Paradoxes and Their Resolutions

A Thematic Compilation by Avi Sion

1. The vanity of the tetralemma


The most radical assault on reason consists in trying to put in doubt the laws of thought, for these are indeed the foundations of all rational discourse. First, the law of identity is denied by saying that things are never quite what they seem to be, or that what they are is closer to grey than black and white. This is, of course, an absurd remark, in that for itself it lays claim to utter certainty and clarity. Then, the laws of non-contradiction and of the excluded middle[1] are denied by saying that things may both be and not-be, or neither be nor not-be. This is the ‘tetralemma’, the fourfold logic which is favored in Indian and Chinese philosophies, in religious mysticism, and which is increasingly referred to among some ‘scientists’. To grasp the vanity of the tetralemma, it is necessary to understand the nature of negation and the role of negation as one of the foundations of human logic.

1.    Phenomena are positive

The first thing to understand is that everything we experience is positive phenomenon. Everything we perceive through our senses, or remember or imagine in our minds, or even cognize through ‘intuition’ – all that has to have some sort of content to be at all perceived. Each sense organ is a window to a distinct type of positive phenomenon. We see the blue sky above, we hear birds sing, we smell the fresh air, we taste a fruit, we feel the earth’s texture and warmth, etc. Similarly, the images and sounds in our heads, whether they come from memory or are produced by imagination, are positive phenomena; and even the objects of intuition must have some content that we can cognize. Secondly, we must realize that many positive phenomena may appear together in space at a given moment. This is true for each phenomenal type. Thus, the blue sky may fill only part of our field of vision, being bounded by green trees and grey buildings; we may at once hear the sounds of birds and cars; and so on. Thirdly, many positive phenomena may at any given time share the space perceived by us. Thus, superimposed on visual phenomena like the sky may be other types of phenomena: the sound of birds in the trees, the smell of traffic in the streets, the feelings in our own body, and so on. We may even hallucinate, seeming to project objects of mental perception onto physical space. For example, the image of one’s eyeglasses may persist for a while after their removal. Fourthly, each positive phenomenon, whatever its type, varies in time, more or less quickly. Thus, the blue sky may turn red or dark, the sounds of birds or traffic may increase or decrease or even stop for a while, and so forth.

2.    There are no negative phenomena

In order to express all these perceptual possibilities – differences in space and in time and in other respects, we need a concept of negation, or more precisely an act of negating. Without ‘negation’, we cannot make sense of the world in a rational manner – it is the very beginning of logical ordering of our experience. Thus, in a given visual field, where (say) blue sky and trees appear, to be able to say ‘the sky ends here, where the trees begin’ we need the idea of ‘negation’ – i.e. that on one side of some boundary sky is apparent and on the other side it is not, whereas on the first side of it trees are not apparent and on the other they are. Likewise, with regard to time, to be able to describe change, e.g. from blue sky to pink sky, we need the idea of ‘negation’ – i.e. that earlier on this part of the sky was blue and not pink, and later on it was pink and not blue. Again, we need the idea of ‘negation’ to express differences in other respects – e.g. to say that ‘the sounds of birds singing seem to emanate from the trees, rather than from buildings’. Thus, negation is one of the very first tools of logic, coming into play already at the level of sorting of experiences.

Moreover, negation continues to have a central role when we begin to deal with abstractions. Conceptual knowledge, which consists of terms and propositions based directly or indirectly on perceptual phenomena, relies for a start on our ability to cognize similarities between objects of perception: ‘this seems to resemble that somewhat’ – so we mentally project the idea of this and that ‘having something in common’, an abstract (i.e. non-phenomenal, not perceived by any means) common property, which we might choose to assign a name to. However, to take this conceptual process further, we must be able to negate – i.e. to say that ‘certain things other than this and that do not have the abstract common property which this and that seem to have’, or to say that ‘this and that do not have everything in common’. That is, we must be able to say not only that one thing resembles another in some way, but also that these or other things do not resemble each other in that way or in another way. Thus, negation is essential for making sense of information also at the conceptual level of consciousness.

