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.

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[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.

Years ago, a friend and I were talking about something, and he mentioned Hitler. I declared

I don't know why you ever liked that guy!

in reply to which he barked

Oh! That is a lie![1]

Well, no, it wasn't a lie. I escalated by betting him dinner on the matter. Then I explained to him that, since the truth of a proposition is a precondition for it to be known, one of the ways that I could not know why he'd ever liked Hitler would be if he'd never liked Hitler. Another way would be if I'd never believed that he'd liked Hitler, regardless of how my friend really felt about Hitler.

Indeed, the contradiction of I don't know why you ever liked that guy! is I know why you at some time liked that guy! Formally,[2] So,

I don't know why you ever liked that guy!

was a truth (though perhaps not a simple truth, as he'd had trouble seeing it).

Having won the wager, I waived the prize; my objectives in betting had all been met. Now, had he won the wager, then I'm sure that he'd have collected; but had I claimed, as he'd thought, that he'd once liked Hitler, then he'd have been quite justified in extracting the dinner; it would have disincentivized my insulting him in such a way, and off-set the felt sting of the calumny.

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[1] That was how he spoke. He often began with Oh!, and when learning English in Hong Kong he had been taught to avoid contractions.

[2] (2015:09/24): I have edited the formal expression, seeking to have it capture more completely the structure of the natural-language expression.

The Eight Amendment to the Constitution of the United States declares

Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.

(Underscore mine.) The constitution of the state of California has a much more complex discussion of bail; but its Article 1, §17 declares

Cruel or unusual punishment may not be inflicted or excessive fines imposed.

(Underscore mine.) Plainly these words are an adaptation from the US Constitution.

The replacement of and with or was apparently to indicate that cruel punishment were not to be deemed acceptable simply by virtue of being usual. Indeed, Article 1, §17 of the constitution of the state of Florida used to declare

Excessive fines, cruel or unusual punishment, attainder, forfeiture of estate, indefinite imprisonment, and unreasonable detention of witnesses are forbidden.

(underscore mine) and the state supreme court made just that interpretation in cases of the death penalty. (The section has since been radically revised.)

However, a hypothetical problem arises from the replacement. Just as cruel punishment is not acceptable regardless of whether it is unusual, unusual punishment is not acceptable regardless of whether it is cruel. And if most or all prevailing punishments were cruel, then any other sort of punishment were unusual; and unusual punishment has been forbidden. Thus, under such circumstance, all punishment were forbidden!

This problem may not be merely hypothetical, in the context of problems such as prison over-crowding. (Of course, when push comes to shove, lawyers and judges tend to shove logic out the door.)

[This entry may be superfluous, in that people who fall for any of the fallacies discussed are unlikely to read the entry, people who employ one of the fallacies are unlikely to reform if they do read the entry, and people who recognize that fallacies are involved may not see much use to analyzing them.]

I often encounter an argument, whose form is

P does A₁;
A₂ is better than A₁;
therefore it would be acceptable/desirable for P to do A₂.

It's easy to find P, A₁, and A₂ such that the intuïtion recoils from the conclusion that

it would be acceptable/desirable for P to do A₂.

and, in the face of such intuïtions, most people will acknowledge the non sequitur (acknowledged or otherwise) in the argument. We could even add a further premise, that

P ought to be persistently active (if not necessarily in their present manner).

and still find P, A₁, and A₂ such that the intuïtion recoils from that conclusion. (Consider that non-profit institutions do facilitate child abuse, and that child abuse is worse than many other things that are still themselves unacceptable.)

Yet one encounters this argument frequently with P as the state, A₂ is something that somebody wants done (such as space exploration) and A₁ is something disturbing that the state is doing or has done recently.

A variation on this can be found with form

P₁ approves when P₂ does A₁;
A₂ is better than A₁;
therefore P₃ should not object to P₂ doing A₂.

A non sequitur is evident in cases where P₃ is plainly no subset of P₁; but this argument is often presented in a manner so as to obscure a distinction, as when an everyone or a no one is used as-if loosely (which is to say inaccurately) in the first premise, but P₃ is some person or group of persons who aren't actually in the set labelled everyone or actually are in a non-empty set labelled no one.

However, this argument is fallacious even when P₃ simply is P₁. There may in fact be an incoherency in approving of A₁ while objecting to A₂, but that inconsistency could be resolved by changing one's position on A₁. For example, if forcing people to pay for birth control is better than forcing them to pay for war with Iraq, then perhaps someone who objects to the former should cease approving of the latter, rather than embracing the former.

(And resistance from P₁ to coherence wouldn't itself license A₂ when A₂ victimizes yet some additional party P₄. One doesn't force atheists to distribute copies of Al Qu'ran on the grounds that neoconservatives would object to such distribution even while supporting worse things.)

Sometimes one even sees an argument of the form

P₁ does not object when P₂ does A₁;
A₂ is better than A₁;
therefore P₃ should not object to P₂ doing A₂.

Variations of this even go so far as to replace objection with more active opposition.

