Part I – Chapter 5

About Paradoxes

1. On the Liar Paradox

2. Making No Claim

3. Nagarjuna’s Trickery

4. Non-apprehension of Non-things

5. A Formal Impossibility

6. The Analytic/Synthetic Dichotomy

7. On the Russell Paradox

8. An Illustration of Russell’s

9. On Grelling’s Paradox

 

1. On the Liar Paradox

Once we grasp that the meaning of words is their intention, singly and collectively – the solution of the liar paradox becomes very obvious. Self-reference is meaningless, because – an intention cannot intend itself, for it does not yet exist; an intention can only intend something that already exists, e.g. another intention directed at some third thing.

In view of this, the proposition “this proposition is false” is meaningless, and so is the proposition “this proposition is true”. Both may freely be declared equally true and false, or neither true nor false – it makes no difference in their case, because the words “this proposition” refer to nothing at all[1].

Although the words used in these sentences are separately meaningful, and the grammatical structure of the sentences is legitimate – the words’ collective lack of content implies their collective logical value to be nil. Self-reference is syntactically cogent, but semantically incoherent. It is like circular argument, up in the air, leading nowhere specific.

Regarding the exclusive proposition “Only this proposition is true”, it implies both: “This proposition is true” and “All other propositions are false” – i.e. it is equivalent to the exceptive proposition “All propositions but this one are false”. The latter is often claimed by some philosophers; e.g. by those who say “all is illusion (except this fact)”.

My point here is that such statements do not only involve the fallacy of self-reference (i.e. “this proposition”). Such statements additionally involve a reference to “all others” which is open to criticism, because:

· To claim knowledge of “all other propositions” is a claim to omniscience, a pretense that one knows everything there is to know, or ever will be. And generally, such statements are made without giving a credible justification, though in contradiction to all prior findings of experience and reason.

· Surely, some other propositions are in fact regarded and admitted as true by such philosophers. They are generally rather talkative, even verbose – they do not consistently only say that one statement and refuse to say anything else.

· And of course, formally, if “this” is meaningless (as previously shown), then “all others”, which means “any other than this” is also meaningless!

The liar paradox, by the way, is attributed to the ancient Greeks, either Eubulides of Miletus (4th Cent. BCE) or the earlier Epimenides of Crete (6th Cent. BCE). I do not know if its resolution was evident to these early logicians, but a (European?) 14th Cent. CE anonymous text reportedly explained that the Liar’s statement is neither true nor false but simply meaningless. Thus, this explanation is historically much earlier than modern logic (Russell et alia, though these late logicians certainly clarified the matter).[2]

 

2. Making No Claim

The Buddhist[3] philosopher Nagarjuna (India, c. 150-250 CE) attacked every thesis he regarded as rational by every means he regarded as logical, and declared his own discourse immune from scrutiny and criticism, by saying (according to one translation):

“If I had a thesis, I would be at fault; since I alone have no thesis, I alone am without fault” (VV 29)[4].

The first aspect of Nagarjuna’s statement is a brazen claim to have no claim. This is of course self-contradictory. Every proposition that claims to be meaningful and true (whether about some experience or about abstraction, whether positive or negative) is an assertion, a claim. To pretend making no claim even as one plainly makes one is a breach of the law of identity: it is denying that a fact is a fact.

There is no logical way to deny or criticize the theses or methodologies of others without opening one’s own discourse to evaluation. All denial or criticism is discourse, and all discourse is subject to logical review. To pretend the logical possibility of dispensation is dishonest (and if such pretense implicitly is bad enough, it is all the more dishonest if made explicitly).

Nagarjuna’s discourse was, in fact (as I show in Buddhist Illogic), shock full of fallacious arguments, a mere parody of logic posing as logic. But he knew that people untrained in logic would fall for it, and he sealed their intellectual fate with the said eyewash claim. To neutralize further discussion, he misled them into believing he had simply shown up the logical absurdity of logic, and all doctrines based on it, but had himself posited no methodology or doctrine of his own.

Not only was his alleged refutation of reason full of errors of reasoning, but his concluding ‘no-claim claim’ was also a mockery of logic and sincerity. He, of course, just says ‘I make no claim’ – and he persistently denies that this statement constitutes a claim. I call that shameless psychological manipulation, motivated by one-upmanship. He cynically takes advantage of the credulity of some people, to dominate them intellectually.

The second aspect of Nagarjuna’s above statement can be viewed as a ‘soft’ version of the liar paradox, since he tells us: everyone but me is in error. Although such a statement is not in itself inconsistent (God could conceivably utter it truthfully) – it is logically open to doubt due to being self-exempting.

Effectively, it says: ‘I am the only human who has knowledge; I know everyone else is incapable of true knowledge’. Only a fool is tricked by such an unsubstantiated claim to privilege. Reason regards all people as technically within range of knowledge given enough effort, even if they do not all fulfill their potential equally. Reason demands that discourse be reasoned and fair – i.e. based on common general norms as to how truth and falsehood are to be determined.

If Nagarjuna were basing his criticism of ordinary human means to knowledge on a claim to have attained a ‘higher level’ of consciousness (i.e. Buddhist enlightenment or Biblical prophesy), we could not convincingly oppose him (being unable to prove or disprove such experiential claims). But he is not using such as claim as his basis – he is attempting to debunk reason through ordinary logical discourse. In that case, he is fair game for logic.

The statement of infallibility is then seen as manifest arrogance, a lack of respect for other thinkers. By saying ‘I alone am exempt from any criticism’ the author aggressively grants himself a special dispensation: he alone is endowed with the way to knowledge; everyone else is an idiot or a dishonest person. It is totalitarian, dictatorial speech.

