Recognizing the Difference

17 January 2017

The word problem is used for multiple concepts:

  • a challenge, especially a mental challenge
  • an unhappy state of affairs that exercises the mind
  • a seemingly insurmountable difficulty

These may actually be very different things, because some states-of-affairs may be changed, and some may not; but it is often hard to distinguish the former from the latter. Thence we use the same word for all three concepts, and thence we get the famous Serenity Prayer of Reinhold Niebuhr:

God, grant me the serenity to accept the things I cannot change, courage to change the things I can, and wisdom to know the difference.

A good economist is going to note that sometimes the costs of changing what we can would be higher than the benefit. Properly understood, that greater costs would mean that we were trading some greater problem for a solution to the problem under immediate consideration. The Serenity Prayer might lead one to mistake sensible calculation for cowardice, though poetry might have to be sacrificed poetry to provide better guidance.

Still, the distinction between that which we can change and that which we cannot is most fundamental, and confusion between the two is a source of frustration and of anguish. That which displeases us and which we can change is a challenge about which we can usefully exercise the mind, until such time as either we develop a reasonable method of changing things or we recognize that it’s not worth the cost. That which displeases us and which we cannot change is an unhappy condition of existence; and — ideally — we ought to come to terms with that condition, to accept it, so that it no longer exercises our minds. (And we ought, similarly, to accept those states the changing of which, though possible, truly would be too costly.)

One seldom-noted root of some of the confusion is a failure to distinguish between what the person in question may change and what someone else may change. The state-of-affairs might be a problem for a person who could change it, or it might not be a problem for that person; but, either way, by someone who cannot change it, it is best treated as a condition of existence.

It may be especially hard to see the distinction when the other person might indeed act to change things and, if that other person did, the first person would want to undertake quickly some course of action. But, likewise, there might be some potential event which would have to be impersonally caused (if at all) and which would call for a prompt response. One should not try to cause that which no person can cause, and one should not try to cause that which only another person can cause, even if there would be things to do should it happen.

It may be too much to hope for serenity from such realizations. It is certainly too much to demand that others be serene in the face of indifference, indecision, or madness on the part of those around them. Sometimes grim acceptance is the best that one can manage.

Deal-Breakers

7 January 2017

Elsewhere, Pierre Lemieux asked In two sentences, what do you think of the Monty Hall paradox? Unless I construct sentences loaded with conjunctions (which would seem to violate the spirit of the request), an answer in just two sentences will be unsatisfactory (though I provided one). Here in my 'blog, I’ll write at greater length.


The first appearance in print of what’s called the Monty Hall Problem seems to have been in a letter by Steve Selvin to The American Statistician v29 (1975) #1. The problem resembles those with which Monty Hall used to present contestants on Let’s Make a Deal, though Hall has asserted that no problem quite like it were presented on that show. The most popular statement of the Monty Hall Problem came in a letter by Craig Whitaker to the Ask Marilyn column of Parade:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, Do you want to pick door No. 2? Is it to your advantage to switch your choice?

(Before we continue, take car and goat to stand, respectively, for something that you want and something that you don’t want, regardless of your actual feelings about cars and about goats.)

There has been considerable controversy about the proper answer, but the text-book answer is that, indeed, one should switch choices. The argument is that, initially, one has a 1/3 probability that the chosen Door has the car, and a 2/3 probability that the car is behind one of the other two Doors. When the host opens one of the other two Doors, the probability remains that the car is behind one of the unchosen Doors, but has gone to 0 for the opened Door, which is to say that the probability is now 2/3 that the car is behind the unchosen, unopened Door.


My first issue with the text-book answer is with its assignment of initial, quantified probabilities. I cannot even see a basis for qualitative probabilities here; which is to say that I don’t see a proper reason for thinking either that the probability of the car being behind a given Door is equal to that for any other Door or that the probability of the car being behind some one Door is greater than that of any other Door. As far as I’m concerned, there is no ordering at all.

The belief that there must be an ordering usually follows upon the even bolder presumption that there must be a quantification. Because quantification has proven to be extremely successful in a great many applications, some people make the inference that it can be successfully applied to any and every question. Others, a bit less rash, take the position that it can be applied everywhere except where it is clearly shown not to be applicable. But even the less rash dogma violates Ockham’s razor. Some believe that they have a direct apprehension of such quantification. However, for most of human history, if people thought that they had such literal intuitions then they were silent about it; a quantified notion of probability did not begin to take hold until the second half of the Seventeenth Century. And appeals to the authority of one’s intuition should carry little if any weight.

