The power of moral non-realist subjectivism

Over on Facebook, Massimo Pigliucci links an NYT article describing a conversion to non-realism. As I usually recommend Dr. Pigliucci as an expert on ethics, I was perturbed to find that he did not understand how moral non-realists could conduct "moral argument." Because after all, he argues, ethical statements are akin to trivial preferences like "milk chocolate versus dark chocolate" in a non-realist account.

This is profoundly misguided, and I was surprised to hear it from Dr. Pigliucci. There is a trivial sense in which it is correct, but there is a very non-trivial sense in which it is deeply confused, a confusion which often leads people to say things like, "but non-realists cannot conduct sensible moral argument."

To shoulder the responsibilities which come with idiosyncrasy, I will give a formal account of how moral argumentation `works' in a non-realist conception. I am convinced that moral non-realism can do almost everything that realists want their metaethics to do, barring the over-confident usage of a few unimportant phrases. I think that, properly understood, moral realism is paltry in light of a developed non-realism. Further, I want to show that moral realisms are in practice a special case of ethical subjectivism.

So how can one capture non-realism in an analytically useful way? For this, we turn to Bayesianism. Recalling that Bayesians assign subjective probabilities to subsets of a sample space, we simply omit `ethical propositions' from the sample space.

Cool. We can still ascribe probabilities to things like `x causes suffering', `I want to do x', and `doing x would conform to principles protecting free expression', but unless statements like `I ought to do x' wholly reduce to `factual' statements, they are not ascribed probabilities. Many `realists' attempt to claim truth-values for ethical propositions by claiming that what we really `mean' by `I ought to do x' reduces to such statements. In this case, they accomplish nothing but a loss of generality. As we will see later, their position can be described using subjective utilities.

So let's say that you are a coherent, well-calibrated Bayesian with probability distribution p who wants to make a choice between two actions, X and ~X. Better yet, let's say you act and let's find out what that means. To capture it, we can say that you prefer X to not ~X by choosing X. We can capture this simple preference with a subjective utility function u_{p,C}, where p is your distribution and C is the choice `X or ~X'. Such binary actions will allow for a trivial ordering relation u(X)>=u(~X) wherever you do X. One can easily see that for a finite set of mutually exclusive choices, a similar ordering can capture the selection of one of the options. So we can assume that this function maps actions into some one-dimensional set of numbers, say the reals. So subjective utilities are functions from actions into the real numbers. If you like, you could let utilities map into the extended reals (i.e. take infinite values), but such functions risk collapsing into triviality unless you are careful to avoid the paradoxes that result

We notice immediately that we are now firmly in the domain of Bayesian decision theory. This is good news; we have a lot of accepted formal tools to use for our moral thinking. Better yet, we have kept things general enough to account for actions as demanded by any other moral system. To see how, let U_p be the set of all possible utility functions u_C=(u(X),u(~X)) where (u(X),u(~X)) is a pair of real numbers. This allows for any preference to any degree. (If moral absolutism-by-degree (say a Kantian imperative) is in play, reintroduce the extended reals.) So the demands of any moral system as applied to a rational agent faced with a choice can be captured. If we have a very clearly defined system S that says something like "X is permissible iff ...", then utility functions can be sorted into a blocks: the u's consistent with S in one, and the u's inconsistent with S in another. More generally, one can define subsets of utility functions consistent with a moral theory. Better yet, we may introduce fuzziness by probabilizing relationships like "u conforms to the principles of free expression", though as far as I can tell uncertainty with regards to conformity with principle may be subsumed in one's utility function. (One may still want to do this for judging other utility functions.)

Cool. So non-realists can even talk about conformity to abstract principles in a principled way without making charges of irrationality concerning preferences. So how does moral argument look? As factual information is introduced, the restrictions on utility functions which do not commit `horrendous' decisions are increased. So a Bayesian who wants to murder for fun could not ascribe a high value to life, or to non-violence, or to the autonomy of humans, or to the prevention of suffering, or...

He may remain perfectly consistent in adopting such a function. But such functions are very rare. Moral argument works so long as we expect humans to have common concerns and interests. There are limits, but it can be done. Further, I think we can expect a non-realist approach to be just as convincing to those who actually do want to murder for fun as a realist approach.

Which is not very. All the realist gains is the satisfaction of calling such a person irrational, instead of a monster or thug or authoritarian.

Chomsky often invokes `elementary principles', like avoidance of hypocrisy, when beginning a speech. He invokes international law and other `moral truisms'. And yet he doesn't strike me as a realist, and if he were to simply demonstrate that US foreign policy and its media defense is hypocritical, I do not think it would be necessary for him to say "and hypocrisy is something we should be avoid." Do we need to insist on truth-values for statements like "we should not tolerate a racist, censorious, unrepresentative, and violent state"?

Our preferences are not drawn out of a hat. Normal people can ascribe meaning to principles and commit to them. That's why moral argument works, even if irreducible `should' statements are not truth-valued.

Now for the inevitable objections. As I put it on Facebook:

People license themselves to every imaginable silliness whenever terms like `moral non-realist' or `ethical subjectivist' show up; and so I, vainly fending off frustration and fury, must patiently answer such profound arguments as "but moral non-realism means morals aren't truth-valued", "but subjective utilities are subjective", and "but people who want to murder for fun might not care about the consequences", all while having to defend newly controversial propositions like, "peoples' preferences do not tend to be randomized".

Objections to non-realism, or moral skepticism (see esp. error theory), are almost always confusions. I am not a nihilist; I think we can and do make meaningful preferences. Or, as seen in the SEP article, there are noises about `presumptions against moral skepticism'. I think that my account, since it can generalize other systems, should not be victim of some prior prejudice. (And in general, I don't like noise about burdens of proof.)

So, let's talk about what subjective utility functions we would like people to have, which have nice properties, and other swell things. Or not. But don't tell me I cannot have that conversation.

What I have not been saying: I have not insisted that all reasonable people be moral non-realists. I'm just pointing out that you can do a lot with a non-realist account. I of course have problems with realist accounts, but that's a different discussion.

Commonplaces and Comments 5

I'm working on a very long post on a topic that's sure to be of local interest. In the meantime, I've found a copy of Richard Jeffrey's Logic of Decision¹, in which his generalization of Bayesian conditioning, probability kinematics, was first published. Decision theory is one of the greatest applications of Bayesian instruments, so I surprise myself in not having discussed it earlier. To motivate interest, I'll discuss what is probably the most famous decision-theoretic argument of all time, Pascal's wager.

As those familiar with the argument know, Pascal's wager is an argument for the strategy of belief in Christian theology, not an argument for its truth. Hitchens dismisses this approach as mere hucksterism, but it need not be so crude. Pascal did not require that we flick on the `believe in God' switch in our heads, nor that we attempt to fool God through insincere belief. Rather, the intended conclusion of the argument is this: that one should by the best strategy available attempt to arrive at sincere Christian belief.

Pascal's argument is often presented using a decision-matrix (see the previous link). If we accept his options - Christianity or atheism - as being exhaustive, the payoff of correctly wagering on Christianity (infinite) is greater than the payoff of correctly wagering against it (finite), just as it is greater than the payoff of incorrect wagers with respect to Christianity (finite). If we denote the payoff of wagering correctly on atheism by x and wagering correctly on Christianity by y, the expected utility of wagering on Christianity is
prob(Christianity)*y+prob(Atheism)*x. So as long as the probability of Christianity is greater than zero, the payoff is infinite. Similarly, wagering against Christianity has `infinitely negative' expected utility. So even if one ascribes prob(Atheism)=1-1/[Graham's number], the infinite payoffs of correct Christian belief swamp the expected utilities.

The most common objection to Pascal's wager - which I think sound - is that the payoff matrix is not comprehensive, i.e. the argument is unsound. So long as we can imagine differing possible `infinite' payoffs against Christian belief, no best strategy follows. But there is another, more interesting objection - which I discovered via Alan Hájek - which disputes the validity of the argument, not the soundness.

