Informative tautologies (updated)

Technically, for a logician, or a semanticist of the Snow-is-White school, tautologies convey no information;  but to linguists and pragmaticians, in context they often do.  In fact, we may state that they usually do, since otherwise why utter them?  “Business is business” is flint-hearted;  “Boys will be boys”, tenderly exculpatory.

The opposite of an utterance that pretends to contain no information (and thus, in particular, to be inexpugnable) but actually does (and often of a trenchant sort), is a definition that, ex cathedra, is all about informing, but which melts to the touch.  Cf. our essay here:

http://worldofdrjustice.blogspot.com/2015/01/pugnaciously-vacuous-definitions_24.html

~

Covertly vacuous uses of technical-sounding terms  have been scouted in histories of science.  As, “Things fall because of gravity, and rise because of levity.” Or Molière’s virtus dormitiva.  But more may be packed into such terms than may be initially apparent.  As, one philosopher pointed out that “The ball rebounded to the height that it did  because of its resiliency.”  But this is informative:  the height is owing to internal characteristics of that ball, rather than from the ball’s having been dropped from a greater height, or having been dropped on a more resilient surface.

Additionally, the initial tautological character of a sentence can  so to speak  “age off”, in accordance with semantic evolution of its terms. Thus, “Atoms are indivisible” was initially as circular as “Bachelors are unmarried”, since they were defined from the outset as indivisibilia (as their name, a-tom, etymologically implies).  But on its current interpretation, the sentence would qualify as false.

A mathematician looks at Newton’s Definition 1, in the Principia:

Quantity of matter is a measure of matter that arises from its density and volume jointly.

Great acumen is hardly needed to realize that this definition is hopelessly circular, since density is normally defined as the ration of mass to volume;  but Newton’s unhelpful phrase  does have some implicit implications.  For example, we expect the mass of an object to remain unchanged if we change its shape.

-- Michael Spivak, Physics for Mathematicians: Mechanics I (2010), p. 9

~

A 2015  example of the uses of tautology:

Robert Buissière on Médi1, re Presidential candidates:

Jeb Bush, frère de son frère,

et Hillary Clinton,  épouse de son époux.

As they stand, these statements are “analytic”; but we understand the import:  Jeb and Hillary got where they are today, largely owing to family association.

Cf. & contrast the common expression “He is his father’s son.”  Normally this means that he takes after his Dad, and not that he is getting any special favors from other people owing to that filiation.  To imply the latter, you might say “Daddy’s little boy” or something.  By contrast, the French phrases in the above context  do not imply that Jeb’s politics are a close match to Dubya’s, let alone that Hillary’s are a close match to Bill’s.

~

A bare tautology like “Business is business”, as a free-standing statement, invites contentful interpretation via a “Gricean implicature” (specifically, the Maxim of Quantity).   The following is a syntactically more complex case, there the tautology is embedded in a subordinate clause:

Ever since self was self, nature been keepin’ folks off of red-hot stoves.

-- Zora Neale Hurston, Their Eyes Were Watching God (1937)

The meaning is:  Common sense has been extant  from time immemorial.  Operationally, the (tongue-in-cheek) interpretation is:  Go back and back in time, sampling as you go. For each sample-point, verify whether  “self = self” holds at that time; and if so, then evaluate “Common Sense is in effect? Y/N”.  The sentence, for all its folksiness, has a kind of philosophy-class spin to it; the moreso as “self = self” calls up First-Order Logic with Identity.

~

Stylistic appreciation

Usually there is a summary “That’s that” finality to tautologies, whether used informatively or not;  stylistically, they are bare-bones.  But consider this:

Herod:  The moon has a strange look tonight. … She reels through the clouds like a drunken woman. … Does she not reel like a drunken woman?  She is like a madwoman, is she not?

Herodias:  No;  the moon is like the moon, that is all.

-- Oscar Wilde, Salomé (1891)

Here the barrenness of the pale white, plain round  far-floating body, is reflected in the unyielding tautological formula.

~

Enten-Eller

Philosophers have scribbled  much ink, and later worn out many a typewriter ribbon, and finally expended great bushels of pixels, discoursing upon the status of “logical truths”; such as, paradigmatically, the following:

            (I) Every man is either married or a bachelor.

(We simplify, since the matter is not really of interest; leaving out of account, for instance, the curious case of Schrödinger’s groom.)

It is agreed that such a sentence tells us nothing about the world, unlike that time-honored exemplar of informativeness,

            (II)  The cat is on the mat.

which has been so oft repeated. down the years, that said cat has achieved the immobility and timelessness of an Egyptian idol.  (Presumably the mat lies in a patch of sunlight, so why ever move?)

            And yet its affordances are quite different from those of another statement of the same logical form; say:

            (II) Every number is either even or odd.

For, although the sentence (I) does not perhaps baldly state anything substantive about the world, its presuppositions speak volumes.  For one thing, it gives us to understand that there is a sharply defined institution, Marriage, into which a man may enter or not; and that his resultant state is either-or. Even so much will give our Martian anthropologists  sufficient grist  for many turns of the mill.

~

The above are mostly individual linguistic parlor-tricks.  Much more generally, there is the matter of the status of the equations of mathematics.  A view put forward by the dessicated and ennervating tendency called formalist (nominalist)  maintained that, being mathematical, they are tautologies, and being tautologies, they are uninformative -- semantically vacuous.  Practical experience shows that doctrine to be false.  As a way to see how such equations manage to be informative, consider the number pi.

Pi can be defined, qualitatively, in a number of different ways, most familiarly as the ratio of a circle’s circumference to is diameter.  It  is, moreover, a Given of the invisibilia, woven into the fabric of the noöspheric pattern;  and -- crucially -- it can be reached in a startling variety of precise quantitative ways, along this strand or that of the warp and the woof -- and the woorp and the wahf, along any of the dimensions of the multidimensional mathematical textile.   Pi is itself ever one, the peak of the Golden Mountain;  but the paths that climb to it are numberless.  It can be expressed as a definite integral; as an infinite continued fraction; as an infinite product; as an infinite sum; with many variations of each.  (Behold some of them at https://en.wikipedia.org/wiki/Pi.)  That fact that any one of these equals π is informative and indeed amazing, for in effect each one constitutes hiking instructions -- a trail-map -- to the summit, each from a wildly different base-camp.  And each one of these infinitely intricate expressions  is equal to any other, in an equation which is as distant from tautological or uninformative as can be imagined.

~

[Update 13 May 2020]  An aborbing article by Evan Osnos, in the 11 May 2020 issue of The New Yorker, maps out the paths by which the Greenwich Connecticut  tennis-and-boating-club crowd  came to support Trump.  The notables of that town have a patrician heritage, in principle at variance with the flashy style of the vulgarian from the Bronx, but as one blue-blood testifies, his conversion came while witnessing an early speech by the candidate:  “He had that line that he would use: ‘Folks, we either have a country or we don’t.’ And I felt the chill .. I’m, like, ‘Oh, my God, this is a really good line.’