Various thinkers have adopted what is sometimes called the principle of indifference or the principle of insufficient reason to argue that, in the absence of any evidence to the contrary, each of n collectively exhaustive and mutually exclusive possibilities must be assigned equal likelihood. But our division of possibilities into n cases, rather than some other number of cases, is an artefact of taxonomy. Perhaps one or more of the Doors is red and the remainder blue; our first division could then be between two possibilities, so that (under the principle of indifference) one Door would have an initial probability of 1/2 and each of the other two would have a probability of 1/4.

Other persons will propose that we have watched the game played many times, and observed that a car has with very nearly equal frequency appeared behind each of the three Doors. But, while that information might be helpful were we to play many times, I’m not aware of any real justification for treating frequencies as decision-theoretic weights in application to isolated events. You won’t be on Monty’s show to-morrow.

Indeed, if a guest player truly thought that the Doors initially represented equal expectations, then that player would be unable to choose amongst them, or even to delegate the choice (as the delegation has an expectation equal to that of each Door); indifference is a strange, limiting case. However, indecision — the aforementioned lack of ordering — allows the guest player to delegate the decision. So, either the Door was picked for the guest player (rather than by the guest player), or the guest player associated the chosen Door with a greater probability than either unchosen Door. That point might seem a mere quibble, but declaring that the guest player picked the Door is part of a rhetorical structure that surreptitiously and fallaciously commits the guest player to a positive judgment of prior probability. If there is no case for such commitment, then the paradox collapses.


Well, okay now, let’s just beg the question, and say not only that you were assigned Door Number 1, but that for some mysterious reason you know that there is an equal probability of the car being behind each of the Doors. The host then opens Door Number 3, and there’s a goat. The problem as stated does not explain why the host opened Door Number 3. The classical statement of the problem does not tell the reader what rule is being used by the host; the presentation tells us that the host knows what’s behind the doors, but says nothing about whether or how he uses that knowledge. Hypothetically, he might always open a Door with a goat, or he might use some other rule, so that there were a possibility that he would open the Door with a car, leaving the guest player to select between two concealed goats.

Nowhere in the statement of the problem are we told that you are the sole guest player. Something seems to go very wrong with the text-book answer if you are not. Imagine that there are many guest players, and that outcomes are duplicated in cases in which more than one guest player selects or is otherwise assigned the same Door. The host opens Door Number 3, and each of the guest players who were assigned that Door trudges away with a goat. As with the scenario in which only one guest player is imagined, more than one rule may govern this choice made by the host. Now, each guest player who was assigned Door Number 1 is permitted to change his or her assignment to Door Number 2, and each guest player who was assigned Door Number 2 is allowed to change his or her assignment to Door Number 1. (Some of you might recall that I proposed a scenario essentially of this sort in a 'blog entry for 1 April 2009.) Their situations appear to be symmetrical, such that if one set of guest players should switch then so should the other; yet if one Door is the better choice for one group then it seems that it ought also to be the better for the other group.

The resolution is in understanding that the text-book solution silently assumed that the host were following a particular rule of selection, and that this rule were known to the guest player, whose up-dating of probabilities thus could be informed by that knowledge. But, in order for the text-book solution to be correct, all players must be targeted in the same manner by the response of the host. When there is only one guest player, it is possible for the host to observe rules that respond to all guest players in ways that are not not possible when there are multiple guest players, unless they are somehow all assigned the same Door. It isn’t even possible to do this for two sets of players each assigned different Doors.


Given the typical presentation of the problem, the typical statement of ostensible solution is wrong; it doesn’t solve the problem that was given, and doesn’t identify the problem that was actually solved.


[No goats were harmed in the writing of this entry.]

Headway

7 January 2017

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.

Dead Celebrities and the Constitution

2 January 2017

The law has long treated the images of living persons as something like trademarks, so that the use of such images without the consent of those persons is restricted in various ways. Over the last few decades, there have been efforts — some successful — to turn these images into property of a sort that may legally be bequeathed upon death, as opposed to such images lapsing into the public domain.

The potential monetary rewards increased simply as celebrity became more culturally important. And the evolution of CGI has made it possible to fabricate the appearance of persons in video, not only increasing opportunities to make money from such images but to use those images in ways that many of us would see as grossly abusive (for example, in pornography).

Reflexively, quite a few people want law at a Federal level to protect the images of the dead. But there is some reasonable question as to whether the US Constitution empowers the Federal government to act.

