S23(a): No object is self-identical at the same non-self-identical instant in time.

S23(b): No object is self-identical at the same and not the same self-identical instant in time.

But, as we have already seen with other dialectical principles, this threatens to explode in the following manner:

S23(c): No object is self-identical at the same and non-same non-self-identical instant in time.

S23(d): No object is self-identical at the same and not the same and not the same and not the same self-identical instant in time.

If objects and processes are never the same, they cannot be the "same, yet different", they can only be "different, yet different". Of course, if they are indeed the "same, yet different" then it cannot also be true that they are never the same. Either way, the use of this IED explodes in the face of whoever attempts to use it.

In that case, it is impossible to refute the LOI in the same crude way attempted by Trotsky (irony intended). As the above shows, this cannot be done without using the LOI in the attempt to do so. Hence, anyone wishing to argue on exactly the same lines as Trotsky (or Hegel) will be forced to use the LOI twice: first, in having to reproduce an identical copy of that argument; second, by using the LOI applied to tensed variable letters (or their equivalent in ordinary language) to establish their case.

The question now is: How might anyone who agrees with Trotsky accomplish the 'very same' task without falling into this double trap?

By means of semaphore, telepathy or Aldis lamp?

Did Trotsky Understand Identity?

Learning Identity In A World With None

As seems reasonably plain, in order to draw conclusions from a putative identity statement like S1, Trotsky must have been able to understand the words it contained -- even if he subsequently claimed that such propositions were not strictly or always true. Clearly, therefore, Trotsky had to be able to comprehend the LOI before he could criticise it.¹⁷

S1: A is equal to A.

But, this raises another serious question: From where did Trotsky's understanding of the LOI originate? It is not easy to see how he could have grasped the concepts involved if he only ever encountered them in false sentences, or if their use was never absolutely legitimate, as was claimed in S9:

S9(a): All bodies change uninterruptedly.  (b) They are never equal to themselves.

On the surface, it would seem impossible for anyone to learn what something meant if all they ever experienced were not examples of the very thing they were being taught, or if they were all incorrect instances of it.¹⁸

The Sting In The Tail

Of course, it could be objected here that Trotsky's ability to understand everyday words for identity is irrelevant since the concepts they express are only valid "within certain limits"; even dialectical concepts only approach the truth "asymptotically".¹⁹

Hence, although Trotsky clearly knew how to use ordinary language, it could be argued that dialectics reveals that the vernacular is in fact unable to depict change adequately. Indeed, the whole point of philosophical criticism is to demonstrate that everyday notions (which are perfectly legitimate in their own sphere of application) are incapable of reflecting fundamental aspects of change or the fluid nature of reality. This has nothing to do with understanding or failing to understand anything. Or so the objection might go.

However, this problem does not just affect the vernacular; the same considerations apply to technical and scientific language. They would be unusable unless it was possible to specify the conditions under which their empirical propositions were true or false (should there be any). [Why this is so will be examined in Essay Twelve Part One.]

But, if identity statements cannot ever be true -- not just as a matter of fact, but of logic --, it would then surely be impossible for anyone to comprehend them. Even assuming identity statements are only ever approximately true, no one would be able to grasp their content, if this were the case. This is because it would be impossible for anyone to comprehend in what way such ordinary identity statements could fall short of something (i.e., "absolute identity") that was never anywhere instantiated in reality; hence, no one would have any idea what "approximate identity" actually approximated to.

This is not easy to see, so some elaboration is necessary.

Trotsky described things as follows:

"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves…. For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing…. Hegel in his Logic established a series of laws: change of quantity into quality, development through contradiction, conflict and form, interruption of continuity, change of possibility into inevitability, etc…." [Trotsky (1971), pp.64-66. Bold emphases added.]

If, as a matter of logic, 'absolute identity' is nowhere instantiated in reality (which seems to be the significance of the word "axiom"; indeed, Trotsky is merely paraphrasing a bogus 'logical' argument of Hegel's), how would anyone be able to declare with any confidence that approximate identity failed to match this ideal by so much or so little? Worse still, if no one had any grasp at all of perfect identity, how could dialecticians be so sure that it is never instantiated in nature or society?

In such circumstances, exactly what is being ruled out?

It is no good appealing to sentences like S1a to answer queries like these since all such the sentences are material objects in their own right, and as such are susceptible to Trotsky 'lens argument'. On that view, no material copy of S1 or S1a would identical to itself, or to any further copy of it, and no copy of S1 or S1a would stay the same over time. In that case, S1a-type sentences cannot inform us exactly what Trotsky and/or Hegel were ruling out. Clearly, in order to do so, their content would have to remain the same, something S9(a/b) and S1a rule out. So, we have as yet no idea what 'absolute', or even 'abstract' identity could possibly be, since no material embodiment of it can fully express this 'concept', given this view.

S1: A is equal to A.

S1a: A is both equal to non-A and to non-non-A.

S9(a): All bodies change uninterruptedly.  (b) They are never equal to themselves.

[Here, in this Hegelianesque 'proposition', the "non-non-A" refers to the negation of the negation of A, as new content 'emerges'. The 'relative stability of words and meanings' response will be considered presently. Also, see Note 15.]

This is because, if we now represent S1a by the propositional letter "P", we are instantly faced with serious problems, as S1a is applied to its own material embodiment -- the very material sentence expressed by P:

S1b: P is equal to non-P and to non-non-P.

And then (replacing each "P" with what it is 'dialectically' identical with) we would have:

S1c: Non-P is equal to non-non-P and to non-non-non-non-P.

S1d: Non-non-P is equal to non-non-non-P and to non-non-non-non-non-non-non-non-P.

And so on, ad infinitem.

And there is little point irritated DM-fans arguing that this is ridiculous, since the Hegelian rule (expressed in S1a) implies the bowl of logical spaghetti contained in S1d, etc. Hence, only the rejection of S1a will stop this logical bindweed from propagating itself.

Worse still, such material embodiments cannot tell us what 'approximate identity' is even an approximation to.

And, if identity statements are never absolutely true (in the material world), how could anyone learn what the implications of accepting or rejecting sentences like S1 amounted to? Again, what exactly would they be accepting or rejecting?

[Note the significance of the italicised word, here. No irony is intended; just a reminder that we cannot evade the use of the LOI as a rule of language if we are to understand one another. If anyone knew exactly what they were rejecting by repudiating absolute identity, then they'd have to use/accept the LOI in the very act of repudiation.]

Even to ask the question requires a use of the dread LOI. If no one has a clue what absolute identity is (and on this theory, just as soon as anyone claimed they did have a clue, that very idea itself would change, if DM is correct), the question naturally arises: What precisely is being rejected in the repudiation of sentences like S1?

If, on the other hand, the LOI encapsulates merely an empirical truth -- even if only an approximate one -- we would still be owed an explanation of what this empirical truth approximates to. And how could that be settled in this world if we never encounter the absolute limit of identity -- and worse: if we do not know what it is?

Questions like these apply equally well to any rejection of propositions expressing identity (as false) -- or even those that question its material application: the negation of an identity statement requires an understanding of what would make it true so that it might legitimately be rejected as false --, even if this is only to point out how limited or "one-sided" it is. But, if there is no way of saying what would make an identity statement true, its denial must lack a sense, for, once more, no one would have the slightest idea what was being ruled out.

Hence, the earlier question about what Trotsky did or did not understand was relevant after all.

It could be argued that it is possible to say what would make an identity statement true; it's just that they are never absolutely true. However, as we will soon see, in order for anyone to be able to say that they are never true, the content of at least one identity statement would have to remain exactly the same over the same length of time, otherwise, anyone who criticised the absolute application of the LOI would not be using/appealing to the same principle from day to day, year to year, century to century. [In here, it is quite easy to see several applications of the LOI -- over many years, too. Once more, a rejection of the LOI self-destructs. (Abstract identity will be discussed below.)]

Again, it could be objected that identity statements are in fact both true and false (this would seem to be the implication of repudiating the LEM as it applies to material processes without qualification -- and this seems to be what lies behind S1a); but it is unclear what that would mean in relation to the LOI. It would either imply that material identity only approximates to absolute identity (since no one would then have a clue what  absolute identity is, on this view), or if it is false then material identity would still only approximate to absolute identity, because, again on this view, identity propositions are mere approximations. Either way, no identity statement would be absolutely true.

S1a: A is both equal to non-A and to non-non-A.

[LEM = Law of Excluded Middle.]

But both of these amount to the same criticism of the LOI (i.e., that it merely captures approximate identity in this material world), and since we still have no idea what ordinary identity approximates to, this objection is completely empty. We still remain in the dark over exactly what is being ruled in or out (that is, we still have no idea what it is that approximate identity approximates to) -- with or without an appeal to the truth or falsity of the LEM.

So: even if the LOI were both true and false, and the LEM was as unsafe to use (unrestrictedly) as we are told, what exactly would the ascriptions contained in any material example of the LOI be both true and false of? But, no answer to that can be given short of a clear grasp of absolute identity, which, alas, no one is supposed to have.

In that case, this new set of DM-difficulties is independent of whether the LEM is safe, unsafe, or both.

Again, it could be pointed out that we do have a clear idea what absolute identity amounts to; it's just that it doesn't apply to anything in reality. And here is what we associate with it:

S1: A is equal to A.

Now, we can have a clear idea of something even if it is never true, or if it is never instantiated in reality. But, in that case, as pointed out above, this clear idea will have to remain the same from year to year, generation to generation, and if that is true, then there is something in reality that instantiates an absolute application of the LOI, namely this clear idea. On the other hand, if this clear idea does not remain the same over many centuries, then we do not have a clear idea of absolute identity, as alleged above.

Once again, it could be objected that propositions about identity could still be true and false, so that one option (i.e., the truth of identity statements) need not necessarily mean that the other must be set in opposition to it. Maybe so, but if we still have no idea what absolute identity is then neither option is viable, whatever anyone does with it. In that case, if no one had a clue what the subject of both this denial and this affirmation amounted to, if no one knew what the 'concept' of absolute identity is, then nothing true or false (or both, or neither) could be said of it.

In order to see this, the reader should try to say something true or false about meskonators (over and above a handful of trivialities, perhaps relating to the spelling of that word, say).

["Meskonator" is a meaningless word; it is a pure invention.]

Plainly, if no one knows what a meskonator is, then no one could possibly know if anything approximated to one, or was absolutely identical to, or totally different from one.

Of course, with identity itself, the situation is even worse, for the last sentence could not even be formed coherently if "meskonator" is replaced by "identity", and no one knew what absolute identity was. The latter term would then be implicated in its own lack of meaning, as opposed to being implicated in that of another word.

To be sure, identity is not an object, but who says 'a' meskonator is? If we have no clear idea of the nature of either of these, or no idea at all, then we can say nothing coherent about them. Worse still, the way that relational terms (such as "identical") are depicted in DM (i.e., as abstract objects -- we saw this in Essay Three Part One, and in Essay Four), then dialecticians least of all can raise this particular objection.

However, the situation is even worse than this might suggest; as hinted at above, if dialecticians are correct, just as soon as anyone claimed they had an idea of absolute identity (via 'abstraction', or whatever), that idea itself (which must be embodied in some form or other in this changing material world, and/or in the central nervous system) must alter, and thus become different. But in that case, how could anyone to tell whether that idea had changed without being able to compare that altered idea of absolute identity with another idea of absolute identity that had not changed? Without that comparison, no one would be able to say whether their idea of absolute identity coincided with absolute identity, fell short of it, or was totally different from it.

And, concentrating on words/concepts 'in the mind'--, contemplated or not by 'speculative reason', a là Hegel --, would be to no avail, either. If DM is correct then nothing remains the same: not words, not thoughts (whether these are or are not about absolute identity, or even a wishy-washy DM-approximation to it), not anything. Now if that is true, then not even the last sentence could be coherently formed, for it too contains a phrase we could never grasp: "absolute identity" -- since it would change just as soon as it was either conceptualised or vocalised.^19a

Furthermore, if words and/or concepts actually stay the same (even if only temporarily), then Trotsky's (and Hegel's) criticisms of the LOI would be seriously compromised. Hence, an appeal to the 'relative' stability of words and/or concepts would be of little assistance to beleaguered DM-fans. If we have no idea what the original phrases "absolute identity" and "approximate identity" relate to (i.e., if we have as yet no clue as to how far short of, or how close to, the intended target they lie), then we would surely have no way of knowing if and when such words remained stable --, i.e., "absolutely identical", or even "approximately identical" with themselves over time. Hence, 'relative stability' is itself parasitic on a concept we do not yet comprehend: identity (in any of its forms).

And it is little use saying that minor changes in words over short periods of time can be ignored, for on this view we would have no way of knowing by how far or how little such words altered if we have as yet no clue what absolute identity is so that we could compare such altered words with an exemplar that did not change.

Still less use would it be any use to argue that it is reasonable to assume our words remain the same from moment to moment, for on this view, we would not yet understand the word "same", nor would we know if anyone was using it in the 'same' way as anyone else, or even in the 'same' way as each one of us had done a few seconds earlier! Once identity is questioned, all such comparisons/contrasts fall apart.

In short, there is no way out of the dialectical hole Hegel dug for all those committed to believing exactly the same about identity as he did (irony intended).