Compare this dismissive ‘you all know nothing’, to the self-inclusive statement ‘I (or we) know nothing’. The latter – even though it implies ‘I know that I know nothing’ and is therefore self-inconsistent – is at least modest; so much so, that such admission is widely considered a mark of wisdom (and it is commendable, in modified form, i.e. as ‘I know close to nothing, very little’).

Self-exemption is a hidden form of self-inconsistency, because it resorts to a double standard. The one making such a claim presents superficially rational arguments against human experience and logic, but does not ask himself or tell us how he (an ordinary human) managed (using the very cognitive means he rejects) to attain such allegedly true knowledge. The author criticizes others, but does not equally well criticize himself.

This is a fallacious mode of thought often found among would-be skeptical philosophers. It comes in many subtle forms. It is wise to always be on the lookout for such practices, applying the reflexive test here demonstrated.

 

3. Nagarjuna’s Trickery

Looking at Nagarjuna’s above statement in more detail, the following may be added.

To begin with, what is meant here by “having a thesis”? This refers to any explicit or even wordless belief, any clear or even vague opinion upheld (considered to constitute knowledge), any proposition one advocates or implicitly logically condones. The subject that Nagarjuna is here discussing is any outcome of human rational cognition, any belief, opinion or doctrine that one may arrive at, rightly or wrongly, by means of ordinary consciousness, i.e. through experience, negation, abstraction, hypothesizing, inductive or deductive argument.

And what is meant here by “being at fault”? This refers to making a mistake in the course of observation or reasoning, so that some thesis one has adhered to is in fact an illusion rather than a reality, false rather than true, erroneous instead of correct.

How do we know the status appropriate to a thesis? We know it (I suggest) by holistic application of the whole science of logic to the totality of the data of experience. Our concepts of cognitive right or wrong are themselves all constructed by logic and experience, without appeal to some extraordinary outside justification (like prophetic revelation or mystical realization, or simply the authority of some great personage or of a religious document or institution).

Now, Nagarjuna is evidently well aware of all that, but is intent on annulling the independent reliability of ordinary experience and reason. His strategy and tactics to this end, in all his discourse, as I have shown throughout my Buddhist Illogic, is to give the impression (however paradoxical) that logic may be invalidated by means of logic. And this twofold sentence of his, “If I had a thesis, I would be at fault; since I alone have no thesis, I alone am without fault”, fits neatly into his destructive philosophical programme.

On the surface, this sentence might be construed as a single argument:

If X (a proposition is proposed), then Y (an error is made)

but not X (no proposition)

therefore not Y (no error)



Although the above apodosis is logically invalid, since it denies the antecedent to deny the consequent, Nagarjuna is not above letting it pass without comment, knowing it will suffice to convince some people, although he is well aware that the logically trained will spot it and object. But for the latter audience, he reserves a more subtle form of manipulation.

It has to be seen that the purpose of this famous Verse 29 in Nagarjuna’s discourse is designed to make a show of logical consistency. He wants by means of it to give the impression that his anti-rational discourse is justifiable, that it has the stamp of approval of logic. Yes, he is actually attacking logic; but at the same time, he has to pretend to use it, because he knows this measure is required to convince people. For most people, a veneer of logic (i.e. mere rhetoric) suffices to put their reason’s critical faculty at rest. We shall now see how he goes about this task.

The first part of Nagarjuna’s statement, viz. “If I had a thesis, I would be at fault”, is not intended (as some have assumed) as a justification for his overall discourse. It is not placed here in his discourse as an argument with intrinsic force, which directly buttresses or proves his philosophy. It is certainly not an obvious logical principle, or axiom, which everyone would agree on without objection, from which his discourse can be inferred or even generalized. No – it is itself an inference and application from Nagarjuna’s main thesis, namely the claim that ‘All human knowledge based on ordinary experience and reason is necessarily erroneous’.

The latter underlying claim is his major premise in a (here tacit) productive eduction, i.e. one that deduces a particular hypothetical proposition from a more general categorical one[5]. This argument is formally valid, running as follows:

All X (opinions) are necessarily Y (erroneous);

therefore,

If this is X (a proposition is proposed), then this is Y (an error is made).



In this way, the first part of Nagarjuna’s statement is made to seem something inferred, rather than an arbitrary claim. It is cunningly presented as an application of already admitted information, rather than as an isolated assertion. Granting the premise, the conclusion indeed logically follows (this is the veneer of logic) – but has the premise already been granted? No. Also note, once the conclusion is seemingly drawn, it can by generalization be used to reinforce the premise; although this is a circularity, it works psychologically.

Moreover, Nagarjuna manages through this implicit productive argument to pretend he is being consistent with himself: he is telling us, effectively: ‘See, I am not just attacking other people’s knowledge, but am prepared to apply the same stringent critique to my own!’ This virtuous declaration is of course dust in your eyes, because he is not here putting the broader principle in doubt but merely reaffirming it. He has nowhere established that ‘All propositions are false’. His is a pseudo-logical posture.

As the next part of his statement clarifies, he does not consider his discourse as falling under the critical rule he has formulated. The proposition “If I had a thesis, I would be at fault” is a counterfactual hypothetical; his own discourse is never made into an issue open to debate. It seems open-minded, but it is a foregone judgment. His intention is to ‘avert all arguments’ and place himself at the outset outside the fray. He seemingly at first admits and then vehemently denies that his own discourse is a product of ordinary consciousness. This convoluted avoidance of cognitive responsibility has fooled many a poor soul.

Moving on, now, to the second part of Nagarjuna’s statement, viz. “since I alone have no thesis, I alone am without fault”. As already pointed out, this can be viewed as the minor premise and conclusion of an invalid apodosis in which the first part of the statement is the major premise. But we could also more generously assume that Nagarjuna intended a valid apodosis, using as its tacit major premise the obvious proposition: ‘If one has no thesis, one cannot make a mistake’.

