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.]

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

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

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

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

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

Towards my next paper, I've been thinking a lot about decision-making where one has uncertainty but not quantified probabilities or even necessarily a total ordering of possible outcomes by plausibility. Most recently, I've tried to formalize the notion of when, without quantified probabilities, one lottery may be said to be fairer than another, and of a simple rule for selecting the fairer of two coins (as in my previous paper I have made considerable use of orderings of coins by entropy).

Yester-day, in the context of such ponderings, I arrived at some interesting, simple complementarity rules. Consider two actions, each paired with one considered outcome. Between these, various plausibility relations may obtain — the first pair may be more plausible than the second, the second may be more plausible than the first, they may be equally plausible, their relative plausibility may be unknown, or the relation may be a union of two or three of the aforementioned (eg one pair may be more-or-equally plausible). In any case, whatever that relation, the same relation will hold if we reverse the order of the pairs and take the logical complement of the outcomes. Here's an example of the formal expression of one of these rules where I'm using the same notation that I did in in my entry of 19 August, and represents the relationship of the left side being more plausible than the right side.

(Common-sense examples are easy to generate. For example: If it is more likely that the Beet Weasel will bite than that the Woman of Interest will stay home, then it is more likely that she will depart than that he will refrain from biting. Or: If we don't know whether a given nickle is more likely to come-up heads than is a given quarter, then we don't know whether the quarter is more likely to come-up tails than is the nickle.)

In the context of an irreflexive, antisymmetric, transitive relation, one can identify closeness without measurement. For example, if A is more plausible than B and B is more plausible than C, then B is closer both to A and to C than they are each to the other.

This abstraction of closeness, along with the principles of complementarity, allow one to identify when one coin is more fair than another, without having any quantification of fairness, so long as one can order the plausibilities of outcomes across coins. One simple rule is to pick the coin whose most likely outcome is less likely than the most likely outcome of other coins; an equivalent rule would be to pick the coin whose least likely outcome is more likely than the least likely outcomes of other coins.

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BTW, the aforementioned previous paper is still in the hands of the editors of the journal to which I submitted it a bit less than four months ago. I've not had any word from them. But, while this journal did not provide a time-frame, other journals give frames such as six months. (A friend recently had one of her submissions rejected at just before the three-month mark.) It is at least somewhat plausible that, by the time that said previous paper is published somewhere, I will have this next paper ready to submit to a journal.

[This entry is based upon a reply to a friend, who requested an explanation of comparative advantage.]

Imagine that you and someone whom you know need each to produce reports that will involve both pages of text and pages of diagrams. Imagine further that you can produce ten pages of text in a day or five pages of diagrams in a day, while this other person can produce five pages of text or three pages of diagrams.

You have an absolute advantage in the production of each good here. None-the-less, if you are able to trade (text for diagrams), both of you can gain.

For every page of diagrams that you produce, you have to forgo production of two pages of text. For every page of diagrams that the other person produces, he must forgo production of one and two-thirds pages of texts.

Slow as he may be at each task, he has a comparative advantage in the production of diagrams. Setting aside transaction costs, if someone will trade text for diagrams at a ratio of better than five-to-three, then he can profitably make diagrams to trade for text. You, meanwhile, have a comparative advantage in the production of text. Setting aside transaction costs, if someone will trade diagrams for text at a ratio of better than one-to-two, then you can profitably make text to trade for diagrams. So trading at something between 1 2/3 pages of text and 2 pages of text per page of diagrams should work for you both.

The only way that each of two parties could not have a comparative advantage in something would be if everyone had exactly the same production trade-off ratios. That's not bloody likely.[1]

We certainly don't require that one party be worse at both things for each party to have a comparative advantage in something. Here

each party has an absolute advantage in something, and a comparative advantage in that same thing. Such examples come freely to mind; and, because in such examples comparative advantage is in the same product as absolute advantage, such examples foster a confusion that absolute advantage determines where one should specialize or (worse) what one should produce. (The latter is worse because it mistakenly implies that one should never trade for something in which one has an absolute advantage.)

Comparative advantage underlies virtually all trade,[2] whether we're talking about two people or two firms or two nations. But it is in international trade that comparative advantage is most often discussed.

This attention is because lay-people are most likely to think that international trade or proper trade policy is instead somehow determined by absolute advantage. The fear that one country can somehow suck up everything through unregulated trade is almost always founded on a belief that absolute advantage (from cheap labor in the undeveloped world or from advanced technology in the developed world) determines who profits.

But explanations in terms of absolute advantage lack coherence. Returning to the original example of producing reports (where you have the absolute advantage in both products), there is no way for you to leave the other person worse-off through trade, unless he can be persuaded to trade at a ratio worse (inclusive of transaction costs) than he can produce for himself. Maybe he's dumb enough for that, but he could be dumb enough for that even if he had the absolute advantage in both.