'Abstract' Versus 'Approximate' Identity And Relative Material Stability

In response, it could be argued that the above comments are completely misguided since Trotsky's argument was aimed at showing how no absolute sense could be made of the LOI when applied to material reality. It is also clear that he only needed to appeal to approximate identity in the first instance -- the sort of identity we meet in everyday life (from which the LOI has been abstracted) -- to underline the limitations of the ideal version of LOI when it is confronted with concrete reality and change. The same perhaps could be said of Hegel (no irony intended).

To that end, therefore, the relative apparent stability of material objects allowed Trotsky to refer coherently to such things as the "same" instant, or the "same" object changing over time, and so on (employing the IED argument, once more). This certainly did not commit him to using the LOI in order to criticise it. Trotsky was obviously arguing dialectically, accessing everyday notions to show how they become contradictory when they are applied beyond the usual boundaries of commonsense.

Or so the argument might go.

However, if sense is to be made of approximate identity, some grasp must surely be had of absolute identity so that a vague idea might be formed of what this watered-down version of the LOI actually approximates to. If this is to be achieved only by a retreat into the abstract then we are no further forward.

Indeed, if there is a problem about material identity in the real world, there must surely be an even more intractable one about abstract identity in an Ideal form. In the absence of a clear account of this abstract notion of identity, we still have no idea what ordinary identity is approximating to. But, how might that be determined without another surreptitious appeal to the LOI?

Hence, we would surely need to know by how much or how little approximate identity was or was not absolutely identical to absolute identity before we could even begin to abstract absolute identity into existence. Waving a phrase about (i.e., "abstract identity") in no way helps anyone understand what concrete identity approximates to.

Consider this sentence:

A1: Approximate identity is approximately identical with absolute identity.

Until we understand absolute identity, any approximation to it must remain an empty notion.

Moreover, and more concretely: Just how is the sense of this abstract notion of identity fixed so that two or more references to it at different times pick out identically the same target, as opposed to nearly the same target? How might even a latter day Hegel determine whether the notion he had formed today of absolute identity was or was not absolutely/approximately identical with the one he had accessed only yesterday?

Consider further these sentences:

A2: Comrade NN means by "abstract identity" exactly the same as comrade NM.

A3: Comrade MM means by "abstract identity" approximately the same as comrade MN.

How could anyone determine what the word "same" meant in these contexts before they knew what the intended goal was (i.e., identity itself)? For all anybody knew, an intentional target like this could be entirely different in the minds of two different abstractors; it is not as if either of them could check inside each other's brain, or 'read' his or her thoughts to monitor their precision. But, how do dialecticians themselves lock on to identical ideas of abstract identity, those they supposedly share exactly with one another (or with Hegel or with Trotsky or with Lenin) -- even before they have determined what that target is (and whether there is in fact one target, or many -- or if there is indeed one at all)? Is it just luck? Or do they know something the rest of us don't?

And we cannot just assume this is possible; if no one knows what absolute identity is, then assuming it means this or that would be about as useful as assuming that a meskonator is a new brand of deodorant. Nor can we appeal to ordinary language for assistance. If this theory undermines the vernacular -- which we have seen it does -- then the latter can't help bail out the very thing that caused the problem.

To be sure, as language users we already know what terms for identity and difference mean. This can be seen from the fact that few readers who have made it this far will have failed to grasp the import of the examples given earlier in this Essay of the use of these terms in everyday contexts. However, in these new and rarefied 'dialectical' contexts, where change and obscurity rule the day, ordinary terms fare rather badly. Even worse, jargonised terms survive not at all. In fact, they commit hara-kiri, as we have just seen.

As we also saw in Essay Three Parts One and Two, theorists who appeal to the existence of "abstract concepts" to justify the "objectivity" of human knowledge find themselves in a serious dilemma at this point. Either (1) they  admit that such concepts already exist, toward which knowledge advances or approximates, or (2) they concede that it is we who construct such notions out of experience by a mysterious process of 'abstraction' (that still awaits explication).²⁰

Option (2), it seems, underlies the proffered 'dialectical' response rehearsed above, while (1) barely conceals its Platonic/mystical provenance. But, whichever horn of this dilemma dialecticians grasp, neither is conducive to the dialectical analysis of the LOI (Trotsky's version or Hegel's alternative).

There are at least two reasons for saying this:

(2) Alternatively, if human knowledge is dialectically conditioned, and there are no abstract concepts that exist independently of our knowledge of them, then there would be no objective way to decide whether or not any two randomly selected dialecticians were aiming at the very same target. Indeed, we would be hard-pressed to say what could count as the same intentional goal in such circumstances (without using the LOI). But, if such dialectical detectives haven't locked on to the very same target, then the second of the above conclusions (i.e., point (1)** above) must surely apply. In that case, their search is, to put it bluntly, aimless. On the other hand, if there is a way of delving into the minds of any two randomly selected dialectical abstractionists, which enables one and all to decide whether they were in pursuit of an identical goal, the first of the above points would plainly follow (i.e., (1)*) -- for it would then be obvious that such truth-seekers had used the LOI to identify exactly the same target, and had done it with equal accuracy.

In fact, if anyone were to advocate or reject either of the above options they would still have to appeal to the very same 'law' in question in order to maintain that approximate identity was more or less identical to abstract identity (whether or not "abstract identity" was understood Platonically or as quasi-Hegelian/dialectical construct). Of course, this concept (i.e., abstract identity) would have to remain rock solid -- frozen in mental or conceptual space -- while it was being approximated to, and in real time, too.

If this wasn't so, that 'target' must surely be misidentified by anyone foolish enough to blaze an intentional trail toward such a mutating 'object'. As was argued in the last sub-section, unless, dialecticians can specify under what conditions their notion of absolute or abstract identity does not change over time -- but remains absolutely self-identical in the minds of supporters and critics alike, over many centuries, for them to be able to say with confidence that they are talking about the same thing --, any reference to 'it' by critic and believer alike would be entirely empty. Otherwise they should acknowledge their irresolvable differences, and cease their pointless blather.

And if the concept of abstract identity were to change (as it is apprehended by one or all), then in order to express this fact, some way must be found to say that this concept was no longer absolutely identical with whatever it used to be absolutely identical with. In that case, access to an unchanging version of absolute identity would still be needed to classify any mutated version of it as just such a mutant. Without that, of course, one would lose the right to say that absolute identity might have changed; indeed, we would need it not to change in order to say that it had!

[Once more, this just underlines the intimate connection there is in language between change and identity, contrary to what dialecticians tell us.]

Moreover, an implicit reference to the LOI would have to be made in each claim that any randomly chosen dialectical bloodhound had a concept of 'abstract identity' which was identical with that of any other, so that it could be said that they were referring to the same 'abstract concept' in making the 'same' point even about "approximate identity" --, even if they were disagreeing.

But, if these assumed ideas of "abstract identity" (or even of "approximate identity") were not exactly the same, then agreement/disagreement over what they were talking about would be illusory, too. On the other hand, of course, if their ideas were 'absolutely identical' then criticisms of the LOI would plainly self-destruct. [Hence, the reference to hara-kiri, above.]

Furthermore, if the supposed subject of enquiry were only 'roughly identical' in the minds of the many LOI-critics sat round the dialectical table, not only would that fact be untestable and unverifiable, it would mean that the topic of discussion would be indeterminate, too -- and for the same reason. Again, this would mean that any and all criticisms levelled against the LOI would be misdirected, for not only would no one know exactly what "abstract identity" was so that it could be criticised equally the same -- and by the use of identical arguments -- by those who do not believe in the absolute validity of the LOI, no two critics of the LOI would be able to say that they had the very same thing in mind when they were even so much as pointing out its limitations. Indeed, they could not even use the word "same" with any clear meaning --, and, annoyingly, for the same reason.

On the other hand, if it were now conceded that any two notions of strict (or even approximate) identity were exactly the same in the minds of any two intrepid dialectical abstractors, so that it could be said of one or both that they were talking about the very same thing, there would be no point in criticising the LOI, for it would be correct -- and admitted to be correct -- at least here by its severest critics.

In that case, when dialecticians presume to tell us that a word (or set of words) in ordinary material language connected with sameness and identity, which we all know how to use, does not mean what we usually take it to mean, then the onus is on them to tell us what they do mean by their new word (or set of words). Until they do, they might as well be talking about schmidentity.²¹

Trotsky's Exact Words Now Dialectically Implode

The claim that our concepts are only approximately true -- if true itself -- would undermine DM more effectively than anything that has been written here.

In order to see this, let us introduce the term "adequate" to describe the language belonging to any theory, but understood in the following manner:

S24: A language is adequate to a theory if, when expressed in that language, the empirical propositions of that theory can be deemed true (by appropriate means).

However, if it is impossible to develop a language adequate to a theory no matter what we do, then it would surely be impossible to grasp that theory's content, or even determine what the theory is about. In the case of identity and DM, this problem is particularly acute.

With respect to the matter in hand, this fatal defect can be highlighted by a consideration of Trotsky's own words:

"The Aristotelian logic of the simple syllogism starts from the proposition that 'A' is equal to 'A'…. In reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different to each other. But one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves." [Trotsky (1971), pp.63-64.]

The question is: re the above words identical to the ones Trotsky actually wrote? Now, even though I have carefully copied the above quotation of Trotsky's words from my own copy of IDM, and checked them against the versions found in TAR and RIRE (and now on-line), as no doubt others have done with their copies --, which copies themselves were copies of successive generations of further copies of the originals --, despite all this, in one sense they are not the very same words that Trotsky committed to paper over 60 years ago: they are just copies! So, in that sense they aren't exactly his words. Hence, if we count words as ink marks on the page, they are manifestly not physically identical with the originals.

[IDM = In Defense of Marxism; i.e., Trotsky (1971); TAR = The Algebra of Revolution (i.e., Rees (1998); RIRE = Reason in Revolt (i.e., Woods and Grant (1995); UO = Unity of Opposites.]

On the other hand, in another perfectly ordinary sense of the term, the words quoted above are identical with the originals; they are Trotsky's words, and that is why his followers constantly quote them. We have other criteria of identity for words (and also for ink marks) that Trotsky had to rely on to put his case to those interested in reading it, upon which his followers also depend.

And I am now about to use his words against him/them.

The fatal consequences mentioned above derive from the import of the last few sentences and from the paradoxical conclusions that arise from rejecting what they say -- as it seems critics of the LOI must do to remain consistent with their own precepts.

For example, any acknowledgement that the above quoted words are identical with Trotsky's own would mean that anyone reading his words now, and accepting his case against the LOI as correct, must have (implicitly) employed that very 'law' in order to criticise it! They would have to say (in effect) that in their copy, these words are exactly the same as the ones Trotsky typed or penned all those years ago, which words now undermine (an absolute application) of the 'law' they had just used to arrive at that conclusion.

Clearly, Trotsky and his epigones failed to take into account this perfectly ordinary (hence material) sense of identity: anyone who reads Trotsky's words today has before them material objects (i.e., ink marks on the page/ pixels on a screen) that are clearly identical with countless other such objects separated from one another in space and time (i.e., other ink marks on paper/pixels on a screen that represent still other copies of the very same article he wrote), by means of which an indefinite number of readers may access the very same ideas that Trotsky intended they should, but which marks are not numerically or materially the same as the ones he originally penned. This is as clear a case of ordinary 'identity-in-difference' as one should wish to find, but one that does not commit us to a belief in those terminally obscure DM-UOs (in fact, here there are none). Unfortunately, this is an example of 'identity-in-difference' that completely undermines Trotsky's criticism of the LOI.

In this case, countless manifestly (and optically) different material objects -- separated in space and time -- are uncontroversially identical. That fact could only be denied by those who possess a defective copy of Trotsky's writings! Even better: the admission of this fact does nothing to undermine anyone's belief in change.

[Annoyingly, that belief itself could remain the same even while agreement with Trotsky on the LOI changed as a result of the above argument.]

Of course, anyone who disagreed with the above line-of-thought would then perhaps be committed to the view that the words in their copy of Trotsky's writings were not identical with the ones Trotsky authored all those years ago. Indeed, that idea might itself have been prompted by Trotsky's writings and the message they conveyed, which these very same words seem to express: that nothing is identical with itself, or with anything else -- including those very words!

Someone could object***: the above considerations actually support Trotsky's case, for here we have several objects that we ordinarily call identical, which are manifestly not the same. So, our ordinary grasp of identity is not as secure as we might think.

In response, it is worth pointing out that we count certain words, phrases and symbols as identical even though they use different letters, or none at all. For example, few of us would say most or all of the following were not the same word/symbol, or didn't mean the same:

B1: Cat and CAT; Trotsky and TROTSKY; colour and color; maths and math; In Defence of Marxism and In Defense of Marxism; Das Kapital and Capital; vixen and female fox; one, eins, un, uno, een, en, egy; 10-9, 2/2 and 1; 3x2 and 6; February 12th and 12th of February; Beta and β; dog and chien; red and red; red and red; red and rot; bold and bold; italic and italic; and and and; empty space and        ; arrow and →; 6 and upside down 9; half full and half empty; non and uou (written upside down); and so on.