Apart from the formally tautological character of that line, it puzzles by its vagueness:  out of context, it is unclear what at all is being hinted at.  Presumably the line is a dog-whistle, a bit of Rorschach rhetoric from which the listener will extract whatever meaning he likes.

The line does nothing for me; but it does recall such successful political antecedents as “The business of America is business.”

~

[Update 14 May 2020] 

Some of you may be familiar with the British sport of trainspotting.  That may or may not be in accordance with current U.K. guidance on coronavirus lockdown.  But here’s a hobby you can practice in the safety and comfort of your own home:

Tautology-Spotting !

As:

Headline in this morning’s New York Times:

The People Behind the Counter Are People

Remember this the next time you order takeout.

That one recalls those sleep-inducing example-sentences from introductory logic class (All brave Athenians are Athenians).  But in this case, it has a punch, and the source of that punch is not logical but lexical.  For, people has a variety of senses;  from the neutrally classificatory

(a)  an instantiation of the species Homo sapiens, near-cousin of Pan troglodytes, sometimes known as “a forked radish”;

to the “pregnant sense”  (I phrase it freely)

(b)  an ensouled being created in the image of the Lord of Hosts, whom Christ died to redeem.

In the cited sentence, the first occurrence of the word people has the (a) meaning; the second occurrence, (b).

For more on perspectival semantics and pregnant senses, check out this essay:

https://worldofdrjustice.blogspot.com/2019/10/words-of-day-sublunary-postlapsarian.html

~

In all the instances above, tautology is used in its traditional logico-philosophical sense.  But technical terms sometimes get picked up by a wider audience, where their use may be lax.  Thus, the literary critic V.S. Pritchett, in his article on the novelist Anthony Powell, wrote:

Mr. Powell is excellent with the raffish…. I think the sententious irony succeeds. … It adds a very English flavor, either of comic tautology or deflation.

The sense of tautology is unclear here.  Perhaps it refers to mechanical repetition, a standard component of broad humor.

Modus Ponens Paradox

Dixit Algorizmi:

* All Elephants are Mortal.

* Babar is an Elephant.

* Yet -- Babar is Eternal.

How do we account for this discrepancy?

For discussion, see:

https://worldofdrjustice.blogspot.com/2015/12/modus-tollens-tollendus-est-iterum-re.html

For additional crispy thought-food, click on any of the Labels below. 

For those who could care a fig for philosophy(**), and are interested only in the mighty Elephant King, click here.

(**) Technically known as the ficus philosophiae.

The Logic of Yea and Nay

The Senate has just shot itself in the foot;  with shrapnel ricocheting onto masses of bystanders.  The move might enter the annals as “Shumer’s Self-mate”.

In the face of such legislative absurdity, the Administration floated the idea of reducing the minimum to pass a Continuing Resolution  from 60%  to a bare majority.    Worth discussing;  but there is a more general logical point behind the whole debate, which seems to have garnered little or no notice in print.

The point is, across the board, the status quo is the default.   As the cock crows, the Senate (to take that as an example, but the point applies to any established body) begins its busy day with various activities.   Everyone shows up for work with teeth brushed and appropriately dressed (men as men, women as women);  there is some sort of opening ceremony (in my primary school, these were Flag Salute and Lord’s Prayer);  a roll-call is taken;  citizens are admitted to the visitors’ balcony; interviews are given to the press;  and so forth.   It would be legitimate to question any one of those activities:  require unisex attire; ban the prayer or the flag salute (some of this has already happened in this venue or that); skip the time-wasting roll-call; exclude spectators (for security, or whatever).  To do that, you introduce a bill, which for passage requires a majority of some weaker or stronger preponderance (say, 60%), depending on the nature of the proposed bill.   What is not required is that,  in order to continue to wear normal clothes, or salute the flag, or what have you, the body must, every morning as the cock crows, pass a brand-new bill (with the required 60%) to continue the status quo.   The burden is on those who would change or disrupt the status quo.

Now, allowing the government to continue putt-putting along, for better or worse, is simply the default, and should be so considered.  It’s the government’s job.   So it is truly not logical that Congress should be obliged to seek a super-majority, simply to keep the government open for business, fulfilling the promises made.    Anyone who would disrupt that scenario, at so fundamental a level, should be obliged to propose doing so, in a measure  -- say, the Government Shutdown Circular Firing-Squad bill -- that itself would require the supermajority.   Instead, the quite radical maneuvre of wrecking the government   is relieved of what should logically be its burden, as though that were the default.

[Update 22 Jan 2018]  The Washington Post today has a good article on the logical/structural parameters of such situations.

https://www.washingtonpost.com/news/worldviews/wp/2018/01/22/why-other-countries-dont-have-government-shutdowns-2/?hpid=hp_hp-top-table-main_wv-shutdown-1105am%3Ahomepage%2Fstory&utm_term=.a67ef5b32d8a

 

 

 

 

 

A side-bet for “Pascal’s Wager”

For the most part, Pascal’s Wager gets little respect these days, whether among believers or agnostics.  But it turns out you can take Pascal’s logic quite seriously, and wind up … backing a horse of another color entirely.

Such are the Yazidis:  devil-worshippers by repute;  but not, it appears, Satanists.  Let us explain.

Historically, Satanists do not form any single actual church, apart from occasional aberrations:

http://www.chicagotribune.com/news/nationworld/ct-satanic-temple-monument-minnesota-20170506-story.html
Rather, they are individuals of a particular bent, who, like Faust or Aleister Crowley, out of diabolical Superbia, dabble in the dark arts.  They generally (like Faust or Crowley) formed part of some ambient mainstream faith, before defecting in service of the Prince of Lies.

As for what is really going on with the Yazidis, it is difficult to know, since they themselves -- a hermetically closed sect -- are mum on the subject, and their neighbors, being all of them detractors, cannot well be relied upon.  But roughly (and since we are dealing here with theological logic, and not with Mideast anthropology, that rough cut will do): 

(Not history, but rational reconstruction):
At some point around the dawn of the sect, its elders reasoned thus:

(1)  God the Father, maker of heaven and earth, is notoriously forgiving.  -- Here they reason with Heinrich Heine, whose reported last-words, trifling with God on his death-bed, were:

Bien sûr, il me pardonnera; c'est son métier.

(2)  Satan, on the other hand, has an evil reputation:  vicious and vindictive.  You don’t want to get on his Enemies List.

(3)  Ergo:  Placate Satan (in words, at least) in this life;  and hope for forgiveness in the next.

Such a prudentially propitiatory policy  may be compared with paying protection-money to Al Capone.  Doesn’t mean you like the guy;  it’s just an expense of doing business.

In summary:  Yazidis, in the old phrase, “hold a candle to the Devil”, but they do not actually (interiorly, in their heart of heartgs) worship hIM.

Se non è vero, è ben trovato;  think of it as a Gedankenexperiment.