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[1] On the other hand, it is quite possible that the ratios could be close-enough that the costs of transaction (including transportation) could swamp-out the potential gains-from-trade.

[2] Off the top of my head, I doubt that there are actually any exceptions. For example, when one buys what may seem an over-priced product, as an act of pity or of charity, which product one could have produced for oneself, either the premium may be viewed as a purchase of something beyond the overt product, or the transaction may be decomposed into a trade coupled with a simple gift.

As an economist, I am especially pleased and amused by the expression make oneself scarce.

Although scarce can mean no more than rare, its principal meaning is of being in insufficient quantity to exhaust desirable use, and it is in this sense that economists employ the term. Actually, something can be quite rare without our having any use for it, and something can be fairly abundant and yet be less than we could use.

A rational decision maker values a potential increase or decrease in the supply of something in terms of what use would be gained or lost. If a resource is not scarce, then it has no further use, and an increase would be valueless. And any decrease that didn't result in scarcity would also be valueless.

Thus, when someone is told Make yourself scarce!, the implication is that, at present levels, there is no further use for him or her; indeed, the suggestion is that there's just too much of him or her as it is. He or she is being told to reduce his or her presence until it has some g_dd_mn'd value.

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It used to be fashionable in some quarters to claim that scarcity as economists understood it were a myth and that we lived in a post-scarcity economy. The essential claim there would be that we couldn't use any more of anything were it to become available. I regard that claim as offensively stupid. Not quite as dreadful were claims that we could soon have a post-scarcity economy. But the implication there would be that humans would be insufficiently clever to think of a further use for anything.

The people who made such assertions should have made themselves scarce.

Consider two propositions:

• The first is that markets are smart, to the extent that they cannot be tricked into anything unless one carefully hides most or all of the contrary evidence.
• The second is that, left unregulated, markets produce some best possible outcomes.

These aren't at all the same proposition. On the one hand, something can be hard to deceive, yet work at purposes contrary to those that one favors. On the other hand, a mechanism can be vulnerable to some sorts of disruption but, in the absence of that disruption, perform some task well. I'm not saying that the propositions are contrary; they could be simultaneously true; none-the-less, they're plainly not identical.

The run-up to the latest economic crisis seems to have been founded in no small part by a confusion of these two distinct propositions. The Bush Administration represented itself — and may well have considered itself — free market, in-so-far as it expected considerable resilience on the part of the market in the face of remarkable levels of state borrowing and considerable other interventions (compassionately conservative or kleptocratic). And Alan Greenspan, who surely considered himself a believer in laissez faire, is these days explaining his optimistic proclamations from before the crisis as stemming from a failure to reälize that investors would not recognize that a boom could not last forever, to which lack of recognition he also attributes the crisis, as if irrational exuberance were simply a Keynesian animal spirit, rather than a product of things such as lending regulations and Federal Reserve interest rate policy.

Meanwhile, many of the Keynesians, socialists, and pragmatic technocrats (long-standing or born-again) are arguing that the fact that the market could be fooled shows that markets aren't clever and that thus various sorts of interventions are needed, as if any defense of free markets must hang upon a belief that markets are simply too clever to be fooled. Left unaddressed is whether the confusion were endogenous or brought on by state intervention, whether those prior interventions that may have been the cause of the confusion produce actual benefits worth the costs of that confusion, and whether more intervention would produce a more clever system or a less clever system.

In fact, there are various long-established free-market schools of thought that attribute economic crises to a propensity of state intervention to fool economic participants. For example, it is difficult to distinguish to what extent interest rates reflect the supply and demand of private savings for future consumption, and to what extent they are an artefact of central bank intervention for other purposes. In the face of Federal Reserve manipulation of interest rates, the market will not be sufficiently smart to see what the price of loanable funds should be, and therefore will almost certainly build too much or too little for the future.

Some years ago, Go Daddy added a back-order service whereby a domain already registered by another party would be monitored and, should its registeration lapse, the domain would be registered in the name of the purchaser of the service.

Some time after that, Go Daddy added an auction service. And, when registration lapses for which Go Daddy is the registrar, then duiring the grace period (42 days in the case of Go Daddy) Go Daddy itself puts the domain up for auction; the auction ends well before the grace period, and the auction results are cancelled if the prior registrant renews before the end of the grace period.

A registrant seeking to have a domain appraised might simply let the registration lapse, watch the auction, and then register before the end of the grace period. A late registration requires a higher fee, but that difference could be viewed as the cost of appraisal.

Now. here's where it gets ugly. Go Daddy holds such auctions even if there is a prior back-order. They hold the auctions even if the domain had a different registrar when the back-order was placed, but then switched to Go Daddy. They hold the auctions even if the back-order was placed before they had an auction service. If the domain should be bid to anything above an opening bid of US$10, the purchaser of the back-order must either pay more or let the domain go to some other party.