This is because our criteria of identity for words, symbols and letters are not the same (no irony intended). With respect to words and symbols, a whole host of criteria apply; physical form is clearly not the only one (as the above objection*** seems to assume). Hence, even though the physical form of the words and symbols used could be different, we nonetheless recognise most or all of them as identical, and despite the fact that they look radically different. B1 above contains a radically shortened list of these.

And it can't be the meaning of the above words in B1 (etc.) that makes them identical, although this is clearly true of some. Hence, we would regard "schmidentity" and "Schmidentity" as identical words even though they have no meaning. That in turn cannot be because they both lack any meaning, otherwise on that basis "schmidentity" and "meskonator" would have to be counted as the same word!

However, when meaning is introduced, the situation becomes even more problematic for dialecticians.

In that case, the meaning of Trotsky's words today must be identical to their meaning 60 years ago, even if their physical form is not, otherwise it would suggest that what Trotsky intended to convey (e.g., that nothing is identical over time) itself meant that that very message was not the same as the one he put out 60 years ago. This is because his message implies not even messages are identical over time! If, therefore, a reader had in fact accessed the identical message put out by Trotsky urging them to come to the conclusion that even Trotsky's message must change over time, then they cannot have understood that message very well if they now agree with a corrupted copy of it -- which original for all they know might support an absolute, unqualified belief in the LOI!

In either case, such a critic would not be able to ascertain exactly what Trotsky had written or meant, even while they feigned assent to the exact import of what he said to arrive at that sceptical conclusion, which had just prompted these very doubts about messages corrupting over time.

Alarmingly, that must mean that such latter-day critics of the LOI must have access to Trotsky's precise thoughts by some other means over and above the physical text they now concede cannot be exactly the same as that which Trotsky had originally authored. This alternative route to Trotsky's thoughts must therefore go beyond the confines of the physical document itself --, since the latter (on this view) exists now in this corruptible world, and in a corrupted form.

This alternative means of communication could only be 'ethereal' or 'telepathic'. So, such telepathic Trotskyists seem to be able to intuit Trotsky's exact meaning -- which now unfortunately prompts them to question just such exactitude!

But, even that would imply that while this 'ethereal' message (issuing somehow from Trotsky) was identical to the original transmitted through the ether all those years ago by the master -- enabling these very doubts about identity to be accessed exactly and with no loss of meaning over the years in this occult manner by contemporary recipients  -- the physical message wasn't!

Naturally, that would still commit such individuals to the accuracy of LOI -- only now applied to 'occult' messages and ethereal identities.

An Attack On The LOI That Is Unequal To The Task

A moment's thought would confirm the fact that the idea that our concepts are somehow inadequate to the tasks we set them cannot be correct, even if the LOI were defective. This is because, if our words are inadequate in some way then so were Trotsky's when he criticized the LOI, and so are those of anyone who echoes such doubts. But that would mean that all such attacks on language would be defective in virtue of that very assault. Indeed, any words that expressed even an abstract disquiet about the adequacy of ordinary words would be inadequate to that task, which would mean that such inadequacies could never be adequately expressed.

Short of saying nothing at all, this impasse will always block the thoughts of anyone who thinks their words are permanently inadequate to any task set them. Of course, that would mean that scepticism about the adequacy of language (howsoever mildly expressed) would either be disingenuous or self-refuting. [Those who doubt that should read this, and then think again.]

[And this includes anyone who tries to impugn ordinary language -- as we will see in Essay Twelve. Some thoughts on this are expressed in Note 19, below, others can be found in Essay Three Part Two.]

Furthermore, if what Trotsky had intended to say about the limitations of the LOI was in principle impossible to express in any language, if even its physical embodiment was not identical with his thoughts as they were taking physical form (when he committed them to paper) -- because of his claim that all objects (including words 'in the mind', or those on paper) are "never equal to themselves", let alone anything else --, then Trotsky himself could not have intended to mean anything by such words! This is because there would have been nothing for him to have intended to have meant in such a scenario; a faded simulacrum of his own thoughts on the LOI would have been of no use even to Trotsky. Hence, even one nanosecond after being thought, Trotsky's own words would be non-self-identical. On paper, or in the mind, they would not be the same as those he had thought seconds earlier, and he would thus have no access to his own precise intentions, for the same reason -- unless, that is, we are supposed to exempt memory, claiming that it is the only thing in the entire universe that does not 'dialectically' change. His thoughts would be forever lost, as if they had never existed.

Hence, if what Trotsky said were correct then not even he could have confirmed whether or not he was criticising the same 'law' from moment to moment -- without surreptitiously appealing to that 'law', or to an identical 'mental' copy of it, while he attempted to do so.

More generally: if it is impossible to specify what it is that is being attacked (on the basis that whatever is thought about 'it' is in principle not identical to what had just been thought about 'it'), no intention to criticise 'it' can crystallise, for there would be no such 'it' to denigrate.

Once again, an appeal to 'approximate identity' might suggest itself here, but that would be to no avail. As was argued above, the latter notion is parasitic on concepts of identity not semantically-challenged in this Idealist manner. Hence, we would need some idea of what was being approximated to if the notion of approximate identity is capable of doing any real work. But, ex hypothesi, this cannot be achieved without an appeal being made to the validity of the LOI as a linguistic rule (and not a metaphysical truth) -- and, moreover, to the same rule applied repeatedly.

Trotsky (or Hegel -- or indeed anyone who agrees exactly with either author), would be forced therefore to use such material criteria to formulate their self-refuting Idealist theories about a 'law' they badly misidentified to begin with!

Yet More Materially-Induced Dialectical Misery

Unfortunately, these fatal defects do not end there: anyone who consults Trotsky's words today, and who agrees with his case against the LOI, owes us an explanation as to why on the one hand the "A"s in S1:

S1: A is equal to A.

are subject to the following objection:

"But in reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different from each other." [Trotsky (1971), p.63.]

when Trotsky's own words (written many years ago) are not. Such a person would in effect be saying:

"I agree that the LOI is defective because of what Trotsky's words say. This is because the marks on paper that I am now reading in IDM express exactly what Trotsky was thinking all those years ago; they convey the very same message he intended. And yet, in at least this respect the LOI must be correct for me to understand Trotsky, agree with him and arrive at exactly the same conclusion he did, and which he intended his readers should. That allows me now to disagree with the LOI that I have just used in arriving at that identical result.

"On the other hand, if my conclusions are only approximately the same as his, I must have some grasp of his exact intentions so that I may truthfully say with what my own opinions are in fact approximating. And this I must know, for I assent to the idea that all identity statements are approximations because I agree with exactly what Trotsky says. Without his words I would still be under the illusion that identity was absolute -- er..., which it must be if I have just used it to get to this point...".

But, if the LOI is correct at least once, and the same person arrives at exactly the same conclusion as Trotsky -- only decades later -- then what Trotsky says in S9, for example, cannot be true:

S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.

In that case, such a person could only agree with Trotsky's criticism of the LOI by appealing (implicitly) to the validity of the very same defective 'law' every time they accessed his words, drawing identical conclusions from it, many times over, throughout their no doubt very confused dialectical career. Naturally, that would mean that anyone agreeing with Trotsky, and who derived the same conclusion, would only be able to do so on condition that they then promptly disagreed with Trotsky in practice, implying that the exact opposite result was correct, a result now not identical with the one Trotsky had obtained!

This further implies that Trotsky's claims are right only if they are not, and that if what he had intended to say were true, it would then become impossible for anyone (including Trotsky) to determine what it was he wanted to say, whether it was right or wrong, or whether it had been committed to paper accurately, transposed correctly between copies -- or even whether these questions were themselves defective or not --, and for the same reasons.

Once more: dialectical confusion is exposed in material practice.

This new (ironic) dialectical inversion (whereby a rejection of the LOI depends on its successful application in practice by anyone trying to ascertain exactly what Trotsky or Hegel thought they wanted to say) just confirms how complex the conventions of material language really are (to paraphrase Wittgenstein), and how it is not possible to criticise those conventions without that attack itself falling apart for want of words with which it might be expressed.

And now we can see why one of the maxims of this site is the following: In this neck of the woods, the Owl of Minerva flies only at its own peril.

[On the Owl of Minerva, see here.]

Clearly, this paradoxical result is a consequence of the cavalier attitude to ordinary language adopted by dialecticians like Trotsky, seriously compounded by a direct assault on the LOI. It is not possible to criticise that 'law' in this manner -- that is, by treating it as a putative truth.

Admittedly, traditional thinkers have always regarded the LOI a deep metaphysical 'thesis' about everything in existence -- that is, as a 'necessary truth'. However, as will be argued in Essay Twelve Part One, all such metaphysical theses are non-sensical. If so, their denial is equally non-sensical.²² Little wonder, then, that Trotsky's 'analysis' collapsed into incoherence.

The weaknesses of the LOI in fact lie elsewhere.²³

The Anti-Dialectical Knock-Out Punch

Finally, it is worth noting that the fact that objects in the world undergo constant change cannot in general be used to refute any of the above points since no matter how fast anything changes whatever it is identical with will change equally quickly. In that case, the LOI is no enemy of change.

With that observation, much of dialectics falls apart.

[The details are spelt out in Note 5b, and in Essay Eight Parts One and Two.]

Traditional Versus Modern FL

There is a serious point at stake here. Traditional AFL, criticised by Trotsky, not only ignored complex inferences inexpressible in syllogisms, it dramatically failed to cope with relational expressions, quantifiers expressing multiple generality, internal and external negation and scope ambiguity. This was partly because of the way that quantifier expressions themselves had been interpreted by earlier logicians --, who, with their slavish adherence to the traditional grammar of subject and predicate, helped cripple logic for over two thousand years. It is no exaggeration to say that much of traditional Philosophy (i.e., Metaphysics) depends on antiquated logic like this. In which case, two millennia of philosophical confusion -- including that found in Hegel -- largely derives from what is in effect Stone Age Logic.²⁴

Now, many of the difficulties outlined in the last three Essays (i.e., Essay Four, Essay Five and this one) were a direct consequence of the crude way that quantifiers, relational expressions and tense operators had been interpreted (or ignored) both by traditional and dialectical logicians. In fact, progress toward unravelling these pseudo-problems could only begin after Frege had completely re-laid the foundations of FL about 120 years ago. As noted earlier, this salient fact has still to register with most dialecticians -- no matter how many times they are told.

At first sight, considerations like these might appear to be dry, impractical and academic, but of no interest to revolutionaries. However, if Marxists plan to use a radically flawed system (DL) in their endeavour to help change the world then this is plainly not irrelevant.

Indeed, our astounding lack of success so far suggests that DL has been a millstone around their necks.

History has so far delivered an unambiguous verdict; DL has been tested in practice and found wanting.

Is DL Superior To FL?

Essay Four began by asking which of these two rival logics (DL or FL) is the more adequate for use in science, and which could most easily accommodate change, identity and motion. It is now quite clear (from Essays Four to Six) that DL is vastly inferior to FL in every single department; it is incapable of handling even the simplest examples of change -- or even of describing them! This is partly because it relies on a garbled version of AFL (and a confused metaphysic crippled by the impenetrable jargon invented by Hegel), and partly because it unwisely attempts to undermine ordinary material language.

In fact, DL is so limited it cannot even cope with a simple bag of sugar -- let alone "long drawn out processes".

Small wonder then that it has so far hindered the scientific development of Marxism.

FL And A Fragmented View Of Reality

It might be felt at this stage that it is not surprising that the views expressed here reach the conclusions they do since they depend on analysis -– that is, they are based on a fragmentary view of reality, one that splits the world and its contents into separate and un-mediated parts. Naturally, when divorced from the whole, such things are going to appear paradoxical. Only against a wider background is it possible to comprehend the world correctly. In broader contexts, the contradictory nature of objects and processes is easier to see, as indeed are the inadequacies of FL and the LOI.

This objection introduces the centrally important DM-concept: Totality. This terminally nebulous idea will be discussed in detail in Essay Eleven Part One.

Notes

1. As we will see, Hegel did at least mention identity in his critique of the LOI, even though it is clear from what he said that not only had he given this topic insufficient thought, he made superficial and erroneous claims about it. [A fuller consideration of Hegel can be found in Essay Twelve (summary here and here), and in Essay Eight Part Three.]

Hegel's alleged denial of the LOI and the LOC are examined in Pippin (1978) and Hanna (1986). Even though these two authors struggle heroically to make Hegel comprehensible on this and other issues, it is difficult to tell whether they have succeeded or not, or both.

It is also worth pointing out that the LOI was unknown to Aristotle.

Now, since this Essay was originally published, the above reference has changed; the one I used earlier can be found here. On this, see also here.

After another edit, the Wikipedia article now attributes this 'law' to Aristotle's Metaphysics, but there it quite clear that it is not stated in its hackneyed form (i.e., A = A). Indeed, if anything, Aristotle actually derides this 'law':

"Now 'why a thing is itself' is a meaningless inquiry (for -- to give meaning to the question 'why' -- the fact or the existence of the thing must already be evident -- e.g., that the moon is eclipsed -- but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical', unless one were to answer 'because each thing is inseparable from itself, and its being one just meant this' this, however, is common to all things and is a short and easy way with the question)." [Metaphysics Book VII, Part 17. This can be found in Aristotle (1984b), p.1643.]