It can be correctly argued that this premise was left tacit simply because it is so obvious to and readily granted by everyone. It is indeed true that if one ventures no utterance, thought or even intention, if one holds no opinion, makes no claim to knowledge, if one remains inwardly and outwardly silent, one will never make any errors. For the status of truth or falsehood is only applicable to meaningful assertions.

A stone is never in error, because it has no thoughts. Likewise, a thoughtless person may by his or her ignorance, blindness or stupidity make many errors of living, but makes no error in the logical sense of having proposed an inappropriate proposition. All that is so obvious (and vacuous) no debating it is necessary. The following apodosis is thus implicit in Nagarjuna’s declaration:

If not X (no proposition is proposed), then Y (no error is made)

but not X (no proposition)

therefore not Y (no error)



This argument has a true major premise, as well as a valid form. This gives his discourse a veneer of logic again, helping him to persuade more victims. However, his minor premise remains well open to doubt, and decisively deniable! (As a consequence of which, his conclusion is of course also open to doubt.) He takes it for granted that he ‘has no thesis’ – but this claim is far from granted already. The tacit major premise acts as a smokescreen for the minor premise.

Moreover, note, although ‘being correct’ implies ‘not being at fault’, the reverse is not necessary. Nagarjuna suggests that his alleged faultlessness implies the correctness of his position, but it does not follow! Only if his criticism of all opposing theses was correct (which is by no stretch of the imagination true), and his thesis was not liable to similar criticism and was therefore the only leftover logical possibility, would such inference be drawn.

Nagarjuna does indeed ‘have a thesis’. His main thesis, the goal of his whole philosophical discourse, is as already mentioned the claim that ‘All human knowledge based on ordinary experience and reason is necessarily erroneous’. This, for a start, qualifies as a thesis – boy, it is a big skeptical thesis, full of negative implications. It is a principle of logic that to deny any thesis is to affirm an opposite thesis. His claim that his doctrine is not a thesis, in the minor premise here, is mere arbitrary assertion.

Furthermore, he ‘has a thesis’ every time he makes a specific assertion of any kind, including the assertion under scrutiny here, viz. “If I had a thesis, I would be at fault; since I alone have no thesis, I alone am without fault”. Note that Nagarjuna thinks that making a negative statement is somehow ‘not having a thesis’ – but the polarity of a statement does not diminish the need for justification; if anything, one can argue that on the contrary negative statements are harder to establish than positive ones!

And we should strictly include as ‘theses’ of his not only such explicit statements, but also all the implicit assumptions and suggestions within his discourse (like the implicit major premise and resulting apodosis we have just highlighted). It makes no difference whether these explicit, or unstated and unadmitted, items constitute information or logical method, content or process.

For all these elements of discourse, be they spoken or otherwise intended, in all fairness fit in our common understanding and definition as to what it means to ‘have a thesis’. For none of these categorical or hypothetical propositions (except perhaps ‘if silence, no error’) is self-evident. They did not arise ex nihilo in Nagarjuna’s mind, ready-made and self-justified.

They are all complex products of ordinary human cognition, based on experience and produced by reason (even if, in Nagarjuna’s case, the mind involved is deranged). They undeniably together form a specific philosophy, a theory of logic, an epistemology and ontology. The mere fact that we can (as here done) at all consider and debate them is proof that they are ‘theses’.

The law of identity (A is A) must be maintained: facts are facts and it is no use pretending otherwise. Nagarjuna may eternally refuse the predicate of “having a thesis”, but we confidently insist on it. His arguments have in no way succeeded in averting this just and true judgment. Consequently, his doctrine is self-contradictory. Not only does he ‘have a thesis’, but since his thesis is that ‘to have a thesis is to be in error’, he has (by its own terms) to be recognized as being in error.

Thus, to end it: Nagarjuna’s statement “If I had a thesis, I would be at fault; since I alone have no thesis, I alone am without fault” weaves a complicated web of deception. It misleads, by means of subtle ambiguities and superficial imitations of logic. Once its dishonesty is revealed, it should be decidedly rejected.

The mere historic fact that Nagarjuna is famous and admired by many does not justify hanging on to his doctrine ad nauseam, trying ex post facto to find ways to make it consistent with logic. Celebrity is not proof of some hidden truth – it is vanity. Most who do so are merely grasping for reflected glory. Anyway, attachment to authority is argument ad hominem. The religious and academic ‘groupies’ who gave him and perpetuate his authority are not logically competent, however numerous they be. It is a case of the blind leading the blind.

 

4. Non-apprehension of Non-things

Nagarjuna defends his ‘non-thesis’ idea in the next verse (VV 30), describing it as “a non-apprehension of non-things” (according to one translation[6]). Now, this is a very funny phrase. To the impressionable, it sounds very deep, pregnant with meaning. It seems to suggest this man has some privileged higher way of knowledge that goes beyond ordinary experience and reasoning.

But in truth, taken literally, we are all quite capable of “non-apprehension of non-things” and daily practice it, for the simple reason that non-things cannot be apprehended! Logically, this is all this phrase means, note well. What then is the old fox up to, here?

Nagarjuna is trying to project his ‘not having a thesis’ position as far as logically possible from our plebian ‘having a thesis’ – i.e. from ordinary consciousness, which consists in ‘the apprehension of things’. He has logically only three alternatives to choose from:

  • the ‘non-apprehension of things’ (unconsciousness);
  • the ‘apprehension of non-things’ (an otherworldly consciousness);
  • or the ‘non-apprehension of non-things’.