To see why, recall that the intended conclusion of the argument is to convince skeptics that a strategy maximizing the likelihood of their arrival at sincere belief is the wisest strategy. Call such a strategy S, say living amongst worshippers and partaking in Christian missionary activities. Let's say the probability that adopting S results in Christian belief is 0.99. Now let's take another strategy, T, which does not likely result in Christian belief, say having a beer, where the probability of `success' is 0.0001. Check the following to your own satisfaction: the utilities of these strategies are the same. In fact, any `mixed strategy' is equally good, so long as there is a non-zero probability that Christian belief will result.

Cool. That's what happens when we play around with infinite utilities, and not just for Christian belief. This is a much more interesting and illuminating objection to Pascal's wager than dry statements about the decision-matrix. See why I think that philosophy of religion is a great `gateway drug' for the rest of philosophy? Whenever I see the Wager being discussed in the future, I'll try boring people with decision theory.

Onto The Logic of Decision, then.

Dominance and Disarmament

Dominance is an introductory topic in The Logic of Decision, like that which appears in Pascal's wager. His more secular example is that of nuclear disarmament (p8). Consider the general 2x2 decision array:



Let the first column correspond to an outbreak of war, and the second to the maintenance of peace. Similarly, let the first row correspond to the outcome of nuclear armament in the first column and disarmament in the second. Then d₁ is the utility of nuclear weapons in the event of war, d₂ is the utility of nuclear weapons in times of peace, e₁ is the utility of disarmament in times of war, and e₂ is the utility of disarmament in times of peace.

To illustrate, let's suppose that these utilities correspond to a naively pacifistic outlook (p2):

.

As Jeffrey notes, disarmament is a very complicated example, but proponents may argue that the values of the utilities are unimportant, so long as e₁>d₁ and e₂>d₂. Given the matrix, this condition does not appear to be overly controversial.²

Yet the argument is fallacious, and not because any utility in question purports to be infinite.

Where the `superdominance' of Pascal's decision-matrix allowed for virtually any strategy to be of equal merit, the `dominance' of this decision matrix excludes certain strategies but not all mixed strategies. As Jeffrey elaborates, deterrence is a sound counter-example to the argument:

The assumption that the dominant act is the better is correct if an extra premise is introduced, i.e., that the probabilities of the conditions are the same, no matter which act is performed. [Emphasis in original] (p9)

A disarmament advocate could preserve the validity of the argument by conceding that the deterrence strategy reduces the probability of war, but that this benefit is countered by the increased probability of accidental war (p10). But the need for that additional premise remains. In general, what is needed for the argument for disarmament and other arguments is a decision-theoretic framework which allows for the effects of the strategies themselves on the probabilities of the outcomes.

And for that, dear reader, you'll have to wait. There are too many goodies in this book to cram into a single blog post. :D

---

1. Richard C. Jeffrey. The Logic of Decision (2nd Edition). UC Press: Chicago, 1983.

2. This is how I have often argued in the past, inspired by such beautiful entreaties as the Russell-Einstein Manifesto. Of course, the REM is not simply an argument for nuclear disarmament, but for the abolition of war. As the decision matrix does not account for, the likely event that disarmed countries will manufacture weapons means that in all probability, disarmament is only useful for the temporary reduction of tensions and the avoidance of horrible accidents. So though I nevertheless support it, it is not with a `golden age' in view.

The Emancipatory Potential of Atheism

[This is addressed to miscellaneous leftist critics of atheism.]

The crime of Thomas Paine was not that he doubted scripture; it was that he doubted it in front of a popular audience. Only recently has he been rehabilitated in American political life as a revolutionary hero; only with the aid of secularist activism - particularly that of Robert G. Ingersoll - has his principled courage been resurrected from a tomb of historical slanders.

In recent history - and, I think, the bulk of distant history - the primary attitude of controlling elites with respect to religion has been in the crudest sense utilitarian. Eisenhower exemplifies this attitude in many oft-cited quotations.

"Our government makes no sense unless it is founded in a deeply felt religious faith, and I don't care what that faith is."

His underlying attitude is in fact shared by Paine - not an atheist - and other `deistic' figures of the enlightenment: the equality of man is necessarily grounded in God, or at least, belief in God. But like Eisenhower, most pursue this thread further than Paine. Atheism may be tolerated, so long as it presents itself quietly and in academic or otherwise elite contexts. It is this longstanding attitude that modern atheism has struggled against and struggles against. I repeat for emphasis: the primary enemy of modern atheists is not religion, but a `religion of religion' or `belief in belief', which is held as holy writ even amongst the secular. Atheists have had some success in this direction; young as I am and in the Southeastern US though I might be, I have seen definite changes.

I do not maintain the untenable: atheism is no guarantor of liberty and welfare, nor is it a sufficient condition for humanitarian care and action. It would be otiose to cite examples here. Nor do I maintain that religious belief precludes humanitarian action. Again, it would be pointless to rattle off cases. I also do not hold that atheists as a group frequently make the most of their advantages. I will not list specifics. I would not dare propose that only atheists be admitted in a progressive party, nor would I propose any internal, rationalist purges of the faithful.

What I do maintain is the emancipatory potential of atheism. Many religions have a similar emancipatory potential, but there are limits and risks. Rather and as always, the decisive question you must ask yourself is this: what is true? Ask yourself carefully; your religious confederates may do the same.

Is exclusive salvation true? Are people divided into the Hell-bound and Heaven-hopeful? Suppose that were the case; do you maintain it has no effect on a utilitarian calculation of action? What if you convince your religious counterparts that religion will diminish or dissolve after the revolution? What does this mean? Should I list more difficulties?

No, I do not suspect that you regard these questions as essential, and regard yourself as fair-minded and priority-focused in disregarding them. But what this masks is a profound contempt for the critical faculties of your fellows, an essential condescension for those `private beliefs', those which you judge false and irrational, and which raise fundamental inconsistencies in the ethics of their actions.

To not see religion - yes, even the watery kind - as a fundamental category is to not understand it. To not be an atheist that is also focused and philosophical, well-read and comprehending, is to cling to inconsistency. Seeing brothers and sisters suffer, a religious person, in compassion and intelligence, may seek to systematically banish as much as she may the pains of her neighbors. In all likelihood, she will be content to treat doctrinal inconsistencies and complications as `mysteries', and ignore them.

And she will raise her children to be religious because she thinks it will make them good. And one day, her or her children will see the paradox, and the internal struggle is as follows: the logic of atheism, or of revival? Revivalism will persist for the foreseeable future, even if a socialist revolution were realized.

What I hold as a minimum is that comfort in an ignorance of religion is necessarily opposed to the principled bettering of society. Today, atheists and the religious are alike content to share this ignorance, yet this is a perverse extension of viewing religious belief as private and subjective - where this is even true. It is not enough to be an atheist; first and foremost, one must be a rationalist and a lover of truth. That we as secular people, hardly a few generations distant from blasphemy laws, forget our tenuous status shocks my conscience.

If you do not wish to distribute polemics against religion as part of your propaganda, I will not be bothered. But the mantra of `privacy' is not enough. Air your opinions as individuals, and if consensus is not to be had, seek the outlines of your disagreements. You do not in being honest have to split the party on religiously sectarian lines. If by airing your opinions about religion you manage to cause a split, ask yourself what the problem really was.

Learnin' me a book at skool

My classes are starting up; my apologies for the few-day break in posting. I at least managed to pester a few people on comments sections. If I learn anything interesting, I'll be sure to tell you folks.

Now that I have a library again, I'll be able to check out interesting stuff which I have been unable to find online. One such book is Jeffrey's The Logic of Decision. As this is a modern classic in Bayesian philosophy, I hope to have a commonplaces post up soon. To prevent my subjective Bayesianism from being lazy, I am also reading Jaynes' Probability Theory: The Logic of Science. I plan on comparing and contrasting these - and possibly Howson and Urbach's Scientific Reasoning: The Bayesian Approach - assuming time constraints from classes allow me. I'm taking a combinatorics [fancy word for counting stuff] course this semester, so doubtless I'll want to bore you about it.

