The Paraconsistent Logic of Synergetics

Contradictions are powerful things.  They can cause strong emotions, from surprised bewilderment to stress and anxiety.  Can contradictions cause us to feel happy or joyful?  Can they enlighten us about the real world and our place in it?  Such is the motivation behind paraconsistent logic, which is a positive, constructive approach to contradictions.  It downplays the importance of consistency (moving ‘beyond’ it) in favour of completeness and coherence. It treats paradox as an invitation to discovery. The Laws of Thought become like an escape room for the mind, which is like trying to escape from over 2 millenia of Western philosophy developed under Aristotle's logic.

The term 'classical logic' refers to modern innovations by thinkers like Frege and Russell in the early 20th century. But classical logic still adopted Aristotle's equivocation of inconsistency with incoherence without question. Both logics teach us that contradictions are absurdities (reductio ad absurdum) and that we should always aim to preserve consistency.  They rely on the Law of Excluded Middle: a sentence is either true or false (P or not-P).  Indeed, this Law matches our experience of the world: it’s either raining (R) or it’s not raining (¬R).  Going beyond these two options brings us into the realm of non-classical logic: it’s both raining and not raining (R & ¬R), and: it is neither raining nor not raining  ¬(R & ¬R).  Generally speaking, the latter is the focus of intuitionist logic while the former is characteristic of paraconsistent logic, which seeks to affirm both truth and falsity (specifically violating the Law of Non-contradiction).

Paraconsistent logic is still contested in philosophy since it challenges some basic rules of reasoning. But it is clearly helpful to think of at least some contradictions as simultaneously true and false.  For example, the Liar’s Paradox (‘this sentence is false’) can be accepted paraconsistently.  And I think this applies to many famous problems of Western philosophy, such as the problem of free will or Kant’s antinomies.  Aside from these perennial dilemmas and tricks of language, what use is paraconsistent logic when dealing with reality itself?  Isn’t this just another empty version of abstract reasoning with no relevance to practical affairs?

Let’s consider the related (but distinct) field of paraconsistent mathematics.  As with logic, it is possible to create a system of axioms that define contradictory mathematical objects.  Infinitesimal calculus has been identified as paraconsistent because it uses points that function both as non-zero values and as zeroes when needed.  A more familiar example is the work of M.C. Escher, which offers us impossible perspectives on inconsistent geometry.  Even though mathematics might not be ‘about the world’, it is clearly a discipline where contradiction is not only permissible but even has a positive or constructive purpose.

Relativity by M.C. Escher (1953)

Just as mathematical contradictions have their place, so too can logical contradictions preserve the conditions for making inferences (known as entailment).  The typical objection to paraconsistent logic stems from the principle of explosion. Explosion occurs when a contradiction is used prove anything as true, opening up the possibility that everything is true (a condition known as trivialism).  Paraconsistent logic accommodates contradictions without explosion. It is like a balancing act between accepting contradictions on one hand while avoiding trivialism on the other.  This does not mean that paraconsistent logic is a creative license for unlimited speculation. Instead, it should be taken as a guide to finding truth on the basis of inconsistent information.

Buckminster Fuller’s Synergetics is a work of inconsistent mathematics that exemplifies this very attitude towards truth.  As a preliminary example, consider the Escher-esque properties of this model.  Like the staircases in Relativity, the square and triangular faces can be perceived in two ways, facilitating convex or concave shapes between the white and grey spaces.

Figure 1032.30 Space Filling of Octahedron and Vector Equilibrium

One of the main principles of Synergetics is the following:

The Unity Principle (UP): Unity is plural and at minimum two.

Taking unity as the value of +1, we can write:

UP: 1=x 

where x is any positive integer ≠ 1.

This principle needs to be unpacked because it violates another fundamental law of logic besides the excluded middle.  This is the Law of Identity: A=A. It simply states that a thing ‘is what it is’, tautologically.  So a triangle is a triangle, the earth is the earth, and 1 is 1.  So Fuller’s UP contradicts the identity of unity, suggesting that the mathematics of Synergetics is inconsistent because it rejects the Law of Identity.  We might hope to fix this by examining the concept of ‘unity’ and distinguishing it from the value of +1. But I think Fuller intended to be contradictory here and that the Unity Principle is meant to violate the Law of Identity in this manner.

