Most people who claim that argument from authority is fallacious would, perversely, argue for that claim by reference to the authority of common knowledge or of what were often taught. A fallacy is actually shown by demonstrating a conflict with a principle of logic or by an empirical counter-example. A case in which an authority proved to be wrong might be taken as the latter, but matters are not so simple.

When one normally makes a formal study of logic, that study is usually of assertoric logic, the logic in which every proposition is treated as if knowable to be true or knowable to be false, even if sometimes the study itself deliberately treats a propostion as false that is true or a proposition as true that is false. In the context of assertoric logic, an argument from authority is indeed fallacious.

But most of the propositions with which we deal are not known or knowable to be true or false; rather, we find that some propositions are relatively more plausible than others. Our everyday logic must be the logic of that ordering. Within that logic, showing that a proposition has one position in the ordering given some information does not show that it did not have a different position without that information. So we cannot show that arguments from authority are fallacious in the logic of plausibility simply by showing that what some particular authority claimed to be likely or even certainly true was later shown to be almost certainly false or simply false.

Arguments from authority, though often not recognized as such, are essential to our everyday reasoning. For example, most of us rely heavily upon the authority of others as to what they have experienced; we even rely heavily upon the authority of n-th-hand reports and distillations of reports of the experiences of others. And none of us has fully explored the theoretic structure of the scientific theories that the vast majority of us accept; instead, we rely upon the authority of those transmitting sketches, gists, or conclusions. Some of those authorities have failed us; some of those authorities will fail us in the future; those failures have not and will not make every such reliance upon authorities fallacious.

However, genuine fallacy would lie in over-reliance upon authorities — putting some authoritative claims higher in the plausibility ordering than any authoritative claims should be, or failing to account for factors that should lower the places in the plausibity ordering associated with authorities of various sorts, such as those with poor histories or with conflicts of interest.

By the way, I have occasionally been accused of arguing from authority when I've done no such thing, but instead have pointed to someone who was in some way important in development or useful in presentation of an argument that I wish to invoke.

As occasionally noted in publicly accessible entries to this 'blog, I have been working on a paper on qualitative probability. A day or so before Christmas, I had a draft that I was willing to promote beyond a circle of friends.

I sent links to a few researchers, some of them quite prominent in the field. One of them responded very quickly in a way that I found very encouraging; and his remarks motivated me to make some improvements in the verbal exposition.

I hoped and still hope to receive responses from others, but as of to-day have not. I'd set to-day as my dead-line to begin the process of submitting the paper to academic journals, and therefore have done so.

The process of submission is emotionally difficult for many authors, and my past experiences have been especially bad, including having a journal fail to reach a decision for more than a year-and-a-half, so that I ultimate withdrew the paper from their consideration. I even abandoned one short paper because the psychological cost of trying to get it accepted in some journal was significantly impeding my development of other work. While there is some possibility that finding acceptance for this latest paper will be less painful, I am likely to be in for a very trying time.

It is to be hoped that, none-the-less, I will be able to make some progress on the next paper in the programme of which my paper on indecision and now this paper on probability are the first two installments. In the presumably forth-coming paper, I will integrate incomplete preferences with incompletely ordered probabilities to arrive at a theory of rational decision-making more generalized and more reälistic than that of expected-utility maximization. A fourth and fifth installment are to follow that.

But the probability paper may be the most important thing that I will ever have written.

I've not heard nor read anyone remarking about a particular contrast between the classical approach to probability theory and the Bayesian subjectivist approach. The classical approach began with a presumption that the formal mathematical principles of probability could be discovered by considering situations that were impossibly good; the Bayesian subjectivist approach was founded on a presumption that those principles could be discovered by considered situations that were implausibly bad.

The classical development of probability theory began in 1654, when Fermat and Pascal took-up a problem of gambling on dice. At that time, the word probability and its cognates from the Latin probabilitas meant plausibility.

Fermat and Pascal developed a theory of the relative plausibility of various sequences of dice-throws. They worked from significant presumptions, including that the dice had a perfect symmetry (except in-so-far as one side could be distinguished from another), so that, with any given throw, it were no more plausible that one face should be upper-most than that any other face should be upper-most. A model of this sort could be be reworked for various other devices. Coins, wheels, and cards could be imagined as perfectly symmetrical. More generally, very similar outcomes could be imagined as each no more probable than any other. If one presumes that to be no more probable is to be equally probable, then a natural quantification arises.