The pros and cons of Pascal’s Wager  are reviewed here:

http://worldofdrjustice.blogspot.com/2017/03/in-defense-of-pascals-wager-expanded.html

The Perfection of Penguins: an inductive proof

In mathematical induction, you begin with the base case a.k.a. Induktionsanfang, traditionally symbolized as P(0) (where P stands for “Penguin”, and  0 depicts the egg).   In the present instance, P(0) is Baby Fuzzy.

Kleiner Kuschel-Induktionsanfang

Now, P(1) = Fuzzy’s Mommy;  and they rejoice in a Perfect Love.  Thus, the Induction Hypothesis is established.

Now consider any arbitrary P(k).  In this case, P(k)  is known as Fluffy;  and P(k +1) (the k-plus-first Penguin) is Fluffy’s perfectly adoring mate.   So once again, the induction goes through.

Thus we have established the Perfection of Penguins.

~

Note:  Using the more powerful technique of Transfinite Induction, we can establish the existence of (the wonderful, the unutterable) Zorn’s Penguin.   This, however, requires the Axiom of Choice, in its strengthened form, “Choose Penguinity”.

Kantian ethics in a pinch

Two items from the morning’s press, illustrate the difficulty of finding guidance in the Categorical Imperative, when the snow hits the fan.

Always keep a copy in your glove-compartment!

(1)  The tragedy of the commons

An op-ed in the wake of “Snowzilla”, followed by a reader’s comment.

http://www.baltimoresun.com/news/opinion/oped/bs-ed-snow-parking-20160125-story.html

Note to Baltimoreans: That parking spot you shoveled out isn't yours

The message was taped to the windshield of a pickup truck parked on a public street:

"This parking space did not magically shovel itself! Be considerate of your neighbors and other people in the neighborhood. DON'T steal parking spaces that people worked for several hours to clear."

The parking space didn't magically pave itself, either — the taxpayers, presumably including the owner of the pickup truck, paid for that — but that point seems to have eluded the anonymous author of this angry missive.

And how can it take "several hours" to shovel the snow out of a parking space? What was the author using to remove the snow, a thimble?

Every time it snows in Baltimore, I am reminded of the words of noted philosopher and humanitarian (and former Sun columnist) Kevin Cowherd: "God forbid we ever get a real emergency around these parts. We'll be eating our children before the sun goes down."

You know the drill. People get out their snow shovels and clear just enough snow to make room for two wheel tracks. Then, after pulling the car out into the street, they put their chairs down in the parking space, to protect the results of "hours" of backbreaking labor, and incidentally making it impossible for snowplows to get through. Then we have a thaw and a freeze, and everything turns into solid ice, with only half as many parking spaces there would be if people had concentrated on getting the snow out of the way instead of guarding "their" parking spaces.

---------

[Comment] Boston dealt with the place markers by picking up everything that was left on the street and carting it away. The natives were outraged but the city stuck to their guns and cleared everything away. Remember too that the mountain of snow they piled up did not melt till last July and it included hundreds of lawn chairs.

Potzteufel !  Who the hell parked in my space ??

https://www.washingtonpost.com/local/the-great-post-blizzard-parking-struggle-pleas-threats-and-lawn-chairs/2016/01/26/16826218-c464-11e5-8965-0607e0e265ce_story.html

In Washington, Nate Bergman spent 96 words stating his case, laser-printed and sealed in plastic.

He’d spent hours digging his car out of a parking place on a Capitol Hill street. He duly acknowledged the legal right of other drivers to take the public space. He appealed to their better natures not to. If that didn’t work, he promised to shovel the snow back to its “original place around your vehicle.” In all, a measured treatise touching on individual liberty and shared responsibility.

In Philadelphia, an unnamed shoveler made the same claim more succinctly: “If you park in my space, I’ll break your [expletive] windows. Have a nice day.”

The death struggle for post-blizzard parking is playing out in different ways in different places all along the East Coast. But few have escaped the plunge into anarchy that erupts when two feet of snow smothers already limited on-street parking.

To the diggers, the moral high ground is clear: Shoveling 300 pounds of snow from a patch of public pavement is like homesteading on the frontier. I cleared it — I own it.

Not so, say the defenders of the commons. Those lawn chairs, pylons and sawhorses standing guard on city streets are squatting on taxpayer property.

“No one owns a parking space,” D.C. Police Chief Cathy L. Lanier said at a news conference Saturday, appealing to citizens to keep their furniture off the streets.

Before the storm even hit, Philadelphia police used a spoof of Drake’s “Hotline Bling” to urge residents to report any illegal cones or trash cans blocking parking spaces. They dubbed it #NoSavesies.

In Boston, which has a long history of snow-related parking disputes, an umbrella group of several South End neighborhood associations banned space saving last year. Now the South End Forum is raising money to compensate drivers when their cars are vandalized by angry space diggers. There were scores of confrontations over parking this week all over the city, including a nonfatal shooting Monday in Dorchester

The moral hair splitting baffled even experts. When does a private citizen earn a claim to a public resource? It’s an ethical whiteout.

Die angewandte "Metaphysik der Sitten"

(2)  Poster-children vs. free riders

That which is held in one half of the brain, is simultaneously rejected in the other.

http://www.nytimes.com/2016/01/26/world/europe/corsico-italy-school-lunch-fees.html

CORSICO, Italy — After being elected last year, Mayor Filippo Errante found that this town, abutting Milan, had accrued an “alarming” debt of more than a million euros in unpaid school lunch fees. So he decided to take what he called an iron-fist approach.

Children whose parents were up to date on payments would be allowed to eat cafeteria-prepared meals. Children whose parents had not paid would not.

“The era of the ‘furbetti’ is over,” Mr. Errante said in a statement on social media last month, using a term that translates to cunning, akin to gaming the system.

Some called the decision a form of blackmail. Others criticized it for creating what they said was a schoolroom apartheid, where some children ate hot meals while the others snacked on homemade panini or a slab of cold pizza.

Petitions and protests ensued. Teachers, principals and many parents rallied in support of those children who would not be given lunches, even as nearly all agreed that school fees should be paid.

The children most affected by the provision belong to families whom Mr. Tortoreto described as “lemons that have already been squeezed dry” by life.  Some are the children of foreigners without legal permission to remain in Italy, others of parents with a history of mental illness.

Refutation vs. Disconfirmation

We continue our observations about Verification (essay here) and Modus Tollens (essay here).

It has been maintained that there is an important asymmetry between the verification and the refutation of a theory in empirical science.   Refutation has been said to be conclusive or decisive, while verification was claimed to be irremediably inconclusive. … The falsity of [the theory] is indeed deductively inferable by modus tollens.

-- “The Falsifiability of Theories”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 62

That is basically because an empirical law is, in general, a universal: ∀x P(x).   Given any  ¬P(a), that is falsified.    Whereas successive confirming instances, while perhaps increasing one’s confidence in the hypothesized law, do not deductively entail it.   Nor need that fact be, in practical terms, a mere quibble, in cases of an infinite domain:  thus, a conjecture in number theory might be verified for all values of n up to a billion -- a trillion -- a googolplex -- yet still turn out to be false.  (Indeed, toy conjectures of this sort  are trivially easy to dream up.  E.g. “No integer is evenly divisible by a googolplex.”)   Taking confirming instances ‘too seriously’, is the Fallacy of Affirming the Consequent.