Having a marked taste for one-upmanship and dramatic extremes, Nagarjuna of course chose the third of these terms as his vehicle. Even though the obvious sense of this phrase is puerile, it has poetic breadth and appeal. It seems to imply ‘knowledge without consciousness’ and ‘consciousness of the unknowable’ all at once.

Thus, his ‘non-apprehension’ is a mix of apprehension and non-apprehension, or something else again. And likewise, his ‘non-things’ are things of some sort as well as non-things, or perhaps something quite other still.

In other words, the negative terms in the phrase “non-apprehension of non-things” are not intended by Nagarjuna nor received by his disciples and students as mere negations of the corresponding positive terms, but as paradoxical terms, which may (in accord with the tetralemma schema) be all at once positive or negative or both or neither.

It is (and isn’t) ‘apprehension/non-apprehension of things/non-things’ all in one.

Nagarjuna stands out in the history of world philosophy as the most unabashed opponent of the laws of thought. Not only does he freely use self-contradictory or middle-including propositions, but he even makes use of terms loaded with contradiction and inclusion of a middle.

Now, some people might say: ‘what is wrong with that?’ They will argue: ‘the real world is extremely subtle and we can only ever hope to express it in thought very approximately; Nagarjuna is only trying to take this uncertainty into consideration within his discourse; the laws of thought are just arbitrary demands, making us force our thoughts into prejudicial straightjackets’.

But logical laxity is not the proper attitude in the face of an extremely complex and hard to express real world. It is precisely because of the great difficulty of the cognitive task at hand that one is called upon to be very clear and careful. Avoiding checks and balances on our judgments does not increase their efficiency but makes them less reliable.

In the case under consideration, if Nagarjuna does indeed have some privileged form of otherworldly consciousness, he can just say so. The laws of thought in no way forbid him to posit such a claim. He does not need to beat about the bush, and pretend to have something unspeakable and not subject to peer review. He can and should be forthright, and defend his position in an equitable way like everyone else.

If he considers the terms ‘apprehension’ and ‘things’ to have some intrinsic logical flaw, he can argue his case openly; he does not need to engage in allusion, suggestion and fallacious argument. Most of us thinkers are open-minded and willing to correct our errors: if these terms are flawed, we are not attached to them; we are flexible, ready to modify or replace them as logically necessary in the light of new evidence and reasoning.

But Nagarjuna is like an accused, who when forced to appear in court refuses to admit his identity, or recognize the authority of the law and the judges, or plead guilty or not guilty, or argue the defense of his case. Worse still, in utter contempt of the court, he does not even admit his refusal to be a refusal – he calls it a ‘non-thesis’. Does that stop court proceedings or make the court declare him innocent? Surely not.

Nagarjuna misunderstands the nature of negation. He thinks that if one person says ‘X’ and another says ‘not X’, the onus of proof is on the first more than on the second. He considers that making a positive statement is more logically demanding than making a negative one. He imagines in his confusion that saying ‘no’ is equivalent to saying nothing, i.e. to not saying anything. Most logicians would disagree with him, and argue that any thesis put forward (even if only by insinuation) is equally in need of proof, whatever its polarity.

I would go further and say that, on the contrary, a negative statement is more demanding than a positive one. You can prove a positive statement easily enough, if you point to sufficient evidence in its favor. But how do you prove a negative statement? It is much more difficult, since negatives are not directly experienced but are only experienced by way of the absence of positives. A negative can ultimately only be proved indirectly, by inability to prove any contrary positive.

Thus, in fact, not only does Nagarjuna’s alleged self-limitation to negatives not exempt him from proofs, but on the contrary it increases the logical burden upon him. He is right in considering negatives as significantly different from positives, but he does not realize that the difference is to his disadvantage. He claims to have no epistemological or ontological basis, and yet to be able to reject offhand all theories of knowledge and reality. Such a grandiose fanciful claim surely requires much more justification than any other!

It should be stressed, incidentally, that Nagarjuna’s “non-apprehension of non-things” should not be interpreted (as some do) as a defense of non-verbal meditative experience or insight. That is not the thrust of his anti-rational philosophy, although its avowed Buddhist affiliation may lead one to suppose so.

If Nagarjuna were a man deeply absorbed in meditation, he would not be writing philosophy. If his intent were to promote meditation, he would simply teach methods of meditation and not stir up verbal disputes. No – this man has philosophical ambitions. Allegedly, these are meant to put into words some of the ‘reasoning’ that he considered the Buddha to have gone through before attaining enlightenment. Nagarjuna assumes from the start that this ‘reasoning’ is necessarily anti-logical, a rejection of reason.

But we must see that this assumption is just a prejudice of his distorted mind. He was a philosophical revolutionary – one who believed that reason has to be overturned, to be transcended. But it is more credible to be evolutionary – and to consider meditation as a way for us to keep moving, beyond the limits of discursive thought, without need to deny such thought within its applicable bounds.

To advocate respect for logic is not to foment endless babble, but rather to require that any thought arising be subjected to responsible cognitive evaluation. Logic is possible entirely without words, by means of silent intentions. Even in deep meditation, some sort of ‘reality check’ by means of logic occurs, and this need not involve any words. It is only by this means, no doubt, that a Buddha-to-be may steer himself well clear of common illusions and insane imaginings, towards to full realization.

Contrary to Nagarjuna’s belief, rationality and spirituality are not necessarily in conflict. Reason and meditation are potentially, to some extent, mutually beneficial. It is not thought as such, much less logic, but only excess of thought, particularly irrelevant chatter, which hinders meditative concentration and contemplation. A certain amount of appropriate thinking is often needed to initially position one’s mind for meditation.

