(Actually, there is some evidence that a quantitative theory of probability developed and then disappeared in ancient India.[10] But Franklin's essential point here is none-the-less well-taken.)

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⁹ Foot-note in the original, citing Decretales Pseudo-Isidorianae, et Capitula Angilramni edited by Paul Hinschius, and recommending comparison with The Collection in Seventy-Four Titles: A Canon Law Manual of the Gregorian Reform edited by John Gilchrist.

[10] In The Story of Nala and Damayanti within the Mahābhārata, there is a character Rtuparna (aka Rituparna, and mistakenly as Rtupama and as Ritupama) who seems to have a marvelous understanding of sampling and is a master of dice-play. I learned about Rtuparna by way of Ian Hacking's outstanding The Emergence of Probability; Hacking seems to have learned of it by way of V.P. Godambe, who noted the apparent implication in A historical perspective of the recent developments in the theory of sampling from actual populations, Journal of the Indian Society of Agricultural Statistics v. 38 #1 (Apr 1976) pp 1-12.

A previous entry quotes a foot-note from Austrian Marginalism and Mathematical Economics by Karl Menger; that foot-note is tied to a sentence that I found particularly striking.

(musings on the relationship of mathematics to economics)

But to assume from the superiority of Galilean principles in the sciences of inanimate nature that they must provide the model for the sciences of animate behaviour is to make a speculative leap, not to enunciate a necessary conclusion.

Whether a decision as such is good or bad is never determined by its actual consequences as such.

Decisions are made before their consequences are reälized (made actual). Instead, decisions are made in the face of possible consequences. There may be an ordering of these consequences in terms of plausibility, in which case that ordering should be incorporated into the making of the decision. Most theories even presume that levels of plausibility may be meaningfully quantified, in which case (ex hypothesi) these quantifications should be incorporated into the process. But even in a case where there were only one outcome possible, while the decision could (and should) be made in response to that unique possibility, it still were possibility of the consequence that informed the decision, and not actuality. (Inevitability is not actuality.)

When the reälized consequences of a decision are undesirable, many people will assert or believe that whoever made the choice (perhaps they themselves) should have done something different. Well, it might be that a bad outcome illustrates that a decision were poor, but that will only be true if the inappropriateness of the decision could have been seen without the illustration. For example, if someone failed to see a possibility as such, then its reälization will show the possibility, but there had to have been some failure of reasoning for a possibility to have ever been deemed impossible. On the other hand, if someone deemed something to be highly unlikely, yet it occurred anyway, that doesn't prove that it were more likely than he or she had thought — in a world with an enormous number of events, many highly unlikely things happen. If an event were highly unlikely but its consequences were so dire that they should have been factored into the decision, and yet were not, the reälization of the event might bring that to one's attention; but, again, that could have been seen without the event actually occurring. The decision was good or bad before its consequences were reälized.

A painter whose canvas is improved by the hand of another is not a better painter for this, and one whose work is slashed by a madman (other than perhaps himself) is not a worse painter for that. Likewise, choosing well is simply not the same thing as being lucky in one's choice, and choosing badly not the same as being unlucky.

Sometimes people say that this-or-that should have been chosen simply as an expression of the wish that more information had been available; in other cases, they are really declaring a change in future policy based upon experience and its new information. In either case, the form of expression is misleading.

Some readers may be thinking that what I'm saying here is obvious (and some of these may have abandoned reading this entry). But people fail to take reasonable risks because they will or fear that they will be thought fools should they be unlucky; some have responded to me as if I were being absurd when I've referred to something as a good idea that didn't work; our culture treats people who attempt heinous acts but fail at them as somehow less wicked than those who succeed at them; and I was drawn to thinking about this matter to-day in considering the debate between those who defend a consequentialist ethics and those who defend a deöntological ethics, and the amount of confusion on this issue of the rôle of consequences in decision-making (especially on the side of the self-identified consequentialists) that underlies that debate.

When someone uses the word random, part of me immediately wants a definition.[1]

One notion of randomness is essentially that of lawlessness. For example, I was recently slogging through a book that rejects the proposition that quantum-level events are determined by hidden variables, and insists that the universe is instead irreducibly random. The problem that I have with such a claim is that it seems incoherent.

There is no being without being something; the idea of existence is no more or less than that of properties in the extreme abstract. And a property is no more or less than a law of behavior.

Our ordinary discourse does not distinguish between claims about a thing and claims about the idea of a thing. Thus, we can seem to talk about unicorns when we are really talking about the idea of unicorns. When we say that unicorns do not exist, we are really talking about the idea of unicorns, which is how unicorns can be this-or-that without unicorns really being anything.

When it is claimed that a behavior is random in the sense of being without law, it seems to me that the behavior and the idea of the behavior have been confused; that, supposedly, there's no property in some dimension, yet it's going to express itself in that dimension.

