constituted, there just is no such thing as a pain that's only a nanosecond long, and there is no such thing as a consciousness of red that's only a nanosecond long.

But this yields the following argument, where we stipulate that a state is "near-instantaneous" provided that it has a temporal dimension of at most a nanosecond:

1. All our intrinsic states are reducible to our near-instantaneous intrinsic states. (Premise)
2. Humans have no near-instantaneous intrinsic state of being conscious of red. (Premise, justified by the thought-experiment)
3. Being conscious of red is an intrinsic state of us. (Premise)
4. So, a human's being conscious of red is reducible to near-instantaneous states that are not themselves consciousnesses of red.

But what goes for consciousness of red goes for all conscious states. So all conscious states are reducible to near-instantaneous states. But, plausibly, we have no near-instantaneous conscious states. So all conscious states are reducible to states that are not conscious states.

If this argument is right, then qualia are reducible. It does not, however, follow that they are reducible to physical states. It could, instead, be the case that conscious states are reducible to fundamental mental states that are not conscious states. The hard problem of mind is the problem of intentionality, perhaps, not the problem of consciousness (a sentiment I think I have heard from more than one person).

But all that said, I am not sure of premise (1) in the argument above. And I am not completely sure that the thought experiment succeeds. Maybe one can say that one is near-instantaneously aware of red but one forgets it right away. I don't know.

Searching for meaning in one's suffering

Here is a logically valid argument:

1. If theism is false, most evils are meaningless.
2. If most evils are meaningless, it is inadvisable for sufferers to put significant effort into searching for meaning in their suffering.
3. It is not inadvisable for sufferers to put significant effort into searching for meaning in their suffering.
4. So, theism is true.

All the conditionals here are material conditionals and I am not claiming any kind of necessity for them. I do find the premises fairly plausible. I am least sure of (2). I not clear on what "meaning in suffering" is, but there does seem to be such a thing—at least, many people report finding it.

More on qualitative probability and regularity

For probabilities that have values, regularity is normally defined as the claim that P(A)>0 whenever A is non-empty. Given finite additivity, this entails that if A is a proper subset of B, then P(A)<P(B). If instead of assigning probability values we deal with qualitative probabilities—i.e., probability comparisons—we now have a choice of how to define regularity:

• the probability comparison ≤ is weakly regular provided that ∅<A whenever A is non-empty
• the probability comparison ≤ is strongly regular provided that A<B whenever A is a proper subset of B.

If one assumes the axiom:

• (Additivity, de Finetti) If A∩C=∅ and B∩C=∅, then A≥B if and only if A∪C≥B∪C,

then weak regularity entails strong regularity.

In a recent post, I showed that there is no rotationally invariant strongly regular qualitative probability defined on all countable subsets of the circle. But perhaps there is a useful weakly regular one that does not satisfy Additivity?

I don't know. Here's a start. Let P be the Bernstein-Wattenberg hyperreal-valued measure on the circle. Define A≤B if and only if for some rotations r and s we have P(rA)≤P(sB). Define A≤B if and only if not B<A. Then ≤ is weakly regular, and rotationally invariant. But I can't prove that it's transitive. Is it?