 

5. A Formal Impossibility

In fact, as I will now show, the sentence “If I had a thesis, I would be at fault”[7] is a formal impossibility. I earlier interpreted and symbolized it as “If X (a proposition is proposed), then Y (an error is made)”, giving the antecedent and consequent two separate symbols, X and Y. But now let us consider these constituents more closely.

What does “making an error” mean here? It is not an ordinary predicate. The consequent Y does not merely refer to some error in general, but specifically to an error in the antecedent X. Y tells us that X is wrong. Therefore, Y formally implies the negation of X, i.e. notX! Granting this, Nagarjuna’s sentence now reads: “If X, then not X”, i.e. “If X is true, then X is false” – a paradoxical hypothetical proposition, whose conclusion would be the categorical “X is false” (as earlier suggested).

However, that is not the end of the matter. If we now consider the meaning of X – viz. “a proposition is proposed” – we may fairly suppose it refers to just any proposition whatsoever. In that case, the proposition concerned might even be the negation of X; so that we may substitute notX for X throughout the hypothesis. So doing, we obtain “If notX, then not notX”, i.e. “If not X, then X”, or in other words “If X is false, then X is true”. This is also, of course, a paradoxical proposition, whose formal conclusion is “X is true”.

We thus – by means of a universal reading of “having a thesis”, as inclusive of “not having a thesis” – now have, not only a single paradox, but a double paradox! That is, our conclusion is not only that X is false, but that X is both true and false. The latter conclusion is of course contrary to the law of non-contradiction, as in the case of the liar paradox.

This means that Nagarjuna’s statement is a formal impossibility: it is a contradiction in terms; it is not only false, but meaningless. It does not constitute legitimate discourse at all, let alone a tenable philosophical position or theory. The words or symbols used in it are logically not even conceivable, so it is as if he is saying nothing. He seems to be saying something intelligible, but it is an illusion.

Now, it may be objected that Y does not necessarily mean that X is wrong, but could merely mean that X could be wrong. That is, “making an error” could be taken to mean that X is uncertain rather than definitely refuted. In that case, we would have the following two hypotheses: “If X, possibly not X” and “If not X, possibly X”; or in one sentence: “Whether X or not X is proposed, the outcome is uncertain”. Indeed, this more modal, ambiguous posture may well be considered as Nagarjuna’s exact intent (which some have interpreted as noncommittal ‘illocution’).

At first sight, due to the use of vague words or of symbols, this objection may seem credible and the contradictory conclusions involved apparently dissolved. But upon reflection, there is still an underlying conflict: to affirm X, or to deny it, is contrary to a position that neither affirms nor denies X. An assertoric statement (affirming or denying X) is incompatible with a problematic statement (saying X may or may not be true). One cannot at once claim to have knowledge (of X, or of not X) and claim to lack it (considering the truth or falsehood issue open). This is as much a contradiction as claiming the same thing (X) true and false.

Someone unacquainted with the logic of hypothetical propositions might now object that X, or notX, is only proposed hypothetically in the antecedent, and so may well be problematic in the consequent. But this is a logically untenable objection, due to the process of addition (described in the chapter on formal logic); i.e. due to the fact that “If X, then Y” implies “If X, then (X and Y)”. In the present case, this means: “If X is asserted, then X is both asserted and uncertain”. It suffices for the contradiction to occur conditionally, as here, for the condition to be disproved; therefore, our conclusion is quite formal: “X cannot be asserted”. QED.

Someone could here, finally, object that the certainty in the antecedent and the uncertainty in the consequent may not be simultaneous, and so not produce a logical conflict. Such objection would be valid, granting that a thought process separated the beginning and end of the hypothetical proposition. However, in the case under scrutiny, Nagarjuna is clearly stating that in the very act of “proposing something”, one would be “making an error”; i.e. the error is nothing other than the proposing, itself. So, no time separation can credibly be argued, and Nagarjuna’s thesis remains illogical.

Note that all the present discussion has concerned only the first part of verse 29, i.e. the major premise “If I had a thesis, I would be at fault”. We have found this hypothetical proposition logically faulty, irrespective of whether Nagarjuna admits or refuses to acknowledge that he “has a thesis”. So, let us now reconsider this minor premise of his, and his conclusion that he “is not at fault”.

We have here introduced a new twist in the analysis, when we realized that “If X, then Y” (understood as “If X, then not X”) implies “If not X, then Y” (since the latter is implied by “If not X, then X”, which is implied by the former by replacing X with notX). So, now we have a new major premise for Nagarjuna, namely “If not X, then Y”, meaning: “If I do not have a thesis, I will be at fault”.

Taking this implied major premise with Nagarjuna’s own minor premise, viz. “I have no thesis” – the conclusion is “I am at fault”. This conclusion is, note, the opposite of his (“I am not at fault”). Thus, even though Nagarjuna boasts his thinking is faultless, it is demonstrably faulty!

For – simply put, leaving aside all his rhetoric – all he is saying is: “no thesis is true”; it is just another version of the liar paradox. And his attempt to mitigate his statement, with the afterthought “except my thesis”, is logically merely an additional statement: a particular case that falls squarely under the general rule. Moreover, before an exception can be applied, the rule itself must be capable of consistent formulation – and this one clearly (as just shown) is not.

Note lastly, none of this refutation implies that silence is impossible or without value. If (as some commentators contend) Nagarjuna’s purpose was to promote cessation of discourse, he sure went about it the wrong way. He did not need to develop a controversial, anti-logical philosophy. It would have been enough for him to posit, as a psychological fact, that (inner and outer) silence is expedient for deep meditation.

 

6. The Analytic/Synthetic Dichotomy

All belief-systems are not on a more or less equal footing. Some are elaborate mazes, concealing numerous self-contradictions. Others more sneakily rely on logical sins of omission, by effectively exempting themselves from scrutiny. The peculiarity of epistemological theorizing, which too many philosophers fail to realize, is the requirement of self-examination, both to develop a realistic methodology and to test one’s theories on one’s own practice.

