16-04 -- Summary Of Essay Four Part One -- Formal Logic And Change
These are Introductory Essays, which have been written for those who find the main Essays either too long, or too difficult. They do not pretend to be comprehensive since they are simply summaries of the core ideas presented at this site. Most of the supporting evidence and argument found in each of the main Essays has been omitted. Anyone wanting more details, or who would like to examine my arguments and evidence in full, should consult the Essay for which each is a précis. [In this particular case, that can be found here.]
Abbreviations Used At This Site
Formal Logic Meets Wilful Ignorance
In Essay Four, I expose the woeful ignorance of FL apparent in the work of the vast majority DM-theorists. Few dialecticians can resist making ill-informed and unsubstantiated criticisms even of AFL, while even fewer appear to know anything at all about MFL.
[DM = Dialectical Materialism; FL = Formal Logic; MFL = Modern Formal Logic; LOI = Law Of Identity; AFL = Aristotelian Formal Logic; LOC = Law of Non-Contradiction; LEM = Law of Excluded Middle; TAR = The Algebra of Revolution; DL = Dialectical Logic.]
One particularly egregious aspect of this self-inflicted ignorance is the fact that most DM-theorists seem to think that FL began and ended with Aristotle, despite being told repeatedly that they are wrong. In fact, as any reasonably decent history of logic would have told them (had they bothered to check), 95% of FL is less than 130 years old. Sixty years ago Burnham tried to tell Trotsky that his knowledge was badly out-of-date, but he might as well have been talking to the cat for all the good it did.
This is one idea that appears to have escaped the Heraclitean Flux.
Even to this day, the 'news' that logic underwent a profound revolution in the late 19th century, at the hands of Frege (easily of the same order of magnitude that Physics underwent in the 17th century) has yet to penetrate most dialectical skulls. Still, they refuse to be told, even though there are now literally scores of sites on the internet that introduce the new logic to those who know very little about it (for example, here and here).
In fact, I raised this with John Rees at Marxism 1990 (an annual gathering of the UK-SWP) -- in front of a large audience -- but there is no evidence in TAR that the message got through. Similar attempts posted on Internet discussion boards are an equal waste of time.
Dialecticians, it seems, are happy to wear this particular badge of ignorance with pride.
Indeed, based on what DM-theorists themselves write about logic, it is clear that the majority of them do not appear to have opened a single logic text ever -- especially before they began pontificating about the subject -- at least, not one written since Hegel misnamed his own particular work, "Logic". And, of the tiny minority who have, few seem to have been able to grasp much of what they rapidly skimmed over. The hackneyed definitions of the three allegedly fundamental 'laws' of logic, which virtually all DM-theorists present their readers, are hopelessly confused; their 'research' in this area has clearly been confined to copying these 'non-definitions' off of one another.
Just to take one example (there are plenty more here) -- the following is one of the worst I have ever seen:
"According to formal logic, the whole is equal to the sum of its parts....
"Let us examine the matter more closely. The basic laws of formal logic are:
"
1) The law of identity ('A' = 'A').
"2) The law of contradiction ('A' does not equal 'not-A').
"3) The law of the excluded middle ('A' does not equal 'B')....
"The law of contradiction merely restates the law of identity in a negative form. The same is true of the law of excluded middle. All we have is a repetition of the first line in different ways. The whole thing stands or falls on the basis of the law of identity ('A' = 'A'). At first sight this is incontrovertible, and, indeed, the source of all rational thought. It is the Holy of Holies of Logic, and not to be called into question. Yet called into question it was, and by one of the greatest minds of all time....
"Similarly with the law of the excluded middle, which assert that is necessarily either to assert or deny, that a thing must be either black or white either alive or dead, either 'A' or 'B'. It cannot be both at the same time. For normal everyday purposes, we can take this to be true. Indeed, without such assumptions, clear and consistent thought would be impossible. Moreover, what appear to be insignificant errors in theory sooner or later make themselves felt in practice, often with disastrous results. In the same way, a hairline crack in the wing of a jumbo jet may seem insignificant, and, indeed, at low speeds may pass unnoticed. At very high speeds, however, this tiny error can provoke a catastrophe. In Anti-Dühring, Engels explains the deficiencies of the so-called law of the excluded middle:
"'To the metaphysician,' wrote Engels, 'things and their mental images, ideas, are isolated, to be considered one after the other and apart from each other, fixed, rigid objects of investigation given once for all. He thinks in absolutely unmediated antitheses. 'His communication is "yea, yea; nay, nay"; for "whatsoever is more than these cometh of evil." For him a thing either exists or does not exist; a thing cannot at the same time be itself and something else. Positive and negative absolutely exclude one another; cause and effect stand in a rigid antithesis one to the other.'" [Woods and Grant (1995), pp.57, 91-93. Quotation marks have been altered to conform to the conventions adopted here.]