Another idea of randomness is one of complexity, especially of hopeless complexity. In this case, there's no denial of underlying lawfulness; there's just a throwing-up of the hands at the difficulty in finding a law or in applying a law once found.

This complexity notion makes awfully good sense to me, but it's not quite the notion that I want to present here. What unites the notion of lawlessness with that of complexity is that of practical unpredictability. But I think that we can usefully look at things from a different perspective.

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After the recognition that space could be usefully conceptualized within a framework of three orthogonal, arithmetic dimensions, there came a recognition that time could be considered as a fourth arithmetic dimension, orthogonal to the other three. But, as an analogy was sensed amongst these four dimensions, a puzzle presented itself. That puzzle is the arrow of time. If time were just like the other dimensions, why cannot we reverse ourselves along that dimension just as along the other three. I don't propose to offer a solution to that puzzle, but I propose to take a critical look at a class of ostensible solutions, reject them, and then pull something from the ashes.

Some authors propose to find the arrow of time in disorder; as they would have it, for a system to move into the future is no more or less than for it to become more disorderly.

One of the implications of this proposition is that time would be macroscopic; in sufficiently small systems, there is no increase nor decrease in order, so time would be said neither to more forward nor backward. And, as some of these authors note, because the propensity of macroscopic systems to become more disorderly is statistical, rather than specifically absolute, it would be possible for time to be reversed, if a macroscopic system happened to become more orderly.

But I immediately want to ask what it would even mean to be reversed here. Reversal is always relative. The universe cannot be pointed in a different direction, unless by universe one means something other than everything. Perhaps we could have a local system become more orderly, and thus be reversed in time relative to some other, except, then, that the local system doesn't seem to be closed. And, since the propensity to disorder is statistical, it's possible for it to be reversed for the universe as a whole, even if the odds are not only against that but astronomically against it. What are we to make of a distinction between a universe flying into reverse and a universe just coming to an end? And what are we to make of a universe in which over-all order increases for some time less than the universe has already existed? Couldn't this be, and yet how could it be if the arrow of time were a consequence of disorder?

But I also have a big problem with notions of disorder. In fact, this heads us back in the direction of notions of randomness.

If I take a deck of cards that has been shuffled, hand it to someone, and ask him or her to put it in order, there are multiple ways that he or she might do so. Numbers could be ascending or descending within suits, suits could be separated or interleaved, &c. There are as many possible orderings as there are possible rules for ordering, and for any sequence, there is some rule to fit it. In a very important sense, the cards are always ordered. To describe anything is to fit a rule to it, to find an order for it. That someone whom I asked to put the cards in order would be perfectly correct to just hand them right back to me, unless I'd specified some order other than that in which they already were.

Time's arrow is not found in real disorder generally, because there is always order. One could focus on specific classes of order, but, for reasons noted earlier, I don't see the explanation of time in, say, thermodynamic entropy.

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But, return to decks of cards. I could present two decks of card, with the individual cards still seeming to be in mint state, with one deck ordered familiarly and with the other in unfamiliar order. Most people would classify the deck in familiar order as ordered and the other as random; and most people would think the ordered deck as more likely straight from the pack than the random deck. Unfamiliar orderings of some things are often the same thing as complex orderings, but the familiar orderings of decks of cards are actually conventional. It's only if we use a mapping from a familiar ordering to an unfamiliar ordering that the unfamiliar ordering seems complex. Yet even people who know this are going to think of the deck in less familiar order as likely having gone through something more than the deck with more familiar order. Perhaps it is less fundamentally complexity than experience of the evolution of orderings that causes us to see the unfamiliar orderings as random. (Note that, in fact, many people insist that unfamiliar things are complicated even when they're quite simple, or that familiar things are simple even when they're quite complex.)

Even if we do not explain the arrow of time with disorder, we associate randomness with the effects of physical processes, which processes take time. Perhaps we could invert the explanation. Perhaps we could operationalize our conception of randomness in terms of what we expect from a class of processes (specifically, those not guided by intelligence) over time.

(Someone might now object that I'm begging the question of the arrow of time, but I didn't propose to explain it, and my readers all have the experience of that arrow; it's not a rabbit pulled from a hat.)

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[1] Other words that cause the same reäction are probability and capitalism.

Sometimes I've simplistically said that invocation of consensus is not a scientific method. A more accurate claim would be that its use is a way of approximating the results of more rigorous methods — a way of approximation that should never be mistaken for the more rigorous methods, and that is often unacceptable as science.

Calling upon consensus is a generalization of calling upon an expert. Using an expert can be analogous to using an electronic calculator. In some sense, using a calculator could be said to be scientific; there are sound empirical reasons for trusting a calculator to give one the right answer — at least for some classes of problems.

But note that, while possibly scientific, the use of the calculator is, itself, not scientifically expert in answering the question actually asked of the calculator (though some scientific expertise may have gone into answering the questions of whether to use a calculator, and of which calculator to use). Likewise, calling upon opinion from a human expert is not itself scientifically expert in answering the question actually asked. That distinction might not matter much, if ultimately scientific expertise from someone (or from some thing) ultimately went into the answer.

