Those who support locking-down in response to SARS-CoV-2 are like weird locusts. Instead of eating the crops; these locusts prevent growth and harvest. That is to say that they prevent economic activity, which is an implicit consumption of an especially perverse sort. In any case, they leave despair and literal starvation in their wake.

To my chagrin, I find that I made a transcription error for an axiom in Formal Qualitative Probability. More specifically, I placed a quantification in the wrong place. Axiom (A6) should read I've corrected this error in the working version.

I found an article that, had I known of it, I would have noted in my probability paper, A Logic of Comparative Support: Qualitative Conditional Probability Relations Represented by Popper Functions by James Allen Hawthorne
in Oxford Handbook of Probabilities and Philosophy, edited by Alan Hájek and Chris Hitchcock

Professor Hawthorne adopts essentially unchanged most of Koopman's axiomata from The Axioms and Algebra of Intuitive Probability, but sets aside Koopman's axiom of Subdivision, noting that it may not seem as intuitively compelling as the others. In my own paper, I showed that Koopman's axiom of Subdivision was a theorem of a much simpler, more general principle in combination with an axiom that is equivalent to two of the axiomata in Koopman's later revision of his system. (The article containing that revision is not listed in Hawthorne's bibliography.) I provided less radically simpler alternatives to other axiomata, and included axiomata that did not apply to Koopman's purposes in his paper but did to the purposes of a general theory of decision-making.

My work and the problems that most interest me are difficult to discuss with friends and even with colleagues because so much infrastructure is unfamiliar to them.

As repeatedly noted by me and by many others, there are multiple theories about the fundamental notion of probability, including (though not restricted to) the notion of probabilities as objective, logical relationships amongst propositions and that of probabilities as degrees of belief.

Though those two notions are distinct, subscribers to each typically agree with subscribers to the other upon a great deal of the axiomatic structure of the logic of probability. Further, in practice the main-stream of the first group and that of the second group both arrive at their estimates of measures of probability by adjusting initial values through repeated application, as observations accumulate, of a principle known as Bayes' theorem. Indeed, the main-stream of one group are called objective Bayesian and the mainstream of the other are often called subjective Bayesian.[1] Where the two main-streams differ in practice is in the source of those initial values.

The objective Bayesians believe that, in the absence of information, one begins with what are called non-informative priors. This notion is evolved from the classical idea of a principle of insufficient reason, which said that one should assign equal probabilities to events or to propositions, in the absence of a reason for assigning different probabilities. (For example, begin by assume that a die is fair.) The objective Bayesians attempt to be more shrewd than the classical theorists, but will often admit that in some cases non-informative priors cannot be found because of a lack of understanding of how to divide the possibilities (in some cases because of complexity).

The subjective Bayesians believe that one may use as a prior whatever initial degree of belief one has, measured on an interval from 0 through 1. As measures of probability are taken to be degrees of belief, any application of Bayes' theorem that results in a new value is supposed to result in a new degree of belief.

I want to suggest what I think to be a new school of thought, with a Bayesian sub-school, not-withstanding that I have no intention of joining this school.

If a set of things is completely ranked, it's possible to proxy that ranking with a quantification, such that if one thing has a higher rank than another then it is assigned a greater quantification, and that if two things have the same rank then they are assigned the same quantification. If all that we have is a ranking, with no further stipulations, then there will be infinitely many possible quantifications that will work as proxies. Often, we may want to tighten-up the rules of quantification (for example, by requiring that all quantities be in the interval from 0 through 1), and yet still it may be the case that infinitely many quantifications would work equally well as proxies.

Sets of measures of probability may be considered as proxies for underlying rankings of propositions or of events by probability. The principles to which most theorists agree when they consider probability rankings as such constrain the sets of possible measures, but so long as only a finite set of propositions or of events is under consideration, there are infinitely many sets of measures that will work as proxies.

A subjectivist feels free to use his or her degrees of belief so long as they fit the constraints, even though someone else may have a different set of degrees of belief that also fit the constraints. However, the argument for the admissibility of the subjectivist's own set of degrees of belief is not that it is believed; the argument is that one's own set of degrees of belief fits the constraints. Belief as such is irrelevant. It might be that one's own belief is colored by private information, but then the argument is not that one believes the private information, but that the information as such is relevant (as indeed it might be); and there would always be some other sets of measures that also conformed to the private information.

Perhaps one might as well use one's own set of degrees of belief, but one also might every bit as well use any conforming set of measures.

So what I now suggest is what I call a libertine school, which regards measures of probability as proxies for probability rankings and which accepts any set of measures that conform to what is known of the probability ranking of propositions or of events, regardless of whether these measures are thought to be the degrees of belief of anyone, and without any concern that these should become the degrees of belief of anyone; and in particular I suggest libertine Bayesianism, which accepts the analytic principles common to the objective Bayesians and to the subjective Bayesians, but which will allow any set of priors that conforms to those principles.

[1] So great a share of subjectivists subscribe to a Bayesian principle of updating that often the subjective Bayesians are simply called subjectivists as if there were no need to distinguish amongst subjectivists. And, until relatively recently, so little recognition was given to the objective Bayesians that Bayesian was often taken as synonymous with subjectivist.

In a previous entry, I asserted that Voigt's Zahl und Mass in der Ökonomik contain[s] more error than insight. Here, I'll discuss one of the more egregious errors. In section V, Voigt writes

An die Spitze der Erörterung dieses vielberufenen Begriffes sollte gestellt werden, dass es Einheiten des Wertes giebt, dass man also untersuchen kann, wievielmal so gross ein Wert als ein anderer ist und Güter gleichen Wertes durch einander ersetzen kann, dass also der Wert ein eigentliches in einer Kardinalzahl ausdrückbares Mass hat.

which may be translated as

At the forefront of discussion of this much used concept should be placed that there are units of value that one thus can investigate how many time as large a value is as another and can replace goods of the same value with each other, that thus the value has a real measure expressible in a cardinal number.