I
t is worth pointing out that these two comrades referenced no logic text as a basis for these 'definitions'. They were happy however to quote Engels, a comrade who wrote before the revolution in logic occurred (so he at least had an excuse), as if he were an expert (when it is obvious from his writing that Engels was worse than a novice in this area).
To be sure, here and there Woods and Grant used a few ideas lifted from two introductory works (i.e., those written many years ago by Luce, and Cohen and Nagel), but they failed to reveal from which lamentably poor logic textbook they dredged-up these prize specimens:
"1) The law of identity ('A' = 'A').
"2) The law of contradiction ('A' does not equal 'not-A').
"3) The law of the excluded middle ('A' does not equal 'B')...." [Ibid.]
Quite what the LOC has to do with whether "A" can or cannot equal "not-A", Woods and Grant failed to say. As we will also find with Hegel, these two have confused the LOC (which is about the truth-functional relation between a proposition/clause and its negation, it is not about objects like "A", still less is it about "equality") with the LOI supposedly stated negatively.
[The lack of any connection between the LOC and the alleged negative version of the LOI is discussed here. On the LOC in general, see here.]
Readers will no doubt note, too, if they check, that these characterisations cannot be found in Aristotle, and he can only be made to say such inane things if what he actually says is ignored, and his words are altered so that they say the opposite of what he intended. In this case, clearly, "Aristotle does not equal Aristotle", according to Woods and Grant. For example, while Woods and Grant are happy to tell us that according to FL "the whole is equal to the sum of its parts", what Aristotle in fact said was this:
"In the case of all things which have several parts and in which the totality is not, as it were, a mere heap, but the whole is something beside the parts...." [Aristotle (1984b), p.1650. I have used the on-line version, here.]
This is hardly an "equal to".
Moreover, their characterisation of the LEM is laughable. What, it may be wondered, has "A is not equal to B" got to do with the LEM -- that is, with whether, of proposition "p", either "p is true or p is false" (or, in some versions, "p v ¬p" -- "¬" being the sign for negation)?
But there is worse to come:
"Even the simplest judgement, as Hegel points out, contains a contradiction. 'Caesar is a man,' 'Fido is a dog,' 'the tree is green,' all state that the particular is the universal. Such sentences seem simple, but in fact are not. This is a closed book for formal logic, which remains determined to banish all contradictions not only from nature and society, but from thought and language itself. Propositional calculus sets out from exactly the same basic postulates as those worked out by Aristotle in the 4th century B.C.,** namely the law of identity, the law of (non-) contradiction, the law of excluded middle, to which is added the law of double negation. Instead of being written with normal letters, they are expressed in symbols thus:
"a) p = p
"b) p = ~p
"c) p V = ~p (sic)
"d) ~(p ~ p) (sic)
"All this looks very nice, but makes not the slightest difference to the content of the syllogism." [Woods and Grant (1995), pp.97-98.]
This is what a)-d) translate out as:
a) p is equal to p
b) p is equal to not-p
c) p or equals not-p (sic)
d) not both p not-p (sic)
a) and b) would be syntactically viable if "p" were an object, but it isn't. [If "p" were an object, it could not be used to say anything. This is precisely the mistake Hegel made, which dialecticians have simply copied; more on that here.]
c) and d) are just gibberish.
Clearly, these two comrades did not copy these prize examples of syntactical confusion from a logic text written anywhere on this planet -- which could mean that they simply made them up. At any rate, this shows that they made no serious effort to comprehend much of what they constantly deride. [Witness the way that they have confused the Propositional Calculus with Aristotelian Syllogistic.** The former was invented by the Stoics (and then largely forgotten until the middle of the 19th century); Aristotle knew nothing of it, as far as we know.]
