Tetravalent logic (Part 1)

Aristotle said: "Being is everything he knows", so that where there is real knowledge - not its appearance or shadow - knowledge and being are one and the same thing.

(R. Guénon, Mélanges, Chap. VI « Connais-toi toi même », 1976, Pp. 56)

All modern knowledge is based on logic. With a few rare exceptions - technical apparatus using fuzzy logic or states of quantum physics - all natural and human sciences (including Western philosophy since at least Plato) are based on classical logic, establishing the veracity or falsity of a proposal.

However, classical logic has limitations that have been known since its inception. These situations, which can be expressed in a few simple sentences, are negligently classified among the paradoxes.

The consequences of refusing to deal with these paradoxical situations are most critical, since this gap forces us to blindly focus on our understanding of the world and ourselves.

Since no one can ignore the fact that the modern world is going through an ultimate crisis, any contribution to broadening the scope of possibilities is welcome, and indeed, we might even say, necessary. This is the purpose of this article.

We will begin with a brief history of logic, going back before the appearance of classical logic, where we will see in passing that logic is born of metaphysics. This will lead us to an interposed dialogue between Heraclitus, Aristotle, Granger and Guénon.

We will then propose a formalization of a tetravalent logic with its truth tables, after having explained its necessity (part 2). We will end by applying the framework of this logic to the semantics of some attributes of the manifested and the unmanifested, where we will find each time the conceptions of Tradition (part 3).

It all starts with a negation

If we take a proposal P and its logical negation not(P) we can place ourselves in one of the following three cases:

•     case 1: P is true or not(P) is true, exclusively;
•     case 2: P and not(P) are both true, simultaneously;
•     case 3: neither P nor not(P) are true, simultaneously.

Heraclitus of Ephesus (~541 - ~480 BC), descendant of the King of Athens and from a priestly family, proclaimed the unity and indivisibility of opposites, that is, the consideration of case 2, as evidenced by its Fragments:

    "All things are born according to the opposition... Change is an upward-descending road and the order of the world occurs according to this road..."

    "All things are mutually contrary. "

    "The god is day-night, winter-summer, war-peace, satiety-hunger. It changes like when you mix perfumes together; then you name it after their smell. "

    "What is cut in the opposite direction is assembled; from what is different is born the most beautiful harmony; everything becomes by discord. "

    "It is the disease that makes health pleasant and good, hunger satiety, fatigue and rest. "

    "What is contrary is useful; what struggles forms the most beautiful harmony; everything is done by discord. "

    "Join what is complete and what is not, what agrees and disagrees, what is in harmony and disagrees; of all things one and one, all things. "

    "They don't understand how what struggles with oneself can fit. The harmony of the world is in opposite tensions, as for the lyre and the bow. "

    "There is a hidden harmony, better than the apparent and where the god has mixed and deeply hidden differences and diversities. "

    "Death of fire, birth for air; death of air, birth for water. "

    "The same thing that lives and that which is dead, that which is awake and that which sleeps, that which is young and that which is old; for the change of one gives the other, and vice versa. "

The principle of non-contradiction rejects case 2: one cannot think P and non(P) true at the same time.

The principle of non-contradiction is an axiom, i.e. it is taken as a first truth that helps to demonstrate other theorems, but itself cannot be deduced or demonstrated. This is a relatively recent invention. It was popularized by Plato (428 - 348 BC) in La République (IV, 436b) and especially by Aristotle (~384 - ~322 BC): "It is impossible that the same attribute belongs and does not belong at the same time and in the same relationship to the same thing" (Metaphysics, Gamma Book, chapter 3, 1005 b 19-20).

The excluded middle was introduced by Aristotle as a consequence of the principle of non-contradiction.

The excluded middle (often wrongly described as a principle, because it is only a law) maintains that either a proposal is true or its negation is true. We cannot think the third hypothetical case, which is therefore rejected. [1]

It is essential to note that since ancient times the knowledge of truth and falsehood has been an experience of thought. The construction of logic derives from the knowledge of being, that is, of metaphysics.  (See in this regard G.G. Granger, Sciences et Réalité, Chap. De l’être au réel : le réel, concept moderne)

The combination of the principle of non-contradiction and the principle of the excluded middle have contributed to the foundation of the so-called classical formal mathematical logic.

In addition to its non-modal syllogistics, Aristotle developed a modal syllogistics in Book I of its Premiers Analytiques (ch. 8-22).

The first formal system of modal logic was developed by Avicenna, who proposed a theory of temporal modal syllogistics. Modal logic as a subject of study owes much to the writings of the Scholastics, in particular Guillaume of Ockham and  Jean Duns Scot, mainly for the analysis of assertions on the essence and the accident.