Quoting this as an example of the use of, or as an allusion to, this 'law' would be to distort what Aristotle says; so I have added the following comment (reproduced here in blue and red) to the Wikipedia article:

And the quotation takes this 'law' out of context, for not only does Aristotle not mention 'identity', he specifically talks about predication (and since identity is a relation, he cannot be talking about identity here):

"Let us state what, '''i.e. what kind of thing, substance should be said to be''', taking once more another starting-point; for perhaps from this we shall get a clear view also of that substance which exists apart from sensible substances. Since, then, substance is a principle and a cause, let us pursue it from this starting-point. The 'why' is always sought in this form -- ''''why does one thing attach to some other?'''' For to inquire why the musical man is a musical man, is either to inquire -- as we have said why the man is musical, or it is something else. Now 'why a thing is itself' is a meaningless inquiry (for (to give meaning to the question 'why') the fact or the existence of the thing must already be evident -- e.g. that the moon is eclipsed -- but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical', unless one were to answer 'because each thing is inseparable from itself, and its being one just meant this'; this, however, is common to all things and is a short and easy way with the question). But we can inquire why man is an animal of such and such a nature. This, then, is plain, that we are not inquiring why he who is a man is a man. We are inquiring, then, '''why something is predicable of something''' (that it is predicable must be clear; for if not, the inquiry is an inquiry into nothing). E.g. why does it thunder? This is the same as 'why is sound produced in the clouds?' Thus the inquiry is about the predication of one thing of another. And why are these things, i.e. bricks and stones, a house? Plainly we are seeking the cause. And this is the essence (to speak abstractly), which in some cases is the end, e.g. perhaps in the case of a house or a bed, and in some cases is the first mover; for this also is a cause. But while the efficient cause is sought in the case of genesis and destruction, the final cause is sought in the case of being also." [Ibid., bold emphasis added.]

So, I think the article needs amending.

Finally, since this 'law' is foreign to Aristotle, how can the author of this article say:

"The law of identity has deep impact on Aristotle's ethics as well. In order for a person to be morally praiseworthy or blameworthy for an action, he or she must be the same person before the act as during the act and after the act. Without the law of identity, Aristotle notes, there can be no responsibility for vice."

Personal identity is not the same as the 'law of identity'.

However, I could not find in the Nicomachean Ethics anything like this reference to personal identity; so perhaps the author of this article will provide an exact quotation?

Nevertheless, the defects of the LOI lie elsewhere; these are outlined in Wittgenstein (1972), pp.97, 105-07, and Wittgenstein (1958), pp.84-85, 91, 111. [Cf., Glock (1996), pp.164-69.]

The best analysis of Wittgenstein's criticisms of identity can be found in White (1978). See also Marion (1998), pp.48-72 for an extended discussion. On identity in general, see Geach (1967, 1970, 1973, 1975, 1990), Griffin (1977), Noonan (1980, 1997, 2006) and Williams (1979, 1989, 1992).

Here are a few examples of the extremely repetitive nature of this part of dialectics:

"[T]he first of [the universal Laws of Thought], the maxim of Identity, reads: Everything is identical with itself, A = A…." [Hegel (1975), p.167.]

"In this remark, I will consider in more detail identity as the law of identity which is usually adduced as the first law of thought.

"This proposition in its positive expression A = A is, in the first instance, nothing more than the expression of an empty tautology." [Hegel (1999), p.413.]

"Abstract Identity (a = a…) is likewise inapplicable in organic nature. The plant, the animal, every cell is at every moment of its life identical with itself and yet becoming distinct from itself….The law of identity in the old metaphysical sense is the fundamental law of the old outlook: a = a." [Engels (1954), pp.214-15.]

"The 'fundamental laws of thinking' are considered to be three in number: 1) The Law of Identity… [which] states that 'A is A' or A = A…." [Plekhanov (1908), p.89.]

"…Hegel elucidates the one-sidedness, the incorrectness of the 'law of identity' (A = A)…." [Lenin (1961), p.134.]

"Formal Logic starts from the proposition that A is always equal to A. We know that this law of identity contains some measure of truth…. Now…when we go to reality and look for evidence of the truth of the proposition: A equals A…we find that the opposite of this axiom is far closer to the truth." [Novack (1971), pp.32-33.]

"Formal Logic asserts: 'A is A'. Dialectical Logic is not saying 'A is not-A'…. It says: A is indeed A, but A is also not-A precisely so far as the proposition 'A is A' is not a tautology but has real content." [Lefebvre (1968), p.41.]

"The Law of identity is usually expressed in the form, A is A. That is, each thing is identical with itself." [Somerville (1946), p.183.]

"The Aristotelian conception of the laws basic to correct thinking may be stated as follows: 1. Law of Identity: Each existence is identical with itself. A is A…." [Somerville (1967), pp.44-45.]

"Classical, Aristotelian logic takes as its fundamental premise the Law of Identity, the statement that a thing is identical with itself. Expressed in a formula: A is A…. In Aristotle's formal logic A is A, and never non-A. In Hegel's dialectics A is A as well as non-A." [Baghavan (1987), pp.75-76.]

"The biggest contradiction of all lies in the fundamental premises of formal logic itself…. The basic laws…are:

1) The law of Identity ('A' = 'A')…." [Woods and Grant (1995), pp.90-91.]

"Dialectics, or the logic of motion, is distinct from formal or static logic. Formal logic is based on three fundamental laws:

"(a) The law of identity: A is equal to A; a thing is always equal to itself." [Mandel (1979), p.160.]

"The laws of logic are based on two main propositions. The first is that of identity or of self-conformity. The proposition very simply states: 'A is A,' that is every concept is equal to itself. A man is a man, a hen is a hen, a potato is a potato. This proposition forms one basis of logic." [Thalheimer (1936), pp.88.]

Examples like these can be multiplied almost indefinitely. Even though there are several minor differences in emphasis between them, the basic point of the above comments is reasonably clear: DM-theorists have fixated on a superficial form of the LOI, one they copy from each other generation after generation; seldom do they bother to check that what they are criticising remotely resembles anything taken from a logic text written in the last 120 years, or even one written by Aristotle!

In at least this respect, DM-authors are depressingly identical. And, as we will see, in response to Hegel (here), identity statements are not tautologies.

As Essay Four showed in detail, this tactic is part of a long and sorry tradition among 'Materialist Dialecticians': define the supposed basics of logic in a completely fanciful way, ridicule them, and then advertise the superiority of DL over this sub-straw man.

Here again is Novack's attempt:

"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 is equal to A.

"…If a thing is always and under all conditions equal or identical with itself, it can never be unequal or different from itself. This conclusion follows logically and inevitably from the law of identity. If A always equals A, it can never equal non-A." [Novack (1971), p.20.]

Clearly Novack failed to consider these counter-examples to his "logical" conclusion:

N1: The number of volumes of Das Kapital is equal to the number of goals in a hat-trick.

N2: There were equal numbers of Union and non-Union members at the meeting last night.

N3: Although NN and MM have different disabilities they came equal first in the 100 metres final at the Para-Olympics, sharing the Gold Medal.

N4: Those two comrades sold equal numbers of papers on two different demonstrations last week.

N5: The author of Novack (1971) is identical to the comrade who penned the words in the last quotation above.

None of these suggests that the items they allude to can never change, but when they do change, anything identical to them will change equally quickly.

Or, of course, it will cease to be identical with whatever it used to be!

Apart from Woods and Grant (1995), none of the above theorists refers his readers to a single logic text (save those written by Hegel and other 'dialectical logicians'); worse, not one of them bothers to quote Aristotle even though many attribute this idea to him.

However, many of the above dialecticians at least mention the word "identity", but they then confuse it with equality; hence most of the criticisms levelled against Trotsky in this Essay apply equally to them (no pun intended) --, except that, at least here, some of the former appear to be slightly less benighted than Trotsky seems to have been.

But, Woods and Grant go further:

"Firstly, let us note that the appearance of a necessary chain of reasoning, in which one step follows from another, is entirely illusory. The law of contradiction merely restates the law of identity in a negative form. The same is true of the law of the 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 (sic).

"There is a story by Hans-Christian Andersen called The Emperor's New Suit of Clothes, in which a rather foolish emperor is sold a new suit by a swindler, which is supposed to be very beautiful, but invisible. The gullible emperor goes about in his fine new suit, which everyone agrees is exquisite, until one day a little boy points out that the emperor is, in fact, stark naked. Hegel performed a comparable service to philosophy in his critique of formal logic. Its defenders have never forgiven him for it." [Woods and Grant (1995), p.91.]

This is typical hyperbole from Woods and Grant, who seem to think that modern logicians will somehow be bothered by the many confusions that litter Hegel's badly misnamed book (on 'logic'). Indeed, the vast majority of logicians pay no more attention to Hegel, or Woods and Grant, than Woods and Grant themselves pay to the work of, say, Frédéric Bastiat. And, based on Woods and Grant's execrable book, they are not likely to change (irony intended) -- even if they could be bothered to read it.

Moreover, as we will see here, the LOC and the LOI are not connected in the way that Woods and Grant say they are; these two have simply copied this error from Hegel, without bothering to check whether the one implies the other. [As we saw in Essay Four, neither AFL nor MFL is based on the LOI, and here the distinction between the two is underlined.] In their haste to blame FL for everything but the Black Death, Woods and Grant failed to notice this.

[LOC = Law of Non-Contradiction; AFL = Aristotelian Formal Logic; MFL = Modern Formal Logic.]

2. In view of the fact that Trotsky most likely read Hegel's Logic, this observation is not entirely correct. Nevertheless, even if questions about the accuracy of the title of Hegel's book on 'logic' are put to one side, it is reasonably clear that apart from this work, Trotsky seems not to have consulted a single logic text before he began issuing ex cathedra pronouncements about it.

[It is, however, possible that Jean van Heijenoort, a member of Trotsky's entourage, and later an expert logician, gave him some advise -- but if he did, there is precious little evidence that any of it sank in. On this, see Van Heijenoort (1978), and Feferman (1993).]

In so doing, Trotsky was not alone; DM-theorists in general are only too happy to regale us with their home-spun ideas about FL -- fables whose pristine simplicity has not been sullied by the arduous task of opening a single book on MFL and then attempting to master what it contains.

Finally, those who think that AFL is based on the LOC (etc.) should consult Lear (1980), pp.98-114, where they will find a more balanced and scholarly account.

3. See Note 8, below.

4. In fact, Trotsky's version was:

S1(a):  'A' is equal to 'A'.

However, since not much seems to hang on Trotsky's use of single quotation marks (over and above his rather odd reference to the microscopic examination of the letters in question), I have ignored them in what follows.

5. Change in, or to NN is irrelevant here; this is because, howsoever much her two roles alter, since NN occupies both at the same time -- unless she resigns from one or both -- it will always be true that "The Unison rep is identical with the STWC Treasurer". The same point applies mutatis mutandis to the other examples listed in the main body of this Essay.

5a. A substantival term is a common noun which in general (but not always) admits of number, e.g., three books, two people, five comrades. Where they do admit of number, they are often called 'count nouns'. The distinguishing mark of such terms centres on the criteria of identity we use in each case.

This is what Professor Lowe had to say:

"...[N]ot all general terms are common names -- for instance, adjectival or characterizing general terms such as 'red' and 'circular' are not, nor are abstract nouns such as 'redness' and 'circularity' (if indeed the latter are deemed to be general terms, for an alternative view is that they are singular terms referring to abstract individuals). The distinguishing feature of common names -- sometimes also called substantival or sortal general terms -- is that they have associated with them, as a component of their meaning, a criterion of identity for the individuals to which they apply (see Lowe 1989, Ch. 2). A criterion of identity for individuals of a kind K is a principle which determines, for any individuals x and y of kind K, whether or not x and y are one and the same K. Thus, the criterion of identity for cities tells us that Paris and London are different cities, since they occupy different locations; and the criterion of identity for rivers tells us that the Isis and the Thames are the same river, since they flow from the same source to the same mouth. Different kinds of individuals, denoted by different sortal terms, very often have different criteria of identity governing them -- and in some cases there is philosophical debate as to precisely what these criteria are (for example, in the case of persons). Credit is once more due to Frege for recognizing the important role that criteria of identity have to play in the semantics of sortal terms." [Lowe, internet resource [2].]

Count nouns are to be distinguished from other common nouns which do not admit of number, e.g. mass nouns (such as chalk, cabbage, meat, etc.). However, some mass nouns are substantivals, e.g., gold, lead, plastic.

Many of the mistakes dialecticians make over identity originate from their failure to notice the different logic that applies to these two sorts of nouns.

On this in general, see here and here.

However, as with most things in philosophical logic, things are never quite so simple. On this, see Geach (1970), pp.39-41.

5b. It could be argued that these examples of identity are not in fact examples of strict identity, since all the items listed will change in small ways, as will their relation to countless other local (and distant) objects. In that case, absolutely nothing in nature will be identical to itself from moment to moment.

This objection has been partially defused in Note 5, above, but will be completely laid to rest below (here,  here, and here), and in the closing part of this Essay.