Now, what is negation? To answer this question we first need to realize that there are no negative phenomena in the realm of experience. Everything we perceive is positive phenomenon – because if it was not we obviously would have nothing to perceive. We can only ‘perceive’ a negative state of affairs by first mentally defining some positive state of affairs that we should look for, and then look for it; if having looked for it assiduously we fail to find it, we then conclude inductively that it is ‘absent’, i.e. ‘not present’. Thus, positive phenomena come before negative ones, and not after. Existence logically precedes non-existence. Negative phenomena are ‘phenomena’ only metaphorically, by analogy to positive phenomena – in truth, negative phenomena are not: they do not exist. ‘Negation’ is not a concept in the sense of an abstraction from many particular experiences having a certain property in common. Negation is a tool of the thinking observer, as above described. It is an act, an intention of his.

3.    A misinterpreted experiment

To illustrate how confused some people – even some scientists – are with regard to negation, I offer you the following example drawn from Richard Dawkins’ The Greatest Show on Earth: The Evidence for Evolution[2]. He describes an experiment by Daniel J. Simons, in which some people are asked to watch a brief video and observe how many times a certain event takes place in it; but at the end they are asked another question entirely, viz. whether they noticed the presence of a man dressed up as a gorilla in the course of the movie, and most of them admit they did not[3]. According to Dawkins, we may infer from this experiment how “eye witness testimony, ‘actual observation’, ‘a datum of experience’ – all are, or at least can be, hopelessly unreliable.”

But this is a wrong inference from the data at hand, because he confuses positive and negative experience. The people who watched the video were too busy looking for what they had been asked to observe to notice the gorilla. Later, when the video was shown them a second time, they did indeed spot the gorilla. There is no reason to expect us to actually experience everything which is presented to our senses. Our sensory experiences are always, necessarily, selective. The validity of sense-perception as such is not put in doubt by the limited scope of particular sense-perceptions. The proof is that it is through further sense-perception that we discover what we missed before. Non-perception of something does not constitute misperception, but merely incomplete perception. ‘I did not see X’ does not deductively imply ‘I saw the absence of X’, even though repetition of the former tends to inductively imply the latter.

A negative ‘phenomenon’ is not like a positive phenomenon, something that can directly be perceived or intuited. A negation is of necessity the product of indirect cognition, i.e. of an inductive (specifically, adductive) process. We mentally hypothesize that such and such a positive phenomenon is absent, and then test and confirm this hypothesis by repeatedly searching-for and not-finding the positive phenomenon[4]. If we were to at any time indeed find the positive phenomenon, the hypothesis of negation would immediately be rejected; for the reliability of a negation is far below that of a positive experience. We would not even formulate the negation, if we already had in the past or present perceived the positive phenomenon. And if we did formulate the negation, we would naturally retract our claim if we later came across the positive phenomenon. Therefore, the content of negative phenomena is necessarily always hypothetical, i.e. tentative to some degree; it is never firm and sure as with (experienced) positive phenomena.

Negative assertions, like positive assertions, can be right or wrong. If one looked diligently for a positive phenomenon and did not find it, then one can logically claim its negation. Such claim is necessarily inductive – it is valid only so long as the positive phenomenon is actively sought and not found. The moment the positive phenomenon is observed, the negation ceases to be justified. If one did not look for the positive phenomenon, or did not look with all due diligence, perhaps because of some distraction (as in the example cited above), then of course the claim of negation is open to doubt; certainly, it is inductively weak, and one is very likely to be proved wrong through some later observation.

4.    Defining negation

How, then, is negation to be defined? We could well say that negation is defined by the laws of non-contradiction and of the excluded middle. That is, with regard to any term ‘X’ and its negation ‘not-X’, the relation between them is by definition the disjunction “Either X or not-X” – which is here taken to mean that these terms (X and not-X) cannot be both true and cannot be both false, i.e. they are exclusive and exhaustive. What do I mean here by ‘definition’? – is that an arbitrary act? No – it is ‘pointing to’ something evident; it is ‘intentional’. Here, it points to the instrument of rational discourse which we need, so as to order experience and produce consistent conceptual derivatives from it. The needed instrument has to be thus and thus constructed; another construct than this one would not do the job we need it to do for us. That is, the only conceivable way for us to logically order our knowledge is by means of negation defined by means of the laws of non-contradiction and of the excluded middle. Without this tool, analysis of experience is impossible.