Of course, the comment these two make about the contradictions allegedly implicit in simple predicative propositions is itself based on a novel piece of grammar (also lifted from Hegel, who borrowed it from Medieval Roman Catholic Logicians). "Caesar is a man" (W1) does not say the particular is the universal, and can only be made to do so by imposing on it a grammatical theory that these two comrades failed to justify. [Indeed, it cannot be justified; on that see Essay Three Part One (summary here).] And even if W1 could be construed in this way, Woods and Grant failed to say why this would be a contradiction, as opposed to being a simple falsehood, or just plain unvarnished nonsense.
[In private correspondence with one of these authors, I have been able to help him correct several of the above errors so that the soon-to-be-released second edition of their book should contain fewer logical screw-ups. However, my advice to excise completely the sections on logic was not taken. The book should, therefore, be renamed: Reason in Remission.]
Dialectical Mayhem
However, in most of the above, the LOI is defined as "A = A", "A is equal to A" -- or even "A is A" (but on this see Essay Six) --, which is said to imply that "A cannot be other than A" (which is incorrect; again on that see Essay Six, here).
The LOC is similarly characterised as "A cannot at the same time be A and not be A" (or even "A cannot be non-A"), which is said to follow from the LOI (but with no proof that it does; again, on that see here), whereas the LEM is depicted rather loosely as "Everything must be A or not A", or even worse, "A does not equal B". [These confusions are dissected here.]
Nevertheless, even if their analysis of the LOC were correct, and it was true that "A is A and at the same time non-A", it would be impossible for DM-theorists even to begin to express their criticisms of their own fabricated AFL-principles. This is because it would be impossible to state the following:
B1: "A is A and at the same time non-A".
If it were indeed true that "A" is at the same time "non-A", then the first half of B1 would have to be re-written as:
B2: "Non-A is non-A".
Or, more accurately, the whole of it as:
B3: "Non-A is non-A and at the same time non-(non-A)".
That is, if each "A" in B1 is replaced with what it is supposed at the same time to be (i.e., "non-A"), B1 would 'dialectically disintegrate' into B3.
Now, this fatal result can only be denied by those who reject the DM-inspired version of the LOC (i.e., those who reject "A is at the same time non-A"), and who thus do not think that the first half of B1 is false, or both false and true, "It depends...".
Even worse still, if every "A" is also "non-A", then these would surely follow from B3:
B4: "Non-(non-A) is non-(non-A) and at the same time non-(non-(non-A))."
B5: "Non-(non-(non-A)) is non-(non-(non-A)) and at the same time non-(non-(non-(non-A)))."
And so on, as each successive "A" in B3 and B4 is replaced by the "non-A" that dialecticians insist they are. Once more, this could only be denied by those who reject standard DM-criticisms of the LOC.
As should now seem apparent, the LOC has an annoying way of hitting back in a most un-dialectical fashion when challenged. Hence, as noted above: it is impossible for dialecticians to say what they mean.
The same problems afflict other DM-inspired criticisms of principles dialecticians claim to have found (some hope!) in textbooks of FL.
In addition, DM-theorists are invariably unclear what the "A"s in these alleged FL-'laws' are supposed to stand for. Based on the above, and on other passages quoted elsewhere at this site, it is obvious that DM theorists regularly confuse these letters with one or more of the following: propositions, judgements, properties, qualities, words, objects, processes, predicates, statements, assertions, type-sentences, token-sentences, concepts, ideas, beliefs, thoughts, phrases, clauses, relations, relational expression, indexicals, places, times, names, and "existences".
The significance of logical disorder of this magnitude lies not so much with the unmitigated confusion it creates, but with the fact that the vast majority of the DL-faithful have not even noticed it!
And when this is pointed out to them, they simply complain about "pedantry"!
2400 years ago (and despite his own confusions) Aristotle was far clearer about such things than all these 'dialectical logicians' put together.
But worse: are we really supposed to believe that this sub-Aristotelian syntactic jumble encapsulates ideas that lie at the very cutting edge of modern science?
Now, anyone tempted to respond to the above on the lines that it gets the DM-view of contradictions (etc.) wrong, and that dialectical contradictions are really this, or they are in effect that, or they are…whatever, need only reflect on the fact that according to the DM-inspired criticism of the LOC, that criticism itself must be this or that, or whatever, while at the same time being not this or that, or whatever -- if we here interpret the "A"s above as "this or that, or whatever", since, on sound DM-principles, these letters can be interpreted in any which way we fancy.