Grünbaum then, however, in a Duhemian vein, weighs in against such modus-tollentic tyranny, with the observation that, in empirical science, you are really assuming (ex hypothesi) not merely a theory, but (in unspoken conjunction therewith) various Auxiliary Assumptions:  and one or more of these, rather than your pet theory, might be the ones to suffer the fury of the tollendum.  Thus,  “Isolated component hypotheses of far-flung theoretical systems are not separately refutable but only contextually disconfirmable” (id., p. 63).

The same idea, more sociologically stated:

Typically  a theory is abandoned, not because of recalcitrant observations on their own, but because of these together with some, perhaps quite complex, reasoning from them:  in the extreme case, the observations may all be quite well known already, and all that is new  is the reasoning.

-- Michael Dummett, Truth and other enigmas (1978), p. 409

Compare:

… that falsity, not truth, is the primary notion.

-- Michael Dummett, Truth and other enigmas (1978), p. xl

~

It is seldom that a theory is decisively k.o.’d by a Popperian uppercut to its predictive jaw.   Often the process happens slowly, like a river silting up;  there is no single Aha! moment (or rather, Oy veh! moment).   In science, as in life,  sometimes love grows old and love grows cold.  As, assessing a certain philosophical program:

The further one goes down the reformist path, the more implausible the consequences of one’s theory are likely to become, until  at some point  the implausibility of the consequences  comes to outweigh the initial attractiveness of the theory.

-- Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. I, p. 299

That gradual, einsickernd, process of disenchantment, lacks the drama of Popperian refutation, or of Kuhnian paradigm-shift;  it is more reminiscent of the dissatisfying phenomenon (if it even is one) of truth decay.

~

The history of science -- though not the historiography of science -- is littered with once-flourishing theories that somehow just … faded, without necessarily ever having been decisively confronted or refuted.  It is no fun to research and write about these, since
(a) they are losers in the lottery (rightly or not), and scarcely known today (so that one’s potential readership is infinitessimal -- and indistinguishable from zero among working scientists); and
(b) there is seldom a clear-cut moral in the tale.

The occasions on which one does get an insight into some defunct theory  tend to be biographies of otherwise-great scientists -- e.g.  Theodore Porter’s fine account of the (as we now reckon him) statistician Karl Pearson, which must needs notice that hero’s period of infatuation with, mm, ahh, … ether squirts (pas devant les enfants!);  or Thomas Hankins‘s  of William Rowan Hamilton.  Or, sometimes, a historian of science rolls up his sleeve and deconstructs/resurrects  some particular thread of experiment.  As, Gerald Holton’s commendable study of “Subelectrons and Presuppositions”, involving the “classic” Millikan experiment, of which we all get a telescoped and sanitized version in high school physics.  The actual saga is more anfractuous, and less rich with easily digestible lessons.

[Millikan] found himself working with the wrong presupposition, but he knew how to rid himself of it eventually.  Millikan launched into that work with the same energy and obstinacy  as into his earlier work on the quantization of the charge of the electron, yet with the opposite assumption.

-- Gerald Holton, The scientific imagination (1978), p. 72

And re the related experiments of Felix Ehrenhaft:

There was never a direct laboratory disproof of Ehrenhaft’s claims.  … Lorentz … remark[ed]:  “The question cannot be said to be wholly elucidated.” In his review of the case, R. Bär noted …: “the experiments left, at the very least, an uncomforable feeling.”  Like most such controversies, this one also faded into obscurity, without anything as dramatic as a specific, generally agreed-upon falsification taking place at all.  Indeed, Ehrenhaft continue to publish on subelectrons into the 1940s, long after everyone else had lost interest in the matter.

-- Gerald Holton, The scientific imagination (1978), p. 79

And that, within (central, classical) physics.  The less rigorous (natural and social) sciences  are much pocked with such as well.

~

Science, not as logically implying some new phenomenon (such as the gravitational bending of light), but as suggesting likely places to look.   Thus:

The search, which has proved so successful, for chemical atoms  having specific nuclear and electronic constitutions, and for chemical molecules having specific atomic constitutions, has been stimulated by the previous construction of theories to explain the consitution of atoms or of molecules already known, theories which had, as it were, ‘gaps’ in them.

-- Richard Braithwaite, Scientific Explanation (1953), p. 70

And indeed, not only the existence of, say, atoms or isotopes with a given atomic weight and atomic number  are thus -- not exactly ‘predicted’ in the deductivist sense, but pointed to -- but additionally, we shall expect certain behaviors, or ranges of behaviors, based upon the place in the periodic table of that species (should it exist).

Thus, when the eka-aluminum that Mendeleev hypothesized  was eventually identified (by a Frenchman, who thus had naming-rights and called it gallium) and turned out to have behavioral values very close to those predicted,  while not proving the theory of atomic chemistry as it then stood, was still much more than just one more danged black raven.

[A footnote on another such predicted discovery, that nicely illustrates consilience in the sciences, combining both forefront atomic physics  and practical geology:

At the time Bohr was developing his theory, the element following lutecium was undiscovered.  But Bohr … asserted that, in this unknown element, the added electron would have to be placed in the level n = 5. This would imply that the unknown element would have an electronic configuration analogous to that of zirconium.  Inasmuch as elements that are chemically analogous  are usually found in the same minerals, Bohr claimed that the missing element should be sought in minerals containing zirconium.  It was indeed in such ores that the missing element, called hafnium [after the Latin name of Bohr’s hometown] was detected.

-- A. D’Abro, The Rise of the New Physics (1939),  p. 549.  ]

This “guiding-light” role of a theory, not being a formal entailment, is likewise not subject to modus tollens (though of course it might be abandoned in the face of an accumulation of recalcitrant instances):

Whereas discovery of observable properties which fill such gaps in a theory  is rightly thought to provide a weighty confirmation of the theory …  failure to discover such properties would not be regarded as weighing against the theory, except perhaps in very special cases.

-- Richard Braithwaite, Scientific Explanation (1953), p. 70

Modus tollens tollendus est ! (iterum re-updated anew)

In philosophy there are very few  and perhaps no valid  logical-impossibility  or reductio ad absurdum  proofs.

-- Alasdair MacIntyre, After Virtue (1981; ²1984), p.  101

Wer A sagt, muss auch B sagen.

-- old folk-saying

Die bürgerliche Stellung des Widerspruchs

-- L. Wittgenstein, Philosophische Untersuchungen, #125

An example of epistemological ‘character armor’:

All scientific research programmes may be characterized by their ‘hard core’.  The negative heuristic of the programme  forbids us to direct the modus tollens at this ‘hard core’.  Instead, we must use our ingenuity to articulate or even invent ‘auxiliary hypotheses’, which form a protective belt around this core, and we must redirect the modus tollens  to these.

