Monday, March 16, 2015

Modus tollens tollendus est ! (re-updated anew)

Die bürgerliche Stellung des Widerspruchs
-- L. Wittgenstein, Philosophische Untersuchungen, #125

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

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.


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; 2nd 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:
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.  

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 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 brains in a vat.   (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 on, 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
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, less 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 simply 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.”

(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:

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:

[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


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