So much to read, so little life!

On more blog-centric matters, I'm declaring the OTF dead. It has been without a pulse for some time, and its corpse is festering. Nothing but vague effluvia comes hence, so it is time to retire the body. It is Zossima, sans the grace and expectations. If you disagree, you should know where to look. I'll be surprised if anyone even attempts an adequate defense, much less a success.

Commonplaces and Comments 4

Some Yudkowskian tidbits on politics.

One.

Every profession has a different way to be smart - different skills to learn and rules to follow. You might therefore think that the study of "rationality", as a general discipline, wouldn't have much to contribute to real-life success. And yet it seems to me that how to not be stupid has a great deal in common across professions. If you set out to teach someone how to not turn little mistakes into big mistakes, it's nearly the same art whether in hedge funds or romance, and one of the keys is this: Be ready to admit you lost.

Two.

Politics is the mind-killer. Arguments are soldiers. Once you know which side you're on, you must support all arguments of that side, and attack all arguments that appear to favor the enemy side; otherwise it's like stabbing your soldiers in the back. If you abide within that pattern, policy debates will also appear one-sided to you - the costs and drawbacks of your favored policy are enemy soldiers, to be attacked by any means necessary.

Like it or not, there's a birth lottery for intelligence - though this is one of the cases where the universe's unfairness is so extreme that many people choose to deny the facts. The experimental evidence for a purely genetic component of 0.6-0.8 is overwhelming, but even if this were to be denied, you don't choose your parental upbringing or your early schools either.

Saying "People who buy dangerous products deserve to get hurt!" is not tough-minded. It is a way of refusing to live in an unfair universe. Real tough-mindedness is saying, "Yes, sulfuric acid is a horrible painful death, and no, that mother of 5 children didn't deserve it, but we're going to keep the shops open anyway because we did this cost-benefit calculation." Can you imagine a politician saying that? Neither can I. But insofar as economists have the power to influence policy, it might help if they could think it privately - maybe even say it in journal articles, suitably dressed up in polysyllabismic obfuscationalization so the media can't quote it.

Three.

If the reactor is more likely to melt down, this seems like a 'point against' the reactor, or a 'point against' someone who argues for building the reactor. And if the reactor produces less waste, this is a 'point for' the reactor, or a 'point for' building it. So are these two facts opposed to each other? No. In the real world, no. These two facts may be cited by different sides of the same debate, but they are logically distinct; the facts don't know whose side they're on. The amount of waste produced by the reactor arises from physical properties of that reactor design. Other physical properties of the reactor make the nuclear reaction more unstable. Even if some of the same design properties are involved, you have to separately consider the probability of meltdown, and the expected annual waste generated. These are two different physical questions with two different factual answers.

A scales is not wholly inappropriate for Lady Justice if she is investigating a strictly factual question of guilt or innocence. Either John Smith killed John Doe, or not. We are taught (by E. T. Jaynes) that all Bayesian evidence consists of probability flows between hypotheses; there is no such thing as evidence that "supports" or "contradicts" a single hypothesis, except insofar as other hypotheses do worse or better. So long as Lady Justice is investigating a single, strictly factual question with a binary answer space, a scales would be an appropriate tool. If Justitia must consider any more complex issue, she should relinquish her scales or relinquish her sword.

So am I Blue or Green on regulation, then? I consider myself neither. Imagine, for a moment, that much of what the Greens said about the downside of the Blue policy was true - that, left to the mercy of the free market, many people would be crushed by powers far beyond their understanding, nor would they deserve it. And imagine that most of what the Blues said about the downside of the Green policy was also true - that regulators were fallible humans with poor incentives, whacking on delicately balanced forces with a sledgehammer.

Close your eyes and imagine it. Extrapolate the result. If that were true, then... then you'd have a big problem and no easy way to fix it, that's what you'd have. Does this universe look familiar?

Four.

A candy bar is a superstimulus: it contains more concentrated sugar, salt, and fat than anything that exists in the ancestral environment. A candy bar matches taste buds that evolved in a hunter-gatherer environment, but it matches those taste buds much more strongly than anything that actually existed in the hunter-gatherer environment. The signal that once reliably correlated to healthy food has been hijacked, blotted out with a point in tastespace that wasn't in the training dataset - an impossibly distant outlier on the old ancestral graphs. Tastiness, formerly representing the evolutionarily identified correlates of healthiness, has been reverse-engineered and perfectly matched with an artificial substance. Unfortunately there's no equally powerful market incentive to make the resulting food item as healthy as it is tasty. We can't taste healthfulness, after all.

Evolution seems to have struck a compromise, or perhaps just aggregated new systems on top of old. Homo sapiens are still tempted by food, but our oversized prefrontal cortices give us a limited ability to resist temptation. Not unlimited ability - our ancestors with too much willpower probably starved themselves to sacrifice to the gods, or failed to commit adultery one too many times. The video game players who died must have exercised willpower (in some sense) to keep playing for so long without food or sleep; the evolutionary hazard of self-control.

ID Quiz

Sure, why not?

1. On a scale of 0 (diehard disbeliever) to 10 (firm believer), how would you rate your level of belief in Intelligent Design? (Minimal Definition of Intelligent Design: The idea that certain features of the universe and of living things are best explained by an intelligent cause, and not by an undirected process.)

I would not assign myself a single number here, since the question is too vague to be properly answered. (I suppose that would make me a `0' by most lights.)

2. What do you regard as the best argument for Intelligent Design?

I've yet to see a properly formulated argument. `Specified complexity' is, I think, beyond salvaging. `Irreducible complexity' could be an argument, if ID theorists (i) did convincing work on the positive side of their theory, and (ii) were more convincing in demonstrating the inadequacy of naturalistic explanations.

3. What do you regard as the best argument against Intelligent Design?

That it is not a theory; rather, it is a collection of poorly-formulated arguments around which a political movement - still agnostic on the designer question? - has not-so-mysteriously built itself for obvious political reasons.

4. I’d like you to think about the arguments for Intelligent Design. Obviously they’re not perfect. Exactly where do you think these arguments need the most work, to make them more effective?

See (i) above. Formally, you want to avoid committing the fallacy of probabilistic modus tollens. The fact that a particular outcome of evolution, e.g. a flagellum, is very improbable given evolution, does not mean that observing this outcome diminishes your odds on evolutionary theory. Rather, one must establish that such a structure should be more probable than that, especially by providing an alternative theory, itself significantly plausible, which makes such structures likely. For IC to be evidence of `something else', you have to have a predictive theory of `something else'. I could add that the difficulties supposed IC structures pose for evolution are themselves not nearly as severe as ID theorists say, but for that to even matter, ID theorists have to present an adequate theory. (No, I don't think that analogies from human designers are going to cut it.)

5. Now I’d like you to think about the arguments against Intelligent Design. Obviously they could be improved. Exactly where do you think these arguments need the most work, to make them more effective?

If you provide a theory which is sufficiently well-formulated as to actually be contrasted with naturalistic theories, we can have that discussion. The spirit of this and my answer to (4) should cover (6).

It's been a while since I've argued about ID, but I haven't seen many changes.

The most beautiful moments in music

I submit that 2:53-4:04 and 5:32-applause are contenders. Any others?

(I have very eclectic tastes; be not afraid.)

Trying to formalize the OTF

Via Victor Reppert, I came across another atheistic critique of the OTF by Thrasymachus. Inspired by its superior clarity, I have decided to further clarify my previous objections. Thrasymachus is also replying to a previous reply by Loftus which is better suited to my purposes than his other writings.

Thrasymachus focuses on the OTF as premised by Loftus in this post:

1. Rational people in distinct geographical locations around the globe overwhelmingly adopt and defend a wide diversity of religious faiths due to their upbringing and cultural heritage. This is the religious diversity thesis.