-- Imre Lakatos, “Morphology of Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 133

In political or religious ideology, there are obvious analogues of this ‘hard core’.  (E.g. 'sacred cows'.)

~

[This exercise originally began as a simple application of cool, unruffled logic  to a sociopolitical topic too hot to handle in less rarified terms.  The point was, If you were logically consistent, you would … (taceo).  But a working-out of the general intellectual idea  led, inevitably, towards that domain (mathematics, logic itself) where the very idea of consistency is most at home;  and led, less obviously, to a kind of backhanded defence of inconsistency  (wherein we follow our countryman Emerson, in his depreciation of that ‘hobgoblin of little minds’).

Modus tollens is that figure whereby, when a proposition entails a falsehood, that proposition is infirmed.   The procedure finds general acceptance among the logically inclined.  This essay considers cases where otherwise logical people nonetheless kick against the pricks of this restriction.

We now return to the original plan.]

~      ~      ~

It should be obvious that the fundamental objections to racism and sexism … apply equally to speciesism.

-- Peter Singer,  Animal Liberation (1975), p. 6; quoted in Stephen Schwartz, A Brief History of Analytic Philosophy (2012), p. 282.

Singer’s proposition has the form  “P => Q”;  we shall call it the “Singer Sentence”.  It is, so presented, a proposition, and no mere proposal, since he claims that it is “obvious”, presumably because founded upon a generally-accepted principle:  basically, What’s sauce for the goose  is sauce for the gander.    Since that principle is reasonable (we have often invoked it ourselves), though extra-logical, we have no a-priori quarrel with the Singer Sentence at all.

Suppose, however, that you yourself eat meat, wear wool and leather, and are content that rabid dogs should be shot.  Suppose further that -- selfish cad that you are -- you do not, like the saintly naked Jainist, wear a veil before your lips when drinking tea, lest you accidentally imbibe some supernatant insect.   Nor (for shame!)  do you enjoin your dentist and your physician  to refrain from administering anaesthia or antibiotics, on the grounds that these, having been developed with the aid of experiments on helpless animals, are fruit from a poisoned tree.   And suppose that (blast your impudence!)  you do not intend to amend your ways.  Well then, in effect, objectively, you hold that “not-Q”.  What follows from this?

What follows (by the most elementary logic -- the rule known as modus tollens) is that, if you assert the Singer Sentence, then you must needs conclude:  not-P.   -- That, of course, would be a political catastrophe.

Short of adopting Jainism, there are only a few ways out:

(1) Have recourse to a kind of meta-tollentic principle, to the effect that any set of propositions which entail not-P  must itself be denied, P being unantastbar.

(2)  Proclaim the truth of Q, even while continuing your carnivorous habits;  shrug apologetically; reference Emerson re the "hobgoblin of little minds".

(3) Boldly hold the joint and several validity of :
P;  P=>Q; and not-Q. 
Then utter not another word upon the subject.

Such a mindset is known as “doublethink”.  It is not so bad once you get used to it, judging by its millions of satisfied customers.

Cf:

Whether the ethic of ‘speciesism’, to use Richard Ryder’s term, can be put on a logical footing  any more sound that that of ‘racism’, I do not know.

-- Richard Dawkins, The Selfish Gene (1976; 2^nd edn. 1989), p. 10

(That ‘any more’ sounds rather half-hearted …)

~

There is a different sort of logic-chopping that may be operative in a case like this.   Someone wishes, for non-logical reasons, to assert that Q; then trumps up (fallacious) reasons for asserting that P => Q.   (Thus, currently, in a certain political fringe, the desired outcome is “It’s all Obama’s fault”.  Whatever P may arise in the world, a hasty P => Q is asserted, to reach the desired conclusion.)   But, for psychological and political reasons, I do not believe that Singer himself is here guilty of this:  he in all likelihood does feel ethico-logically compelled to deduce Q, however unwelcome that conclusion may be in many quarters.   For he likewise (by different steps) reaches a quite distinct and socially unmentionable conclusion (this time concerning deformed fetuses or the disabled), and one which is precisely of a nature to enrage that segment of society which would embrace his earlier conclusion about animals.  So, no, Singer was not trying to win any popularity contests.

[Footnote] For those of you in the quandary (2), here is the place 4 U to shop (courtesy of Garrison Keillor):

People’s Meats

Most of us accept strict vegetarianism as the best way,  but many find it difficult to change their eating habits.  People’s Meats is an interim solution.  All of our meat comes from animals who were unable to care for themselves any longer.  Hoping to phase out the operation, we do not advertise hours, prices, or location.  We do not deliver.

~            ~            ~

The analysis above was clad in sociopolitical raiment;  but its skeleton is logical, which is subject-matter-neutral.  Consider the following (which is skeletally somewhat distinct, but in ways  unimportant  for our purpose):

Let P be standard mathematical praxis.   (And here -- as seldom -- we actually are referring to the human practice of mathematizing, rather than to the timeless and species-independent truths of mathematics itself, whatever these may be.   For more on the distinction, see here: http://worldofdrjustice.blogspot.com/2012/08/on-nature-of-mathematical-knowledge.html.)

Something similar to the  P => Q step was broached about a century ago; it concludes (while using reasonable background metamathematical assumptions comparable to the “what’s sauce for the goose is sauce for the gander” enthymeme above) that this standard practice leads to paradox.  As John von Neumann put it,

A closer study of the merita of the case, undertaken by Russell and Weyl, and concluded by Brouwer, showed that the way in which  not only set theory  but also most of modern mathematics  used the concepts of ‘general validity’ and of ‘existence’  was philosophically objectionable.

-- quoted in James R. Newman, ed. World of Mathematics (1956), p. 2058

Must we then give up P?   Brouwer (a Dutch mathematician who had previously proved important results) now plays the role of Singer, and went on to his logical conclusion:

A system of mathematics which was free of these undesirable traits, “intuitionism”, was developed by Brouwer.  In this system  the difficulties and contradiction of set theory  did not arise.  However, a good fifty per cent of modern mathematics, in its most vital -- and up to then unquestioned -- parts, especially in analysis, were also affected by this “purge”:  they either became invalid, or had to be justified by very complicated subsidiary considerations.

-- id.

(That final clause is a far more dreadful consequence than might be apparent to those outside mathematics, since mathematicians prize elegance and generality of proof. )

Du muss dein Leben aendern ...

So -- shall we bow to these strictures, and surrender our mathematical meat? 

Von Neumann goes on:

Only very few mathematicians were willing to accept the new, exigent standards for their own daily use.  Very many admitted that Weyl and Brouwer were prima facie right, but they themselves continue to trespass, that is, to do their own mathematics in the old, “easy” fashion -- probably in the hope that somebody else, at some other time, might find the answer to the intuitionistic critique and thereby justify them a posteriori.

In short, the bulk of mathematicians adopted strategy (3) above.

Brouwer, like Singer, went on to make a pest of himself for many years.  

