In his Philosophical Theories of Probability, Donald Gillies proposes what he calls an intersubjective theory of probability. A better name for it would be group-strategy model of probability.

Subjectivists such as Bruno di Finetti ask the reader to consider the following sort of game:

• Some potential event is identified.
• Our hero must choose a real number (negative or positive) q, a betting quotient.
• The nemesis, who is rational, must choose a stake S, which is a positive or negative sum of money or zero.
• Our hero must, under any circumstance, pay the nemesis q·S. (If the product q·S is negative, then this amounts to the nemesis paying money to our hero.)
• If the identified event occurs, then the nemesis must pay our hero S (which, if S is negative, then amounts to taking money from our hero). If it does not occur, then our hero gets nothing.

Di Finetti argues that a rational betting quotient will capture a rational degree of personal belief, and that a probability is exactly and only a degree of personal belief.

Gillies asks us to consider games of the very same sort, except that the betting quotients must be chosen jointly amongst a team of players. Such betting quotients would be at least examples of what Gillies calls intersubjective probabilities. Gillies tells us that these are the probabilities of rational consensus. For example, these are ostensibly the probabilities of scientific consensus.

Opponents of subjectivists such as di Finetti have long argued that the sort of game that he proposes fails in one way or another to be formally identical to the general problem for the application of personal degrees of belief. Gillies doesn't even try to show how the game, if played by a team, is formally identical to the general problem of group commitment to propositions. He instead belabors a different point, which should already be obvious to all of his readers, that teamwork is sometimes in the interest of the individual.

Amongst other things, scientific method is about best approximation of the truth. There are some genuine, difficult questions about just what makes one approximation better than another, but an approximation isn't relevantly better for promoting such things as the social standing as such or material wealth as such of a particular clique. It isn't at all clear who or what, in the formation of genuinely scientific consensus, would play a rôle that corresponds to that of the nemesis in the betting game.

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Karl Popper, who proposed to explain probabilities in terms of objective propensities (rather than in terms of judgmental orderings or in terms of frequencies), asserted that

Causation is just a special case of propensity: the case of propensity equal to 1, a determining demand, or force, for realization.

Gillies joins others in taking him to task for the simple reason that probabilities can be inverted — one can talk both about the probability of A given B and that of B given A, whereäs presumably if A caused B then B cannot have caused A.

Later, for his own propensity theory, Gillies proposes to define probability to apply only to events that display a sort of independence. Thus, flips of coins might be described by probabilities, but the value of a random-walk process (where changes are independent but present value is a sum of past changes) would not itself have a probability. None-the-less, while the value of a random walk and similar processes would not themselves have probabilities, they'd still be subject to compositions of probabilities which we would previously have called probabilities.

In other words, Gillies has basically taken the liberty of employing a foundational notion of probability, and permitting its extension; he chooses not to call the extension probability, but that's just notation. Well, Popper had a foundational notion of propensity, which is a generalization of causality. He identified this notion with probability, and implicitly extended the notion to include inversions.

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Later, Gillies offers dreadful criticism of Keynes. Keynes's judgmental theory of probability implies that every rational person with sufficient intellect and the same information set would ascribe exactly the same probability to a proposition. Gillies asserts

[…] different individuals may come to quite different conclusions even though they have the same background knowledge and expertise in the relevant area, and even though they are all quite rational. A single rational degree of belief on which all rational being should agree seems to be a myth.

So much for the logical interpretation of probability, […].

No two human beings have or could have the same information set. (I am reminded of infuriating claims that monozygotic children raised by the same parents have both the same heredity and the same environment.) Gillies writes of the relevant area, but in the formation of judgments about uncertain matters, we may and as I believe should be informed by a very extensive body of knowledge. Awareness that others might dismiss as irrelevant can provide support for general relationships. And I don't recall Keynes ever suggesting that there would be real-world cases of two people having the same information set and hence not disagreeing unless one of them were of inferior intellect.

After objecting that the traditional subjective theory doesn't satisfactorily cover all manner of judgmental probability, and claiming that his intersubjective notion can describe probabilities imputed by groups, Gillies takes another shot at Keynes:

When Keynes propounded his logical theory of probability, he was a member of an elite group of logically minded Cambridge intellectuals (the Apostles). In these circumstances, what he regarded as a single rational degree of belief valid for the whole of humanity may have been no more than the consensus belief of the Apostles. However admirable the Apostles, their consensus beliefs were very far from being shared by the rest of humanity. This became obvious in the 1930s when the Apostles developed a consensus belief in Soviet communism, a belief which was certainly not shared by everyone else.

Note the insinuation that Keynes thought that there were a single rational degree of belief valid for the whole of humanity, whereäs there is no indication that Keynes felt that everyone did, should, or could have the same information set. Rather than becoming obvious to him in the 1930s, it would have been evident to Keynes much earlier that many of his own beliefs and those of the other Apostles were at odds with those of most of mankind. Gillies' reference to embrace of Marxism in the '30s by most of the Apostles simply looks like irrelevant, Red-baiting ad hominem to me. One doesn't have to like Keynes (as I don't), Marxism (as I don't) or the Apostles (as I don't) to be appalled by this passage (as I am).