However, it is worth pointing out that the examples in the main body of this Essay are merely being used to show that abstract and material identity are not the same as abstract and material equality, and that ordinary material language (but not the technical jargon of philosophers) is our best guide to what we mean by identity, sameness, equality and difference.

[The primacy of ordinary language is taken for granted here; in Essay Twelve this stance will be defended in depth (summary here); but see here, also.]

As far as the objection that relational changes alter the objects in question is concerned, that itself depends on several other DM-theses being true (for example, DM-Holism and the doctrine of "internal relations"). Since these are taken apart in Essays Three Part Three and Eleven Part One and Part Two, no more will be said about this here.

Nevertheless, a few things need to be said about the doctrine of universal change.

Naturally, it would be perverse to deny that things change; not only is this given in ordinary language and common understanding (a highly truncated list of ordinary words for change can be found here) it is a familiar feature of everyday life, and highly confirmed by science.

However, even if the evidence we now have were to be multiplied by several million orders of magnitude (i.e., by a factor of, say, 10^1000000  -- or more), that would still not be enough to justify the sort of mad dog Heracliteanism we find in DM-texts:

"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves.... [E]verything exists in time; and existence itself is an uninterrupted process of transformation….

"For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing….

"Dialectical thinking analyses all things and phenomena in their continuous change….

"Dialectics…teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality." [Trotsky (1971), pp.64-66. Italic emphases added.]

"Dialectics…prevails throughout nature…. [T]he motion through opposites which asserts itself everywhere in nature, and which by the continual conflict of the opposites…determines the life of nature." [Engels (1954), p.211. Italic emphases added.]

"[Among the elements of dialectics are the following:]…internally contradictory tendencies…in this [totality]…and unity of opposites…. [E]ach thing…is connected with every other…[this involves] not only the unity of opposites, but the transitions of every determination, quality, feature, side, property into every other….

"In brief, dialectics can be defined as the doctrine of the unity of opposites. This embodies the essence of dialectics….

"The splitting of the whole and the cognition of its contradictory parts…is the essence (one of the 'essentials', one of the principal, if not the principal, characteristic features) of dialectics….

"The identity of opposites…is the recognition…of the contradictory, mutually exclusive, opposite tendencies in all phenomena and processes of nature…. Development is the 'struggle' of opposites.

"…The unity…of opposites is conditional, temporary, transitory, relative. The struggle of mutually exclusive opposites is absolute, just as development and motion are absolute." [Lenin (1961), pp.221-22, 357-58. Emphases in the original.]

"According to Hegel, dialectics is the principle of all life…. [M]an has two qualities: first being alive, and secondly of also being mortal. But on closer examination it turns out that life itself bears in itself the germ of death, and that in general any phenomenon is contradictory, in the sense that it develops out of itself the elements which, sooner or later, will put an end to its existence and will transform it into its opposite. Everything flows, everything changes; and there is no force capable of holding back this constant flux, or arresting its eternal movement. There is no force capable of resisting the dialectics of phenomena….

"At a particular moment a moving body is at a particular spot, but at the same time it is outside it as well because, if it were only in that spot, it would, at least for that moment, become motionless. Every motion is a dialectical process, a living contradiction, and as there is not a single phenomenon of nature in explaining which we do not have in the long run to appeal to motion, we have to agree with Hegel, who said that dialectics is the soul of any scientific cognition. And this applies not only to cognition of nature….

"And so every phenomenon, by the action of those same forces which condition its existence, sooner or later, but inevitably, is transformed into its own opposite….

"When you apply the dialectical method to the study of phenomena, you need to remember that forms change eternally in consequence of the 'higher development of their content'….

"In the words of Engels, Hegel's merit consists in the fact that he was the first to regard all phenomena from the point of view of their development, from the point of view of their origin and destruction…." [Plekhanov (1956), pp.74-77, 88, 163. Bold emphases alone added.]

"'All is flux, nothing is stationary,' said the ancient thinker from Ephesus. The combinations we call objects are in a state of constant and more or less rapid change….

"…[M]otion does not only make objects…, it is constantly changing them. It is for this reason that the logic of motion (the 'logic of contradiction') never relinquishes its rights over the objects created by motion….

"With Hegel, thinking progresses in consequence of the uncovering and resolution of the contradictions inclosed (sic) in concepts. According to our doctrine…the contradictions embodied in concepts are merely reflections, translations into the language of thought, of those contradictions that are embodied in phenomena owing to the contradictory nature of their common basis, i.e., motion….

"…[T]he overwhelming majority of phenomena that come within the compass of the natural and the social sciences are among 'objects' of this kind…[:ones in which there is a coincidence of opposites]. Diametrically opposite phenomena are united in the simplest globule of protoplasm, and the life of the most undeveloped society…." [Plekhanov (1908), pp.93-96. Bold emphases alone added.]

"There are two possible ways of regarding everything in nature and in society; in the eyes of some everything is constantly at rest, immutable…. To others, however, it appears that there is nothing unchanging in nature or in society…. This second point of view is called the dynamic point of view…; the former point of view is called static. Which is the correct position?... Even a hasty glance at nature will at once convince us that there is nothing immutable about it….

"Evidently…there is nothing immutable and rigid in the universe…. Matter in motion: such is the stuff of this world…. This dynamic point of view is also called the dialectic (sic) point of view….

"The world being in constant motion, we must consider phenomena in their mutual relations, and not as isolated cases. All portions of the universe are actually related to each other and exert an influence on each other…. All things in the universe are connected with an indissoluble bond; nothing exists as an isolated object, independent of its surroundings….

"In the first place, therefore, the dialectic (sic) method of interpretation demands that all phenomena be considered in their indissoluble relations; in the second place, that they be considered in their state of motion….

"Since everything in the world is in a state of change, and indissolubly connected with everything else, we must draw the necessary conclusions for the social sciences….

"The basis of all things is therefore the law of change, the law of constant motion. Two philosophers particularly (the ancient Heraclitus and the modern Hegel…) formulated this law of change, but they did not stop there. They also set up the question of the manner in which the process operates. The answer they discovered was that changes are produced by constant internal contradictions, internal struggle. Thus, Heraclitus declared: 'Conflict is the mother of all happenings,' while Hegel said: 'Contradiction is the power that moves things.'

"...As we already know that all things change, all things are 'in flux', it is certain that such an absolute state of rest cannot possibly exist. We must therefore reject a condition in which there is no 'contradiction between opposing and colliding forces' no disturbance of equilibrium, but only an absolute immutability….

"In other words, the world consists of forces, acting many ways, opposing each other. These forces are balanced for a moment in exceptional cases only. We then have a state of 'rest', i.e., their actual 'conflict' is concealed. But if we change only one of these forces, immediately the 'internal contradictions' will be revealed, equilibrium will be disturbed, and if a new equilibrium is again established, it will be on a new basis, i.e., with a new combination of forces, etc. It follows that the 'conflict,' the 'contradiction,' i.e., the antagonism of forces acting in various directions, determines the motion of the system…." [Bukharin (1925), pp.63-67, 72-74. Bold emphases added.]

Now, the evidence supporting these recklessly bold claims is conspicuous by its absence, as we discovered in Essay Two. Even so, and despite this, the authority of assorted Idealists (like Heraclitus and Hegel) seems to be sufficient for the above comrades.

However, the uncontroversial admission that most things change does not amount to any sort of concession to DM, since dialectical change is supposed to be the result of 'internal contradictions'. Now, that doctrine is demolished in Essay Eight Parts One and Two. In Essay Eleven Part One, Heraclitean change will be destructively criticised, too.

So, non-dialecticians can agree with dialecticians on the reality of change; where they differ is over the cause of change: 'internal contradictions'.

It is worth underlining here that the denial of universal change does not imply that everything is changeless; just that some things might be (and probably are) changeless, some things not; on this see Notes 11 and 12, below. Clearly, this is an empirical matter that cannot be settled by the authority of a handful Idealist Philosophers, or by dogmatic assertion.

Nevertheless, let us suppose that an object B has the following properties, qualities or relations: B₁, B₂, B₃,..., B_n.

Now, according to several of the above dialectical worthies, all of these properties, qualities and relations must change all the time (into what they do not say, but presumably it is into not (B₁, B₂, B₃,..., B_n)). [However that possibility is closed off in Essay Eleven Part One. It is also blocked in Essay Seven.]

Nevertheless, even as B changes it is still identical with itself. In order to see this, let us suppose that as each property B_i changes it becomes B_i*, in the first instance, and then B_i** in the next, and so on. But at any moment, B's identity will be given by its set of properties, qualities or relations (if we must view identity traditionally); in the first case, for example, this will be: {B₁*, B₂*, B₃*,..., B_n*}. So even though B changes, it retains its changed identity, and hence, as long as it exists it is identical to itself (albeit, its changed self). Hence, when viewed this way, identity is no enemy of change.

[Dialecticians often appeal to the existence of UOs to defuse this sort of objection; these are examined in Essays Seven and Eight Parts One and Two.]

[UO = Unity of Opposites.]

Of course, the above scenario (which is called Maximal Heracliteanism (or MAH) in Essay Eleven -- link below) might not be the option that most dialecticians would want to adopt (even though DM-classicists quoted above seem to be sold on it). If so, they should pause for thought before finally deciding. This is because, if just one of the properties, qualities or relations B enjoys (say B_k) remains the same for a few seconds then the LOI must apply to it, and the dialectical game is up -- for here we would have something that remained the same, and was identical to itself, even if momentarily.

In contrast, the maximalist option (i.e., MAH) has even worse consequences for DM; these are spelt-out in detail in Essay Eleven Part One).

Either way, Heraclitus is no friend of DM -- or if he is, he is also its enemy. A nice unity of opposites, that!

6. Hegel's egregious logical blunders are exposed here and here (and will be fully exposed in Essay Twelve, Parts Five and Six -- summary here).

However, the use of ordinary words for identity and difference is surprisingly varied, and disconcertingly complex. Consider the following (greatly shortened) list of examples:

E1: The same letter can appear in the same word in different places, and in a different word in the same place (e.g., "t" can appear first and fourth in both "trite" and "trot"); a different letter can appear in the same word, in the same or in a different place (e.g., if "chien" and "dog" are counted as the same word in different languages, "c" appears in the first place in the French word, and "d" in the same (i.e., first) place in the English word, and the different letters "d" and "g" appear in the same English word). The same word, in the same or different places in a sentence, can mean the same or different things (e.g., in Chomsky's example, "Pretty little girls' school", the word "pretty" can be taken in several ways, depending on how the whole phrase is read, as can each sub-phrase: "Pretty little", "little girls'" and "Pretty little girls'", to name but three), and different words, in the same or different places, can mean the same or different things (as in, "The striker hit the scab" and "The scab was hit by the striker" (where the same words mean the same in different places in two different sentences with the same sense); and "The striker hit the ball", where the same word could mean different things (i.e., "striker" could mean a player on a Football field or someone engaged in a strike)). Moreover, the same word can mean different things at one and the same time to two different people (e.g., if one of them reads it as a code word, on one occasion), and different things to the same person at different times (if, say, their facility with the language concerned improves). Naturally, permutations like this can be knitted together endlessly to form complex identity/equality sentences that we can all understand, given the right level of concentration. For example, the same word could mean different things to the same person at different times, but the same thing in different places, while it could mean the same thing at the same time or at different times in the same place or in different places to the same or different people (etc.).

E2: The same numeral can appear in the same place in the same number in different places at the same or different times (e.g., the figure "9" in a mathematics book, or on a bank statement), or in the same place different numbers (as in 191 and 1911). Not only that, identically the same numeral can appear in the same number in different places, where it will have a different mode of signification (e.g., in 2500, 2450 and 2445; here the same numeral "5" means something different in each case, or in 191 and 1911 where the "9" appears in the same place (i.e., second from the left) but means something different, or where it appears in different places (in the tens column and in the hundreds column) but could mean the same (i.e., if the "9" in 191 stood for 90 ten pence pieces, and the "9" in 1911 stood for 900 one pence pieces)). Furthermore, the same numeral can appear in the same sign in the same place and mean something different, depending on how it is read (e.g., the numeral "1" in "10" could mean "one" written in the tens column, or it could mean "one" written in the unitary power of two column in binary code, with the first "one" signifying "ten" and the second indicating "two"). Or the very same "2" on a clock face could signify 2am or 2pm. Or think of the way that "1" can mean something different if it occurs in the same place in 01/02 and 01/02; in the first it could mean the 1st of February (if read by a UK citizen), in the second, the 2nd of January (if read by a US citizen). So, in the last few cases, the very same thing could be identical in certain respects while being different or unequal in others. Examples are easy to multiply. The same points (or different ones) can be made about the same (or different) musical notes, dance steps, gestures, works of art, signs, signals, symbols and noises.

E3: The same day of the week occurs in the same place in different weeks, and for 24 hours on the same day in the same week. And it can occur in the same place in different weeks of the same or different months. The reader, no doubt, can supply his/her own complex permutations as the temporal vocabulary used is changed -- as in: same/different second, minute, hour, year, decade, century, millennium, geological time period, eon…

E4: The same book can appear in different libraries in the same place, or in different libraries in different places, and a different book can appear in the same or different libraries in the same or different places. The same copy of The New York Times can be read by different people in the same place at the same time, or in different places at the same time, or in the same place at different times -- and it can be read by the same person in different places at the same or different times, and so on. The same can happen with TV programmes, films, works of art and plays.