P₁ does not actively oppose P₂ doing A₁;
A₂ is better than A₁;
therefore P₃ should not actively oppose P₂ doing A₂.

The appeal for those who present these arguments is that, if they were accepted, then almost no A₂ could be practicably challenged, as the objector could be dismissed for not having tackled each and every greater evil.

Of course, if this argument held, then it could virtually always be turned around against the claimant. For every P₂ and A₂, there is a P'₂, A'₁, and A'₂ such that A'₂ is the supposed ill addressed by A₂, P'₂ effects A'₂, and A'₁ is some greater ill effected by P'₂. In other words, even if A₂ were good, it would itself almost never address the greatest evil, so that there would always be something else that one would be required to do before ever getting to A₂.

One claim about Libertarians that won't withstand any real scrutiny — yet is very common amongst journalists and educators — is that Libertarians don't believe in doing anything to address the immediate needs of the poor. If asked to defend the claim, those who make it will either note Libertarian opposition to various state programmes, and with a crude induction draw the inference that Libertarians don't believe in doing anything to achieve the ostensible goals of those programmes, or they'll note the Libertarian objection in principle to any state programme with such goals, and treat this as QED.

Well, let's lay the form of that out:

L does not believe that X should be done by S,
therefore
L does not believe that X should be done.

Oooops! That isn't really very logical, is it? I mean that we can find plenty of X and S where this won't work, when we make ourselves L.

Libertarians don't believe that the state should do a lot of things, including farming, financial intermediation, and managing roads. Genuine anarchists go further, to claim that the state shouldn't do anything. That hardly means that they don't think that these things should be done by someone. It doesn't even mean that they won't agree that they should be those who do these things. (Indeed, people who rely upon the state are most likely to say that it ought to do whatever it does at the expense of someone else, as when they call for higher taxes on those who make more money.)

This point of logic ought to be obvious. Well, many journalists and educators are such damn'd fools that they truly don't see it, and an awful lot are knaves, who see it but don't want it to be seen by others.

One way that I see the eristicism effected is by the specious society-state equation — by treating the state as if it is society, which is to say as if it is us. Formally, this would be

L does not believe that X should be done by the state,
which is to say that
L does not believe that X should be done by society,
which is to say that
L does not believe that X should be done by any of us.

except that it's not explicitly expanded in this way, else the jig would be up. One place you'll see this eristic equation employed is in many quizzes that purport to tell the taker what his or her political classification is. If he or she answers affirmatively to a claim such as that society should help the poor then the typical quiz will score that towards state socialism and away from classical liberalism (of which Libertarianism is the extreme).

(Actually, one needs to be very careful whenever encountering the word society. In practice, it is often used to mean everyone else. Sometimes it's used to refer to some hypothetical entity which is somehow more than a group of people and their system of interaction; this latter notion tends to operationalize, again, as everyone else. Equating society with the state, and coupling this with demands for the state to make greater demands on other people is a popular way of making society mean everyone else.)

The fact is that one simply cannot tell, one way or another, from the datum that a person is a Libertarian whether he or she thinks that some goal ought to be pursued, unless the goal involves what a Libertarian would label coercion; because Libertarianism itself is no more than a belief that one ought not to initiate the class of behaviors to which they apply this label. A person can be a Libertarian and be all for voluntary redistribution, or that person might indeed be someone who embraced some of the more callous proclamations of Ayn Rand, or the Libertarian might hold some intermediate postion. Libertarianism itself is neutral.

(Within the Randian camp, there has been a willful confusion of the fact that Libertarianism itself has limited scope with the proposition that any given person who is a Libertarian must somehow have no view about matters not within that scope, or with the claim that a Libertarian must think that anything not prohibitable is good.)

Parallels can be found here with the claim that atheists do not believe in morality of any sort. Not only is the underlying fallacy very similar, but the implication in each case is that, should the persons in question believe that something ought to be done, they are more likely to see themselves as the someone who ought to do it.

14 A corroboration of this point of view seems to come from some of the so-called tests for logical thinking, which include questions such as What is the next number in the (!) sequence 1, 3, 6, 10 … ? and (in slightly simplified form) Which of the following four figures differs from the other three: a square; a cross; a circle; a triangle? Questions of this kind do indeed test a valuable ability, viz. the ability to guess what the man who formulated them had in mind, e.g. the circle, because it is the only figure that is round (although, of course, the square is the only one with four corners; the cross the only one with a ramification point; the triangle the only one with exactly three corners). What in this way certainly is not tested is logical thinking. Yet the marks youngsters make in such tests highly correlate with their achievements in elementary mathematics! Far from demonstrating that those test questions have more to do with logic than with guessing and empathy this correlation rather seems to indicate that the presentation of elementary mathematics has more to do with guessing and empathy than with logical thinking.

(Karl Menger, an eminent mathematician of the 20th Century, was the son of Carl Menger, one of the preceptors of the Marginal Revolution in economics, and founder of the Austrian School of economics.)

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.]