Now, let's see those who accuse careful logicians of "pedantry" try to squirm their way out of that one!
[I
n Essay Eight Part Three, we shall see that this serious difficulty afflicts, and thus neutralises, the best account there is (or, at least, the best account I have so far seen) of the nature of 'dialectical contradictions'.]
In that case, the radically imprecise nature of the DM-inspired criticism of the LOC (which sees everything as "this or that, or whatever, and not this or that, or whatever" -- where each "this or that, or whatever" is just left undefined, so it can be anything you like) must itself be "both a criticism and not a criticism" of the LOC. This must be so unless, of course, criticisms are themselves exempt from their own criticism -- and cannot thus ever aspire to become one of these wishy-washy dialectical letter "A"s.
Alas, this means that dialecticians' own criticism of the LOC must now self-destruct. So, for example, any attempt made by DL-fans to define the LOC must be "a definition and not a definition" -- if their own 'analysis' of the LOC and the LOI is invoked against any such attempt.
Hence, using "D" to stand for the MAD-'definition' of the LOC (whatever that 'definition' is, and whatever it means, if we are ever told), it must be the case that "D is at the same time non-D". Clearly, that would mean that the MAD-inspired criticism of the LOC undermines its own definition of it! Or, at least, it does and it doesn't.
It is at this point that even DL-fans might just begin to see how devilish their own Diabolical Logic really is.
[MAD = Materialist Dialectics/Dialecticians; BAD = Buddhist Dialectics/Dialecticians; DL = Dialectical Logic.]
However, long experience 'debating' with comrades who think Hegel is the best thing since sliced Aristotle suggests that one should never underestimate a dialectician's capacity for ignoring anything he or she does not like. 'Debating' with those whose brains have been compromised by this Hermetic virus is like debating with Buddhists -- except the latter are at least fair. With respect to both sets of mystics (the BAD or the MAD), whatever argument is deployed against their system simply doubles back to prove their case. The fact that BAD-ies can tell us absolutely nothing about 'Nirvana' phases them not (since it is 'Nothing'!), just as it scarcely registers with MAD-ies that they cannot say what their "Totality" is, either.
And it is no use pointing out to MAD-ies or BAD-ies that their belief in universal contradictions is self-contradictory, for to do so would merely be to feed this monster, and thus lend it strength.
Now, it could be objected once more that DM-theorists do not object to the use of the LOC, the LOI or the LEM in their proper field of application. These principles fall short when they are applied to processes in the world, to change and movement. This hackneyed response will be tested to destruction in Essays Five, Six and Eight Parts One and Two (where consideration will be given to Engels's 'analysis' of motion, Hegel and Trotsky's attempt to criticise the LOI, and the claim that change is the result of 'internal contradictions').
In the meantime, it is worth pointing out that these DM-inspired criticisms of FL are themselves phenomenal/material objects (i.e., they have to be written in ink on a page somewhere (etc.), or propagated in the air as sound waves at some point), and as such they are surely subject to change (if everything is). In that case, they "are never equal to themselves". If so, the above DM-inspired criticisms of FL must apply to each material copy of any DM-inspired criticism of FL.
In that case, no materially-configured DM-criticism of the LOC is equal to itself, and hence each phenomenal example of a DM-criticism is at the same moment both "a criticism and not a criticism".
The rest follows as before.
The counter-argument to this (that dialecticians only need to appeal to the 'relative stability' of material objects/processes to make their point) is neutralised in Essay Six. The other counter-argument -- that this ignores Hegel's use of identity to derive the alleged fact that everything is related to, or 'reflects', its 'own other', and not merely to everything that it is 'not' --, is defused in Essays Seven and Eight Part Three.
Logic And Change
Now, it's rare to find a dialectician who fails to say the following:
"Formal logic regards things as fixed and motionless." [Rob Sewell.]
"Formal categories, putting things in labelled boxes, will always be an inadequate way of looking at change and development…because a static definition cannot cope with the way in which a new content emerges from old conditions." [Rees (1998), p.59.]
"There are three fundamental laws of formal logic. First and most important is the law of identity. This law can be stated in various ways such as: A thing is always equal to or identical with itself. In algebraic terms: A equals A.