The generalization of invoking consensus proceeds in at least one direction, and perhaps in two. First, using consensus generalizes from using one expert to using n experts. But, second, invoking consensus often generalizes from invoking the views of experts to invoking the views of those who are less expert, or even not expert at all.

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Individual human experts, like individual electronic calculators, may not be perfectly reliable for answers to some sorts of questions. One response to this problem is the generalization of getting an answer from more than one, and, using a sort of probabilistic reasoning, going with the answer given by a majority of the respondents, or with some weighted sum of the answers. However, this approach goes astray when a common error prevails amongst most of the experts. If one returns to the analogy of digital calculators, various limitations and defects are typical, but not universal; a minority of calculators will answer some questions correctly, even as the majority agree on an incorrect answer. Likewise with human experts. That's not to say that being in the minority somehow proves a calculator or a human being to be correct, but it does indicate that one should be careful in how one responds to minority views as such. (In particular, mocking an answer for being unpopular amongst experts is like mocking an answer for being unpopular amongst calculators.) Counting the votes is a poor substitute for doing the math.

A hugely important special case of the problem of common design flaws obtains when most specialists form their opinions by reference to the opinions of other specialists. In this case, the expert opinion is not itself scientifically expert. Its foundation might be in perfectly sound work by some scientists, or it might be in unsound work, in misreading, in intuïtion and in guess-work, or in wishful thinking; but, in any case, what is taken to be the scientifically expert opinion of n experts proves instead to be that of some smaller number, or of none at all! In such cases, consensus may be little better, or nothing other, than a leap-of-faith. It isn't made more scientific by being a consensus.

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In a world in which expert opinion were always scientifically expert, broadening the pool to include those less expert would typically be seeking the center of opinion in less reliable opinion. However, as noted above, a field of expertise isn't necessarily dominated by scientific experts, in which case, people less expert but more scientific may move the center of opinion to a better approximation of a scientific opinion.

Additionally, for an outsider in seeking the opinion of experts, there is the problem of identifying who counts as an expert. The relevant knowledge and the relevant focus do not necessarily reside in the same people. As well as experts failing to behave like scientists, there are often people instead focussed on other matters who yet have as much relevant knowledge as any of those focussed on the subject in question.

So a case can be made for sometimes looking at the opinions of more than those most specialized around the questions. None-the-less, as the pool is broadened, the ultimate tendency is for the consensus to be ever less reliable as an approximation of scientific opinion. One should become wary of a consensus of broadly defined groups, and one should especially be wary if evidence can be shown of consensus shopping, where different pools were examined until a pool was found that gave an optimal threshold of conviction for whatever proposition is being advocated.

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What I've really been trying to convey when I've said that invocation of consensus is not a scientific method is that a scientist, acting as a scientist, would never treat invocation of consensus — not even the consensus of bona fide experts — within his or her own area of expertise as scientific method, and that everyone else needs to see consensus for no more than what it is: a second-hand approximation that can fail grotesquely, sometimes even by design.

When some party attempts to communicate, there are conceptual differences amongst

• what symbols were transmitted
• what conceptual content is appropriately associated with those symbols
• what conceptual content the party desired to convey

Put more colloquially,

• what someone literally said is one thing
• what the words mean is another
• what someone intended to say is still another

though, ideally, perfect agreement of a sort would obtain amongst them.

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People who won't distinguish amongst these are a bane. They'll claim that they said something that they didn't; that you said something that you didn't, that their words meant something that they couldn't; that your words meant something that they couldn't. They expect a declaration That's not what I meant! to shift all responsibility for misstatement to the other person. They expect to be able to declare That's not what you said! when it's exactly what you said but not what they had thought you intended or not what they had wanted you to say.

It's of course perfectly fair to admit that one misspoke with That's not what I meant!, so long as one is not thus disavowing the responsibility for one's actual words. I'm writing of those who avoid responsibility by the device of refusing to acknowledge anything but intentions or supposèd intentions.

Some of them are even more abusive, attempting to use That's not what I meant! to smuggle ad hoc revisions into their positions. By keeping obscured the difference between what was actually said and what was intended, they can implicitly invoke the fact that intent is less knowable than actual words, while keeping misstatement unthinkable, so that the plausibility that there was a misstatement cannot be examined.

One thing that I certainly like about the 'Net (and about recording equipment) is that it has made it more difficult for people to refuse to acknowledge what they have or another party has actually said. They'll still try, though. I've repeatedly participated in threads where someone has denied saying something when it's still in the display of the thread. (And, oddly enough, it seems that I'm often the only person who catches this point. I don't presently have much of a theory as to why others so frequently do not.)

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Setting aside those who won't distinguish amongst these three, there are people who more innocently often don't distinguish amongst them. I was provoked here to note the differences as they will be relevant to a later entry.

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).

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

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.]