Harold Walsby: Aristotle’s Principle

Art does not exactly imitate that which can be seen by the eyes, but goes back to that element of reason of which Nature consists and according to which Nature acts. (Goethe)

For generations now many artists, poets and mystics have expressed their deep conviction that human reason cannot be bounded by what has often been called “the narrow logic of the Schools.” Not being logicians, this is about as far as they could go in the matter. They meant, of course, something rather more radical than what the past reformers of logic would mean by that phrase. However, expressing deep conviction, and doing something about it, are two very different things!

It is, indeed, true that of all the reforms which have taken place in logic, from the earliest times up to the present, none of them has amounted to what could be properly regarded as the beginning of a radical revolution within the subject itself. This includes the liberation of logic from the confines of verbalism (with all that entails) when it acquired mathematical form – and a new image – during the latter half of the 19th Century.

On the other hand, the vast revolution which has taken place within mathematics – beginning with geometry and algebra in the first half of the same century – inevitably caught up with the somewhat slow mathematical trend in logic and swept over it like a torrent Writers on these subjects therefore tend to speak only of “the reform in logic” whereas “the revolution in mathematics” is the familiar standard phrase. From time to time there have been high hopes for a real revolution within logic comparable with that which began in 1825, when Lobachewsky dropped his bombshell by challenging the “universal Truth” of Euclid’s famous parallel postulate. (Logic was then still floundering in the prison of syllogistic rules despite a number of abortive attempts at mathematical rescue). Even as recently as 1959, in his introduction to the Dover edition of Lewis and Langford’s classic “Symbolic Logic,” S.P. Lamprecht voiced these revolutionary hopes: “The matrix ‘method’ … is employed to defend the idea of plurality, of logical truth … and thus supports the supposition of a variety of non-Aristotelian logics akin to the variety of non-euclidean geometries.’ (My emphasis – HW)

Whether or not such a variety of non-Aristotelian logics has already begun to appear depends, of course, on what one is prepared to accept as “non-Aristotelian.” It is not uncommon, for example, to confine the term “Aristotelian” to the traditional, verbal or syllogistic logic, and therefore to regard all logic in mathematical dress as non Aristotelian. This, however, is by no means generally accepted. Some writers confine the term “Aristotelian” to the syllogistic logic, which was actually formulated by Aristotle and which – almost unaltered – held undisputed sway for nearly 2,000 years until the time of Francis Bacon. Again, the fundamental law of the first workable mathematical logic to be published (that of George Boole, in 1854) was actually claimed by its author to be thoroughly Aristotelian! Obviously in the sphere of logic, there is no important and clear-cut criterion, as exists for geometry, to mark off the classical system from new revolutionary ones. As yet there is no historical equivalent of Euclid’s parallel postulate.

This is not for the want of a basic postulate which could be challenged. It is due simply to the fact that no successful system of logic has yet been produced which directly and overtly challenges such a postulate.

If we now approach the traditional verbal, logic with this viewpoint, we note at once that the famous “laws of thought” have, at various times, come under strong suspicion – and even more than suspicion as possible assumptions to be challenged. These so-called primary laws of thought are three: the Law of Identity (X = X, or everything is identical with itself); the Law of Contradiction (nothing can both be X and non-X); and the Law of Excluded Middle (everything is either X or non-X).

It is not necessary to discuss these three candidates for challengeable assumption in any detail at this stage. Theoretically, they are, of course, no longer generally regarded as immutable laws of thought. This is because of the growth of “the postulational method,” now pervading mathematics and stemming directly from Lobachewsky’s revolution.

Proposals were made in 1933 by Alfred Korzybski for the development of a “non-Aristotelian” system based on challenging the universal truth of the Law of Identity. These were not really carried through, however. Certain developments have taken place in mathematics (initiated by L.E.J. Brouwer in 1907, followed by K. Goedel in 1931) that challenge the universal validity of the Law of Excluded Middle upon which much mathematical proof still depends. The mis-named “Law of Contradiction” has been obliquely or indirectly challenged many times, but most notably by German philosophers, Fichte, Schelling and Hegel, around the turn of the 18th to the 19th centuries. The main point to be made here concerning these “primary laws of thought” is that the huge structures of mathematics and science have, in the main, been built – and moreover, are still being built – with their aid.

We are thus brought to consider a clear-cut criterion of what, in the field of logic, we are to regard as “non-Aristotelian.” Aristotle himself regarded the Law of Contradiction as the most fundamental of all axioms, from which all others are derived. Moreover, it is precisely this Law which, through the ages, has been contrasted by the philosophers with the metaphysical law of dialectic (the unity or coincidence of opposites) – the latter traditionally associated with the Pythagoreans, Megarians, Platonists and Stoics, the former with Aristotle, the Peripatetics and Scholastics. In addition, it is this particular Law which is especially identified with “the narrow logic of the Schools” by the aforementioned poets, mystics, etc. Further, and most important of all, though it is now treated theoretically as being an arbitrary postulate or axiom (in the modern sense) this Law is in practice still treated as a universal or immutable law of thought throughout all existing systems of logic and mathematics!

Let us spell it out in the words of E.T. Bell, on postulate systems generally. The Laws of Contradiction and Excluded Middle, he writes, “were (almost universally) accepted until 1907-1912 in all sane reasoning, but both, be it observed, are postulates … In practically all mathematics since 1912, however, the whole machinery of common, classical logic has been included in the postulates of all mathematical systems … Are the postulates then completely arbitrary? They are not, and the one stringent condition they must meet has wrecked more than one promising set and the whole edifice reared upon it. The postulates must never lead to an inconsistency. Otherwise they are worthless. If by a rigid application of the laws of logic a set of postulates leads to a contradiction … the set must either be amended … or it must be thrown away … and we must start all over again.” (Mathematics: Queen and Servant of Science, 1958). Our comment must be: When is an arbitrary postulate not an arbitrary postulate, but a universally valid law? Answer – when it is the Law of Contradiction! (Indeed, we have here the Achilles’ heel of the postulational method, which we shall pursue elsewhere).

Clearly, despite all the reams which have been written, and pompous words uttered, on the non-universality of postulates and the treatment of logic as a postulate system, the bare fact remains, as Professor Bell implies, that the “postulate of contradiction” is not treated as other than a completely universal law throughout all non-trivial and viable mathematics.

We have here then, prima facie, our most worthy candidate, not only for a criterion to mark the great divide between what we can properly regard as “Aristotelian'” and what “non-Aristotelian,” but also as a challengeable basic postulate of logic comparable with Euclid’s parallel postulate. If the challenging of this Law as completely universal should lead to a new non-Aristotelian system capable of mathematical expression and having widespread application to the real world, then it would be the first of its kind.

That this particular Law (which divides every field of thought into X and non-X) should prove the most suitable touchstone of the division between Aristotelian and non-Aristotelian, surely implies some thing deeper than mere poetic justice? With no more ado I shall adopt this Principle as that criterion, and shall henceforth apply the term “Aristotelian” to all systems, whether mathematical or verbal, which include the Principle of Contradiction without also explicitly including a more general principle of which it is a special case.

Henceforth, also, I shall not call the “Law of Contradiction” by this misnomer but by its less-frequently-used, yet more proper title: “the Law of Non-contradiction” (or just “Aristotle’s Principle”)

George Walford International Essay Prize

Harold Walsby: Aristotle’s Principle

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