E5: The same worker could join the same strike at different times, or different strikes at the same time (if he/she has two jobs and both are in dispute). And different workers could join the same or different strikes at the same or different times in the same or different places. And the same strike could spread to different places, involving different workers at the same or different times. The same or different cheques could be made valueless if the same Bank goes bust, and the same person could be made an orphan and a lone child at the same time if both its parents are killed in the same or different accidents at the same or different times.

E6: The same element in the periodic table can appear in different parts of the universe at the same or different times, and in the same or different compounds at the same or different times. The same geodesic can be traversed by different particles, at the same or different times. The same inertial frame can contain the same or different objects at the same or different times, and different inertial frames can contain the same or different objects at the same or different times. The same (or different) goes for cars, taxis, trains, planes, ships and buses.

And so on, ad nauseam. [Try expressing any of that in Hegel-speak!]

As noted elsewhere, ordinary (and technical/semi-technical) language has a seemingly limitless capacity for allowing its users to express complex and subtle differences in meaning way beyond that permitted by the lifeless, non-material language found in Hegel. This is not surprising: ordinary and technical/semi-technical languages were formed over tens of thousands of years by working people/materially-motivated scientists in their interaction with the world and with one another; these systems of communication reflect our species' complex inter-relationship with changing reality -- and contain our best guide to identity, sameness and difference, and much else besides.

In contrast, Hegel's lifeless, jargon-bound language reflects alienated ruling-class consciousness (cobbled-together in a dubious and class-compromised tradition of thought that stretches back over two thousand years), and (in Hegel's case) was invented by a man who, in his theoretical activity, was more concerned with his relation to the world of ideas than with his ordinary interaction with objects and processes in the material world. Small wonder then that his ideas cannot cope with living, material reality.

7. However, as with other philosophical issues, Trotsky is in good company here; Philosophers and modern Logicians also manage to confuse the two. In fact, this is a highly neglected area in the Philosophy of Logic. There are signs, though, that this is beginning to change; on this see Sanford (2005).

In fact, astute logicians have been aware of such complexities for years, but have been opposed by traditionalists reluctant to change (no irony intended). On this, see Geach (1967, 1973, 1975, 1990), Griffin (1977), and Noonan (1980, 1997).

8. As pointed out above, DM-theorists' comments on FL are exceedingly repetitive (Graham Priest's work being a notable exception here). In marked contrast to their attempts to understand other areas of bourgeois 'science' -- for example, classical and modern economics, history, politics and current affairs --, dialecticians display little or no comprehension even of elementary Logic. Indeed, their writings almost invariably contain highly superficial and inaccurate characterisations of what turns out to be obsolete FL. Not surprisingly, such 'logical straw men' are quite easy to knock over. [On this, see Essay Four, here and here.]

It is instructive to compare this dishonest approach with the justifiable condemnation that the very same DM-theorists level at analogously distorted views of Marxism found in the writings of most of Marx's bourgeois critics.

[For some reason, the words "sauce", "goose" and "gander" come to mind here.]

In this regard, it is quite clear that John Rees cannot possibly have checked a single logic text (other than that written by Hegel, perhaps) before he wrote what he did about FL and the LOI in TAR. As seems also to be the case with most other Marxist critics of FL, Rees appears to have confined his 'research' on this topic to reading only what previous dialecticians had written and simply copying what he found. Indeed, a supporter of this site tried to point this out to him at a large public meeting in London in 1990, but this was clearly a waste of breath.

In modern symbols, one form of Leibniz's Law is as follows:

[1] (∀x)(∀y)((x = y) º (Fx ® Fy)).

[1] is otherwise known as the "Indiscernibility of Identicals". Translated, it reads roughly: "Any two objects are identical if and only if they share the same properties" -– or, "…whatever is true of one is true of the other." This particular 'law' will not be defended here for reasons outlined in Note 1, above. Its translation into ordinary language is not happy on any reading. That alone shows it is not equivalent to the ordinary use of such phrases as "equal to", "the very same as", "identical with", or even "numerically identical with".

Nevertheless, it is important to note that the use of the "=" sign in [1] is strengthened by the presence of the bi-conditional "º", hence, it is not identical with Trotsky's use of the former sign, either (no irony intended).

Contrast [1] with the following version of the same 'Law':

[2] (∀x)(∀y)((∀F)(Fx º Fy) ® (x = y)).

[2] is otherwise known as the "Identity of Indiscernibles". Loosely translated it reads: "Any two objects that share every property in common are identical."

One of Trotsky's mistakes was to suppose that this 'law' was empirically testable. He did this when, for example, he supposed that the truth of S1 [i.e., "A is equal to A"] could be checked directly with an eyeglass, and when he referred to the weighing of bags of sugar. He clearly regarded S1's failure to pass such tests as sufficient grounds for rejecting it. However, it did not seem to occur to Trotsky that an empirical test of Leibniz's Law is wholly inappropriate -– it would be just as misguided as an empirical test would be of, say, "a + b = b + a" (i.e., commutativity over addition) in Mathematics. Anyone who thought to test such a rule in this manner would be regarded as hopelessly confused, and rightly so.

[Of course, if this 'law' is regarded as an expression of a rule, then the temptation to think it can be tested simply vanishes. On this see Note 10, below.

Incidentally, on this issue it will not do to point out that certain operators in mathematics do not commute since no one in their right or left mind would test these empirically, either. Of course, tests might be performed to see whether or not certain systems in nature observed/obeyed commutativity (for example, in Matrix Mechanics), but no test would or could be run on the principle itself. And despite what dialecticians say, the same goes for the LEM, as it is allegedly used in QM. More on this later.]

[QM = Quantum Mechanics; LEM = Law of Excluded Middle.]

Compare the two versions of Identity outlined above with the following:

[3] j(y) º [(∃x)((x = y) & j(x))].

[3] appears in Griffin (1977), p.1, which also contains a strengthened version of Leibniz's Law:

[4] (∀x)(∀y)[(x = y) º (∀j)(j(x) º j(y))].

[4] roughly reads: "Any two objects are identical if and only if for any property, one has it if and only if the other has it." Griffin also gives other versions of [1] and [2] above, (ibid., p.2). Incidentally, [3] roughly says "Anything true of some object is equivalently true of any object identical to it." [In fact, I use this version in the present Essay to show that identity is no enemy of change.]

["∀" is the universal quantifier, equivalent to "All" or "Every"; "∃" is the existential quantifier, equivalent to "Some" or "At least one"; "º" is the sign for logical equivalence, i.e., "If and only if"; "j" and "F" are predicate variables; "®" is the implication arrow, equivalent to "if...then"; "x" and "y" are bound variables. For more on these, see Priest (2000) and Tomassi (1999). On quantifiers, see note 13a.]

8a. And the same could be said -- with slightly less justification --, about Hegel; at least he used the right word, even if it is clear that he failed to grasp the complexity of this logical 'law', just as he failed to do justice to the ordinary words we have for identity, etc.

9. This from the UK Guardian newspaper (Wednesday, 18/10/95):

"K2 appeared over the 40 million years or so that India has been colliding with greater Asia. It was 'discovered' (i.e. by the British) and designated K2 (Karakoram Peak 2) in 1856. The peak was granted the name [Mount Godwin-Austen] in 1888, after its first surveyor, Col Henry Haversham Godwin-Austen (1834-1923). The previous title is now preferred as being less imperialistic. Ordinarily a mountain would revert to its local name, but K2 is so remote that it appears never to have gained one." [Letter from one S McDiarmid, reprinted in "Notes and Queries". See also here.]

This is perhaps a more interesting example of ordinary ascriptions of identity than that considered by Frege (with respect to the Evening Star and the Morning Star, etc.), in Frege (1892). [This topic will not be explored any further in this Essay.]

And this is from the same newspaper:

"In Nepal they call it Sagarmatha. To the people of Tibet, it is Chomolungma, though the ruling Chinese prefer the variant Qomolangma. When the British first began mapping India, they knew it as Peak B, then as Peak XV. But in 1865, to honour the surveyor-general of India who first mapped it, Peak XV was given the name Mount Everest. And Everest the mountain has remained throughout much of the rest of the world to this day.

"Now China is launching a fresh effort to outlaw the name Everest. Accusing British colonialists of 'raping the sacred mountain of Tibetans by giving it a false name', Chinese newspapers are calling on the world to 'respect Tibetans' by using the 50th anniversary of the first ascent next year to recognise the mountain henceforward as Qomolangma.

"At first sight, the proposal does not seem unreasonable. There are, after all, lofty precedents for such renaming. The highest point in Africa, the summit of Kilimanjaro, which was once known as Kaiser Wilhelm Spitze, properly became Uhuru Peak. The world's second-highest peak, once Mount Godwin-Austen to the British, has become K2 (ironically this also dates from imperial survey days). But doubts about the new proposal soon creep in. If Everest is unacceptable, why should the world not prefer the Nepalese name to the Chinese or Tibetan one? Who are the Chinese, of all people, to accuse others of raping Tibet? And how is the 'English language hegemonism' of which China complains worse than its Chinese language equivalent?

"We hold no great brief for the name Everest, though it has to be said that the word has a fine ring to it. But the answer is to live and let live. If people prefer Chomolungma, let them use that name. If others want to stick with Everest, let them do so too. We have no problem with diversity, though the Chinese may. In the end, the world's greatest mountain is surely more important than any name that mere mortals give to it." [Guardian, 20/11/02. Italics added; quotation marks altered to conform to the conventions adopted at the site.]

So, here we have several perfectly ordinary identities: Everest is identical to Chomolungma, Qomolangma, Peak B and Peak XV; Kilimanjaro is identical to Kaiser Wilhelm Spitze, and Uhuru Peak, and so on.

Instead of reading Trotsky -- or far worse, Hegel -- on identity, comrades would do far better to begin with Frege. [Frege (1892).] [It has to be said, even Frege confuses identity with equality!]

Excellent introductions to Frege's thought can be found in Kenny (1995), Noonan (2000) and Weiner (1990, 1999); for a useful guide to the philosophical issues involved, see Linsky (1977).

And we find this in a recent issue of the New Scientist:

"In 2003, at team...[in] Moscow discovered two distant elliptical galaxies just a whisker apart. Detailed analysis of the twins known as CSL-1, suggested that they were images of the same galaxy.

"The team suggested that the duplicate images were being created by a "cosmic string".... If one of these cosmic strings were to pass between Earth and a giant galaxy, the warping of space-time by the string would create a gravitational lens and form two identical images of the galaxy -- exactly like CSL-1....

"Unfortunately for the proponents of cosmic strings, observations with Hubble on 12 January have revealed that CSL-1 is actually two different galaxies...." [New Scientist 189, 2537, 04/02/06, p.21.]

Now, this is a clear example of science at work. Instead of 'solving' this problem in an a priori manner (a là Trotsky or Hegel), declaring that things are never equal to themselves, astronomers have been able to show that these images were not of the same object, but of two different galaxies. [On gravitational lensing, see here.]

However, one wonders what dialecticians would have said if these two images had been shown to be of the same galaxy.

In the article posted at the above link several examples are given of multiple images of identically the same object. In relation to this consider the following options:

G1: Image I₁ is identical to image I₂.

G2: Image I₁ and image I₂ are both of the same object.

G3: But, image I₁ is an image of object O₁.

G4: And, image I₂ is an image of object O₂.

G5: Therefore, object O₁ is identical to object O₂.

It is not easy to see how dialecticians would be able to tell G1 and G2 apart (had these scientist found they were of the same object), nor account for the scientific conclusion recorded in G5.

And it would be no use pointing to the alleged limitations of the LOI here, since, no matter how much objects O₁ and O₂ changed they would still be identical, since they would change at an identical rate as 'one another' (being one object, not two)! Certainly, the images of these objects may or may not be identical, but these 'two' objects cannot fail to be such.

Of course, if it were now claimed that, on the basis of what Trotsky or Hegel said, objects O₁ and O₂ were nonetheless not identical (or they were both identical and not identical(!)), then that would fatally undermine this part of Astrophysics, since it would nullify the application of gravitational lensing in the above manner. If the objects here are non-identical, then images I₁ and I₂ would plainly be of two different objects; the above inference would falter and the theory would fail.

As noted above, it is not easy to see how dialecticians can hold on to their criticism of the LOI without undermining at least this part of science.

And, as we will see later, an appeal to "approximate identity" here would be to no avail.

10. This observation, of course, depends on the said weighing scales changing at the same rate as the sugar being weighed, which, while unlikely, could happen. The point is, of course, that this is an empirical matter that cannot be settled a priori, as Trotsky attempted to do.

Nevertheless, it is worth remembering that Trotsky's argument here revolves around the accuracy of measuring a pound bag of sugar (etc.); he merely extrapolated from these few (theoretical) observations about local conditions to bold general claims about all objects for all of time:

"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves…. [T]he axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is, if it does not exist…. For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing…. Hegel in his Logic established a series of laws: change of quantity into quality, development through contradiction, conflict and form, interruption of continuity, change of possibility into inevitability, etc…. All this demonstrates, in passing, that our methods of thought, both formal logic and the dialectic, are not arbitrary constructions of our reason but rather expressions of the actual inter-relationships in nature itself. In this sense the universe is permeated with 'unconscious' dialectics." [Trotsky (1971), pp.64-66; 107. Emphases added.]

