Logical and Spiritual REFLECTIONS
Book 3. In Defense of Aristotle’s Laws of Thought
Chapter 8. The game of one-upmanship
People who think the law of non-contradiction and/or the law of the excluded middle is/are expendable have simply not sufficiently observed and analyzed the formation of knowledge within themselves. They think it is just a matter of playing with words, and they are free to assert that some things might be “both A and not A” and/or “neither A nor not A”. But they do not pay attention to how that judgment arises and is itself judged.
They view “A is A”, etc., as verbal statements like any other, and think they can negate such statements like all others, saying “A is not A”, etc. But in fact, negation is not possible as a rational act without acceptance of the significance of negation inherent in the second and third laws of thought, in comparison to the first law of thought. To say “not” at all meaningfully, I must first accept that “A cannot be not A” and that “there’s no third alternative to A and not A”.
To try to introduce some other (less demanding) definition of negation is impossible, for true negation would still have to be thought of (in a hidden manner or using other words). Inventing a “many-valued logic” or a “fuzzy logic” cannot to do away with standard two-valued logic – the latter still remains operative, even if without words, on a subconscious level. We have no way to think conceptually without affirmation and denial; we can only pretend to do so.
Many “modern” logicians are so imprisoned by symbolic logic that they have lost contact with the intended meanings of their symbols. For this reason, the symbols ‘X’ and ‘not X’ seem equivalent to them, like ‘X’ and ‘Y’. But for classical logicians, a term and its negation have a special relationship. The negation of X refers to all but X, i.e. everything that is or might be in the whole universe other than X.
Figure 3 Visualizations of negation.
The diagram above illustrates how differently these people effectively visualize negation:
Obviously, if a person mentally regards ‘X’ and ‘not X’ as commensurate, he will not understand why they cannot both be affirmed or both be denied at once; the second and third laws of thought will seem to him prejudicial and conventional. To return to a rational viewpoint, that person has to become conscious of the radical intent of the act of negation; it leaves no space for mixtures or for additional concoctions.
Bipolar logic is not a mere “convention”, for the simple reason that making a convention presupposes we have a choice of two or more alternatives, whereas bipolarity is the only way rational thought can at all proceed. We do not arbitrarily agree bipolarity, because it is inherent in the very asking of the question. To claim something to be conventional is already to acknowledge the conflict between it and the negation of it, and the lack of anything intelligible in between the two.
The motive behind the attempts of some thinkers to deny the laws of thought (i.e. the laws of proper affirmation and denial) is simply an ego ambition to “beat the system”, or more specifically (in the case of Western philosophers) to surpass Aristotle (the one who first made these laws explicit objects of study). “You say X? I will ‘up the ante’ and say Not X (etc.) – and thus show I am the greatest!”
This is not mere perversity – but a sort of natural denial instinct gone mad. For, funnily enough, to deny some suggestion (including the suggestion there are three laws of thought) is in the very nature of conceptual knowing, a protective mechanism to make sure all alternative interpretations of fact are taken into consideration. This is precisely the faculty of negation – the very one which gives rise to the need for the laws of thought! The problem here is that it is being turned on itself – it is being over-applied, applied in an absurd way.
This can go on and on ad infinitum. Suppose I say “A” (meaning “A but not notA”), you answer “not A” (meaning “notA but not A”); I reply “both A and notA”, you oppose “neither A nor notA”; what have we said or achieved? Perhaps I will now say: “all of these four alternatives”; and you will reply: “none of these four alternatives”. Then I trump you, asserting: “both these last two alternatives” and you answer: “neither of them”. And so forth. Whither and what for?
A more complex version of the same game of one-upmanship can be played with reference to the laws of thought:
- A is A (affirming the law of identity).
- A is not A (denying the law of identity).
- Both (1) and (2). A is A, and A is not A. (disregarding the law of non-contradiction).
- Neither (1) nor (2). A is not A, and A is not not A (disregarding the law of the excluded middle).
- Both (3) and (4).
- Neither (3) nor (4).
- Both (5) and (6).
- Neither (5) nor (6).
- And so on and so forth.
Thus for the first law of thought; and similarly for the other two. We do not merely have a choice of four alternatives (the first four in the above list), a so-called ‘tetralemma’, but an infinite choice of denials of denials of denials… How would we even evaluate the meaning of all these alternatives without using the laws of thought? They would all be meaningless, because every proposed interpretation would be in turn deniable.
Thus, the attempt to propose a radically “alternative logic”, instead of the standard (Aristotelian) logic, is really the end of all intelligible logic, the dissolution of all rationality. It is not a meaningful option but a useless manipulation of meaningless symbols. None of it makes any sense; it is just piling up words to give an optical illusion of depth. People who engage in such moronic games should clearly not be granted the status of “logicians”.
 Incidentally, I notice people on the Internet nowadays labeling the three laws of thought (LOT): LOI, LNC and LEM, for brevity’s sake. Sure, why not?
 Some logicians accept the law of non-contradiction as unavoidable, but consider the law of the excluded middle as expendable: this modern notion is quite foolish. Both laws are needed and appealed to in both deductive logic and in inductive logic. They do not only serve for validation (e.g. of syllogisms or of factorial inductions), but they generate questions and research (e.g. what does this imply? or what causative relation can be induced from that?). Moreover, they are mirror images of each other, meant to complement each other so as to exhaust all possibilities, and they ultimately imply each other, and both imply and are implied by the law of identity.
 Note that difference does not imply incompatibility. Two things, say X and Y, may be different, yet compatible – or even imply each other. We are well able to distinguish two things (or characteristics of some thing(s)), even if they always occur in tandem and are never found elsewhere. Their invariable co-incidence does not prevent their having some empirical or intellectual difference that allows and incites us to name them differently, and say that X is not the same thing as Y. In such case, X as such will exclude Y, and not X as such will include Y, even though we can say that X implies Y, and not X implies not Y.
 Note that if we start admitting the logical possibility of “A and notA” (or of “not A and not notA”), then we can no longer mention “A” (or “notA”) alone, for then it is not clear whether we mean “A with notA” or “A without notA” (etc.). This just goes to show that normally, when we think “A” we mean “as against notA” – we do not consider contradictory terms as compatible.