"...If a thing is always and under all conditions equal to or identical with itself, it can never be unequal to or different from itself. This conclusion follows logically and inevitably from the law of identity. If A equals A, it can never equal non-A." [Novack (1971), p.20.]
Once again, the bemused reader will search long and hard, and to no avail, to find a reference to a single logic text (in the writings of dialecticians) that supports these contentions.
In fact, Formal Logic uses variables -- that is, it employs letters to stand for named objects, designated expressions (some of these are called "predicates"), and the like -- all of which can and do change.
This handy device was introduced by Aristotle, who experimented with variables approximately 1500 years before the same tactic was extended into mathematics by Muslim Algebraists -- who in turn used them several centuries before René Descartes (1596-1650) began employing them, in the 'West'.
This is what Professor Nidditch had to say:
"One has to give Aristotle great credit for being fully conscious of this [i.e., of the need for a general account of inference -- RL] and for seeing that the way to general laws is by the use of variables, that is letters which are signs for every and any thing whatever in a certain range of things: a range of qualities, substances, relations, numbers or of any other sort or form of existence....
"If one keeps in mind that the Greeks were very uncertain about and very far from letting variables take the place of numbers or number words in algebra, which is why they made little headway in that branch of mathematics...then there will be less danger of Aristotle's invention of variables for use in Syllogistic being overlooked or undervalued. Because of this idea of his, logic was sent off from the very start on the right lines." [Nidditch (1998), pp.8-9. Italic emphasis in the original.]
Now, Engels himself admitted that the introduction of variables into mathematics allowed mathematicians to depict change:
"The turning point in mathematics was Descartes' variable magnitude. With that came motion and hence dialectics in mathematics, and at once, too, of necessity the differential and integral calculus…." [Engels (1954), p.258.]
But, if variables allow mathematicians to handle change, why can't this be true of logicians?
Nevertheless, a few counterexamples to the claim that FL cannot cope with change should at least give neutral observers pause for thought (though not dialecticians, who ignore awkward details like these, and continue to make the same baseless assertions about FL and change whatever facts are lobbed in their general direction).
In the following examples, I have kept the arguments very simple (this is a summary, after all!):
Argument One:
Premiss One: All dialecticians are human beings.
Premiss Two: All human beings age.
Conclusion: All dialecticians age.
Argument Two:
Premiss One: All dialecticians believe Formal Logic cannot cope with change.
Conclusion: The refutation of a dialectician is the refutation of someone who believes Formal Logic cannot cope with change.
[A refutation is a form of change.]
Argument Three:
Premiss One: For all objects/processes x, for some time t, and for some time t' (where t'>t), if Fx(t) and ¬Fx(t'), then x has changed.
Premiss Two: Fa(t) and ¬Fa(t'),
Conclusion: a has changed.
[Many more example can be found in Essay Four.]
Now, the above examples will not alter the minds of dialecticians (who, unlike formal logicians, cannot cope with a change of opinion), perhaps since these seem rather trite. But they were introduced specifically to show that FL can handle change, and they were kept simple since most DM-fans find it hard to follow complex arguments expressed in modern logic, anyway. If the full range devices available to modern logicians had been used, and other branches of logic had been employed (of which dialecticians seem unaware -- such as Modal and Temporal Logic), countless complex examples could be given to show how MFL can cope with change.
[However, on this, see the last section of this Essay.]
Moreover, when it is remembered that DL itself cannot cope with change (the proof of that allegation can be found here), the bankruptcy of at least this area of DM should be apparent to all but True Believers.
DL -- A Superior Logic?
At first sight, it would seem obvious that a logical system based on a static view of the world -- as it is alleged of FL -- would have few if any practical consequences. On the other hand, it would appear equally clear that a different logical system based on the opposite view of reality -- as is also claimed of DL -- should have countless practical applications in science and technology.
Oddly enough, the exact opposite of this is the case: DL has no discernible practical or scientific applications, and has featured in none of the advances in the natural or physical sciences (and arguably none even in the social sciences) -- ever. Worse still: DL has made no contribution to technological innovation. [The argument to the contrary is neutralised in Essay Four, here, here and here.]
In stark contrast, FL has played an invaluable role in the development of science and mathematics, and has featured in countless applications in technology and the applied sciences.