There are more examples of paraconsistency in Synergetics.  Consider the following principle: the ‘minimum subdivision of Universe’ is the tetrahedron.  Now examine how Fuller revised Euclid’s axioms of geometry according to this principle:

• Euclid: A point is that of which there is no part. [Book I, Definition I]

• Fuller: A ‘point’ is a tetrahedron of negligible altitude and base dimensions. [240.05, Synergetics]

  
  

• Euclid: And a line is a length without breadth. [Book I, Definition II]

• Fuller: A ‘line’ (or trajectory) is a tetrahedron of negligible base dimension and significant altitude.  [240.06, Synergetics]

  
  

• Euclid: And a surface is that which has length and breadth only. [Book I, Definition V]

• Fuller: A ‘plane’ (or opening) is a tetrahedron of negligible altitude and significant base dimensionality. [240.07, Synergetics]

According to the Law of Identity, a point is a point (P=P), a line is a line (L=L), a surface is a surface (S=S) and a tetrahedron is a tetrahedron (T=T).  Yet all of these identities are contradicted by the logical sentences T=P, T=L and T=S.  We might even add P=L, L=S, and S=P while we’re at it.  The problem is obvious: everything is made from everything else, turning the very notion of ‘thing’ into a vacuous term and the very act of ‘making’ into a meaningless process.  We seem to be on the verge of explosion.

Granting these contradictions of identity, how does the paraconsistent logic of Synergetics avoid exploding into trivialism?  Althouh Fuller had arguments in his defense (which are worth considering), I don't believe he ever produced a succinct and persuasive answer to this question. As someone who could lecture for many hours at a time, he’s not going to convince anyone that a certain degree of logical explosion has not occurred.  If anything, Fuller’s approach challenges us to rethink ‘explosion’ as the revelation of unlimited possibilities, and ‘trivialism’ as the discovery of omnidirectional structure (the word comes from trivium).  These wider insights of Fuller's method transcend the domain of logic proper and belong instead to metaphysics and epistemology (as with the view known as dialetheism).

In my opinion, the problem of logical explosion in Synergetics has a mathematical solution in the branch of geometry known as topology. Topology studies properties of mathematical objects that do not change even when the objects themselves change. Identity in Synergetics is always topological identity, which is what prevents its paraconsistent logic from exploding into trivialism.  I'm basing this claim on statements like the one below, where Fuller hinted at the 'coping' involved:

• 400.20  Comprehensibility of Systems: All systems are subject to comprehension, and their mathematical integrity of topological characteristics and trigonometric interfunctioning can be coped with by systematic logic.

Fuller also described topology as ‘deceptive’, which seems odd but is really quite apt. There’s a joke that a topologist can’t tell the difference between a coffee mug and a donut - this is because both shapes have the same topological identity (they are homeomorphic).  It makes sense topologically but is highly deceptive from a functional point of view. A mug can hold coffee but a donut can’t.

Synergetics is a work of geometric models.  These models, whether materially constructed or formally illustrated, are always topological.  This means that they have distinct ostensive features that identify them as what they are. In my previous blog post, I suggested that ostensive definitions are required for the justification of inductions in Synergetics. Now we have one reason why this is so - the 'argument from topological identity'. But the context of this argument is not the familiar one of classical logic. It is intended to function as a bridge between a generalized principle of reasoning and the inchoate demands of inconsistent premises. I offer this argument from topological identity as the justification for Fuller's systematic theorizing about the limits of conceptual thought, which are themselves treated as true contradictions (dialetheias).

Where Western philosophy evolved under the rules of Aristotelian logic (culminating in the classical logic of the 20th century), Eastern philosophers developed an early paraconsistent formula for truth values known as the catuskoti. By transcending the habits of classical logic, Western philosophers are rediscovering a deep connection with their Eastern counterparts. Graham Priest is one of the most progressive philosophers in this area, both in his defense of dialetheism and his analysis of Eastern metaphysics. I don't know where Fuller fits into this emerging integration of East and West but I believe that he deserves a place at the table.

---

About the Author:

Dante Diotallevi is an independent scholar living in Canada. He holds a BSc. in Biology and an M.A. in Philosophy from Queen's University in Kingston, Ontario.

See the TRIMTABS page for more information.