The principle of included middle

Principle of included middle

(Methodos III)

Marc Jacquinet

 

La  difficulté pour la pensée à s’ouvrir à une architecture plus fluente et morcelée de ses représentations ne vient pas seulement de ses habitudes ritualisées ou de sa méconnaissance, renforcée de refoulements, de la polysémie du monde.  La simplification du problème de la différence résulte souvent d’un double conditionnement intellectuel et même culturel : celui d’une part de l’orthodoxie logique de l’identité, de la non-contradiction et du tiers exclu, c’est-à-dire de l’appareil formel qui commande de manière latente toutes les activités du savoir []

Jean-Jacques Wunenburger, op. cit. p 137

 

This principle is of paramount importance for the whole edifice I construct.  At one general level, the principle of included middle is integrated in the general process of thought.  This is what Jean-Jacques Wunenburger (Wunenburger 1990) defends when he talks of going beyond the classical mode of thinking, what he calls the “Aristotelian and Cartesian orthodoxy” (p. 10), and recognizes that the two great types of knowledge configuration, the analytical and the dichotomical methods of abstract thinking, are worth enquiring (p. 23).   The dichotomical mode accentuates the differences or the similarities so that the diverse elements tend to fall into the desired categories.  Aristotle did investigate the question of the categories and the logic of reasoning.  The first book of his treatise on logic and reasoning, the Organon (Organon), is called the Categories (Kateoriai) and constitutes the first step of the Aristotelian system of logic: well defined categories; its other elements are: the three principles of identity, non-contradiction and excluded middle; and the syllogism, all treated in the whole aforementioned opus on reasoning and also in other texts, namely the Rhetoric and Metaphysics

What is of interest to us is the Aristotelian legacy in formal logic, mathematics and science which rests on the principle of identity (A = A), the principle of non-contradiction (A = non-A, both A and non-A cannot be true at the same time), and the principle of excluded middle, let A and C be two different statements so that C cannot satisfy the two following propositions simultaneously:  A=C and non-A=C. 

These principles appear to be simple.  But they are not, if they appear quite clear in the abstract, their application in science, logic and mathematics has proven to be highly problematic: either they are often ignored, so the scientific reasoning is not always that rigorous; or they are challenged from different perspectives by mathematicians, logicians and philosophers like Brouwer, Weyl, Lupasco, Heisenberg, Zadeh, etc.[1]

To admit the included middle in traditional formal logic, this is tantamount to admit a principle of contradiction.  But this is not necessarily the case in some non-standard logic, such as the one developed by Stephane Lupasco and further refined by Basarab Nicolescu or J.-J. Wunenburger.[2]  We can “integrate without annihilate the classical principle of logic” (Wunenburger, 1990: 154).  The idea of identity, determining the quality of the statements reveals a drive to fix an “indicator of certainty (ibid. p. 154), “espousing the complex structure of the becoming and the multiple” (ibid. p. 154). 

Aristotle states the principle of non-contradiction in the Book IV, section 3, of his Metaphysics, as:

[…] the most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known (for all men may be mistaken about things which they do not know), and non-hypothetical. For a principle which every one must have who understands anything that is, is not a hypothesis; and that which every one must know who knows anything, he must already have when he comes to a special study. Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect; we must presuppose, to guard against dialectical objections, any further qualifications which might be added. This, then, is the most certain of all principles, since it answers to the definition given above. For it is impossible for any one to believe the same thing to be and not to be, as some think Heraclitus says. For what a man says, he does not necessarily believe; and if it is impossible that contrary attributes should belong at the same time to the same subject […]

 

He further insists, in the fifth section of the same book, on the principle of non-contradiction:

For the same thing can be potentially at the same time two contraries, but it cannot actually. And again we shall ask them to believe that among existing things there is also another kind of substance to which neither movement nor destruction nor generation at all belongs.