[Footnote]

Intuitionism -- initially a sort of mathematical vegetarianism -- is by no means dead.  Michael Dummet, no crank, espouses intuitionistic logic (I am currently painfully working my way through his essay on the subject, line by line.)   And it has subsequently morphed in ways that are quite beyond me, e.g. in topos theory.

[Footnote 2, a half hour later]  In his essay “The Philosophical Basis of Intuitionistic Logic” (1973), Michael Dummet writes (for our present purposes, the context is unimportant), concerning a proposal that he has just put forward:

What is involved is a thesis in the theory of meaning  of the highest possible level of generality.  Such a thesis is vulnerable in many places:  if it should prove that it cannot be coherently applied  to any one region of discourse,  to any one class of statements, then the thesis cannot be generally true,  and the general argument in favor of it  must be fallacious.  [dbj:  That last phrase has rather a Sherlockian cast to it.]  Construed in this way, therefore, a position in the philosophy of mathematics  will be capable of being undermined by considerations which have nothing directly to do with mathematics at all.

This amounts to offering a hostage to fortune -- specifically, a hostage to modus tollens, in its strong quantified form:    P => ∀x Q(x):  the existence of but a single exception (∃x  ¬Q(x) )  blows the whole game.

~

We need not have recourse to anything so rarified as ethics or metamathematics  to be confronted with an apparently (morally, intellectually) scandalous state of affairs:  It stares us in the face in that favorite tow-headed boy of the philosophy of science, physics.   And here, in that most venerable part of it,  Maxwell’s theory of electromagnetism.  Rock-solid in its own day, it survived (and even helped inspire) the introduction of Special Relativity.  It fit smoothly with the new developments in non-EM forces, graciously uniting with the Weak force to form the Electro-weak, and so forth.  And yet, in the context of the atomic theory, it was (if only in whispers) an intellectual scandal, since it predicted that atoms were unstable:  the whirling electrons would radiate energy away and quickly collapse into the nucleus.  Thus, the material world as we know it, could not exist.

Now, one stance of reaction in the face of so bald a challenge, is to say:  “R-r-right!  Science cannot err;  ergo, the world as we know it  does not exist.  Meta-ergo, we all are just brains in a vat.”   Such, in effect, is the path taken by contemporary eliminative materialists -- a shuffling tribe of hunchbacked ne’er-do-wells, who, faced with their irrefutable failure to derive free will, faith, thought, beauty, aspiration, or much of anything of value, from their prolonged and proctoscopic vivisections of sea-slugs and the like (to which the World of Dr Justice, friend to all creatures great and small, responds with the evisceration of eliminative materialists)  -- these gentlemen (stretching that word to its breaking-point), these … bipeds, conclude that Free Will is an illusion, faith and beliefs and reason  all one great gigantic joke, and that we are all just robots, brains marinating in a vat of simulation, or (barely) glorified sea-slugs.   (In this they are partly correct:  the eliminativists -- mark the name -- are but brains in a chamber-pot.)

So, are those who do not take that bold blind path, whenever some contradiction turns up, in a state of intellectual bad faith?

The answer is a subtle and qualified mmmm….n-n-no-o-o …..  For relief, we turn to Quine.

~

We frequently do turn to Quine for relief,  to savor his elegant pellucid prose, when the cacophony of the agora grows too rebarbative.    Yet it is not for his literary qualities that we seek him now, but for his (jointly with Pierre Duhem) holism.  Specifically, his celebrated doctrine that theses face the tribunal of experience, not single-spies, but as a corporate body.  There are (so to speak) concentric shells of propositions relatively dispensible and relatively central:  but in principle, none is immune to revision.  (Intuitionists have even gone dicking about with the Law of the Excluded Middle, and you don’t get more central than that. -- In that case, Quine was unimpressed:  “When you change the logic, you are only changing the subject.”  Rather a Platonist remark, that, Van.)

Thus, consider again the plight of electromagnetic theory, faced with the atomic paradox.  Its bacon was eventually saved, not by any refinement of that theory itself, but by an entirely unanticipated development:  quantum mechanics.

Now, there is no sense in which a pre-quantum physicist could have said “We saw that coming” or “Toleja so”.   Until the quantum theory arrived from nowhere, physicists had been content to simply live with the contradiction, in the classic fashion of Walt Whitman (“Do I contradict myself?  Very well then, I contradict myself.”)   And their quietism was justifiable, even during the years when no resolution was in sight.  For, Maxwell’s electromagnetic theory had done sterling service  both practically and theoretically, in a host of ways.  [This, I am aware, is intellectually comparable to the classic defense of Mussolini, that he made the trains run on time.]  The fact that it predicted an anomaly on the atomic level … well okay, an anomaly involving THE ENTIRE UNIVERSE BLOWING TO BITS -- but still, an atomic anomaly, not a macroscopic one (save secondarily), suggesting that one might, for the duration (until this beast be slain), simply wall-off the subatomic level (“there be dragons”) and get on with our lives.  It certainly did not make sense to throw out the Maxwellian baby with the anomalous bathwater.

In the case of EM, it turned out that there wasn’t even any bad bathwater to discard:  electromagnetism survived intact.   In the case of the aether, the anomaly was the Michelson-Morley experiment, whose results were later explained by yet another where-did-that-come-from new revolutionary theory, Special Relativity;  and in this case, the aether theory had to be discarded.   But both cases illustrate the thesis of Quine-Duhem, that when a body of doctrine is challenged, it is not initially evident which pieces must eventually give, and some may be close to the core of the structure.   In the case of Relativity and quantum theory, the transmogrification went deeply into the core indeed, upending our notions of space, time, causality, and continuity.   In retrospect, those curiously stable atoms seem not so bad;  the explanation is harder to live with than was the thing that was unexplained.

~

A word on our less-than-effusive approval of Quinean holism.

It is all too easy to imagine self-serving uses of such a principle.  As, Dennis the Menace, caught with his hand in the cookie-jar, exclaims:

“Mother, do not prejudge!  Granted, your B-fibres seem to present an image of someone resembling young Master Dennis with his arm hovering above a receptacle of some sort.  The hand itself -- which you suspect of larceny -- is not visible;  perhaps it has been tragically amputated, in which case the young fellow is more to be pitied than blamed.  Yet, how are we to reconcile this dubious alleged image with the far more desireable thesis that his character is pure as the driven snow?  Remember:  Propositions face the tribunal of experience as a corporate body!  Perhaps 'tis but an illusion of swamp-gas;  nay, perchance the fault lies somewhere in that oft-critiqued principle of Induction …”

(Later, as he sits with his pookie-bear in the familiar corner, he steams:  “But I had her epistemologically …”)

~

Of all human endeavors, surely mathematics is the most sensitive to refutations, however slight.    Yet behold this brawny scoffing attitude, specifically as regards the central and indispensible Calculus (famously the target of Bishop Berkeley’s barbs -- which it shrugged off):

If the calculus had not been ‘justified’ Weierstrass-style, it would have been ‘justified’ anyway.  The point is that the real justification of the calculus  is its success.

-- Hilary Putnam, “What is Mathematical Truth?”