The "polyvalent logics" question the excluded middle since Lukasiewicz in 1910, which returns to the ancient question of "futurs contingents": if a proposal concerning the future could already be characterized in the present as true or false, we should admit that the course of events is determined in advance. Versatile logic challenges the principle of the excluded middle. They recognize values other than the true and the false, they admit modalities such as the possible, or, below that, the impossible (which is a reinforced false), and beyond the necessary (higher degree of the true).

Modal algebras developed since the 20th century provide models for the propositional calculation of modal logic in the same way that Boolean algebras are models for classical logic. [2]

There is now a whole spectra of intermediate logics, ranging from intuitionist to classical logic, depending in particular on the number of modalities (or truth values) and axioms chosen to constitute the logical system.

Which system is the most relevant to use to rebuild our knowledge, after binary logic?

Do sciences really make us discover the reality of things? Do they build the world from scratch in their laboratories, and then force us to believe it?

G.G. Granger, a professor at the Collège de France, argued that scientific reality was only a way of accessing a certain type of object:

    "Validation is first exercised by self-control of the application of operating rules, which are themselves formal objects of a logic. But what fundamentally distinguishes this self-control is precisely that it ultimately concerns not isolated elements of the logical-mathematical formal object, but the whole system. It is then characterized by the prefix "meta": it is no longer simply a question of reasoning at the level of logic or mathematics, but at a higher meta-mathematical level. A formal system is then tested in its total structure, in terms of its non-contradiction, completeness and fertility. The first criterion normally expected of its reality is certainly the establishment of its non-contradiction. But it has been understood since Tarski and Gödel during the century that has just ended that the establishment by meta-mathematical means of the independence of certain parts of the system, the truth of certain proposals at the same time as the impossibility of demonstrating them in the system, and even the impossibility of demonstrating at its own mathematical level the non-contradiction of the system, were also in a new sense the attributes of its reality...[.] We see therefore that in all cases scientific reality is necessarily dependent on a use of the conceptual imagination. " (Science et Réalité, Ed. Odile Jacob, 2001, Chap. 8 "Systèmes et réalité", Pp. 241-242)

We have stressed that the construction of logic derives from the knowledge of being, that is, from metaphysics. It is therefore mandatory to look at what Tradition has to offer in this regard. We use a capital letter T to indicate that we go back further than Aristotle and Plato. We started with Heraclitus, we continue in the footsteps of René Guénon, starting by explaining the importance of the ternaries.

After 2 there are 3, which give birth to 4

"What we have just said already determines the meaning of the Triad, at the same time as it shows the need to make a clear distinction between ternaries of different genres;[...]

One of these two genres is the one where the ternary is constituted by a first principle (at least in a relative sense) from which two opposite terms derive, or rather complementary terms, because, even where the opposition is in appearances and has its raison d'être at a certain level or in a certain domain, complementarism always responds to a deeper point of view, and therefore no longer truly in conformity with the real nature of what it is about; such a ternary can be represented by a triangle whose vertex is placed at the top (fig. 1).

The other genus is the one where the ternary is formed, as we said earlier, by two complementary terms and by their product or resultant, and it is to this genus that the Far Eastern Triad belongs; unlike the previous one, this ternary can be represented by a triangle whose base is on the contrary placed at the top (fig. 2).

If we compare these two triangles, the second one appears to be a reflection of the first, which indicates that, between the corresponding ternaries, there is an analogy in the true meaning of this word, i. e. to be applied in the opposite direction; and, indeed, if we start from the consideration of the two complementary terms, between which there is necessarily symmetry, we see that the ternary is completed in the first case by their principle, and in the second, on the contrary, by their resultant, so that the two complementary terms are respectively after and before the term which, being of a different order, is almost isolated from them. This is further clarified in both figures by the direction of the arrows, going, in the first, from the upper vertex to the bottom, and, in the second, from the bottom to the lower vertex;[...]; and it is obvious that, in any case, it is the consideration of this third term that gives the ternary as such its full meaning.

Now, what must be clearly understood before going any further is that there could only be "dualism" in any doctrine if two opposed or complementary terms (and then they would rather be conceived as opposed) were first posed and considered as ultimate and irreducible, without any derivation from a common principle, which obviously excludes the consideration of any ternary of the first kind; one could therefore only find in such a doctrine ternaries of the second kind[...]

The consideration of two ternaries like the ones we have just mentioned, having in common the two complementary principles of each other, leads us to some other important remarks: the two inverted triangles which represent them respectively can be considered as having the same base, and, if we figure them united by this common base, we see first that the set of the two ternaries forms a quaternary, since, two terms being the same in both, there are in all only four distinct terms, and then only the last term of this quaternary, located on the vertical line resulting from the first term and symmetrically to it with respect to the base, appears as a reflection of this first term, the reflection plane being represented by the base itself, i. e. being only the median plane where the two complementary terms resulting from the first term are located and producing the last one (fig. 3).