A Go Daddy back-order on a domain is worse than useless to its buyer if the domain may be expected to be registered with Go Daddy at the time whenever registration lapses. If the domain is sufficiently attractive that a back-order would be useful without an auction, then there will be competitive bidding in an auction.

Now, I'm sure that Go Daddy sent a notice that the old back-orders were going to be subjected to the new protocol, and that refunds were offered. But few if any customers would have understood the implications of a change, otherwise there would have been a lawsuit that Go Daddy would have lost, as simple refunds wouldn't have covered the economic loss avoidably being placed on these customers.

I recently read someone defending socialism on the ground that socialism has the same root as does society. Well, I don't object to society. And I venture to guess that she doesn't object to fathers, yet I go further to guess that she does object to what's called patriarchy. One mustn't over-reach with etymology, with dear old dad, nor with society.

I've previous explained the economic calculation problem of socialism: Rational allocation of resources requires trade-off signals that reflect as much relevant information as practicable. Most of the relevant information is highly decentralized, and some of it (such as the expectations and preferences of participants) is intrinsically so. A market brings that information into play by way of prices (trade-off signals) developed by the give-and-take of would-be consumers and of would-be sellers. Socialists haven't developed an alternative; they correct the market only at the cost of over-all misallocations with their own costs in human welfare.

This point is as true in the delivery of health care as anywhere else. Almost everyone agrees that American health-care delivery is in appalling shape, but there are those who ignore that the problems have grown as state interventions have increased. Commentators frequently note that costs have exploded in the last fourteen years, but then most of these commentators are silent on the fact that the period followed upon the last round of reforms. Of course, the period before those reforms wasn't itself some sort of golden age; the reforms were effected because many things were seen to be worse than once they were, and getting worse still. But, again, due attention was not paid to the rôle of prior state intervention in effecting that worsening. This routine of blaming what remains of a market for the mounting problems of an increasingly state-controlled system began well before I was born.

Many people, even defenders of socialized medicine for the United States, admit that the socialized systems elsewhere have some dramatic flaws. The belief of the defenders is that the United States can develop a better system, perhaps in part by learning from the problems of other states. But the deep problem is, again, that of trade-off signals. And one of the seldom-recognized implications of that is that greater state control here has led and will lead to a worsening of systems elsewhere. A state-controlled system can somewhat compensate for its own inability to formulate rational trade-off signals by being guided (directly or indirectly) by prices generated elsewhere. (This solution is imperfect because the prices of one region cannot be expected to be ideal for another; and, if they were, using them fully would generate exactly the same out-comes as would be effected by a free market, rendering the socialism absurd.) Implicitly, production and distribution of health care in the industrialized nations with more socialized medicine has been significantly guided by the choices made in the United States. To the extent that our prices as well continue to become the guesses of bureaucrats rather that the outcomes of interaction between free consumers and free producers, socialized medicine everywhere will be shooting in an ever-growing darkness.

Even assuming that morality can somehow ignore such practical problems, the morality of the claims for socialized medicine strikes me as utterly bogus. Many people declare health care to be a fundamental right, but that's plainly incoherent as one could exercise any fundamental right without the presence and assistance of other people. There have been very few attempts to build ground-up cases for a moral entitlement to health care — identifying some actual fundamental right from which a right to health care is derived in a social context — and every one with which I'm familiar has been exploded on logical grounds. Mostly people just confuse the appealing proposition that it would be a very fine thing if no one was denied health care for simple lack of resources with there being a right to health care. There are a great many hypotheticals that would be very fine things. I know people such that it would be a very fine thing if they had the companionship of someone of the desired sex, and such that they would like that even more than access to medical care; I hardly think that we should force someone else to provide that companionship though.

Some very fine things become very vile things when delivered by virtue of confiscations, regardless of whether we imagine that the confiscation is effected by society, or recognize that it is by a state or by a gang or by a mob.

Last night, I finished the clean-up of a LAΤΕΧ version of my paper on incomplete preferences. From remarks by a person more knowledgeable about ΤΕΧ than I, it seemed that my best option in dealing with the under-sized angle brackets was to just fall back to using only parentheses, square brackets, and braces for taller delimiters. And most width problems were resolved by expressing formulæ over more lines. Unfortunately, these changes leave the formulæ harder to read than in the original.

This after-noon, I completed the submission process to one of the two specialized journals recommended by the advising editor who rejected it at the previous journal to which I submitted it. The submission process for this latest journal required that I name the other journals to which I'd submitted the paper. As simultaneous submissions are disallowed, basically they were asking for a list of which journals has rejected the paper. I gave it. (I didn't tell them that the third had been suggested by the second, nor that theirs had been suggested by the fourth.)

Anyway, I'm back to waiting for a response.