The system proposed by Immanuel Kant (Prussia, 1724-1804) is a case in point. The analytic/synthetic” dichotomy, in spite of the prestige of its inventor and later defenders, is full of logically arbitrary declarations and circular arguments. The dichotomy is nonsensical, i.e. not a viable philosophical construct, because it fails to explain and justify itself, i.e. its own genesis.

Kant’s analysis, rather than being a priori and necessary (as he claims), is quite a posteriori and contingent. Moreover, it proposes a static ordering of knowledge, whereas knowledge can only be understood and validated by consideration of its dynamic aspects, its conceptual genesis and development.

Knowledge is not established by linguistic analyses of axiomatic tautologies, or by syntheses of particular empirical data – but by an active, flexible combination of all one’s experience and the full range of logical techniques. It is a holistic, ongoing enterprise, depending on the whole of one’s knowledge context and all our rational means.

Language plays an important technical and creative role in this genesis, by locking our attention onto a clearly pointed-at or a vaguely known and still-unfolding phenomenon or abstraction. Logic is used to rationalize experience, but it is not arbitrary. Experience is a sine qua non of all conceptual work – i.e. all propositions, even ‘logical’ ones are to some degree ‘synthetic’.

What is missing in the ‘knowledge is either analytic or synthetic’ proposal is the full realization of the inductive nature of knowledge. Many philosophers seem to understand the term ‘logic’ only in its sense of ‘deduction’, but the truth is that deduction is only one tool within logic as a whole, which is essentially ‘induction’! Induction too has its rules[8].

In this perspective, different items may indeed be assigned varying degrees of “immunity from revision[9]”, which may change under appropriate conditions. For example, the laws of thought are most immune. The law of conservation of matter and energy is more immune than the finding that water boils at 100 deg. C, say. All depends on the amount of data an assumption is based on and how much a change in such assumption would affect the rest of knowledge.

Although the ‘analytic’ notion was proposed as an explanation of logical necessity, it of course does not follow that its rejection constitutes rejection of logical necessity (let alone of natural necessity, i.e. that of empirical “laws”, which it implies but is not implied by). Necessity is a valid, accessible and unavoidable concept.

Logical certainty is possible not only by logical insight (when the negation of a proposition is contradictory, for instance; or again, when a notion is seen to be based on circular arguments), but also by generalization or adductive argument from natural necessity, itself based on previous generalizations or adductive arguments, and ultimately on experiences.

All such knowledge remains in principle revisable, but that does not mean that we indeed always find convincing reason to revise it! The choice of our ultimate principles is thus not purely arbitrary or relative, but depends on sincere and conscientious application of logical methodology, including for a start careful observation.

 

7. On the Russell Paradox

A class may be viewed as an imaginary envelope, which flexibly wraps around all the class’ purported members, however dispersed in place and time, to the exclusion of all other things. The question arises, can the figurative envelope of the class “classes” wrap itself too, or not?

Reviewing the Russell paradox[10], we must conclude that not all ‘word-objects’ are ‘things’ – measures of things are not themselves to be counted as things. Since classification is an expression of our measurement of things, it cannot itself be counted as a thing. To do so gives rise to a paradox, we should avoid it.

In other words, the problem involved is that the iterative form (“class of classes”) is not identical with the simple form (“class”), except very superficially and verbally – so the former cannot logically be subsumed under the latter. There is a sufficiently significant modification of the subject-predicate relation involved, caused by the iteration of the same term, to exclude the reflex of subsumption. The paradox arising if we do not restrain this impulse is precisely what teaches us to exercise such restraint.

The word ‘things’, note, has many meanings. Sometimes, we intend by it all possible objects of thought. Sometimes, we mean to exclude words from it[11]. Sometimes, we mean to exclude classes; or more narrowly, as just pointed out, classes of classes; ditto, with regard to concepts or to concepts of concepts. Sometimes, the word ‘things’ includes only material objects, whatever their category. Sometimes, we mean by it ‘entities[12]’ (material, mental or spiritual bodies, or delimited substances, individual cases of which are generally subjects of propositions) in contrast to their ‘properties’ (the predicates of place, time, quality, action, quantity, relation, and so forth). Sometimes, in everyday discourse, we refer to ‘things’ in contrast to ‘persons’ – i.e. ‘things’ here means inanimate or non-volitional entities. And there are yet more senses of the word.

Thus, whenever logicians refer to ‘things’, they ought to try and first make clear just what is to be included under that heading.

Incidentally, even worse than ‘self-membership’ as a concept to swallow, is the notion of “classes that seem contradictory to what they include” – the latter seems inconceivable at the outset, at least in verbal appearance! Thus, for instances: “no relationship” is a relationship of sorts; “non-classes” is in a sense a class. There has to be some fallacy involved in such terms, which needs to be clarified. Perhaps the problem is a hyperbole or misnomer?

The answer to this question would be that we are here again dealing with classes of classes, and these need not be outwardly consistent with their member classes. Thus, the class of non-relationships still involves a relationship. The class of non-classes is nonetheless a class. The class of empty or null classes does have members. The class of meaningless or self-contradictory classes is itself neither meaningless nor self-contradictory. And so forth.

 

8. An Illustration of Russell’s

More on the Russell paradox: Bertrand Russell illustrates his paradox with reference to:

(a) a catalogue of all books that mention themselves, and

(b) a catalogue of all books that do not mention themselves.