I'll deal first with the point that it seems that one can investigate how many times as large a value is as another.

Numbers are used in many ways. Depending upon the use, what is revealed by arithmetic may be a great deal or very little. Sometimes numbers are ascribed with so little meaning that we may as well consider them strings of numerals, the characters that we use for numbers, and not numbers at all. Sometimes numbers do nothing but provide an arbitrary order, good for something such as a look-up table but nothing else. Sometimes they provide a meaningful order, but one in which the results of most arithmetic operations are meaningless, as when items produced at irregular intervals are given sequential serial numbers. (The difference between any two such numbers tells one which was produced before the other, but little else.) Sometimes the differences between the differences are meaningful, as when items are produce at regular intervals and given sequential serial numbers. And so forth.

Monetary prices are quantities, but they are more specifically quantities of money; that does not make them quantities of value nor proxies of quantities of value. One would have to show that the results of every arithmetic operation on such a quantity of money said something about value for it to be shown that value were itself a quantity.

The second part of Voigt's claim is that one Güter gleichen Wertes durch einander ersetzen kann [can replace goods of the same value with each other]. But an equivalence between things corresponding to the same numbers doesn't make results of the application of arithmetic to those numbers meaningful. (Consider lots of items produced at irregular intervals, with each item in the lot given the same serial number, unique to the lot but otherwise random.) And we should ask ourselves under just what circumstances we can and cannot ersetzen one set of commodities of a given price with another of the same price.

Nor does somehow combining the use of quantities of money for prices with a property of equivalence imply that value is a quantity.

Voigt is unusual not in making this unwarranted inference, but in so clearly expressing himself as he does. From the observation that prices are usually quantities of something, which quantities increase as value increases, most people, and even most economists blithely infer that value itself behaves as a quantity.

In early 2013, I made freely available a transcription of Zahl und Mass in der Ökonomik: Eine kritische Untersuchung der mathematischen Methode und der mathematischen Preistheorie (1893) by Andreas Heinrich Voigt. I have to-day completed a first pass of a translation of this as Number and Measure in Economics: A Critical Examination of Mathematical Method and of Mathematical Price Theory. Although I believe that there are many errors to be corrected in that translation, I am making it available. I do not plan to use a different URI for corrected versions.

I have been very disappointed by my reading of Voigt's article. I regard it as containing more error than insight.

In the course of translation, I found and corrected extremely minor errors in the transcription of the original. A name was at one point misspelled by me, and I failed to capitalize a word beginning a sentence. I also marked a die die as questionable which I've since concluded was deliberate. I do not believe that anyone could have been led to a mistaken reading as a result of those errors, but I have naturally corrected them.

I may change the URI for the transcription, moving it from another domain to place it amongst the uploads for this 'blog. If so, then I will edit entries to reflect that change.

On 20 February 2020, a year to-the-day after I submitted my paper Formal Qualitiative Probability to The Review of Symbolic Logic and nearly five months after I was notified that a revised version had been accepted, Cambridge University Press published the manuscript on-line.

(I believe that an unchanging DOI 10.1017/S1755020319000480 will be used for whatever is the latest version of the article, as it is type-set for paper publication and eventually assigned to a specific issue.)

This work was badly treated across journals of philosophy. Regardless of whether any of my future work is perhaps best regarded as philosophic, I will henceforth avoid submitting to such journals.

I recently started reading Psychology from an Empirical Standpoint, a translation of Psychologie vom empirischen Standpunkt[e] by Franz Brentano. My copy happens to be of the 1973 Humanities Press edition.

In the translation of the 1874 foreword, I hit a sentence

I was prompted to undertake a rather detailed study of these opinions because at the present time they enjoy an undue popularity and exert a lamentable influence upon a public which, in matters of psychology even less than in other fields, has not yet learned to demand scientific cogency.

This sentence is a muddle, with restrictions and negations working to say something contrary to what Brentano must surely have intended. A scan of the original is available on-line, the German reads

Und was mich dazu trieb, auch auf sie weitläufiger einzugehen, waren nur eine ungebührliche Verbreitung und ein beklagenswerther Einfluss, welche sie gegenwärtig auf ein Publicum gewonnen haben, das in sachen der Psychologie weniger noch als anderwärts auf wissenschaftliche Strenge Anspruch zu machen gelernt hat.

That refers to

a public, who in matters of psychology less still than elsewhere have learned to make a demand for scientific rigor

So the muddle is not in the original, but is an artefact of translation that doesn't strive to be as close to the original as possible while conforming to the conventions of the target language.

I checked a scan of the 1995 second edition of the translation, and found the same muddle as in the 1973 edition.

Naturally, I'm wondering to what extent I can reasonably trust the remainder of the translation. I reälize that passages recognized as crucial will probably have been treated with greater care and received more scrutiny, but there may be passages the importance of which has not been recognized. And even passages whose importance were recognized might be poorly translated. (I certainly saw such cases in translations of the work of Aristoteles.)

On 22 September, I was informed that my article Formal Qualitative Probability had been accepted for publication by The Review of Symbolic Logic. I do not yet know in what issue it is to be published.

Up-Date (2019:11/03): I have not yet received a galley-proof of my paper nor a copyright-assignment form. I have learned that the journal publishes papers on-line before scheduling them to a specific issue of the journal, so I will probably be in the odd position of having a DOI and a URI before I know the issue in which the paper is to appear.