There is a hugely important difference between consent within a system and consent to that system.

Some examples:

• Many people object to some matters being decided by bullet; they think that such violence is a Bad Thing. But that doesn't make them hypocrites if they respond to shooting or to the threat of shooting with bullets of their own.
• Some of us object to some things being decided by ballot; we feel that some things (such as the ability of consenting adults to marry) are rights that cannot be taken (though power may be taken) no matter how many people object. But that doesn't make us hypocrites when we respond to voting or to the threat of voting with ballots of our own.
• Quite a few people think that Social Security is a Bad Idea. But that doesn't make them hypocrites if they accept it when offered; they were forced to pay into the system, and they may conclude that refusing to take the money may have no marginal effect on whether it continues.

It is a separate issue whether returning fire, voting in elections that one feels should not be held, or consuming entitlements that one believes should not exist would be wise practical responses; the point is that none of these actions is hypocritical.

Okay, I'm going to presume that all my readers recognize the class of distinction upon which I'm focussing.

So, to-night, I saw NBC News present a report on United States Senators and Representatives who have voted against stimulus bills, yet had subsequently sought to get some of the monies therefrom for their respective districts. The report treated these people as hypocrites. The reporter repeatedly claimed that they'd somehow reversed themselves, and quoted others representing them as hypocrites; and no one was quoted offering any sort of explanation of why this would not be hypocrisy. The only defense quoted was merely that of one congressman, allowed to explain that he thought that seeking monies for which his constituents had paid was in their interests.

There's a saying that reäctions depend upon whose ox gets gored, but they also depend upon whose bull does the goring. Had such a programme been suggested by a high-ranking member of the previous Administration, the main-stream media would have directed considerable attention to it and to objections. There are people who will be quite silent now, or will even defend the proposal, who made a habit of furiously denouncing that previous Administration for the mere possibility that it might do such things when third parties suggested that they would.

Probability is one elephant, not two or more formally identical or formally similar elephants.

There's a huge amount of utterly useless meta-discussion in Gillies' book. He writes about his writing without providing any meaningful enlightenment whatsoëver.

An editor should have put his or her foot down, and told Gillies to trim away all this fat. But, again, there doesn't seem to be much editing of books these days, except when done by authors themselves.

A lot was and is again being made of Alexander Haig's declaration on 30 March 1981, when the President was shot and rushed to hospital, I'm in charge here. Frankly, I think that his remark at that time was willfully misinterpretted by various hostile parties. On the other hand, I remain disgusted by something that Haig said just twelve days earlier, on 18 March.

In 1980, El Salvador was caught-up in civil war, with the United States supporting the Salvadoran state against left-wing insurgents. On 2 December, four American church-women (a lay missionary and three nuns) were beaten, raped, and murdered by members of the Salvadoran National Guard.

As these events were being uncovered, Haig was called before the Foreign Affairs Committee of the United States House of Representatives. Haig told the Committee

I'd like to suggest to you that some of the investigation would lead one to believe that perhaps the vehicles the nuns were riding in may have tried to run a roadblock, or was accidentally perceived to have been running a roadblock and there may have been an exchange of gunfire and perhaps those who inflicted the casualities may have tried to cover it up, and this could have been at a very low level of both competence and motivation in the context of the issue itself.

(Underscore mine.) By referring to an exchange of gun-fire, Haig tried to creäte an impression that these women had tried to shoot it out with the National Guard.

Haig went to his grave mocked for attempting to tell the nation that the White House was under active leadership while the President was incapacitated and the Vice President was in-transit. Haig faced minimal consequences for having attempted to depict four unarmed victims of torture and murder as guerillas who had simply died a gun-fight.

I first encountered symbolic logic when I was a teenager. Unfortunately, I had great trouble following the ostensible explanations that I encountered, and I didn't recognize that my perplexity was not because the underlying subject were intrinsically difficult for me, but because the explanations that I'd found simply weren't very well written. Symbolic logic remained mysterious, and hence became intimidating. And it wasn't clear what would be its peculiar virtue over logic expressed in natural language, with which I was quite able, so I didn't focus on it. I was perhaps 16 years old before I picked-up any real understanding of any of it, and it wasn't until years after that before I became comfortable not simply with Boolean expression but with processing it as an algebra.

But, by the time that I was pursuing a master's degree, it was often how I generated my work in economics or in mathematics, and at the core of how I presented the vast majority of that work, unless I were directed otherwise. My notion of an ideal paper was and remains one with relatively little natural language.

Partly I have that notion because I like the idea that people who know mathematics shouldn't have to learn or apply much more than minimal English to read a technical paper. I have plenty of praise for English, but there are an awful lot of clever people who don't much know it.