As noted earlier, I've been reading Subjective Probability: The Real Thing by Richard C. Jeffrey. It's a short book, but I've been distracted by other things, and I've also been slowed by the condition of the book; it's full of errors. For example,

It seems evident that black ravens confirm (H) All ravens are black and that nonblack nonravens do not. Yet H is equivalent to All nonravens are nonblack.

Uhm, no: In words, that all ravens are black is equivalent to that all non-black things are non-ravens.[1]

The bobbled expressions and at least one expositional omission sometimes had me wondering if he and his felllows were barking mad. Some of the notational errors have really thrown me, as my first reäction was to wonder if I'd missed something.

Authors make mistakes. That's principally why there are editors. But it appears that Cambridge University Press did little or no real editting of this book. (A link to a PDF file of the manuscript may be found at Jeffrey's website, and used for comparison.) Granted that the book is posthumous, and that Jeffrey was dead more than a year before publication, so they couldn't ask him about various things. But someone should have read this thing carefully enough to spot all these errors. In most of the cases that I've seen, I can identify the appropriate correction. Perhaps in some cases the best that could be done would be to alert the reader that there was a problem. In any case, it seems that Cambridge University Press wouldn't be bothered.

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[1]The question, then, is of why, say, a red flower (a non-black non-raven) isn't taken as confirmation that all ravens are black. The answer, of course, lies principally in the difference between reasoning from plausibility versus reasoning from certainty.

I have been reading Gut Feelings: The Intelligence of the Unconscious by Gerd Gigerenzer. Gut Feelings seeks to explain — and in large part to vindicate — some of the processes of intuïtive thinking.

Years ago, I became something of a fan of Gigerenzer when I read a very able critique that he wrote of some work by Kahneman and Tversky. And there are things in Gut Feelings that make it worth reading. But there are also a number of active deficiencies in the book.

Gigerenzer leans heavily on undocumented anecdotal evidence, and an unlikely share of these anecdotes are perfectly structured to his purpose.

Gigerenzer writes of how using simple heuristics in stock-market investment has worked as well or better than use of more involved models, and sees this as an argument for the heuristics, but completely ignores the efficient-markets hypothesis. The efficient-markets hypothesis basically says that, almost as soon as relevant information is available, profit-seeking arbitrage causes prices to reflect that information, and then there isn't much profit left to be made, except by luck — unpredictable change. (And one can lose through such change as easily as one might win.) If this theory is correct, then one will do as well picking stocks with a dart board as by listening to an investment counselor. In the face of the efficient-markets hypothesis, the evidence that he presents might simply illustrate the futility of any sort of deliberation.

Gigerenzer makes a point of noting where better decisions seem often to be made by altogether ignoring some information, and provides some good examples and explanations. But he fails to properly locate a significant part of the problem, and very much appears to mislocate it. Specifically, a simple, incorrectly-specified model may predict more accurately that a complex, incorrectly-specified model. Gigerenzer (who makes no reference to misspecification) writes

In an uncertain environment, good intuitions must ignore information

but uncertainty (as such) isn't to-the-point; the consequences of misspecification are what may justify ignoring information. It's very true that misspecification is more likely in the context of uncertainty, but one system which is intrinsically less predictable than another may none-the-less have been better specified.

I am very irked by the latest chapter that I've read, Why Good Intuitions Shouldn't Be Logical. In note 2 to this chapter, one reads

Tversky and Kahneman, 1982, 98. Note that here and in the following the term logic is used to refer to the laws of first-order logic.[1]

The peculiar definition has been tucked behind a bibliographical reference. Further, the notes appear at the end of the volume (rather than as actual foot-notes), And this particular note appears well after Gigerenzer has already begun using the word logic (and its adjectival form) baldly. If Gigerenzer didn't want to monkey dance, then he could have found an better term, or kept logic (and derivative forms) in quotes. As it is, he didn't even associate the explanatory note with the chapter title.

Further, Gigerenzer again mislocates errors. Kahneman and Tversky (like many others) mistakenly thought that natural language and, or, and probable simply map to logical conjunction, logical disjunction, and something-or-another fitting the Kolmogorov axiomata; they don't. Translations that presume such simple mappings in fact result in absurdities, as when

She petted the cat and the cat bit her.

is presumed to mean the same thing as

The cat bit her and she petted the cat.

because conjunction is commutative.[2] Gigerenzer writes as if the lack of correspondence is a failure of the formal system, when it's instead a failure of translation. Greek δε should sometimes be translated and, but not always, and vice versa; likewise, ∧ shouldn't always be translated as and nor vice versa. The fact that such translations can be in error does not exhibit an inadequacy in Greek, in English, nor in the formal system.

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[1]The term first-order logic refers not to a comprehensive notion of abstract principles of reasoning, but to a limited formal system. Perhaps the simplest formal system to be called a logic is propositional logic, which applies negation, conjunction, and disjunction to propositions under a set of axiomata. First-order logic adds quantifiers (for all, for some) and rules therefor to facilitate handling propositional functions. Higher-order logics extend the range of what may be treated as variable.

[2]That is to say that