Once more: how could Trotsky possibly have known all this? His only 'evidence' appears to be a few badly worded thought experiments.

Again, it could be objected that Trotsky was merely drawing out the consequences of accumulated human experience of change, deriving reasonable conclusions from thousands of years of our growing knowledge of the world.

In response to this (and in addition to the points made in this section) it might be worth trying to reconstruct the reasoning behind Trotsky's claims. Presumably he argued as follows: anyone who performs a weighing experiment exactly as specified -- who repeats the same procedures -- will find that no two measurements are exactly the same. But this imputed argument -- if it is his -- would embroil Trotsky in having to apply the LOI yet again, only now applied to his own instructions and the measurements he would recommend. Clearly, these would have to be carried out identically the same each time if they were to count as a verification of what Trotsky said -- but now aimed at demonstrating that the LOI is defective after all!

Of course, given Trotsky's strictures on the LOI, no two attempts to carry out any set of instructions would ever be the same. In that case there would be no accurate way to test his or anyone else's predictions.

On the other hand, if Trotsky had been faced with someone who claimed that at least two of their results were identical, he could only have responded in one or more of the following ways:

(1) Insisting that this experimenter must have been mistaken.

(2) Pointing out that the machines used were not accurate enough.

(3) Maintaining that his instructions had not been carried out exactly to the letter.

(4) Arguing that identically the same experiments had not been performed each time.

In other words, in the absence of a mistake (and if the same results were recorded on more accurate scales) -- i.e., ruling out (1) and (2) above --, Trotsky would only be able to criticise the above reported experimental verification of the LOI by an appeal to that very same 'law', but now applied to his own instructions! Hence, in order to counter results that would disconfirm his forecast (about varying weights) he would have to argue that only those who followed his instructions identically and to the letter could disprove the LOI!

The irony is thus quite plain: identically performed experiments are required to prove that nothing is identical with anything else -- including experiments!

To be sure, anyone who only roughly followed his instructions (who was perhaps content with a wishy-washy, "approximate-within-certain-limits" dialectical-sort-of-equality) would probably find that many (if not most) of their measurements gave identical results for the weights of bags of sugar, confirming this 'law'!

In which case, Trotsky's predictions about such objects would end up being refuted by anyone who adopted this diluted version of the LOI!

Such experimenters would succeed in confirming the absolute version of the 'LOI by adopting a weaker variant of it!

Conversely, the more exactly an experimenter adhered to Trotsky's instructions, the more likely it would be that they detected non-identical weights. In that case, they would succeed in disconfirming the absolute version of this 'law' by applying an exact copy of Trotsky's instructions! So, by reverse irony, they would refute Trotsky in practice by doing exactly as he instructed, using the LOI applied to instructions in order to disconfirm it as applied to bags of sugar!

Relying on evidence alone, therefore, Trotsky was certainly not justified in projecting his conclusions as far as he thought he could --, i.e., across the entire universe and for all of time -- not least because he performed no experiments himself.

[Indeed, the development of science has justified this more cautious approach. On that, see Notes 11 and 12, below.]

Hence, Trotsky's claim that all objects everywhere change all the time, if extrapolated beyond the aforementioned conventions and scientific facts, would transform the LOI into a metaphysical truth. There could be no body of evidence large enough to support an extrapolation as bold as this -- that is, there could be none that wasn't also based on those very same conventions relating to identically performed experiments, and the use of ordinary words for identity, etc. Extrapolation beyond these -- by means of them -- to universal theses that are applicable everywhere and for all of time would convert them into universal truths -- the very thing Trotsky affected to disavow. In that case, Trotsky would have to appeal to the LOI as a universal truth to justify his general conclusions about everything in existence behaving exactly as he said it did, with every human being measuring objects identically throughout all of human history.

So, any evidence from either the past or the present used (by DM-theorists) to undermine this 'law' would automatically call into question the methods by which it had been collected, processed and checked. Without the LOI applied as a rule of language, or as a rule guiding practice, no one would be able repeat the same experiments to verify or refute earlier results, check measurements and confirm the accuracy of predictions. Nor would they be able to learn to use the very same theories as any one else ever had (nor even make sense of that possibility), or appeal to the very same 'law' (in the way that, say, Trotsky and Hegel did). Without this 'law' applied as a rule of language, or of practice, there would be no conceptual space within which science or ordinary practice could develop --, and thus no reliable data, and no settled theories --, for anyone even to begin to confirm the DM-hypothesis of universal change.

In that case, no viable science/philosophy could question the application of this 'law' as a rule of language (or of practice) while remaining viable.

[The problems in fact go much deeper than this, but their consideration would take us too far into areas covered by other Essays posted at this site.]

This of course, explains the origin of the difficulties highlighted above (in connection with the postulated refutation of Trotsky's predictions about the weights of bags of sugar) -- and it also reveals why Hegel got into such a tangle in his attempt both to half accept and to half reject this 'law', and why this entire topic was such a puzzle to him. If this 'law' is treated as a metaphysical truth (which has generally been the case with traditional theorists) -- i.e., as a 'necessary truth' --, then its falsehood becomes impossible to state (as we have seen throughout this Essay) --, as does its truth --, at least in comprehensible language, or in language that does not implicitly rely on this very 'law' to state what was intended of it.

[The 'truth' if this 'law' is no less problematical than is its 'falsehood', for both of these depend on treating it as a sort of empirical thesis; more on this in Essay Twelve Part One.

Why this is so is connected with Wittgenstein's comments on 'rule-following'; again, on this see the references given in Essay Twelve Part One, and in Essays on the nature of language to be published here at a later date, for example, here.]

On the other hand, the 'truth' of this 'law' does not seem (to some) to be at all trivial. If it is viewed (traditionally) as a 'law' which is said to determine or depict the nature of everything in existence, then it appears (to Hegelians and DM-fans) to rule out change. But even then, this supposed defect proves impossible to express in language which does not also rely on this very 'law', only now featuring as a rule governing the use of words to make that very point! [As we have also seen.]

Alternatively, if this 'law' is viewed rightly as a rule of language. or of practice (without which rule humanity could not have developed a single coherent idea), then these knotty problems simply vanish. On that basis, this 'law' is not a universal truth because of what Hegel or anyone else has said about it; it is not a truth to begin with! Because it is a rule, it cannot be true or false, only practical or impractical, applicable or not.

In that case, without a clear idea of how to use words for identity (etc.), it would be impossible even for DM-theorists to begin to wonder whether our words were approximately stable from moment to moment, or if they altered in other alarming ways. If the LOI is rejected as a rule of language --, or if it is held to be a truth or an approximate truth -- then all of the above points go by the board.

For example, in order to be able to say whether or not something was true (or partially true), we would need to know how to use the word "truth" in the same way from moment to moment, just as we would also have to know the same with respect to our words for identity, sameness and difference. Without some notion as to what counts as identically the same employment in all these cases, we could not even begin to say what would constitute an approximation to anything whatsoever, or, indeed, in what way anything fell short of a standard that itself presupposed the applicability of such words for identity.

Now, Hegel missed this fundamentally important point; so did Lenin, Trotsky and other DM-theorists. They regarded the LOI as a truth (even if it was one that only the "abstract understanding" employed, or which was valid "only within certain limits"). But it cannot be a truth (for the reasons outlined above); our use of words for identity express rules that lie behind our capacity to utter sentences that we can even begin to regard as true or false, or a bit of both (should we so please).

[This is not a 'transcendental argument', merely a reminder that we have to use words to express our thoughts, and we may only do certain things with such words given contingent facts about our history and our social nature.]

So, this 'law', applied as a rule of language, has to be employed even to make the point that anything is approximately true; in which case that 'law' can neither be true nor false itself.

And this is what makes the comments of dialecticians in this area valueless, and it is also why their ideas collapse so readily into incoherence.

All this shows why an appeal to human experience since the beginning of time on its own is irrelevant: without a reference to the LOI applied as a rule (or convention) to experiments, instruments, to Trotsky's own writings and to those of experimenters, etc., empirical evidence cannot be used to attack the LOI. Hence, the moment the LOI is deployed in an attempt to reveal its own empirical limitations (a là Trotsky) -- or its theoretical short-comings (a là Hegel) --, then that attempt itself must self-destruct. Which is indeed what we have seen. [More on this in Note 15, below. See also Note 13.]

Naturally, a grudging acceptance of the above conventions would have the reverse effect; it would involve dialecticians in using criteria that delineate the conditions required for the performance of identical but real experiments (etc.) -- those that would undermine their own ideal 'thought experiments' aimed at revealing the alleged deficiencies of FL, and of the LOI.

Now, DM-theorists may believe with all their might that all objects change constantly, but that is all this will ever remain: a mere belief, an act of faith. There could be no conceivable body of evidence in favour of this act of faith that was not itself dependent on conventions of measurement (counting and comparing) that are not themselves subject to Trotsky's (or Hegel's) strictures -- should the latter continue to be accepted. And that is why both Hegel and Trotsky had to use this 'law' in their futile endeavour to undermine it, and why they both wound up in practice refuting their own criticisms of it.

Beyond this, the idea that reality is in the grip of a universal 'Heraclitean Flux' is supported by nothing more than an unfounded extrapolation from a few badly-worded 'thought experiments', themselves based on a laughably superficial understanding of a seriously mis-identified 'law'. [Irony intended.]

11. Physics And Identical Objects

However, it is very easy to make two identical objects -- everyone does so when they throw a light switch. Here is some material devoted to this topic, copied from another Essay posted at this site:

Physicists tell us that every photon, for example, is identical to every other photon. This how Steven French puts things:

"It should be emphasised, first of all, that quantal particles are indistinguishable in a much stronger sense than classical particles. It is not just that two or more electrons, say, possess all intrinsic properties in common but that -- on the standard understanding -- no measurement whatsoever could in principle determine which one is which." [French (2006), 'Identity and Individuality in Quantum Theory', p.5, The Stanford Encyclopedia of Philosophy (Spring 2000 Edition), Edward N. Zalta (ed.); available here.]

And Paul Dirac made the same point this way:

"If a system in atomic physics contains a number of particles of the same kind, e.g., a number of electrons, the particles are absolutely indistinguishable. No observable change is made when two of them are interchanged…." [P. Dirac, The Principles Of Quantum Mechanics (Oxford University Press, 4th ed., 1967), p.207.]

In that case, every time a worker turns on a light, he or she makes/generates countless trillion identical objects per second -- which must mean that such workers are "unconscious" anti-dialecticians, if we apply the same sort of reasoning here as Trotsky.

Naturally, contentious claims like these can only be neutralised by an a priori stipulation to the effect that every photon in existence (past, present and future) must be non-identical -- despite what scientists say, and in abeyance of the almost infinite amount of data that would be needed to support such a cosmically ambitious claim. At this point, perhaps, even hardnosed dialecticians might just be able to see in this a blatant attempt to impose DM on reality.

A recent discussion of these issues can be found in Brading and Castellani (2003), and Castellani (1998). An even more recent discussion can be found in Saunders (2006), and particularly French and Krause (2006). See also the Wikipedia entry here.

It could be objected that Trotsky would surely have been unaware of these recent developments in Physics -- but as the references given show, such facts were largely true of classical particles; quantum particles merely present a more extreme form of strict identity. And Lenin it was who reminded us that science is ever revisable; hence no dialectician (who happens to agree with Lenin) could consistently rule out the possibility that scientists would one day discover identical particles -- as indeed they have.

Even so, Trotsky was quite happy to impose this view on nature before all the evidence was in.

12. Once more, this is material that has been posted in another Essay at this site:

It is now thought that certain sub-atomic particles are equal to themselves for unimaginably long periods of time. Protons, for instance, have an estimated half-life in excess of 10³⁶ years (but estimates vary). Apparently, electrons are even less 'dialectical'. During that time protons do not change (as far as is known), and as such they are surely equal to themselves.

As far as protons are concerned, we are told the following:

"Along with neutrons, protons make up the nucleus, held together by the strong force. The proton is a baryon and is considered to be composed of two up quarks and one down quark.

"It has long been considered to be a stable particle, but recent developments of grand unification models have suggested that it might decay with a half-life of about 10³² years. Experiments are underway to see if such decays can be detected. Decay of the proton would violate the conservation of baryon number, and in doing so would be the only known process in nature which does so." [Quoted from here.]

Wikipedia adds:

"In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, usually a neutral pion and a positron. Proton decay has not been observed. There is currently no evidence that proton decay occurs."

Of course, it could be objected to this that particles such as protons (i.e., hadrons) are composed of even more fundamental particles, which do enjoy a contradictory life of their own 'inside' each host 'particle'; their machinations would therefore mean that apparently changeless protons are in fact changing 'internally' all the time. But, this response simply pushes the problem further back, for these other, more fundamental particles (i.e., quarks --, in the case of protons, two "up" and one "down" quark), are themselves changeless, as far as is known. Moreover, since protons are baryons -- i.e., they are composed of three quarks --, it is not easy to interpret their inner lives as in any way "contradictory" (with three terms?). Even more difficult to account for dialectically are electrons and photons (which are leptons and gauge bosons respectively), since they have no known internal structure. Unless acted upon externally, their 'lifespan' is, so we are told, infinite; hence, if they change, it is not because of any "internal contradictions".