Partly I have that notion because it is easier to demonstrate logical rigor by using symbolic logic. I want to emphasize that word demonstrate because it is possible to be just as logically rigorous while expressing oneself in natural language. Natural language is just a notation; thinking that it is intrinsically less rigorous than one of the symbolic notations is like thinking that Łukasiewicz Polish notation is less rigorous than infixing notation or vice versa. I'll admit that some people may be less inclined to various sorts of errors using one notation as opposed to another, but which notation will vary amongst these people. However, other people don't necessarily see that rigor when natural language is used, and those who are inclined to be obstinate are more likely to exploit the lack of simplicity in natural language.

But, while it may be more practicable to lay doubts to rest when an argument is presented in symbolic form, that doesn't mean that it will be easy for readers to follow whatever argument is being presented. Conventional academic economists use a considerable amount of fairly high-level mathematics, but they tend to use natural language for the purely logical work.[1] And it seems that most of them are distinctly uncomfortable with extensive use of symbolic logic. It's fairly rare to find it heavily used in a paper. I've had baffled professors ask me to explain elementary logical transformations. And, at least once, a fellow graduate student didn't come to me for help, for fear that I'd immediately start writing symbolic logic on the chalk-board. (And perhaps I would have done so, if not asked otherwise.)

The stuff truly isn't that hard, at least when it comes to the sort of application that I make of it. There is a tool-kit of a relatively few simple rules, some of them beautiful, which are used for the lion's share of the work. And, mostly, I want to use this entry to high-light some of those tools, and some heuristics for their use.

First, though, I want to mention a rule that I don't use. This proposition, normally expressed in natural language as A is A and called the Law of Identity, is declared by various philosophers to be one of the three Principles of logic. But I have no g_dd_mn'd idea what to do with it. It's not that I would ever want to violate it; it's just that I literally don't see anything useful to it. Ayn Rand and many of those for whom she is preceptrix treat it as an essential insight, but I think that it's just a dummy proposition, telling me that any thing can stand where that thing can stand.[2]

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[1] There's an idiotic notion amongst a great many mainstream economists that the Austrian School tradition is somehow less rigorous simply because some of its most significant members eschew overt mathematics in favor of logical deduction expressed in natural language. But most of the mainstream is likewise not using symbolic logic; neither is necessarily being less rigorous than otherwise. The meaning of variables with names such as q_t can be every bit as muddled as those called something such as the quantity exchanged at this time. There are good reasons to object to the rather wholesale rejection of overt mathematics by many Austrian School economists, but rigor is not amongst the good reasons.

[2] [Read more.]

Essai philosophique sur les probabilitiés was published a couple of years after Théorie analytique des probabilitiés, as a popular introduction to that earlier work. Objecting that Essai is not in fact a very original work, given that Théorie was the summary of more than a century and a half of mathematical research together with important developments by the author, is a bit absurd.

An editor should have brought this dissonance to Gillies' attention. I don't quite know what editors do these days, beyond deciding whether a given work may be expected to sell.

The Gallup Organization has acknowledged that the President's disapproval rating and approval rating are now matched, at 47%. I am highly skeptical that it took until now for that to happen.

[Correction and Up-Date (2010:01/24): It seems that initial reports were slightly off, that Gallup reported a 48% approval level and 47% disapproval level. And to-day they report both as simultaneously holding steady, skating against each other. (2010:01/25): Well, no, the Gallup Orgainzation indeed had them equal in their report for 20-22 Jan; then they showed his approval rating ticking up to 48% while the disapproval rating held steady.]

The Financial Times explains that things are almost certainly going to become more awkward for the President. The Republicans smell blood; moderate Democrats feel more free (or obliged) to say no. The Secretary of the Treasury is enmeshed in scandal over his actions when with the Federal Reserve; the Chairman of the Fed may not have the votes for reconfirmation.

I'd note other things. Unemployment has stayed high; some of the President's defenders say that there isn't much that he can do about that, but he and his party hugely increased the deficit on the claim that they could, with few people now believing that the money were well spent. The American automotive manufacturing industry has no real prospects for long-term health. The two wars that so many voters expected to be neatly or quickly resolved (one of which Obama said must be fought to victory) are still grinding-up American soldiers. Russia and China want Iran to continue to be a problem. The Guantanamo naval prison is still unclosed, and the ACLU has denounced the plan to continue holding prisoners without trial once they are relocated. Skepticism about anthropogenic climate change is growing, and supposed points of no return have been passed. The world still treats America with disdain, and much of it is on the cusp of telling us that Obama is a failure or that he's better than we deserve or both.

It is very likely that the Gallup Organization will one day report that this President's disapproval rating has passed 50%.

Readers of this 'blog might recall the Decimator. Well, according to Hugo Chávez, the United States has one.

I'm just hoping that it doesn't fall into the wrong hands, and get directed at the Amsterdam Fault. Meanwhile, maybe I can become one of the Rocket Men. At my age, hopes of becoming a super-hero have dimmed, but I at least look younger than Jeff King.