The rub comes by way of Article I §8, which enumerates the powers of Congress. It specifically empowers Congress

To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries;

As I’ve explained, this enumeration did not creäte copyrights; it instead gave Congress limited power to protect them; without that empowerment, the matter would simply have been left in the hands of common law and of the legislation of the constituent states. But neither §8 nor any other part of the unamended Constitution makes explicit mention of intellectual property of any other sort, and it is an unreasonable stretch to suggest that there is implicit reference to similar such property, exactly because there had been a felt need for explicit mention of the rights of authors and of inventors. Any prerogative to protect the images of the dead was left in the hands of the constituent states.

I used the word unamended advisedly, because one might try to make a case that the Fourteenth Amendment empowers the Congress to treat personal images as continuing to be property beyond personal death. But one can legitimately make that claim only if somehow the associated rights are basic rights of persons (or of US citizens as such) or can be shown to derive from basic rights; the Fourteenth Amendment doesn’t empower Congress to reconfigure legal rights ad libitum.

There’s a wide-spread propensity to confuse great desirability with moral necessity. But the penalties of law are always ultimately acts of violence, and there’s a need for more than emotional appeal when the law forbids something. So, while it would be greatly desirable for the images of the celebrated dead to be treated with something like the respect that they would have received when alive, a solid case should be made for regarding them as property, or indeed they should be regarded as not property.

Some efforts to make personal images into property that remains such beyond the death of the persons have focussed on getting constituent states to have it treated as such in law. It’s perhaps worth noting that the Fourteenth Amendment might be read to block the constituent states from converting these images to property (though I am sure that no court to-day would accept that reading).

The Present Is Another Country

31 December 2016

To-day in the bistro, an unfamiliar girl perhaps six years old sat near me on the same bench. She was joined after a few minutes by a woman, probably her mother, who sat on the other side of their table. The little girl was interested in me and in my attention. Years ago, the drill for me would have been to engage her in brief conversation, asking her name and so forth, and probably answering questions about what I were doing. To-day, it was to turn briefly to give a friendly smile to her, and then turn quickly back to my business, all without pause.

Our notions of proper behavior have changed because Americans are generally far more concerned and otherwise anxious about threats to children, especially sexual threats from adult men. I share that concern and anxiety. I don’t know whether things have become more or less dangerous than once they were, but honestly I think that question is secondary. Even if our world is less dangerous for children than in the past, I want us to be concerned; if it was once less perilous, none-the-less I wish that we’d even then been more vigilant.

(The Economist used to make a practice of mocking Americans for our fear for children. Perhaps they still do. I don’t expect that it has occurred to them, in the wake of the various scandals concerning the sexual exploitation of children in the UK, that perhaps they have once again been mistaken in their presumptions of superiority.)

Yes, I’m saddened that I couldn’t have had that short conversation. But I don’t live in the world that ought to be, and in this case I cannot make ours a better world by acting as I would if it were. Had I interacted with the child in a more inviting way, I would have helped to foster norms and expectations that are exploited by predators. She didn’t much need to talk to me. She does need to be spared some of the awful possible outcomes of the world in which we live.

Holiday Wishes

25 December 2016

Buon Natale! Possa il nuovo anno essere meraviglioso per voi.

Not My Supergirl

22 December 2016

Because my father watches Supergirl, I have now seen a few episodes. I note that they were written on the presumption that Hillary Clinton would be elected President, and thus that the character of the President is played by Lynda Carter imitating the voice and speaking patterns of Ms Clinton — so far, no cackle — albeït with the character named Olivia Marsdin.

And here’s what is truly telling: In this fantasy, people who do not freely support Madam President are physically assaulted by state officials. For saying that she is Not my President. a fellow has his thumb wrenched by DEO agent Alex[andra] Danvers, who later begins to beat information out of him even though he is not even so much as a suspect.

It seems not to have occurred to writers, to producers, to directors, nor to starring actors that their heroes in an imagined world guided by Hillary Clinton were doing things that would seem thuggery if done by champions of a social order directed by Donald Trump.

Yet the shoes were merely on the other feet — or, in this case, mistakenly presumed to be so.

On the Economists' use of Rent*

21 December 2016

Ordinary language typically uses rent to mean a recurring payment for the use of some good or service, especially for the use of land. The word has various other meanings in ordinary language; but, in economics, rent is used to mean something given in exchange, above and beyond what would have been the minimum necessary to effect the exchange in the absence of some restraint of trade. Usually, this concept is applied to pecuniary payment for a good or service above and beyond the minimum necessary to get that good or service, but one should see that exchanges without money might still involve such increased payment of one good or service for another, and that the same basic idea could be applied to cases in which someone were compelled to deliver more of a good or service than the minimum otherwise necessary to secure some amount of money.