Breezy, that!  Huey Long couldn’t have put it more pithily.

~

And now let us bring it all on home:  confronting the challenge of modus tollens, when a refutation or contradiction or paradox is met, in its home territory of mathematical Logic…

Celebratedly, the great German logician Frege  fell into despair (his masterwork already in galleys), upon being informed by Russell  of the latter’s eponymous Paradox. 

There we see  the logical conscience  at its most delicate.  For  Russell’s paradox, worthy though it be, is rather far-fetched, involving sets-that-are-members-or-not-members-of-themselves  (to which your average mathematician, let alone physicist, will say:  Huh??), all too reminiscent of the well-known but trifling Barber Paradox, involving a purported barber who “shaves everyone who does not shave himself”.  Paradox:  Does he shave himself???   Answer:  Fageddaboudit;  ain’t no such barber.

Rather other was the case of Quine’s Mathematical Logic.   In the version of its first edition (1940), this was shown to entail a contradiction.

Now:  an axiomatic system of logic, such as ML, is so tightly knit, that one bad apple really does spoil the whole barrel -- you can’t just shrug and say, “Nobody’s perfect.”  The situation in question, is generally held to be a catastrophe -- that any system which can derive a contradiction, can derive any proposition at all.

However, Hao Wang stepped in (ever the gentleman), and tidied things up, and all was well:  put right  in the second edition.

Haec fabula docet:  Contradictions are a bummer,  but don’t commit suicide  on their account.

[Footnote]   The Duhem-Quinean corporate-body doctrine  can be stated in terms of modus tollens, thus.  Given

            P1 + P2 + … Pn => Q

and

            ¬Q

we conclude

            ¬P1  ∨¬ P2 …  ∨  ¬Pn

That is, at least one of the co-conspirators whose conjunction led to a falsity, must itself be false.

But this does not imply  that the eventual valid conjunction will involve most or even any of those Pi;  we might even toss the whole lot of them overboard, and usher in a whole different set of conjuncts, to accomplish what we previously attempted with the P’s.  Such, roughly, describes the introduction of quantum mechanics, or the refutation of astrology, or any other paradigm-switch between incommensurables.

In his section on Quine-Duhem, Lakatos writes: 

Some people felt intuitively that the modus tollens from refutation  may ‘hit’ very distant premisses  in our total knowledge,  and therefore were trapped in the idea that the ‘ceteris paribus clause’ is a premiss which is joined conjunctively with the obvious premisses.  But this ‘hit’ is achieved, not by modus tollens, but as a result of our subsequent replacement of our original deductive model.

-- Imre Lakatos, “Morphology of Scientific Research Programs”, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 186

~

On a related note, another look at the notion that, should you ever derive a contradiction -- not as a tollentic hypothetical, but as the result of a deduction -- it’s Game Over for everything you’ve ever done.  A noted logician writes:

In an inconsistent system, every proposition is a theorem.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 42

Yet a few pages later  he qualifies this:

There is a generally accepted principle  that a contradiction implies everything.  We may yet distinguish proofs of the system which go through contradictions  from those which do not.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 47

~

Purely linguistic note:  A savorsome turn of phrase for the modal-tollentic move, one which I had not previously encountered:

Assume  towards a contradiction  that there is a set U  ∈ T that is not in T’.

-- Volker Runde, A Taste of Topology (2005),

~    ~     ~

There is a curious parallel, or at least an analogy, between, on the one hand,

(1) the foundational challenges to mathematics  alluded to above, together with, not a refutation of that challenge, but a broadly dismissive (and, arguably, pragmatically well justified) response on the part of professionals in the field;

and on the other hand

(2) recent broad-bore philosophical challenges to adaptationism (neo-Darwinism), along with the overall reaction of evolutionism’s professionals: indifference or -- the socio- and noö-politics of the thing being what they are -- foaming hostility.

Consider in particular  a recent (2010) volume co-authored by Jerry Fodor and Piattelli-Palmarelli, the well-written if pugnaciously titled What Darwin Got Wrong.

Their basic thesis is extensively and subtlely argued; hopefully I don’t overmuch crush it by fitting it into the following nutshell:

=>  The (currently hegemonic) gene-centered adaptationist theory  empirically does not work; indeed, for quite general reasons (having little to do with biology per se) it could not work even in principle;

=> A holistic, phenotype-cum-ecosystem-cum-kitchen-sink-centered approach  might -- might -- prove valid, at least in principle;  but in practice, the problem is intractable from a nomologically explanatory standpoint, apart from occasional lucky breaks.

Or, in the author’s own undistorted prose (p. 127):

To be sure, none of that actually shows that there aren’t laws of selection:  there may be, on the one hand, units of phenotypic change, and, on the other hand, units of ecological change;  and there may be laws that connect the two.  But there’s no reasons to suppose, as adaptationists routinely do, that the units of phenotypic change  are anything like what we generally think of as individual phenotypic traits.

(For connoisseurs of academic-polemical rhetoric, there is actually a sly move here.   While rhetorically conceding the possibility of a pheno-unit/eco-unit correlation, thus retrodictively legitimizing the latter two theoretical posits, these supposititious entities “units of ecological change” are by no means as familiar as other entities in the ontology of biology, and on the face of it  sound sort of bogus…)

~

We must distinguish two categories of challenge to any received body of doctrine:  The Anomaly; vs. Foundational.   The former (speaking just psychologically now) can either represent a mere annoyance  (or even:  Something to be hidden from the fickle general public at all costs, lest they overestimate its importance -- e.g.,  Evolution, Climate Change, anything medical), or a (possibly career-making) challenge.   The latter, to almost everyone, from the man in the street to the faculty lounge, tend to be just d*mned annoying.

Thus, consider  the Lasting Atom Scandal of the pre-quantum years, a poster-child example of Anomaly. (Nobody called it that, but  in all candor  they should have.)  This was phenomenologically egregious (in ways even a layman could understand), and had to be resolved somehow (some day, by someone else) -- indeed, imagine that the problem existed today, rather than a hundred years ago:  You’d have Republicans calling for the de-funding of physics, demanding e-mails archives, etc.  But it did not necessarily challenge -- certainly it did not set out to challenge, chin-foremost -- the nature of (say) Time, or Energy, or  for that matter  the existence of atoms (which had indeed been doubted by scientists, much later in history than most folks realize, but on quite different and less sophisticated grounds).

This anomaly, as we have seen, turned out to be handled in the most gratifying way possible:   We got to retain all the Maxwellian E-M we had laboriously learned, and now the new Quantum Mechanics stepped in (Jeeves-like)  to handle the anomaly.

More recently, there actually have been some challenges to physical theory  at a more basic level:  Time, which hoped to have escaped further challenge  by being subsumed into Space-Time, is once again on the carpet, from string theorists and others, who maintain that such parameters should fall out of a final theory, and not be input to it.  

Had such a challenge been posed, say, in the nineteenth century, it would have been an impudent kick at the foundations.  But now it (allegedly)  grows out of theory:  The theory may be mistaken, but it calls Time into question because it thinks it has something better. 