2. Consequently, it seems very likely that adopting one’s religious faith is not merely a matter of independent rational judgment but is causally dependent on cultural conditions to an overwhelming degree. This is the religious dependency thesis.

3. Hence the odds are highly likely that any given adopted religious faith is false.

4. So the best way to test one’s adopted religious faith is from the perspective of an outsider with the same level of skepticism used to evaluate other religious faiths. This ex-presses the OTF.

Loftus uses the phrase "the odds are highly likely" in response to the observation that a deductive equivalent of the above is invalid. But as Thrasymachus points out, it still is not clear that (3) follows from (1)/(2).

First, let me clear some fumes: I am assuming that everyone involved agrees that certainty in religious beliefs is unwarranted. I am also assuming that after this is recognized, the religious beliefs in question can be probabilized. This is not always obvious: some claims are not obviously susceptible to forceful probabilities. The doctrine of the Trinity, for example, has other conceptual issues to clear up before this may be done. Instead of throwing up our hands, we can focus on the subset of putative truths essential to Christianity (C) which can be probabilized, e.g. the Resurrection. It is the probability of these claims in conjunction that is represented by prob(C).

Second, (1) assumes that differing religious people are or can be rational, at least in the sense that their beliefs are internally consistent. Else, we have no need of the OTF, as incoherency arguments would more than suffice.

Now we can see what would be required for (3) to follow from (1) and (2). I will set as a threshold that (3)/(4) translate as requiring, at a minimum, that Christianity is not more likely to be true than not, i.e. 0.5>p(C). Denote the religious diversity thesis by Div, the religious dependency thesis by Dep, and p the prior probability of some unspecified Christian.¹ The odds form of Bayes' Theorem is as follows:

.

To get (3)/(4) as I interpret them, we need

.

In order for this to be the case, we need to know three different quantities. p(C)=1-p(~C), p(Div & Dep|C), and p(Div & Dep|~C). All we can say about p(C) is that it is greater than 1/2, as we are talking about a believer's prior. So we need something at least as strong as p(Div & Dep|~C)>p(Div & Dep|C). But as I pointed out in a previous post, not even this inequality must hold.

At the end of my last post, I asked a question: what would Calvin think of the OTF? It wasn't an idle question. If you are a Christian who believes in the predestination of the Elect and the Fallen world, the fact that your religion is one amongst many is not a surprise. That few have the right faith may not bother you at all. As far as I am aware, nothing about your religion says that it should not appear to an outsider as one among many. Strange then that advocates of the OTF tell you that the existence of other religions discredits your religion. You can reasonably say, "my religion looks like one of many to you? Swell, your point being? We agree about this, and it bothers me not. For chances are that you are not one of the Elect and are not destined for Salvation and understanding. That you and others do not believe as I do does not surprise me in the least; if anything, I would be surprised if outsiders readily understood the Truth and could easily aspire to it, as I understand otherwise."

One can argue against such a person, but the appearance of his beliefs to a skeptic should not itself constitute an argument.

The case is different whenever we look at more common evangelical versions of Christianity, in which it is asserted that God intervenes or has intervened to aid Christianity and that the Holy Spirit works on the consciences of most or all to guide them to Truth. Free will. All that jazz. If a supernatural agency is at work in the Christian sociology of Christianity, then it is surprisingly hidden in the actual sociological details concerning Christianity. Here, the fact that Christian belief is largely a function of geography and parenting is very surprising. To a person who thinks that Christianity is a natural phenomenon, it should not be. I think that this is a very powerful argument against evangelical Christianity.

Notice then that there are at least two possible outcomes of "Christianity is like other religions to an outsider": it is irrelevant to some Christians, and it constitutes a challenging argument to others. So what we can not do is treat the motivations for the OTF as legitimizing it against religions generally, since the observations motivating the OTF are in no way an argument against certain religions. To pretend otherwise is to do nothing more than pomo an important, but narrow, point.

But it gets worse: even in the cases where the requisite inequality does hold, it may not be large enough to require our believer to make further arguments so as to defend his faith. This is because we still need to know what values of p(C) are warranted. Sure, it's less than 1, but is it less than 0.999 or 0.8?

And so we come to the reason why I did not attempt to formalize the OTF much earlier: it simply isn't a probabilistic inference; it is a demand about priors. I think this is why Loftus has yet to put an argument about probabilities in terms of formal probabilities, as far as I can find. This is not a case of updating a prior set of rational beliefs to a new probability by reasoned argument. Instead, it is an attempt to force a reworking of priors based on evidence.² Again, I do not see why Christians need to accept this; intellectual consistency only requires that they account for Div and Dep by calculating their effects on their beliefs through conditioning.

Here we depart from the most accepted form of Bayesianism, i.e. subjective Bayesianism, entirely. We are encountering a curious version of objective Bayesianism. `Normal' objective Bayesians calculate `informationless' priors by equivocating across possibilities. What Loftus appears to want, as I noted in my previous posts, is that we gauge p(C) in something like the following way:

a. p(C)=1/N where N is the number of possible, mutually contradictory religions.
b. p(C)=1/N where N is the number of mutually contradictory religions in human history.
c. p(C)=1/N where N is the number of existing, mutually contradictory religions.
[Each of the above has an analogue where `religions' is replaced by `Christian sects'.]
d. p(C)=x where x is the frequency of the occurrence of Christians with respect to the general population. (Of the country, or world, or something.)
e. p(C)=A/B where B is the number of rational people and A is the number of rational people who are Christians.

And so on. Before moving on, the first response our Christian might deploy to any combination of the above is a simple one: No.

He is presumed to be rational and he can account for (1)/(2) in the usual way. Sorry to wax tautological, but he simply cannot be convicted of irrationality or unreasonableness whenever he is being both rational and reasonable, as judged by standard philosophical criteria. To go further with this, Loftus will have to mount a convincing attack on Bayesianism itself.

And of course we run into the earlier problem yet again; the argument Loftus presents cannot be probabilized. None of the above statements follows, or can follow, deductively or probabilistically, from (1)/(2).

I could continue on about the other problems, especially as they pertain to Loftus' desire to demand priors about religion but not about secular claims, or that this approach would most likely result in a weaker case against Christianity than the traditional arguments, but I've said this already, and Thrasymachus has done a better job explicating it. I could repeat why `skepticism' is not a sort of default, and that positive claims will be necessary to argue against Christianity. (Otherwise, it's the fallacy of probabilistic Modus Tollens all the way down.) Or, I could reiterate some of Reppert's objections; for example, (1) and (2) are not so undeniably true as Loftus suggests, and Christians may account for differing religions using faith-based claims. The Pharaoh's magicians did not perform wonders so great as Aaron's, but they still made a snake out of a staff. Also, demons and sinful nature.

I pause. Is the argument really this straightforwardly awful? How does Loftus defend it?

One...option for the Christian might be to argue that I have not shown there is a direct causal relationship between RDPT (i.e. the Religious Dependency Thesis) (or 1) and the RDVT (i.e., the Religious Diversity Thesis) (or 2). Just because there is religious diversity doesn’t mean that religious views are overwhelmingly dependent on social and geographical factors, they might argue. Reminiscent of David Hume, who argued that we do not see cause and effect, they might try to argue I have not shown it exists between the RDPT and the RDVT. After all, if Hume can say he never sees one billiard ball “causing” another one to move just because they do so after making contact, then maybe there is no direct causal relationship between the RDPT and the RDVT. Is it possible, they might ask, that just because people have different religious faiths which are separated into distinct geographical locations on our planet, that “when and where” people are born has little to do with what they believe? My answer is that if this is possible, it is an exceedingly small possibility. Do Christians really want to hang their faith on such a slender reed as this? I’ve shown from sociological, geographical and psychological studies that what we believe is strongly influenced by “the accidents of history.” That’s all anyone can ask me to show.