[An appeal to antiquarks here, to save the dialectical day, would be to no avail, either. This is because quarks do not turn into antiquarks, nor vice versa, which is what the Dialectical Holy Books tell us should happen to all such 'opposites'. On that, see here and here. (Anyway, since there is apparently very little antimatter in the entire universe, so we are told, this is an academic question.)]

So, it seems that the picture of reality painted by dialecticians is more a Jackson Pollock than it is a Van Eyck.

Figure One: DM-style 'Realism'    

Figure Two: Non-DM-style Realism

On protons, see here, here, and here; on electrons, here; Leptons here; photons, here. On this topic in general, cf., Perkins (2000), and French and Krause (2008) -- but more specifically Saunders (2006).

In fact there appear to be two schools here, those who hold that all such particles are identical and indiscernible (rather like the dollars/pounds in your bank account, not the dollars/pounds in your wallet or pocket), and those who claim they are identical and discernible. On this see the above references and Muller and Seevinick (2009), and Muller and Saunders (2008).

Naturally, dialecticians might want to object to the above on the lines that electrons, for example, are not really particles --, or that they are probability waves, or that they are this or they are that. Perhaps so, but then once again whatever they are, they are identical with that, and they change equally quickly as they themselves do. Furthermore, if they change, they do not do so as a result of their 'internal contradictions'. [More on this here.]

This comment puts paid to much of the confused ruminations on sub-atomic 'particles' found in, for example, Woods and Grant (1995). More details on this will be posted in Essay Seven Part Two at a later date. On change though 'internal contradiction, see Essay Eight Parts One and Two.

13. The reader will no doubt have noticed that in order to interpret Trotsky, we have to use the dread LOI. In that case, we have to decide whether S8 is equivalent to (i.e., identical with) S11 or S12.

S8: A pound of sugar is equal to itself.

S9(b): [All bodies] are never equal to themselves.

S10: Let A₁ be a pound of sugar at time T₁.

S11: Let A₂ be a pound of sugar at time T₂.

S12: S8 means A₁ is equal to A₁.

S13: S8 means A₁ is equal to A₂.

Of course, it is true that Trotsky believed that a complete understanding of change (even if humanity never actually attained to one) would require the employment of concepts more adequate to the task -– i.e., those found in DL. Many of these 'concepts' are examined throughout this site, where they are shown to be no less confused.

[IDM = In Defense of Marxism (i.e., Trotsky (1971); DL = Dialectical  Logic.]

However, in his Notebooks Trotsky added a number of important qualifications to his comments on the LOI found in IDM. Among which were the following:

"a = a is only a particular case of the law of a ¹ a…. Formal Logic involves stationary and unchanging quantities: a = a. Dialectics retorts: a ¹ a. Both are correct. A = a at every given moment. A ¹ a at two different moments. Everything flows, everything is changing." [Trotsky (1986), pp.86-87.]

This suggests that Trotsky might have accepted a version of S13 or S21(a):

S13: S8 implies A₁ is equal to A₂.

S21(a): There is an A and a time t₁ such that, A at t₁ is not equal to A at t₂.

S13 was in turn dependent on S8, S10 and S11:

S8: A pound of sugar is equal to itself.

S10: Let A₁ be a pound of sugar at time T₁.

S11: Let A₂ be a pound of sugar at time T₂.

[UO = Unity of Opposites.]

This would appear to mean that Trotsky was committed merely to the idea that an object is not self-identical at some later time, as opposed to adhering to the stricter principle that objects are not self-identical at any given moment -–, which belief is in turn based on the doctrine that all objects are UOs. If so, the above passage seems to suggest that Trotsky was in fact rejecting a core DM-idea: that UOs exist in every object and process, and which drive change because of such 'internal contradictions'. Clearly, it is highly unlikely that Trotsky denied this core DM-thesis. For example, his emphasis on the contradictory nature of the former USSR (elsewhere in IDM) strongly suggests he accepted this doctrine.

On the other hand, since Trotsky nowhere (to my knowledge) refers to the idea that change is the result of the struggle between UOs, it is possible that he did reject, or did not fully accept this doctrine.

Nevertheless, since this quotation is taken from notebooks not intended for publication it would be unwise to rely too heavily on what they say as an accurate indication of Trotsky's intentions. This is especially so since it appears to contradict what was said in IDM:

A1: "In reality 'A' is not equal to 'A'…. [O]bserve these two letters under a lens -- they are quite different from each other." [Trotsky (1971), pp.63-64.]

S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.

Compare this with a passage from the Notebooks (quoted above) where Trotsky now seems to say the opposite:

A2: "A = a at every given moment." [Trotsky (1986), p.87.]

If A1 and S9 were correct, A2 couldn't be; at best, it would represent only half the story.

For example, in A2, the two letter "A"s are easy to distinguish without the aid of a lens: the second letter is in the lower, while the first is in the upper, case. But here in A2, Trotsky now argues that these "A"s are equal at "every given moment" -- even though they look different to the naked eye!

Conversely, in A1 Trotsky claims the opposite of this is true with respect to two letter "A"s that not only look identical but also are in the same upper case! He claims that the two capital letters in A1 look different if examined under a lens, while a lower case "a" and a capital "A" in A2 are equal at every moment!

If A2 were correct, then Trotsky's reference in A1 to the physical appearance of these two letter "A"s when viewed under an eyeglass would be entirely pointless. The only reasonable conclusion here seems to be that since A1 was intended for publication it must contain Trotsky's more considered thoughts.

Furthermore, as noted above, A2 seems to be inconsistent with the claim that change is the result of internal contradictions: that is, with the idea that at any given moment an object both is and is not self-identical, thus constituting a UO -- in this case presumably a unity of "A and not A" (i.e., "pound bag of sugar and not pound bag of sugar"), or that a pound bag of sugar is both identical and not identical with its 'other', as Hegel might have put it. [On this, see Essay Eight Part Three, and Essay Twelve (summary here). On the confusions Hegel's view must always introduce in this area, see here.]

Precisely what the 'other' is of a pound bag of sugar is somewhat unclear. A pound bag of tea? A half pound bag of tea? A pound bag of Quinine? But if a pound bag of sugar has no 'other' (and no logical 'other', either) then, according to DM, it cannot change. To be sure, sugar is highly complex; there are any number of things it can and does change into, so it must have countless 'others' (which fact rather makes a mockery of Hegel's 'analysis' of change). More on that here.

In addition, A2 is itself rather badly worded. When Trotsky wrote:

A3: "A = a at every given moment" (emphasis added)

he must have meant:

A4: "A = a at any given moment."

This is because the wording of A3 implies that "A" never changes; i.e., that at all times "A = a" -- something Trotsky certainly did not believe.

On the other hand, A3 might contain an indirect allusion to Trotsky's point about abstract moments in time:

"A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is if it does not exist." [Trotsky (1971), p.64.]

But, according to this, if Trotsky was referring to abstract moments in time in A3, it would mean that the items he mentioned could not exist. If so, it would be unclear how A3 could ever be true -– that is, always assuming it was about such non-existent things.

Of course, if A2 and A3 were merely about letter variables (not their supposed referents, what they supposedly refer to) it might prove possible to re-interpret them in a more viable form. One such re-configuration could see them recording the fact that while objects in the world change, letters depicting them do not. But that would make Trotsky's other assertions about the "A"s in A1 decidedly odd, for the aim there had been to argue that these letters were not in fact "equal" irrespective of what they referred to. That was the whole point of Trotsky's appeal to ocular inspection. And since variable letters are physical objects in their own right, his claim surely was that they are just as susceptible to change and diversity as are the things to which they supposedly refer. That option therefore does not look at all promising.

On the other hand, if Trotsky had wanted to argue for something more complex in this regard it would definitely be impossible to comprehend his point. For example, if he had meant something like the following:

(1) Variable letters and what they refer to both change, and that they do so as follows:

(2) Each letter "A" no longer refers to whatever it used to refer to moments earlier, and,

(3) The object that each old letter "A" once denoted is no longer the same as it was when first identified, and,

(4) Earlier and concurrent manifestations of any and all letter "A"s are never the same as the 'same' new letter "A" now on the page/screen (which page/screen also changes), and,

(5) Any two or more concurrent letter "A"s on the 'same' line (which also changes) are not only different from each other, they change at different rates, and,

(6) Each letter individually denotes in a different and changing manner objects in reality, which objects are also different and are all changing at different rates themselves.

If something like this had been Trotsky's intention then his entire point would become too obscure to assess for we wouldn't have a clue what he was banging on about. But, if all things change uninterruptedly in every respect (as Trotsky himself claimed) then he must have 'meant' this!

It could be objected that Trotsky only needs to appeal to the relative stability of medium-sized objects in reality to neutralise criticisms like this. Hence, if language and most medium-sized objects are relatively stable, points (1) to (6) above do not apply.

But, how could anyone committed to this theory know whether or not language is 'relatively stable' -- especially if they also believe that everything is in the grip of the Heraclitean Flux? In fact, as soon as language itself is implicated in this Flux, everything semantically solid melts into thin air. Hence, it would be no good appealing to evidence (drawn from dictionaries, textbooks, memory, common usage, etc.) in support of the claim that language is 'relatively stable', for if everything is changing then so is the language in which this evidence is expressed, so are the notebooks from which it has been trawled, and so are the memories on which all of these depend. Given this way of looking at things, then for all anyone knows, every single word could change its meaning every fraction of a second (along with any and all memories of or about the objects that seem familiar to us, etc.). This is almost certainly what Voloshinov believed:

"[T]heme must be unitary, otherwise we would have no basis for talking about any one utterance. The theme of an utterance is individual and unreproducible, just as the utterance itself is individual and unreproducible. The theme is the expression of the concrete, historical situation that engendered the utterance. The utterance 'What time is it?' has a different meaning each time it is used, and hence, in accordance with our terminology, has a different theme, depending on the concrete historical situation ('historical' here in microscopic dimensions) during which it is enunciated and of which, in essence, it is a part." [Voloshinov (1973), p.99. Bold emphases added. Quotation marks altered to conform to the conventions adopted here.]

And, as we will see in Essay Thirteen Part Three, several other comrades say the same sort of thing.

However, as Plato himself was quick to recognise, the Heraclitean Flux is no respecter of theories; in fact it thoroughly mangles them. [On this, see Note 10 above, and Note 15 below -- and, in this case, the aforementioned Essay on language, where Voloshinov's ideas will be subjected to sustained and destructive criticism.]

Furthermore, the reasoning in the Notebooks appears to be somewhat confused. For example, it is not easy to see how "a = a" could be a particular case of the 'law' "a ¹ a", any more than "a + b = c", for instance, could be a particular case of the rule "a + b ¹ c". If "a ¹ a" is a 'law', then "a = a" refutes it; it does not instantiate it. Of course, this is so unless the word "refute" has suddenly changed its sense (perhaps because of the local effects of the pesky Heraclitean Flux). Naturally, in such a madcap Heraclitean world it is not easy to see exactly what would, or could, either stay the same or change -- or, indeed, for how long a decision about even that possibility would remain stable, too!

Moreover, other things that Trotsky said in IDM indicate that the above passage from his Notebooks is not a reliable guide to his thinking:

"A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is if it does not exist." [Trotsky (1971), p.64.]

This at least confirms the accuracy of the interpretation put on Trotsky's analysis of the LOI in this Essay -– that is, in so far as any sense can be made of what he was trying to say.

Finally, what TAR itself says about Trotsky's argument seems to agree with the interpretation given here. [Cf., Rees (1998), p.273.]

13a. Quantifiers in language are words like "all", "every", "some" and "none", etc. Tensed quantifiers are words like "always", "never" and "sometimes". On this, see here, here and here.

14. It could be objected to this that moments in time are not objects, and that time is simply one of the "modes of existence of matter". But, if time can be measured then Trotsky's criticisms must apply to it, too. Moreover, if we accept what Lenin said about matter (that it is whatever exists "objectively outside the mind" (a doctrine examined in detail in Essay Thirteen Part One)), time must be material, too -- unless, of course, it 'exists' only in the mind.

14a. It could be argued that since moments in time follow on from each other, it is not possible to measure one of them in order to compare it with any other -- since the earlier one won't now exist to make the comparison! This is not the case with respect to objects that have to be weighed; they clearly exist side by side during the entire process, facilitating comparison. Or so the objection might go.

But can't moments differ even if we are unaware of it? And can't two objects be weighed simultaneously, and the duration of each weighing instance timed simultaneously too, with the latter durations compared just like their weights?

15. Again, it could be insisted that all Trotsky needs is the relative stability of the words he used, which won't have changed significantly during the short intervals involved.

Unfortunately, Trotsky holed that response well below the water line by declaring that:

"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves… But everything exists in time; and existence itself is an uninterrupted process of transformation…. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is if it does not exist." [Trotsky (1971), p.64.]