Much of the history of physics has been like that -- which is partly why it has been (politically) such smooth sailing.   Michaelson-Morley presented an Anomaly to the aether theory -- but no-one had been going around snarkily dissing ideas like distance and simultaneity until Special Relativity came along and re-conceived these.  This was (to use the Hegelian terminology, which here does fit) not a destruction, but a sublation  of the old ideas.

Sometimes there will be a simple anomaly, such as the discovery of continuous-but-nowhere-differentiable-functions, or space-filling curves.  These are eventually gobbled up and incorprated into a more robust and sophisticated mathematics.

Often an original sui-generis gnarly anomaly  will suggest a more general research program:  As, We need to pay closer attention to matterns of continuity and convergence (pointwise, uniform, almost-everywhere, etc., on the analytic side; and eventually the full exfoliation into the neighborhood-systems of topology).

Quite otherwise are challenges that come out of nowhere and that threaten to knock the stilts out from under you.  Russell’s Parodox, reaching Frege at press-time, did not strike him as another delightful puzzle to wrestle with some Sunday morning, but as a poisoned dart in his life-work.   Though they individually made numerous important positive contributions, the lytic work of Gödel and of Brouwer can be read this way, challenging the very notion of validity, truth, and proof.   Denial of the Excluded Middle indeed!  “Sirs, there you go too far!”  (“When you change the logic, you are merely changing the subject.” -- Quine, dyspeptically.)

Schematically (in your worst nightmare):

“Miller has reduced mathematics to set-theory;  and Spiller has shown that set-theory is rotten at its foundations.  Therefore everything that you are doing, or have ever done, or ever could do in your sorry life, is utterly worthless.”

Or:

(Pontius Pilate; Brouwer; post-Modernists):  “What is truth, anyway?  Huh? -- Meh.”

~

[Afternote]   It is a highly useful feature of the Blogspot interface, that it allows hot-linked Labels.    So for instance, were you disposed to read more by or about Mr. Hao Wang, for example, you would simply click on his name in the Labels field, and you will see every post so Labeled, in reverse chronological of posting.   But, annoyingly, there is a stringent limit on how many Labels any given essay is allowed.  We filled this one up with mathy stuff and now have no slack left over for the Darwiny bits.  So here you go:

http://worldofdrjustice.blogspot.com/search/label/adaptationism

http://worldofdrjustice.blogspot.com/search/label/philosophy%20%20of%20science

http://worldofdrjustice.blogspot.com/search/label/ultra-Darwinism

http://worldofdrjustice.blogspot.com/search/label/Charles%20Darwin

Note:  The last two are not redundant upon each other, and indeed have (I believe) zero overlap.  Darwin was not an ultra-Darwinist;  “Je ne suis pas marxiste” -- Marx.

Additional relevant Labels, which (boo, blogspot) would not fit into the Label field  for this post:
http://worldofdrjustice.blogspot.com/search/label/Ernest%20Gellner

[Post-Afternote]  Other examples of modus drasticus tollens.

A mittel-europäischer rationalist recalls “those golden, and, all in all very peaceful final decades of the colonial system”, and adverts ad the hermeneuts:

The argument seems to be -- Descartes led to Kipling.  We repudiate Kipling, so we must repudiate Descartes as well.   The expiation of colonialism must include the repudiation of clarity, for that had been but the tool of domination.

-- Ernest Gellner, Language and Solitude (posthum. 1998), p. 176

We, by contrast, raise a glass to Kipling;  so if this Descartes fellow had anything to do with it, chap can’t be all bad.

“During five literary generations, every enlightened person has despised him,  and at the end of that time  nine-tenths of those enlightened persons are forgotten, and Kipling is  in some sense  still there.” -- George Orwell, 1942

Modus tollens has surprisingly many enemies. Intuitionists, indeed, reject it:

The intuitionist position is that one can only state “P or Q” when one can give either a constructive proof of P  or a constructive proof of Q.  This standpoint has the consequence that proofs by contradiction (reductio ad absurdum) are not valid.

-- José Ferrerós, “The Crisis in the Foundations of Mathematics”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 150

Holism à la Duhem-Quine  skirts it with a flanking maneuver:

… arguing that a given observational consequence is deduced  not from an empirical hypothesis alone, but rather from the conjunction of this hypothesis with the relevant set of auxiliary assumptions.   Hence, the failure of the observational consequence does not deductively refute the hypothesis by modus tollens, when taken by itself, but discredits only its conjunction with the pertinent auxiliaries.  

--  Th. Kupka, “Introduction”, in: Adolf Grünbaum, Collected Works, vol. I (2013), p. 2

~

Another example from a political context, where propositions are surrounded by swarms of sensitivities --  Re series of premises and conclusions relating ethnicity, nationalism, and industrialism:

The argument is impeccable.  Its premises are valid.  How can a valid inference from true premises  yield a conclusion which appears to be wholly refuted by historical reality?

Ernest Gellner, Nationalism (1997), p. 32

~

Quite different in purpose and detail from the classical modus tollens, is the assumption of a premise known to be contrary to fact, but where you have antecedently proven (at great expense of elbow-grease) that the assumption of this simple premise does not effect the results of calculations which otherwise must be carried out laboriously.  As, when (having slogged through a bit of integral calculus) you prove that, in calculating the gravitational effects of a ball, you can pretend that the entire mass of the ball is concentrated at the center:  an enormous simplification.  Or again:

We can get the correct answer for the probability of partial reflection  by imagining (falsely) that all reflection comes from only the front and back surfaces.

-- Richard Feynman, QED (1985), p. 107

The elegant idiom for introducing such a foredoomed hypothesis is “Suppose, per impossibile, ...”

~

At the antipodes from the “tollendus” camp, are the celebrants of falsification or falsifiability, associated  in particular  with the name of Karl Popper. ...

~

Miscellaneous additions: 

(1) Psychological observations

People seldom intuit what is unpalatable to them.  [Moreover, one can] eliminate any undesirable indirect implication of their special insight  by means of an additional hilfs-intuition, liquidating the embassassing logical relation.

-- Ernest Gellner, The Devil in Modern Philosophy (1974), p. 95

In short, one ‘answers’ the sceptic  by striding across logical gaps  to conclusions inconsistent with premises that one does not contest.

-- John Watkins, Science and Scepticism (1984), p. 34

(2) The reductio ad absurdum /”self-mate” gambit:

The traditional argument for the primacy of acceleration-retardation  rests on the absurdity of denying it.

-- Stephen Jay Gould, Ontogeny and Phylogeny (1977), p. 216

Chomsky’s book Syntactic Structures, which is regarded by some as a foundation-stone for this kind of activity, has been described by no less an authority than Roman Jakobson  as an argumentum a contrario (Jakobson, 1959), showing the impossibility of the whole enterprise.

-- Hilary Putnam, “Some Issues in the Theory of Grammar”, in  Mind, Language, and Reality (1975), p. 85

.