Actually, we can ask for a valid argument. This is simply a genetic fallacy. The deductive genetic fallacy remains a fallacy, even if you argue for odds instead of certainties.

What else can I say? Nothing about this argument works, nor could it conceivably be reworked to capture what Loftus wants. There's a reason for this: it isn't actually an argument. It is a symptom of Loftus' assumption that he objectively and most accurately views the world in a culture-transcendent way.

I might have spoken too soon: if a Christian happens to trust Loftus more than God, there may be an opening for the OTF.

One last quibble to anticipate an objection: Loftus may claim that he is not addressing Calvinists, only evangelical Christians. That doesn't change the fact that his argument is not even an argument of that form. For this discrepancy to matter, he must restate his argument so as to account for variations in prior probability and variations in the Bayes factor specific to the religion in which he is interested. That is, he must pursue normal argumentation.

If he does so, I'll be more than happy.

---

1. It has to be this way, as we are interested in whether or not warrant for religion can be retained, not just how a skeptic feels about religion.

2. This is much weirder than anything attempted by normal objective Bayesians. I do not know of any accepted precedent for an approach like this.

Edit 8/8/11: I've been having a blast with acronyms lately. Please plagiarize the hell out of this excerpt from a comment at Reppert's place:

I should mention that I've seen John's post that he's on a blogging break, so I do not expect any response soon.

To be honest, I don't expect a serious response. Here's what he said to Thrasymachus' post back in January:

"I see nothing here I need to respond to."

Oh, my argument is invalid, cannot be reworked to convincingly get what I want out of it, and my approach in general is a failure. Where's the problem?

Staggering. And this is followed by another unhesitant shift:

"You can insert the word “skeptical” for “outsider” if you wish. And being skeptical means doubting or rejecting anything that the sciences say otherwise."

And we return to the uniqueness problems and question-begging again...

So I'll have fun at his expense until he or others get back to me with a real argument. A satisfactory response will do the following things:

1. Restatement: the precise structure and intended conclusion(s) of the OTF must be clearly stated, along with any contested background assumptions.

2. Support: The structure and conclusions of the argument must be corroborated. Is it deductive? If so, state exactly where and why. Is it a probabilistic argument? Then capture the argument using the formal tools of probabilism and defend it. Is it an argument about prior distributions? Then state clearly why it is that a coherent agent must adopt, prior to evidence, a specific distribution based on an observation which can already accounted for by a religious person or may be calibrated in a traditional, probabilistic manner (conditionalization).

3. Comprehensiveness: Clearly state outstanding objections and why they fail or are otherwise innocuous.

I call it the Simple Test For Understanding, or STFU, because proponents of the OTF should STFU already or move on.

An unconfirmability argument

[A caution: This post is long, verbose, and overly technical. Read the dialogue I posted at the end. If it is comprehensible to you, you should have an adequate grasp of the contents of this and the previous post.]

In my last post, I interpreted Hume's argument against the confirmability of miracles and found it to be unsound. Now I want to affirmatively answer another question: given that Hume's argument is unconvincing, is there another unconfirmability argument which applies to paradigm miracles like the Resurrection?

If so, we are not doomed to Earman's (apparent) conclusion, that we must analyze the details of every miracle claim if we are to safely reject miracles (p.3). Rather, we can specify categories of claims, narrower than `miracle', that cannot be confirmed by certain categories of evidence. Though we might not safely assert something like "no evidence you have should convince me of a miracle," we may be able to say something like "by itself, the evidence you present is by nature incapable of overcoming the prior probability of the type of miracle you assert." I will leave open the possibility that miracles like the Resurrection (R) can be confirmed; I am only closing a class of potential means of doing so. I do not think that I am providing or can provide "an everlasting check to all kinds of superstitious delusion" which "will be useful as long as the world endures" (EPHU, p.169); rather, I am giving something which, if successful, would obviate any non-pedagogical, rational need for any detailed Bayesian analysis of the Resurrection like this one, so long as we lack other significant arguments in favor of Christianity. One notices the number of qualifiers required to invest in such an argument, and there will be more. This form of argument does not constitute a license for ignorance, and it will require creative adaption to specific miracle claims.

A lot hinges on the problem of determining appropriate priors in subjective Bayesianism. In order to provide some convincing estimate of the prior odds on R, a trick - I think novel to me - must be applied.

To motivate this trick, I'll cite its precedent and inspiration: the method of reparation, originally due to Richard Jeffrey. This tool was invented for a specific problem, the problem of old evidence. Roughly, the problem is as follows: whenever a new theory is crafted, its ability to explain known phenomenon is considered to be of epistemic significance. For example, relativity theory's ability to `predict' the perihelion precession of Mercury - a phenomenon which had long defied explanation in classical physics - is considered powerful evidence for that theory. But the hypothetico-deductive principle, which states that verification of an uncertain prediction of an uncertain hypothesis increases the probability of that hypothesis, is incapable of yielding this well-founded intuition; the `prediction' is already known, i.e. not uncertain.

Reparation solves this problem by positing a hypothetical prior probability, an ur-distribution or ur-prior, in which the theoretical prediction itself in addition to the corresponding observation is treated as uncertain. The `discovery' of that implication and the evidence then raises the probability of the hypothesis in question in a straightforward way. I want to extend this to a problem which plagues estimating the prior probability of miracles, i.e. the problem of old evidence and explanation, or if you prefer, the old everything problem. When priors are not in dispute, this method is unnecessary; this is not an actual case of confirmation. But on the very safe assumption that the prior probability of R, in the absence of other argument, is calibrated with respect to confirmation of natural principles, it is quite useful. We `know' that the prior ratio is small, but we do not know how small. This is important when disputes depend on whether or not that prior is greater than 1/1000 or less than 10^-44. We need something better than the `vague intuitions' rightly deplored by the McGrews (CCRJ, p.50).

The idea is to posit an ur-prior u which is (partially) devoid of current background knowledge. The voided background knowledge is treated as uncertain in u, the recapturing of which yields through conditionalization a suitable prior p. In this context, the `recaptured' knowledge is the bulk of the `uniform evidence of sense' discussed by Hume, which is assumed to be wholly confirmatory of the law L. As I am working with the Resurrection, L is the principle that dead people remain dead. By definition and subset rule, we have for any probability pr that

.

By quick algebra, we yield the following inequality:

.

So by calculating our confidence in L, we set a maximum confidence in R.

As I mentioned, we want to `recapture' p using u by conditioning, that is,

,

where O is the set of observations confirming the putative law. But here I seem to have merely shifted the problem elsewhere, since we now need the u-odds on ~L. This regress is halted by the following abstract consideration: at some theoretical point of time, a hypothetical rational agent should have crossed the `more likely than not' threshold in favor of the law. For this reason, we may assume that the u-odds on ~L is 1. From there we must estimate the raw confirmatory force of the remaining experience as captured by the u-Bayes factor. Cleaning up the previous mess, we want to analyze

.

Now here is the crucial question: what does ~L look like after this partial recapitulation of background knowledge? An `anti-law' which states that all dead people resurrect will have probability zero, but this is an extreme; the preexisting information will select for those non-laws which ascribe higher probability to individual non-resurrections. These include, amongst other things, statements like "everybody who dies remains dead except under very rare conditions." However, even this will be made less probable by O so long as the further prediction of its elements are uncertain. Unless one claims to be able to specify in advance who should be exempted from death based on limited confirmation, the effect remains strong as regards statements like "everybody stays dead except for Jesus." (Lazarus and other proposed resurrections are not part of the background knowledge being recaptured here.)

To go further, I assume that O contains N instances of confirmation beyond those which lead our hypothetical agent to equivocate, each of which are of equal weight¹, i.e.



where I use the combinatorial notation [N]={1,...,N} for the sake of cleanliness. Supposing that each element of O is independent modulo ~O, we have that

.