Suppose now that someone comes along and nevertheless objects to the preceding assertion. Well, he says, how do you know that the dilemma “either X or not-X” is true? You just arbitrarily defined things that way, but it does not mean it is a fact! Could we not equally well claim the tetralemma “Either X or not-X or both or neither” to be true? The reply to that objection is very simple. Suppose I accept this criticism and agree to the tetralemma. Now, let me divide this fourfold disjunction, putting on the one side the single alternative ‘X’ and on the other side the triple alternative ‘not-X or both or neither’. I now again have a dilemma, viz. “either ‘X’ or ‘not-X or both or neither’.” Let me next define a new concept of negation on this basis, such that we get a disjunction of two alternatives instead of four. Let us call the complex second alternative ‘not-X or both or neither’ of this disjunction ‘NOT-X’ and call it ‘the super-negation of X’.

Thus, now, the objector and I agree that the disjunction “either X or NOT-X” is exclusive and exhaustive. We agree, presumably, that this new dilemma cannot in turn be opposed by a tetralemma of the form “Either X or NOT-X or both or neither” – for if such opposition was tried again it could surely be countered by another division and redefinition. We cannot reasonably repeat that process ad infinitum; to do so would be tantamount to blocking all rational thought forever. Having thus blocked all avenues to thought, the objector could not claim to have a better thought, or any thought at all. There is thus no profit in further objection. Thus, the tetralemma is merely a tease, for we were quite able to parry the blow. Having come to an agreement that the new disjunction “Either X or NOT-X” is logically unassailable, we must admit that the original disjunction “Either X or not-X” was logically sound from the first. For I can tell you that what I meant by not-X, or the ‘negation of X’, was from the beginning what is now intended by NOT-X, or the ‘super-negation of X’!

I was never interested in a relative, weak negation, but from the start sought an absolute, strong negation. For such utter negation, and nothing less radical, is the tool we all need to order experience and develop conceptual knowledge in a consistent and effective manner. In other words, whatever weaker version of negation someone tries to invent[5], we can still propose a strong version such that both the laws of non-contradiction and of the excluded middle are applicable without doubt to it. If such negation did not exist, it would have to be invented. No one can destroy it by denying it or diluting it. Those who try to are merely sophists who do not understand the source, nature and function of negation in human discourse. They think it is a matter of symbolic manipulation, and fail to realize that its role in human discourse is far more fundamental and complex than that. Negation is the indispensable instrument for any attempt at knowledge beyond pure perception.


From A Fortiori Logic, Appendix 7.3.


[1]              I inadvertently left out the words “and of the excluded middle” in the original editions of A Fortiori Logic, although it is clear from the rest of the sentence that I intended them.

[2]              New York: Free Press, 2009. Pp. 13-14.

[3]              The video can be seen at:

[4]              Not-finding is the non-occurrence of the positive act of finding. Objectively, note well, not-finding is itself a negative phenomenon, and not a positive one. But subjectively, something positive may occur within us – perhaps a sense of disappointment or continued relief. See more on this topic in my Ruminations, chapter 9.

[5]              There are people who say that the law of non-contradiction is logically necessary, but the law of the excluded middle is not. Clearly, this claim can be refuted in the same way. If they claim the three alternatives “Either X or not-X or ‘neither X nor not-X’” – we can again split the disjunction into two, with on one side “X” and on the other side “not-X or ‘neither X nor not-X’” – and then proceed as we did for the tetralemma. The same can be done if anyone accepts the law of the excluded middle but rejects the law of non-contradiction. All such attempts are fallacious nonsense.


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