We have just seen that the two extreme terms of the quaternary, which are at the same time respectively the first term of the first ternary and the last of the second, are both, by their nature, intermediaries in a way between the other two, although for the opposite reason: in both cases, they unite and reconcile in themselves the elements of complementarism, but one as a principle, and the other as a result. »
(The Great Triad, 1946, Chap. II - Different kinds of ternaries)

René Guénon developed in more detail the meaning of the quaternary in Le Symbolisme de la croix (1931), which is entirely consistent with the above passage.

The square of Aristotle

In classical modal logic, four modalities are identified for evaluating a proposal:

•     necessary (if and only if the proposal is not possibly false; "what cannot but be true");
•    contingent (if and only if the proposal is not necessarily false and not necessarily true; "what may be false");
•     possible (if and only if the proposal is not necessarily false; "what may be true");
•     impossible (if and only if the proposition is not possibly true; "what cannot but be false").

« Another way of showing the relationships between the four modalities is to arrange them in a square configuration, which goes back at least to Aristotle. The square of the modalities is isomorphic to the logical square of the terms and to that of the form A (universal affirmative), E (universal negative), I (particular affirmative) and O (particular negative) proposals of Aristotelian logic (fig. 4).

Fig. 4

There are, in each of these squares, four fundamental types of relationships between their respective four terms: contradiction or negation (on the diagonals) contradiction or incompatibility (on the upper horizontal side) subordinacy or non-exclusive disjunction (on the lower side) subordination or implication (on the vertical sides). » (Lucien Scubla, L’aporie de Diodore Cronos et les paradoxes de la temporalité. Jean-Pierre Dupuy et la philosophie, 2007)

These 4 modalities are linked, only one is needed to define the other three. The interpretation in intuitive logic is as follows:

•     impossible ≡ necessary that does not... not...;
•     possible ≡ no impossible ≡ no need that does not... not
•     contingent ≡ not necessary ≡ possible that not... not;

In this interpretation, the possible and the impossible derive from the necessary (or its negation). The quota also results from the necessary, but also from the possible (and therefore from the impossible). The modal square is thus only a reduced deformation of the primordial diamond as described by Guénon, which is constructed on equilateral triangles instead of isosceles rectangular triangles (Fig. 5). This reduction, an example of a progressive loss of meaning, is a special case whose generality has been perfectly explained by Guénon (cf. La crise du monde moderne, Le règne de la quantité et les signes des temps). [3]

Fig. 5

Let us note how much the conceptualization described by Guénon agrees with and extends G. G. Granger's conclusion :

« The reality of science objects would therefore, according to our analyses, mean a certain relationship between a virtual aspect and a current aspect of the representation of the experience. This would be the meaning of a unity of science, a unity that it would not be useful to simply designate as a "concordance" of the two aspects. It is therefore legitimate to use the plural to refer to scientific realities, insofar as, as we have tried to show, the thinking of science, in each of its fields, determines specific criteria for its achievement of reality. » (Sciences et réalité, Conclusion, Pp. 243)

This essay is extended by the formal propositional calculation in tetravalent logic, detailed in the second part. 

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This article is a translation of the original article published in French.

[1] In classical binary logic, the excluded third theorem is derived from the principle of non-contradiction by introducing the relationship of equality or equivalence, the Boolean negation operator, by accepting the additional axioms:

•     identity principle: P = P
•     elimination of double negation: not(not(P)) = P

then by establishing the values of the truth tables of the operators NOT, AND and OR and by first demonstrating the equality not(A and B) = not(A) or not(B).

The principle of non-contradiction is the proposal "P and not (P) = FALSE".
implies the proposition: "not(P and not (P)) = TRUE"
implies the proposition:" not(P) or not (no (P)) = TRUE"
implies the proposition: "not(P) or P = TRUE"
implies that "P is TRUE" or "not(P) is TRUE"
so the proposal of the excluded middle is verified.

[2] Lewis founded modern modal logic in his 1910 Harvard thesis and in a series of articles published between 1912 and 1932, when his book Symbolic Logic, written in collaboration with Langford, was published.

The logicians Brouwer and Heyting in 1930 criticized, in the name of "intuitionist logic", a certain type of reasoning held according to the principle of the excluded middle applied to finite sets. They believe that one has no logical right to infer the truth of a proposal from the falsity of its negation. Heyting does not say that the excluded middle principle is always wrong, but it limits its scope.

The mathematical structure of modal logic, i. e. Boolean algebras augmented by unary operations (often called modal algebras), began to emerge with McKinsey who showed in 1941 that Lewis' S2 and S4 systems were decidable.

[3] Why a representation in the form of a diamond and not a tetrahedron? The latter would place the Contingency and the areas of the Possible and the Impossible on the same level. It would "shorten" the distance between the Contingency and Necessity. There is no way to justify these assertions. On the contrary, we will see in the last part how useful it is to keep the diamond model.