So if N=10⁶, even a quite large value for u(O₁|~L) yields an absurdly small upper bound on the prior odds on R. If this assumption of independence is to fail in favor of the theists, there must be in the domain of u some significant set of propositions in ~L that correlate the elements of O. In other words, there must be plausible causal processes not implying L which probabilistically tend to keep dead people dead.

And here is where the remainder of our scientific knowledge and experience come into play. We do not consider L to be a fundamental feature of the universe but a consequence of other properties. Were a technology developed that allowed for the reanimation of corpses, we need not say that a putative law of nature had been overturned; instead, we would say that the `ordinary course' had been altered by the introduction of a new yet naturalistic element. Without such technology, we expect other principles to produce L. These include thermodynamic principles. When Jesus died, there are three sorts of miraculous possibilities: either his vital tissues never fatally decayed by supernatural sustenance, his vital tissues decayed but were reconstituted gradually or suddenly prior to his reanimation, or he reanimated without the function of his vital organs. Each of these possibilities runs against other established regularities. Properly speaking, then, L is not merely the generalized rule that dead people remain dead but also the set of combinations of rules which produce that outcome. For the Resurrection to occur, every one of those combinations must fail to hold.

So a theory which advantageously counters the above independence assumption must probabilistically correlate the elements of O, have significant plausibility in its own right, and fail to assume or imply any element in L, be it the putative law itself or any combination of principles which produce that law. In this case and in the case of other miracles which violate mass/energy conservation and/or thermodynamic principles, we must toss out modern science before examining any evidence proposed against its principles. This is an absurd task, but that absurdity is consequent to proposing a suspension of the natural order.

In such a manner I tentatively maintain that the independent assumption as employed is valid in establishing an estimated upper bound on the prior odds on R. So long as we agree that it is disgustingly low, a precise value is unimportant for the strengthening of the argument. Since as you may have noticed this method of unconfirmability argument requires some detailed information concerning the relevant miracle, specifying any particular value is unimportant. For an approximate value's import to materialize, we must also delve into the category of evidence in play.

Hume appears to recognize the need for categorical bounds on the strength of evidence in his essay:

When any one tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till then, can he pretend to command my belief or opinion. (EPHU, p.174)

I think that whenever Hume attempts to place such a bound on `testimonial evidence', he overreaches by virtue of that category's broadness. "I would not believe such a story were it told to me by Cato" (p.172) does not a sufficiently general argument make, nor are vague allusions to particular failures of human reporting which remain in the background, as these do not necessarily apply to all claims that humans make. Instead, it is better to focus on more task-specific categories. It might be possible to do so, but there is no reason to argue for more than is required.

Now I must specify what I mean by a categorical bound in formal terms: roughly, a categorical bound B is a real number such whenever evidence E for an uncertain hypothesis H lies in a set C, any Bayes factor produced by E is less than or equal to B.² That is,

.

If H is a miracle claim violating a putative law L supported by recaptured observations O, then tautologically the argument is successful if

.

What remains is to show that the evidence proposed by the miracle-claimant lies in such a set C.

What would C look like? If we want to be trivial, we can use uncontroversial examples. For example, two gamblers who have vastly differing prior odds on the fairness of a coin should not be able to resolve their dispute by tossing the coin a small number of times to check the outcomes. With precise values available, one could derive the minimum number of tosses which could possibly result in agreement.

One may extend this notion by thinking of the epistemic limitations on experiments like "tossing a coin a certain number times" as instances of methodological bounds. In the medical sciences, anecdotal and testimonial evidence as applied to broad categories of claims is almost completely useless for confirmation; careful study is required. What this reflects is a well-founded confidence in systematic errors frequently present in anecdotal reports which cannot be sufficiently discounted in the absence of controlled study. Anecdotal evidence for a novel, popular medical hypothesis cannot by nature discount those established theories and credences, a task which is necessary in order to confirm the new remedy against a significant prior implausibility.

Similar considerations apply to many classes of potential miracles and the evidence most commonly presented for them. With the Resurrection, we need not attempt the task Hume appears to set for himself in arguing that any set of witness testimony is incapable of producing convincingly calibrated Bayes factors above something like 10¹⁰⁰⁰. In this case, we may instead suggest that the methods of historians in establishing the historicity of those reports and their background, while possibly very good, are not so good as to allow a confident assertion that the historical record produces a Bayes factor above 10¹⁰⁰⁰. Myth and utter fabrication, even if wildly improbable in this case, have precedents, and I do not think they are capable of being discarded with absolute or nearly absolute confidence given the difficulties in method.

Hoping my proposals have been convincing thus far, I conclude by giving the sort of conversation which two Bayesians could have.

Christian: Ah, skeptic! Just the person I was wanting to see! I have crafted a convincing case that the Resurrection did in fact occur, and I was wanting your feedback.

Skeptic: That sounds very interesting, but before we go into the details, would you consider the Resurrection, had it occurred, to have constituted a suspension of the natural order?

Christian: Certainly, as you well know. Otherwise, the Resurrection would be meaningless. If Jesus' Resurrection were, say, a mere product of absurdly unlikely but possible quantum fluctuations, then any argument to theism or Christianity from that event would be undermined. It would be an isolated physical anomaly; nothing more, nothing less, and surely not a communicative sign of divine endorsement of the validity of Jesus' teachings.

Skeptic: I'm glad that you and I agree. And within this possibly suspended natural order, would you admit that local, epistemic generalizations hold and that tremendous confidence in those generalizations is yielded by what Hume would call `uniform sense data'?

Christian: While I obviously do not accept Hume's argument, I agree that incredulity prior to the examination of the evidence is wholly reasonable. That is why I have analyzed the evidence; one cannot say prior to analysis whether or not the evidence is sufficient or insufficient. You and I both know that this is merely a matter of probability.

Skeptic: I do not accept Hume's argument either, but one may with certain conditions be able to obviate the need to examine all of the details in advance by grounding a posteriori bounds on Bayes factors produced by certain types of evidence.

Christian: Part of this worries me, as it sounds like an excuse to avoid examining the evidence, which, as a good skeptic and Bayesian, you should be interested in doing. In any case, I suspect that such an approach would undermine important areas of scientific research, were it to be accepted.

Skeptic: I admit that this idea is a time-saver and has its ideological attractions, but allow me to specify some of those conditions. Hopefully, when you are satisfied with their stringency, your worries will vanish. But before I may do so, I nevertheless must know the nature of the argument that you are proposing. As I said before, the bounds to which I refer would be a posteriori, not some analytic consequence of Kolmogorov's axioms, uncontroversial metaphysical theses, or sound subjective Bayesian principles. I could not pretend to Hume's rhetoric and claim "an everlasting check to all kinds of superstitious delusion," or claim to have silenced any potential reasoned argument on your part or the part of your comrades in arms, be they future comrades or present confederates. In order to state exactly what I can say in advance of detailed analysis, I have to consider at least some of the details.

Christian: That at least sounds more interesting than another platitudinous regurgitation of Hume's breathless meanderings. Fine, I will play along. I am arguing, as against many prominent skeptics, to and from the historicity of the texts with respect to several key facts, especially those facts concerning the secular claims of witness testimony.

Skeptic: Have you accorded these facts certainty in your analysis as opposed to a more general analysis, for example using Jeffrey conditioning or classical conditioning on a partition of the historical possibilities?

Christian: For the facts concerning the witnesses, I strengthened the relevant arguments so as to make those facts not only plausible, but so overwhelmingly likely as to ensure that errors of omission do not seriously undermine the strength of the argument.

Skeptic: How overwhelmingly likely?

Christian: I think that I see roughly where you are heading with this. By your earlier hintings, it is clear that you are relying on some estimate of the prior odds of the Resurrection. Riddle me this: How do you propose to estimate prior odds on the Resurrection in any convincing way? You and I are both critics of equivocation and objective Bayesianism. You and I both acknowledge the limitations of current theories of calibration, especially as applied to claims like the Resurrection.

Skeptic: Properly speaking, I do no such thing.

Christian: Help me out here.