Case (a) presents no problem: the catalogue can list itself without contradicting its own definition; whereas, if it does not list itself, it betrays that definition. Case (b), on the other hand, is a problem: if it does not list itself, in accord with its own definition, it thereby becomes eligible for inclusion in itself; but, if it does indeed list itself, it contradicts its own definition. The latter is the double paradox under discussion.

Now, my first objection would be as follows. The catalogue’s title (and even, perhaps, a brief description of its contents, an abstract) could perhaps be listed within the book itself– but such a book would not and cannot include a reproduction of the whole book inside itself (not to mention all the other books it lists or reproduces), for the simple reason that the task would be infinite (a book within a book within a book… etc., or the same in the plural).

The book is therefore not itself a member of itself; strictly speaking, only words about the book are mentionable in it. The terms inclusion or membership, as used here, then, have a very limited meaning. Thus, the plausibility of Russell’s example is very superficial, spurious; he is being fallacious, sophistical, suggesting something impossible.

Moreover, every book “includes itself” in the sense that it consists of whatever contents it has and no more. But if a book is conceived as including a number of other books, defined by some statement (e.g. all English books), the book cannot include itself in the sense that this content is only part of itself. This would not only signify infinite regression (a book with other books plus itself in it, the latter in turn with other books plus itself in it, and so forth), and infinite size, but it would constitute a contradiction within the definition. The book cannot both be all its content and only part of its content.

In this perspective, defining the book as ‘the catalogue of all books that do not include themselves’, the Russell paradox is akin to the liar paradox, since the projected book is an entity that has no finite dimension; it can never be pinned down.

A second objection would be the following. Even if we take Russell’s construct as a mere list of books, defined as ‘the catalogue of all books that do not mention themselves’, the definition is absurd, since it cannot logically be realized. We simply cannot write a book listing all books that do not mention themselves (Conrad’s Lord Jim, Hugo’s Notre Dame, etc.), in view of the stated dilemma, that whether we list or not list the book itself in it we are in a contradiction. Therefore, this concept is of necessity a null-class and meaningless.

Logic has not been stumped by the paradox, but has precisely just been taught that the proposed concept is unsound and unusable; it must therefore simply be dropped or at least changed somewhat. There is nothing dramatic in the paradox; it represents one of the functions of Logic. We might try to propose a modified concept, as follows. Perhaps we should instead refer to a library.

(a) Consider a catalogue of all books in a certain library, which is to be placed in that same library. If the book lists itself, it presents no problem. If the book does not mention itself as being in the library, it is simply incomplete and should be expanded; or its title is incorrect and should be modified (“all books but this one”); or it should be left out of the library.

(b) Now, with regard to a catalogue of all books not in our library: such a book cannot both mention itself and be put in the library. If we want to keep it in our library, we must erase its mention of itself. If we want it to mention itself, we must leave it out of the library. These are practical alternatives, which present no problem.

In this perspective, as we seek a practical expression for it, the Russell paradox becomes more akin to the Barber paradox.

 

9. On Grelling’s Paradox

Grelling[13] labels a word ‘homological’, if it has the quality it refers to (e.g. the word “short” is short, or the word “polysyllabic” is polysyllabic), or ‘heterological’, if it lacks the quality it refers to (e.g. “long” lacks length, or again “monosyllabic” is not monosyllabic). He then asks whether these two words, themselves, are to be categorized this way or that, arguing:

· If “heterological” is homological, then it is heterological (contradictory predicates).

· If “heterological” is heterological, then it is homological (contradictory predicates).

But it is a misapprehension of the meanings of these words to even try to apply them to themselves. In their case, the references are too abstract to have visible or audible concomitants. Neither term is applicable to either of them.

Note first that the apparent contradictions in predication either way apply to the word “heterological” only. For, using similar reasoning with regard to the word “homological”, although it might seem more consistent to say that “homological” is homological than to say that it is heterological, the sequence of predicates would seem consistent both ways, i.e.:

· If “homological” is homological, then it is homological (consistent predicates).

· If “homological” is heterological, then it is heterological (consistent predicates).

This could be taken to suggest that the term homological is somehow better constructed, while the term heterological has a structural fault. But this is not the real issue here.

The real issue is distinguishing between the physical words “homological” and “heterological” and their respective intended meanings, viz. homological and heterological. When we intend a word as such, we traditionally place it in inverted commas; and when we intend its assigned meaning we use it simply. In the above propositions, through which a paradox apparently arises, the subjects are words as such (in inverted commas) and the predicates are the meanings of such words.

In this perspective, there is no basis for the claim that “heterological” is heterological implies “heterological” is homological, or vice versa. The inference is very superficial, because it confuses the word as such (intended as the subject) with the meaning of the word (intended as the predicate). That is, the inverted commas in the subject are not used sincerely, but we secretly intend the underlying meaning as our subject.

How did we draw out the consequents from the antecedents? Could we see at a glance that the first thesis implies the second? Let us look at the hypothetical propositions in question more closely:

If in the antecedent we place the emphasis on the property referred to by the word “heterological”, viz. some presumed quality called heterologicality, we would formulate the paradoxes as follows:

· If the word “heterological” has the property it refers to (i.e. it is heterological), then it apparently lacks the property it refers to (i.e. is homological).

· If the word “heterological” lacks the property it refers to (i.e. it is homological), then it apparently has the property it refers to (i.e. is heterological).

If on the other hand, in the antecedent we place the emphasis on the word “heterological” having or lacking the property it refers to, we would instead formulate the paradoxes as follows:

· If the word “heterological” has the property it refers to (i.e. it is homological), then it apparently lacks the property it refers to (i.e. is heterological).

· If the word “heterological” lacks the property it refers to (i.e. it is heterological), then it apparently has the property it refers to (i.e. is homological).