Now, the preceptors did understand that most or all of the things that they were treating as perfectly symmetrical were no such thing. Even the most sincere efforts wouldn't produce a perfectly balanced die, coin, or roulette wheel, and so forth. But these theorists were very sure that consideration of these idealized cases had revealed the proper mathematics for use across all cases. Some were so sure of that mathematics that they inferred that it must be possible to describe the world in terms of cases that were somehow equally likely, without prior investigation positively revealing them as such. (The problem for this theory was that different descriptions divide the world into different cases; it would take some sort of investigation to reveal which of these descriptions, if any, results in division into cases of equal likelihood. Indeed, even with the notion of perfectly balanced dice, one is implicitly calling upon experience to understand what it means for a die to be more or less balanced; likewise for other devices.)

As subjectivists have it, to say that one thing is more probable than another is to say that that first thing is more believed than is the other. (GLS Shackle proposed that the probability of something might be measured by how surprised one would be if that something were discovered not to be true.)

But most subjectivists insist that there are rationality constraints that must be followed in forming these beliefs, so that for example if X is more probable than Y and Y more probable than Z, then X must be more probable than Z. And the Bayesian subjectivists make a particular demand for what they call coherence. These subjectivists imagine that one assigns quantifications of belief to outcomes; the quantifications are coherent if they could be used as gambling ratios without an opponent finding some combination of gambles with those ratios that would guarantee that one suffered a net loss. Such a combination is known as a Dutch book.

But, while quantifications can in theory be chosen that insulate one against the possibility of a Dutch book, it would only be under extraordinary circumstances that one could not avoid a Dutch book by some other means, such as simply rejecting complex contracts to gamble, and instead deciding on gambles one-at-a-time, without losing sight of the gambles to which one had already agreed. In the absence of complex contracts or something like them, it is not clear that one would need a preëstablished set of quantifications or even could justify committing to such a set. (It is also not clear why, if one's beliefs correspond to measures, one may not use different measures for gambling ratios.) Indeed, it is only under rather unusual circumstances that one is confronted by opponents who would attempt to get one to agree to a Dutch book. (I don't believe that anyone has ever tried to present me with such a combination, except hypothetically.) None-the-less, these theorists have been very sure that consideration of antagonistic cases of this class has revealed the proper mathematics for use across all cases.

The impossible goodness imagined by the classical theorists was of a different aspect than is the implausible badness of the Bayesian subjectivists. A fair coin is not a friendly coin. Still, one framework is that of the Ivory Tower, and the other is that of Murphy's Law.

One of the principles often suggested as an axiom of probability is that of additivity. The additivity here is a generalization of arithmetic addivity — which generalization, with other assumptions, will imply the arithmetic case.

The classic formulation of this principle came from Bruno di Finetti. Di Finetti was a subjectivist. A typical subjectivist is amongst those who prefer to think in terms of the probability of events, rather than in terms of the probability of propositions. And subjectivists like to found their theory of probability in terms of unconditional probabilities. Using somewhat different notation from that here, the classic formulation of the principle of additivity is in which X, Y, and Z are sets of events. The underscored arrowhead is again my notation for weak supraprobability, the union of strict supraprobability with equiprobability.

One of the things that I noticed when considering this proposition is that the condition that Y∩Z be empty is superfluous. I tried to get a note published on that issue, but journals were not receptive. I had bigger fish to fry other than that one, so I threw-up my hands and moved onward.

When it comes to probability, I'm a logicist. I see probability as primarily about relations amongst propositions (though every event corresponds to a proposition that the event happen and every proposition corresponds to the event that the proposition is true), and I see each thing about which we state a probability as a compound proposition of the form X given c in which X and c are themselves propositions (though if c is a tautology, then the proposition operationalizes as unconditional). I've long pondered what would be a proper generalized restatement of the principle of additivity. If you've looked at the set of axiomata on which I've been working, then you've seen one or more of my efforts. Last night, I clearly saw what I think to be the proper statement: To get di Finetti's principle from it, set c₂ = c₁ and make it a tautology, and set X₂ = Z = Y₂. Note that the condition of (X₂ | c₁) being weakly supraprobable to (Y₂ | c₂) is automatically met when the two are the same thing. By itself, this generalization implies my previous generalization and part of another principle that I was treating as an axiom; the remainder of that other principle can be got by applying basic properties of equiprobability and the principle that strict supraprobability and equiprobability are mutually exclusive to this generalization. The principle that is thus demoted was awkward; the axiom that was recast as acceptable as it was, but the new version is elegant.