It’s mostly because of David Ricardo (18 April 1772 – 11 September 1823), an influential economist, that rent has this meaning. Ricardo used the word rent to refer to payment for the indestructible powers of land. That is to say that rent meant payment for the use of those properties of land that were not diminished by use; payment for harvesting things such as preëxisting plants and minerals would not, strictly speaking, be rent. Perhaps the only thing that could literally meet this criterion would be specific area, but Ricardo favored simplified models, so one might consider highly durable or naturally renewed properties as indestructible.

In Ricardo’s mind, if someone renting land were given a right to cut down trees already on the land, or to quarry marble from it, then this right were essentially of the same sort as the right of someone buying lumber or stone at a mill; we certainly don’t label the prices of such commodities as rent. Ricardo wanted to identify what were distinctive and essential in what we called rent, and to reserve the word rent just for those components of payments.

Ricardo imagined land as of different qualities, even when viewed only in terms of indestructible properties, but imagined the existing quantities of lands of each quality as not something ever increased by human action. And, in the context of his models, he concluded that rent were not determined by its cost of manufacture nor otherwise by some minimum below which the land-owner could not afford to produce it; rent, according to Ricardo, were purely an artefact of monopoly in the provision of land.

So long as this belief prevailed, it was natural for economists to extend the use of the word rent to other payments which they regarded as perfectly analogous. The word rent came to stand for any payment above and beyond the minimum necessary to effect an exchange were there a non-monopolistic market. And, even when most economists moved away from Ricardo’s theory, as classical economics yielded to more thorough-going marginalism, they held onto this extended use of the term rent.


Because this use of rent can certainly be confusing, both when economists are interacting with lay-persons and when economists are attempting to discuss markets in which commodities are leased, sometimes instead of the bald word rent, they use the term economic rent. (This solution is imperfect, as that term can in ordinary language refer to a leasing payment which is in some sense reasonable for a lessee or potential lessee to bear.)


Economic rent, and the pursuit of rent — called rent-seeking — are considered quite important by most economists.

Rent-seeking, perhaps often unconscious but certainly almost never acknowledged, is wide-spread, and explains a very great deal of the political process.

Rents cannot be paid unless they are paid by someone; often, those paying them do not even realize that they are bearing these costs, or mistake their sources. Worse, rents typically transfer wealth inefficiently, with more lost by whoever is made to pay the rent than would be lost by some other transfer mechanism. Still worse, rent-seeking imposes costs even when it is unsuccessful.


Within a neo-classical framework, rent (or economic rent) might be alternately be defined as something given in exchange, above and beyond the opportunity cost of producing that for which it is exchanged. When the frequent assumptions of complete preferences, continuous divisibility of goods and of services, and ignorability of small costs hold, this new definition is equivalent to the original definition. But, exactly because one ought not to over-commit to those assumptions, it is best not to adopt the alternate definition as such.

I have encountered professors of economics asserting things to be rents that certainly were not. For example, one simply declared that the salary of a specific basketball player were a rent, but the professor had lost himself in a taxonomy which imagined the player simply as playing basketball or as idling. In fact, the player could have played for teams other than the one that hired him, and could have applied his abilities to something other than playing in professional basketball. If there were a rent in his salary, it may not even have been the major portion.


* This entry is primarily infrastructural. I want to be able to refer to rent and to rent-seeking in later entries, without there defining the associated terms.

Humpty Dumpty, Prescriptivism, and Linguistic Evolution

13 December 2016

In Chapter 6 of Through the Looking Glass by Charles Lutwidge Dodgson (writing as Lewis Carroll), a famous and rather popular position on language is taken:

When I use a word, Humpty Dumpty said, in rather a scornful tone, it means just what I choose it to mean — neither more nor less.

If Mr Dumpty’s words simply mean whatever he intends them to mean, then the rest of us are not in a position to understand them. If he provides us with verbal definitions, we must know what the defining words mean. He could not even declare in a manner intelligible to us that he meant most words in the same sense as do you or I. We might attempt to tease-out meanings by looking for correlations, but then we would be finding meanings as correlations, which assumes properties (such as stability) that represent more than pure choice on the part of Mr Dumpty. Having been made perfectly private, his vocabulary as such would have no practical value except for internal dialogue. There is a paradox here, which Dodgson surely saw, yet which so very many people don’t: If Mr Dumpty’s apparent declaration were true, then it could not be understood by us. He might actually just be making some claim about breakfast. We might take (or mistake) his claim for a true proposition (that his vocabulary were purely idiosyncratic), but any co-incidence between his intention and our interpretation would be a result of chance. We could not actually recognize it for whatever proposition it actually expressed.