In any of these cases, the consequent is constructed by comparing the subject “heterological” to the antecedent predicate heterological or homological; if they are the same word, we ‘infer’ homological as our consequent predicate, while if they verbally differ, we ‘infer’ heterological. But in truth, in making these comparisons between antecedent subject and predicate, we have not spotted any quality in the word “heterological” as such, but have tacitly referred to its underlying meaning, and faced that off against the hypothesized predicate.

In other words, the statement that “heterological” is homological (or for that matter that “homological” is heterological) is not as self-contradictory as it appears at first glance; it could conceivably be consistent. In truth, it is indeterminate and therefore meaningless.

More precisely, to resolve the paradox we have to remember how our terms were induced in the first place. We can tell that “short” is short merely by seeing or hearing the word “short” (supposing that any one syllable, however written or pronounced, counts as short). But in the case of a term like heterological, you cannot tell whether the word has or lacks the property it refers to, because that property is not a concrete (visible or audible) quality of the word, but something abstract that we apply to visible or audible components of words. If the quality sought is not visible or audible, it is unknowable and there is no way for us to tell which predicate applies.

That is, our initial definitions of those terms, which mention “a word having/lacking a certain quality it refers to”, are not clear and precise, because they do not specify as they should that the qualities intended are phenomenal, i.e. perceptible aspects of the word. If the word labels something not included in its physical aspects, the terms homological and heterological simply do not apply. To apply them is to play verbal tricks. Thus, neither of these predicates is applicable to either of these words as such.

It might be objected that words do have non-phenomenal attributes. For example, we often consider a word useful or useless. In such case, we might ask: is the word “useful” useful or not? Yes, I’d reply to that. Therefore, “useful” is homological. Likewise, “useless” is useful, therefore “useless” is heterological. In this perspective, one may doubt the exactitude of what we have just proposed, that homological and heterological are terms that presuppose concrete (rather than abstract) predicates.

But to this objection, one could counter that the utility of a word is ultimately something concrete: a word is useful if it makes a perceptible practical difference in the development of knowledge. In that case, our definition could be modified slightly, specifying that the terms homological or heterological are only applicable when we can first directly or indirectly anchor them to some concrete property.

In sum, these terms must refer to something other than themselves before they can at all be used. The fallacy involved is similar to that in the liar paradox, where the term “this” is used with reference to itself, whereas it only acquires meaning when it has something else to refer to. Such terms are relational, and so cannot refer to other relations in a circular manner or ad infinitum: they need to eventually be anchored to some non-relational term.

Notice, by the way, that if we changed the word “short” to say “shortissimo”, with reference to the same meaning, the word would change status and become heterological, since “shortissimo” is not shortissimo. On the other hand, whatever other word we substitute for the word “heterological”, Grelling’s paradox in relation to it remains apparent. This test shows that in the latter case it is not purely the word that we are thinking of, but rather its underlying meaning. With regard to the word “useful”, we could also say that it is useful by virtue of its content, or at most by virtue of its being a word (a unit of language), and not because of its specific shape or sound.



[1] See Future Logic, chapter 32.2.

[2] See Future Logic, chapter 63, sections 3 and 6.

[3] Needless to say the following comments are not an attack on Buddhism, but on the rhetoric of Nagarjuna. Buddhism is not well served by such games. I think of Nagarjuna whenever I read v. 306 of the Dhammapada: “He who says what is not… and he who says he has not done what he knows well he has done… sinned against truth”. For me, he is just a philosopher like any other; his interest in Buddhism is incidental (as is his saintly status in the eyes of many).

[4] Nagarjuna in Vigraha Vyavartani (Averting the Arguments), verse 29. The translation used here is given by ‘Namdrol’ in the E-Sangha Buddhism Forum (http://www.lioncity.net/buddhism/

index.php?s=d8946a5bcb1f56f3e9e21a108125823f&showtopic=5604&st=100&#entry82577). Note however that the word “alone” in this translation may not be in the original, judging by other translations I have seen, even though it does seem to be Nagarjuna’s intent.

[5] See Future Logic, chapter 29.3.

[6] By Frederick J. Streng. The full text of his translation seems to be that posted in the Internet at: http://www.orientalia.org/article491.html. Note that the phrase “non-apprehension of non-things” is considered an incorrect translation by Plamen Gradinarov. However, while willing to admit the latter’s objection, I do not agree that Streng’s freer translation is entirely inadmissible. In my view, it may not be literally precise, but it captures Nagarjuna’s paradoxical spirit and intent. See our discussion of this issue at http://nyaya.darsana.org/topic3.html. In any case, even if the phrase “non-apprehension of non-things” is best not relied on, the criticisms of Nagarjuna in the present section can still be proposed on other grounds.

[7] Two other translations of this sentence confirm and amplify this reading. “If I would make any proposition whatever, then by that I would have a logical error” (Streng). “Should I have put forward any thesis, then the logical defect would have been mine” (Gradinarov).

[8] As I believe I convincingly demonstrated in Future Logic.

[9] I took this term (in 2003) from an essay called Revisionary Immunity, by a Dr. Greg Bahnsen (d. 1995), posted on the Internet at www.cmfnow.com/articles/pa018.htm, in the website of the (Christian) Covenant Media Foundation. This essay is on the whole a brilliant and important piece of work, an excellent example of logical criticism of confused notions – although the author, motivated by an agenda of religious apologetics (Christian), seems ultimately to advocate a rejection (or rather, an excessive relativism) of empiricism and logic.

[10] See Future Logic, chapters 43-45, on class logic.

[11] Though of course, this distinction may be paradoxical, since the word ‘word’ refers to words.

[12] The word ‘entity’, of course, is sometimes meant more generally, with reference to any existent.

[13] See Dict. of Philo. p. 135.

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