Skeptic: I rely on the notion that miracles, to occur, require a suspension of the natural order. As you have probably anticipated, I rely on the epistemic status of that natural order with respect to any potential exemption to gauge a suitable prior on the Resurrection...

Christian: Sorry to interrupt, but clearly you seem to be contradicting yourself.

Skeptic: Only if you assume that I need a specific range of prior odds. Instead, I use the deductive implications of putative laws to straightforwardly derive inequalities via the subset rule which by basic algebra translate into an upper bound on the prior odds of the Resurrection. I only need inequalities and bounds, not specific, well-defined ranges of reasonable discussion.

Christian: Ah, I see. You're assuming that the only relevant calibrating factor is the relation of a potential suspension of the natural order to the epistemic status of the natural order, of course.

Skeptic: That's right; hence why I do not claim that my approach, even if valid, would constitute the end of the discussion. One may still need to engage the evidence, but only if an adequate, well-established natural theology is formulated so as to calibrate the priors differently.

Christian: Which of course would present a serious difficulty, since the primary and standard means of evidentially filtering Christianity out of the more general category of theism is by arguing for the Resurrection. Now I am curious: supposing you could bound the prior odds on the Resurrection below 10^-1000, what would you say to me if I claimed to have produced a Bayes factor based on a confidence some salient facts which is greater than 10¹⁰⁰⁰?

Skeptic: I would say that you have proposed the a posteriori equivalent of proving the rationality of the square root of two, Euler's number, or pi.

Christian: That's quite a strong statement; how do you mean it?

Skeptic: I might agree that your proposed facts are plausible, even extremely convincing. But I would nevertheless insist that they cannot be sufficiently plausible as to yield such a factor. Formally, I would put evidence like that you have proposed into a set of similar evidences and claim that Bayes factors in favor of the Resurrection produced by an element in that set are below 10¹⁰⁰⁰.

Christian: In which case, your argument would be tautological or trivial unless you can convincingly establish, before engaging all of the details, that my textual evidence cannot be stronger. Again, I do not see how you are avoiding the shortcomings of Hume.

Skeptic: Well, you have surely noted my insistence on your specifying the type of evidence in question. I doubt you fail to imagine how that might be relevant.

Christian: I have a rough idea: are you proposing theses, like those of Hume, against testimonial evidence? Just because testimonial evidence is always subject to some precedented, possible counter-thesis, that does not mean that one can say that testimonial evidence as a category must be at least this or that weak by that virtue. The details decide how significant those considerations need to be.

Skeptic: I agree, which is another reason why I do not claim to be vindicating Hume's essay. `Testimonial evidence' is perhaps too broad a category to be subject to sufficiently small, convincing categorical boundaries. As I said before, some specifics are required. Allow me to motivate those which apply to the textual record on which you plan to rely: you can envision cases where an experiment, by its nature, cannot overcome discrepancies in prior odds so as to yield agreement between two rational agents, correct?

Christian: In highly idealized scenarios like fair dice rolls or well-understood machines and programs, sure, but I do not see the relevance to a scenario so complex and multivariate as eyewitness testimony.

Skeptic: You may at least be able to anticipate a generalization of simple and uncontroversial lessons to broader notions like `historical methodology', correct?

Christian: Not exactly, as I see such a category as too vague to easily bound.

Skeptic: Again, it depends on specifics. What is the method which you used to arrive at your initial, secular factual claims? Presumably, you do not claim to have directly observed the events in question.

Christian: Of course not.

Skeptic: And so there is some significant uncertainty in the indirect inference methods, i.e. historical methods, which you employ?

Christian: At least in a trivial sense, but that need not translate into any boundary.

Skeptic: Actually, it does, unless you claim that there is no minimally significant alternative to your facts which your methods can not diminish to an arbitrary degree. For example, can you rule out as strongly as you like the possibilities of fraud and later myth-making with respect to these secular facts?

Christian: I wouldn't say that, but again, I see no reason why, in advance, I can not devalue such possibilities sufficiently as to overcome the prior implausibility of the Resurrection.

Skeptic: If by `in advance' you mean in advance of all background knowledge, surely you are correct, but I mean the reliability of your methods with respect to our current knowledge about its reliability. If that reliability is such that the probabilities of hypotheses like frauds and myths cannot be convincingly grounded below 10^-1000, I have established my case. For such extreme values, I would say that this can be said further in advance than I need to argue, but to firmly secure your methods into the category which I require, I will need to know a few more specifics.

Christian: I think I understand now, and I think that I see how you will be able to secure your conclusions were I to spell out more details. I suppose that I will have to qualify my paper with a placeholder for the time being and play with the formalisms to double-check your statements.

Skeptic: That sounds fine. In the meantime, I would be happy to read your paper. After all, you might be able to calibrate the relevant priors differently. It is still worth reading, for this and other reasons, if your conclusions are as strongly supported as you have suggested.

Christian: I look forward to your review. However, I hope you only resume technical blogging after all that wine you just drank leaves your system.

Skeptic: You're breaking the fourth wall.

---

1. The simplifying assumption of equal-weightedness is not generally true. If alternatives to the law include something like `dead people remain dead unless you perform a certain magic ritual', then only failures of that ritual will contrast the hypothesis with L. We can recapture the plausibility of the assumption by stipulating that a theoretical agent at this theoretical threshold point has effectively ruled any particular such hypothesis.

2. I've been playing with this notion for some time, and I know of several generalizations if anyone is interested.

Hume and the confirmability of miracles

While discussing the McGrews' Bayesian analysis of the Resurrection (CCRJ), I frequently mentioned that the McGrews and myself were not attempting to derive odds on the Resurrection. Rather, we were focusing on the Bayes factor - aka the likelihood ratio - which if you recall, is the number by which the prior odds ratio p(`thing')/p(`other thing') is multiplied to yield the posterior odds ratio q(`thing')/q(`other thing'). So at the very minimum, one needs to estimate what the prior odds should be in order to derive the final odds. Since neither of our approaches were sufficiently general to capture a truly cumulative Bayes factor, even this may be inadequate, but since the factor I derived - 10⁶ - was calibrated given generous textual assumptions in favor of the Resurrection, we may be able to tentatively estimate an upper bound on reasonable posterior odds using that factor if we have an upper bound on reasonable priors. If that upper bound is less than 10^-6, we may conclude that the Resurrection probably did not occur, i.e. 0.5>q(R).

I opine that barring other arguments in the context of a natural theology, such an upper bound exists. That is, the Resurrection can not reasonably be confirmed from the textual record alone with respect to convincing background knowledge which is shared by skeptics and Christians alike. But I am interested in Hume's more general thesis, which is that miracles by their nature can not reasonably be confirmed. The details of an analysis of the Resurrection are merely an instance of a more general unconfirmability argument.

I pause to obviate a potential objection: I am fully aware that the proper interpretation of Hume's Enquiry concerning Human Understanding, especially Of Miracles, is a hotly disputed topic. I am attributing an unconfirmability argument to Hume; I doubt he can be convincingly interpreted as not making such an argument. But I am not here interested in any historical exoneration or conviction of Hume of philosophical crimes. Rather, I want to work with his apparent argument by recasting it in formal terms, propose that it is inadequate, and attempt to shape a more satisfactory unconfirmability argument.

I am neither discussing nor proposing a definition-dependent impossibility argument, e.g., "A miracle is the violation of mathematical, divine, immutable, eternal laws. By the very exposition itself, a miracle is a contradiction in terms: a law cannot at the same time be immutable and violated." Rather, I am interested in miracles loosely defined as particular exceptions to otherwise exceptionless laws or regularities of nature. In this context, the regularities of importance are putative laws, and their importance is epistemic, not ontological. Whether or not a putative law is true is largely irrelevant: what matters is how well-supported it is.