In order to communicate thoughts with language to other persons, we must have shared presumptions not only about definitions of individual words, but also about grammar. The more that such presumptions are shared, the more that we may communicate; the more fine-grained the presumptions, the more precise the communication possible. In the context of such presumptions, there are right ways of using language in attempt to communicate — though any one of these ways may not be uniquely right or even uniquely best — and there are ways that are wrong.

Those who believe that there are right ways and wrong ways to use language are often called prescriptivist, and generally by those who wish to treat prescriptism as wrong-headed or as simply a position in no way superior to the alternatives. Yet, while one could find or imagine specific cases where the beliefs concerning what is right or wrong in language-use were indeed wrong-headed, forms of prescriptivism follows logically from a belief that it is desirable for people to communicate, and especially from a belief that communication is, typically speaking, something rather a lot of which is desirable. As a practical matter, altogether rejecting prescriptivism is thoughtless.

To the extent that the same presumptions of meaning are shared across persons, the meanings of words are independent of the intentions of any one person. Meanings may be treated as adhering to the words themselves. Should Mr Dumpty take a great fall, from which recovery were not possible, still his words would mean exactly what they meant when he uttered them. A very weak prescriptivism would settle there, with the meaning of expressions simply being whatever were common intention in the relevant population. This prescriptivism is so weak as not often to be recognized as prescriptivism at all; but even it says that there is a right and wrong within the use of language.

Those more widely recognized as prescriptivists want something rather different from rude democracy. In the eyes of their detractors, these prescriptivists are dogmatic traditionalists or seeking to creäte or to maintain artificial elites; such prescriptivists have existed and do exist. But, more typically, prescriptivism is founded on the belief that language should be a powerful tool for communication as such. When a typical prescriptivist encounters and considers a linguistic pattern, his or her response is conditioned by concern for how it may be expected to affect the ability to communicate, and not merely in the moment, but how its acceptance or rejection will affect our ability to understand what has been said in the past and what will be said in the future. (Such effects are not confined to the repetition of specific pattern; other specific patterns may arise from analogy; which is to say that general patterns may be repeated.) Being understood is not considered as licensing patterns that will cause future misunderstandings.

In opposing the replacement of can with the negative can’t in can hardly, the typical prescriptivist isn’t fighting dogmatically nor to oppress the downtrodden, nor merely concerned to protect our ability to refer to the odd-ball cases to which can’t hardly with its original sense applies; rather, the prescriptivist is trying to ward-off a more general chaos in which we can hardly distinguish negation from affirmation. (Likewise for the positive could care less standing where the negative couldn’t care less would be proper.) When the prescriptivist objects to using podium to refer to a lectern, it’s so that we continue to understand prior use and so that we don’t lose a word for the exact meaning that podium has had. We already have a word for lecterns, and we can coin new words if there is a felt need for more.

The usual attempt to rebut prescriptivism of all sorts notes that language evolves. Indeed it does, but prescriptivisms themselves — of all sorts — play rôles in that evolution. When a prescriptivist objects to can’t hardly being used where can hardly would be proper, he or she isn’t fighting evolution itself but participating in an evolutionary struggle. Sometimes traditional forms are successfully defended; sometimes old forms are resurrected; sometimes deliberate innovations (as opposed to spontaneous innovations) are widely adopted. Sometimes the results have benefitted our ability to communicate; sometimes they have not; but all these cases are part of the dynamic of real-world linguistic evolution.

The Evolution Card is not a good one to play in any event. Linguistic evolution may be inevitable, but it doesn’t always represent progress. It will not even tend to progress without an appropriate context. Indeed, sometimes linguistic evolution reverses course. For example: English arose from Germanic languages, in which some words were formed by compounding. But English largely abandoned this characteristic for a time, only to have it reïntroduced by scholarly contact with Classical Greek and Latin. (That’s largely why our compounds are so often built of Greek or Latin roots, whereäs those of Modern German are more likely to be constructed with Germanic roots.) It was evolution when compounding was abandoned, and evolution when it was reädopted. If compounding were good, then evolution were wrong to abandon it; if compounding were bad, then evolution were wrong to reëstablish it. And one cannot logically leap from the insight that evolution is both inevitable and neither necessarily good nor necessarily bad to the conclusion that any aspect of linguistic practice is a matter of indifference, that nothing of linguistic practice is good or bad. One should especially not attempt to apply such an inference peculiarly to views on practice that one dislikes.

Nihil ex Nihilo

6 December 2016

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 ∧ [(¬YY)∀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.