 

and in the book IV, section 7:

 

But on the other hand there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate. This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false; but neither what is nor what is not is said to be or not to be.-Again, the intermediate between the contradictories will be so either in the way in which grey is between black and white, or as that which is neither man nor horse is between man and horse. (a) If it were of the latter kind, it could not change into the extremes (for change is from not-good to good, or from good to not-good), but as a matter of fact when there is an intermediate it is always observed to change into the extremes. For there is no change except to opposites and to their intermediates. (b) But if it is really intermediate, in this way too there would have to be a change to white, which was not from not-white; but as it is, this is never seen.

 

 

The leading astray from the principle of non-contradiction has serious implications.  In one sense, there cannot be contradictions in science.  In the Cartesian ideal, this is formally rejected.  In other paradigms, like here, the included middle is accepted and the non-contradiction principle is respected.

 

Second, in principle, to admit the included middle in traditional formal logic, this is tantamount to admit a principle of contradiction.  But this is not necessarily the case in some non-standard logic, such as the one developed by Stephane Lupasco and further refined by Basarab Nicolescu or J.-J. Wunenburger.[3]  We can “integrate without annihilate the classical principle of logic” (Wunenburger, 1990: 154).  The idea of identity, determining the quality of the statements reveals a drive to fix an “indicator of certainty (ibid. p. 154), “espousing the complex structure of the becoming and the multiple” (ibid. p. 154).

The included middle can respect the principle of non-contradiction if we distinguish different levels of reality, so that the contradiction at the same level is not present when we consider the elements at two distinct levels. 

 

Anyway, the included middle is no panacea and leads to great modesty to the results obtain and to prudence in the research design.  Indeed, a non-rigorous use of the principle of the included middle and the one of contradiction can lead to non sequitur without realizing it.  Moreover, the abuse of if may lead to a kind of rhetoric where everything is possible, then nothing is explainable and no causal distinctions are possible.

The included-middle anyway is worth the try with caution and rigor.  The reality often appears as more complex than the dualities and dichotomies.  Human phenomena and the very social sciences are imprecise.  This justifies the inclusion of fuzzy logic or imprecise logic into our toolbox.  Furthermore, the notion of imprecise probabilities and the possibility theory are two aspects of the same problem: an imprecise reality embedded in radical uncertainty and partial ignorance of our knowledge constructions, even the scientific ones.

I will base my approach to the tertium datur on the works of Lupasco, Nicolescu and Wunenburger. With regard to Lupasco, I think that there is more than a logic of contradiction.  The relations can be of contradiction, competition and complementarity.  Furthermore, the higher caution we can have with definitions, the lower the necessity to recourse to the logic of contradiction (Gonseth 1924/1974b, Gonseth 1998).  On the other and, the possibility of diverse levels of reality permits other potentialities for the formal logic that are further away from the reductive and mutilating Aristotelian logic of the complex phenomena, and in this vain, the levels of reality is promising.

[1] The point of view defended by Brouwer or Weyl is not supported by other mathematicians like Hilbert or Hadamard and Gonseth.  See the book by Gonseth, Fernand (1924/1974a): Les fondements des mathématiques. De la géométrie d’Euclide à la relativité générale et à l’intuitionisme. Paris: Albert Blanchard (Librairie scientifique et technique)

[2] In the next section, the concept of levels of reality has been advanced by Nicolas Nicolescu to go beyond the logic of contradiction of Lupasco.

[3] In the next section, the concept of levels of reality has been advanced by Nicolas Nicolescu to go beyond the logic of contradiction of Lupasco.


Cite this article as: {Jacquinet, Marc}, The principle of included middle,"{Management Critique}," in {https://critiquema.hypotheses.org}, 16/12/2019, https://critiquema.hypotheses.org/266.

Marc Jacquinet

https://orcid.org/0000-0003-1157-060X

Vous aimerez aussi...

Laisser un commentaire

Votre adresse de messagerie ne sera pas publiée. Les champs obligatoires sont indiqués avec *

Ce site utilise Akismet pour réduire les indésirables. En savoir plus sur comment les données de vos commentaires sont utilisées.

Rechercher dans OpenEdition Search

Vous allez être redirigé vers OpenEdition Search