Indeed, one excellent example (among the many) of the impact FL on technology is the development of computers. Their origin goes back many centuries, but advances in mathematical logic (post 1850) proved to be decisive. The invention of Boolean and Fregean Logic, the mathematical logic of Russell, Whitehead, Hilbert, Peano, von Neumann and Church (etc.) -- along with the logico-mathematical work of Alan Turing -- all helped to make the development of computers possible. FL has not only contributed to the evolution of software and of computer languages, the principles of Propositional Calculus govern the operation of all standard processors (etc.).
Moreover, there are numerous other examples of the practical applications of FL, ranging from Cybernetics to Code Theory and from Linguistics to Game Theory and Discrete Mathematics. The question is: Can dialecticians point to a single successful application of DL in technology, or in the natural and physical sciences? The answer is reasonably plain; they can't. But this glaring failure becomes all the more revealing when it is remembered that DM-fans repeatedly claim that their 'logic' is superior to FL when it is applied to the material world.
Now, only the terminally naive will imagine that any of the above will have the slightest effect on dialecticians, or stop them saying the same erroneous things about FL, year in, year out.
As we noted above, the Heraclitean Flux has no control over these, its most inconsistent progeny.
Accounting For Change
When faced with the sorts of points made above, dialecticians have often responded as follows: "Well, how do you account for change?" Here is my reply:
1) Historical Materialism (minus the Hegelian jargon) is well-equipped to account for change. So we do not need DL.
2) Ordinary language contains countless words that can be used to depict every conceivable change imaginable. Hence, Hegelian jargon is not needed.
And this is no mere dogma; it is easily confirmed. Here is a greatly shortened list of ordinary words (restricted to modern English) that allow speakers to refer to changes of unbounded complexity:
Vary, alter, adjust, amend, make, produce, revise, improve, deteriorate, edit, bend, straighten, weave, twist, turn, tighten, loosen, relax, slacken, bind, wrap, pluck, tear, mend, repair, damage, mutate, metamorphose, transmute, sharpen, modify, develop, expand, contract, constrict, constrain, widen, lock, unlock, swell, flow, differentiate, divide, partition, unite, amalgamate, connect, fast, slow, swift, rapid, hasty, heat up, melt, harden, cool down, drip, cascade, drop, pick up, fade, darken, wind, unwind, meander, peel, scrape, graze, file, scour, dislodge, is, was, will be, will have been, had, will have had, went, go, going, gone, return, lost, age, flood, crumble, disintegrate, erode, corrode, rust, flake, shatter, percolate, seep, tumble, mix, separate, cut, chop, crush, grind, shred, slice, dice, saw, spread, fall, climb, rise, ascend, descend, slide, slip, roll, spin, revolve, oscillate, undulate, rotate, wave, conjure, quickly, slowly, instantaneously, suddenly, gradually, rapidly, hastily, inadvertently, accidentally, snap, join, resign, part, sell, buy, lose, find, search, explore, cover, uncover, stretch, compress, lift, put down, win, ripen, germinate, conceive, gestate, abort, die, rot, perish, grow, decay, fold, many, more, less, fewer, steady, steadily, jerkily, smoothly, quickly, very, extremely, exceedingly, intermittent, continuous, continual, push, pull, slide, jump, run, walk, swim, drown, immerse, break, charge, retreat, assault, dismantle, pulverise, disintegrate, dismember, replace, undo, reverse, repeal, enact, quash, throw, catch, hour, minute, second, instant, invent, innovate, rescind, destroy, annihilate, boil, freeze, thaw, cook, liquefy, solidify, congeal, neutralise, flatten, crimple, evaporate, condense, dissolve, mollify, pacify, calm down, terminate, initiate, instigate, enrage, inflame, protest, challenge, expel, eject, remove, overthrow, expropriate, scatter, gather, assemble, defeat, strike, revolt, riot, march, demonstrate, rebel, campaign, agitate, organise…
Naturally, it would not be difficult to extend this list until it contains literally tens of thousands of words all capable of depicting countless changes in limitless detail (especially if it is augmented with the language of mathematics). It is only a myth put about by Hegel and DM-theorists that ordinary language cannot express change. On the contrary, it performs this task far better than the incomprehensible and impenetrably obscure jargon Hegel invented in order to fix something that wasn't broken.
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