My paper on indecision is part of a much larger project. The next step in that project is to provide a formal theory of probability in which it is not always possible to say of outcomes either that one is more probable than another or that they are equality likely. That theory needs to be sufficient to explain the behavior of rational economic agents.

I began struggling actively with this problem before the paper on indecision was published. What I've had is an evolving set of axiomata that resembles the nest of a rat. I've thought that the set has been sufficient; but the axiomata have made over-lapping assertions, there have been rather a lot of them, and one of them has been complex to a degree that made me uncomfortable. Were I better at mathematics, then things might have been put in good order long ago. (I am more able at mathematics than is the typical economist, but I wish that I were considerably still better.) On the other hand, while there are certainly people better at mathematics than am I, no one seems to have accomplished what I seek to do. Economics is, after all, more than its mathematics.

What has most bothered me has been that complex axiom. There hasn't seemed much hope of resolving the general over-lap and of reducing the number of axiomata without first reducing that particular axiom. On 2 January, I was able to do just that, dissolving that axiom into two axiomata, each of which is acceptably simple. Granted that the number of axiomata increased by one, but now that the parts are each simple, I can begin to see how to reduce their overlap. Eliminating that overlap should either pare or vindicate the number of axiomata.

I don't know whether, upon getting results completed and a paper written around them, I would be able to get my work published in a respectable journal. I don't know whether, upon my work's getting published, it would find a significant readership. But the work is deeply important.

In his foundational work on probability,[1] Bernard Osgood Koopman would write something of form α /κ for a suggested observation α in the context of a presumption κ. That's not how I proceed, but I don't actively object to his having done so, and he had a reason for it. Though Koopman well understood that real-life rarely offered a basis for completely ordering such things by likelihood, let alone associating them with quantities, he was concerned to explore the cases in which quantification were possible, and he wanted his readers to see something rather like division there. Indeed, he would call the left-hand element α a numerator, and the right-hand element κ the denominator.

He would further use 0 to represent that which were impossible. This notation is usable, but I think that he got a bit lost because of it. In his presentation of axiomata, Osgood verbally imposes a tacit assumption that no denominator were 0. This attempt at assumption disturbs me, not because I think that a denominator could be 0, but because it doesn't bear assuming. And, as Koopman believed that probability theory were essentially a generalization of logic (as do I), I think that he should have seen that the proposition didn't bear assuming. Since Koopman was a logicist, the only thing that he should associate with a denominator of 0 would be a system of assumptions that entailed a self-contradiction; anything else is more plausible than that.

In formal logic, it is normally accepted that anything can follow if one allows a self-contradiction into a system, so that any conclusion as such is uninteresting. If faced by something such as X ∨ (Y ∧ ¬Y) (ie X or both Y and not-Y), one throws away the (Y ∧ ¬Y), leaving just the X; if faced with a conclusion Y ∧ ¬Y then one throws away whatever forced that awful thing upon one.[2] Thus, the formalist approach wouldn't so much forbid a denominator of 0 as declare everything that followed from it to be uninteresting, of no worth. A formal expression that no contradiction is entailed by the presumption κ would have the form ¬(κ ⇒ [(Y ∧ ¬Y)∃Y]) but this just dissolves uselessly ¬(¬κ ∨ [(Y ∧ ¬Y)∃Y])
¬¬κ ∧ ¬[(Y ∧ ¬Y)∃Y]
κ ∧ [¬(Y ∧ ¬Y)∀Y]
κ ∧ [(¬Y ∨ ¬¬Y)∀Y]
κ ∧ [(¬Y ∨ Y)∀Y]
κ (because (X ⇔ [X ∧ (Y ∨ ¬Y)∀Y])∀X).

In classical logic, the principle of non-contradiction is seen as the bedrock principle, not an assumption (tacit or otherwise), because no alternative can actually be assumed instead.[3]. From that perspective, one should call the absence of 0-valued denominators simply a principle.

[1] Koopman, Bernard Osgood; The Axioms and Algebra of Intuitive Probability, The Annals of Mathematics, Series 2 Vol 41 #2, pp 269-292; and The Bases of Probability, Bulletin of the American Mathematical Society, Vol 46 #10, pp 763-774.

[2] Indeed, that principle of rejection is the basis of proof by contradiction, which method baffles so many people!

[3] Aristoteles, The Metaphysics, Bk 4, Ch 3, 1005b15-22.

I've previously touched on the matter of there being markèdly differing notions all associated with the word probability. Various attempts have been made by various writers to catalogue and to coördinate these notions; this will be one of my own attempts.