The proper definition of a miracle is also in dispute. As regards this item, I follow Tim and Lydia McGrew in treating the Resurrection as a paradigm (CCRJ, p.4). As I will explain in the course of this discussion, working with paradigm cases like the Resurrection is all that should be required. Failures of consensus on miracles are relevant to Hume's argument, but they will not prove necessary to reject his conclusion. Which is roughly as follows:

Hume's Argument: There cannot exist evidence E for a miracle M such that

.

With conditionalization, this is equivalent to stating that the posterior odds on a miracle can never be greater than 1/2.¹

The initial prospects of this statement are rather dim. As is commonly pointed out, there are no a priori boundaries on the size of Bayes factors: no matter how small the prior odds on a miracle, there exist finite Bayes factors which can overcome them. Similarly, disputes concerning the definition of a miracle make it impossible to have confidence in such a general statement.

I would also add that we should not be overly interested in such an argument as employed to justify ignoring any potential evidence for miracles. The debate is worthwhile. As I have noted in a slightly different context:

It is very often said, by e.g. PZ Myers and Massimo Pigliucci, that one cannot evidence Christianity or gods because they are not coherent hypotheses. More needs to be said about this, but I would at least suggest the following: if the evidence for the Resurrection really is extremely convincing to reasonable people on the assumption of coherency, we should take an attitude similar to that which I think we take to science: some conceptual fuzziness is to be tolerated, barring flat contradiction, where overwhelming evidence for an aspect of a theory is available. Were I to find the evidence for the Resurrection convincing, I know I would be working very diligently to craft a coherent Christian philosophy to accommodate it. So to me, the coherency difficulty is in many ways secondary, unless that difficulty is so severe that one cannot even begin to discuss relevant evidence. I think we usually manage to do so. Wouldn't you agree?

The argument also conflicts with the empirical, tentative nature of skeptical inquiry. I think we should wish to avoid such categorical statements.

The temptation to fashion such an argument is understandable. But it should be resisted. Any epistemology that does not allow for the possibility that evidence, whether from eyewitness testimony or from some other source, can establish the credibility of a UFO landing, a walking on water, or a resurrection is inadequate. (Earman, p.4)

As Earman also notes, there are events which, were they to occur, surely amount to convincing evidence for a miracle claim:

Suppose, for the sake of illustration, that there is a well developed theology based on the existence of a god called Emuh. who promises an afterlife in return for certain religious observances in this life. Suppose that this theology predicts that on such-and-such a day Emuh will send a sign in the sky. And suppose that on the appointed day, the clouds over America clearly spell out in English the words “Believe in Emuh and you will have everlasting life,” while the same message is spelled out in French over France, in Deutsch over Germany, etc. Then even though these cloud formations may not contravene any of the general principles taken at the time in question to be laws of nature and, indeed, may be explicable in terms of those principles, it would not be untoward to take these extraordinary occurrences to be support for Emuh theology. (p.11)

So Hume's argument, were it valid and coherent, would prove too much. It also ignores the effect of evidence for a theology and its implications for the proposed miracle claim. Were the gloating fiction of LaHaye's Left Behind series to be actualized, it would confirm the Resurrection. I am unsure as to how or why someone would seriously argue otherwise, even if the various details of Christian theology are unclear.

I will not second another common objection: I do not think that Hume's argument, interpreted as I have interpreted it, would destroy the possibility of overturning laws in science. His "straight rule of induction" is problematic in this context (Earman, pp.31-2), but laws are not to my knowledge overturned in the way that a putative miracle claim would overturn them.

Take the Conservation of Mass. Did measurements of nuclear reactions overturn a uniform experience? No, because what changed was not the article where uniform experience applied, but where a novelty was being analyzed. If for example I react 50 grams of sodium with 70 grams of chlorine gas to form salt (NaCl) at approximately standard temperature and pressure and measure a net change in mass of 20 grams, I or my instruments screwed up. Mass is still conserved within significant margins of instrumental error for `ordinary' chemical reactions. The implications of such experience have not been contradicted, but superseded. With miracles, where the putative law needs to otherwise be intact for theological reasons, no such consideration applies. We are not talking supercessions or the overturning of laws as done in the sciences; we are talking about flat-out, singular violations of an otherwise sound natural order. We are talking about an experiment incapable of replication. Were mass conservation to always hold, and the only exception were to occur in one apparently sound experiment, we should have discounted the experiment as flawed if replication failed.

I judge Hume's argument to be a failure and its conclusion to be unsound. But this need not be the end of the story. We may yet build a better monument by clearing away the noisy rubble of Hume's rhetoric and picking out the useful pieces. That will be the subject of my next post.

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1. This interpretation, though disputed, has a lot of support, perhaps apart from the target threshold 0.5>q(M). Were Hume making an impossibility argument, it is odd that he should emphasize the relative strength of evidences (p.169) and the uncertainty of the relevant propositions (pp.169-70); that he discusses evidence at all would also be strange. In addition to his use of probabilistic terminology, he also casts his argument in terms of degrees-of-confidence: "Some events are found, in all countries and all ages, to have been constantly conjoined together: Others are found to have been more variable, and sometimes to disappoint our expectations; so that, in our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence." And he continues famously: "A wise man, therefore, proportions his belief to the evidence. In such conclusions as are founded on an infallible experience, he expects the event with the last degree of assurance, and regards his past experience as a full proof of the future existence of that event" (p.170).

With terms like `infallible' and `proof' in play, I think that Hume may be interpreted as arguing for a prior probability of zero for miracles, or perhaps an infinitesimal probability (p.171). But I do not think that such a prior is convincing to all - or many - concerned, and it is therefore useless. Many Bayesians accept - here the terminology is unfortunate - the principle of regularity, which states that all possibilities have probability greater than 0, assuming that those possibilities are uncertain and assigned any probability whatever. In any case, we are presumably inviting Christians to the discussion, so we must at least assume that non-zero priors are in play.

There are other ambiguities, and I am lead to second Earman's hostile conclusion (p.20):

I defy the reader to give a short, simple, and accurate summary of the argumentation in "Of Miracles." What on first reading appears to be a seamless argument is actually a collection of considerations that sometimes mesh and sometimes don't. It will take much work to tease out the components of Hume's argument and to evaluate the soundness of individual components and the effectiveness of the entire package.

Immediately after bringing up `proofs' of experience, Hume dives right back into emphasizing the fallibility of evidence, particularly witness testimony (EPHU, p.171). There are other deficiencies in his presentation. After defining miracles as putative exceptions to uniformly evidenced laws, he states the following:

There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appelation. And as a uniform experience amounts to a proof, there is here a direct and full proof, from the nature of the fact, against the existence of any miracle; nor can such a proof be destroyed, or the miracle rendered credible, but by an opposite proof, which is superior. (p.173)

With this stringency, the mere proposal of evidence for an event disqualifies that event's being a miracle, as experience is no longer uniformly against it. And then, the presentation of additional evidence should leave an opening. I'll stick with the Bayesian interpretation because it is the only plausible interpretation to be found.

Hello lurkers!

I hope that there are lurkers, as there are currently no commenters. That's alright; I did not start this blog in order to accrue a following. But if readers would introduce themselves and their interests, I would be quite happy! I could turn my attention to your interests. Hopefully, I will manage to deter you.

I've been away for several days to help a friend move. My apologies. Thanks to a slight injury, I have further perfected the very-much-not-recommended art of self-surgery. I'm pretty proud of myself at the moment.

On the way to the new place, I decided to put my rationality skills to the test. Boredly approaching the Atlantic shoreline in the Carolinas, I made an estimate about the frequency of pine trees as one approaches the beach. I came up with the following estimate as derived from highway observations:

- Approx or above 50% pines, less than 50% deciduous within 150 miles
- Approx or about 75% pines, less than 25% deciduous within 50 miles

where this estimate completely fails close to riversides.

Now here is the question I sought to intuitively answer (and mostly failed to correctly intuit): Why is it that pines/needle-trees become increasingly common once one approaches the Southeast US coast? I recall seeing this before across the Carolinas, Florida, and etc.

Any ideas?