Oppositional logic and its requisite polyhedral geometry

Framing Cognitive Space for Higher Order Coherence (Part #7)

4D Tesseract: The challenge to comprehension is all the greater in that any dynamic suggests that it is at least in 4D that a 3D framework merits consideration. With respect to the Logic Alphabet Tesseract (below left), as noted by Louis Kauffman:

Shea Zellweger did an extensive study of the sixteen binary connectives in Boolean logic ( "and", "or" and their relatives -- all the Boolean functions of two variables), starting from Peirce's own study of these patterns. He discovered a host of iconic notations for the connectives and a way to map them and their symmetries to the vertices of a four dimensional cube and to a three dimensional projection of that cube in the form of a rhombic dodecahedron. Symmetries of the connectives become, for Zellweger, mirror symmetries in planes perpendicular to the axes of the rhombic dodecahedron... Zellweger uses his own iconic notations for the connectives to label the rhombic dodecahedron, which he calls the "Logical Garnet". This is a remarkable connection of polyhedral geometry with basic logic. The meaning and application of this connection is yet to be fully appreciated. It is a significant linkage of domains. On the one hand, we have logic embedded in everyday speech. One does not expect to find direct connections of the structure of logical speech with the symmetries of Euclidean Geometry. It is the surprise of this connection that appeals to the intuition. Logic and reasoning are properties of language/mind in action. Geometry and symmetry are part of the mindset that would discover eternal forms and grasp the world as a whole. To find, by going to the source of logic, that we build simultaneously a world of reason and a world of geometry incites a vision of the full combination of the temporal and the eternal, a unification of action and contemplation. The relationship of logic and geometry demands a deep investigation. This investigation is in its infancy (The Mathematics of Charles Sanders Peirce, Cybernetics and Human Knowing, 8, 2001)

Zellweger's depiction is usefully complemented by that of the 4D tesseract as in the other images below, as discussed separately (Oppositional logic? 2018; Tony Phillips, Topology of Venn Diagrams, AMS, June 2005).

With regard to the argument above, one provocative approach is to consider how the polyhedral forms employed in the extensively developed literature on logical geometry might be presented in relation to a 3D representation of the Tao symbol. An important key to the significance for governance of hyperdimensionality is that work on logical geometry. This notably featured in the presentations at the IV International Congress on: The Square of Opposition -- Vatican City, 2014, compiled by Jean-Yves Béziau and Gianfranco Basti (The Square of Opposition: a cornerstone of thought, 2017):. This constitutes a collection of new investigations and discoveries on the theory of opposition (square, hexagon, octagon, polyhedra of opposition), ranging from historical considerations to new mathematical developments of the theory of opposition including applications to theology, theory of argumentation and metalogic.

Of particular relevance, beyond the Aristotelian square of opposition is its relationship to the rhombic dodecahedron featuring in Hasse diagrams -- both involving discussion of the hypercube, as featured in the work of Lorenz Demey and Hans Smessaert (The Relationship between Aristotelian and Hasse Diagrams, 2014; Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation, Symmetry, 9, 2017, 204; Geometric and Cognitive Differences between Logical Diagrams for the Boolean Algebra B₄) and by Hans Smessaert (On the 3D Visualisation of Logical Relations, Logica Universalis, 3, 2009, 2).

Arguably if there is one characteristic of psychosocial reality which is a fundamental challenge to governance it is that of "opposition" and the framework within which it can be appropriately integrated. The argument for doing so is that literature is particularly focused on the geometrical representation of opposition as articulated in truth tables through the set of Boolean connectives. A key polyhedron used to map the 16 (-2) Boolean logical connectives in that approach is the rhombic dodecahedron of 14 vertices and 12 faces.

The Logic Alphabet Tesseract - a four-dimensional cube (see coding). by Shea Zellweger	Tesseract animation simulating requisite 4-dimensionality?	Topologically faithful 4-statement Venn diagram is the graph of edges of a 4-dimensional cube as described by Tony Phillips	Organization of contingent bitstrings on a rhombic dodecahedron

Diagram by Warren Tschantz (reproduced from the Institute of Figuring) .	by Jason Hise [CC0], via Wikimedia Commons	A vertex is labeled by its coordinates (0 or 1) in the A, B, C and D directions; the 4-cube is drawn as projected into 3-space; edges going off in the 4th dimension are shown in green.	Adapted from Lorenz Demey and Hans Smessaert (2017)

The images above recall that of the diamond cubic crystal structure with its repeating pattern of 8 atoms (presented above). This is all the more intriguing because of the value attached to diamonds in society and the particular value associated with it in Buddhism. Major symbolic importance is associated with the diamond, notably in Buddhist traditions, as a metaphor of a particular emergent order of the mind and the understanding of that order as a 'vehicle', or 'body', for the spirit (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). The terms 'diamond mind' and 'diamond body' are widely used in Buddhism and are notably a focus for Diamond Way Buddhism. This metaphor seems however to focus on the individual and not on the ordering of society.

Other experimental animations may be used to suggest other ways of comprehending the cognitive dynamics. As shown below, the moving cubes could be understood as the "cognitive tanks" discussed in the preceding argument (Tank Warfare Challenges for Global Governance: extending the "think tank" metaphor to include other cognitive modalities, 2019). That on the left is reproduced from an earlier discussion (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016; Neglected recognition of logical patterns -- especially of opposition, 2017).

The point can be developed further with respect to oppositional logic (Guoping Du, Hongguang Wang and Jie Shen, Oppositional Logic, Logic, Rationality, and Interaction, Springer, 2009, pp 319-319) and discussed separately in terms of a 4-dimensional polyhedral configuration of directions (Neglected recognition of logical patterns -- especially of opposition, 2017).

Of particular relevance to the argument here, relating to a fundamental pattern of distinctions, are the commentaries of Louis Kauffman on the much-cited study by George Spencer-Brown (Laws of Form, 1969) and the remarkable connection between Laws of Form, polyhedral geometry, mirror symmetry and the work of Zellweger (indicated above). As noted in an insightful review by Kauffman (Laws of Form: an exploration in mathematics and foundations):

... this is an approach to mathematics (and to epistemology) that begins and ends with the notion of a distinction. Nothing could be simpler. A distinction is seen to cleave a domain. A distinction makes a distinction

Although Kauffman specifically indicates that the further implications of such fundamental insights remain to be explored, it is far from clear whether appropriate effort has been devoted to their relevance to the "poisonous" distinctions which are such a prominent feature of psychosocial dynamics at this time. Specifically there is the question of how such a subtle understanding of distinction and difference might help to reframe relations between partisan political extremes, religions in conflict ("interfaith discourse"), and disciplines cultivating fragmentation ("interdisciplinarity"). Given the many related studies of Kauffman on knot theory, the relevance of such insights have yet to be applied to the sense in which society has "tied itself into a complex knot" -- even an archetypal Gordian knot, as discussed separately (Engaging globally with knots and riddles -- Gordian and otherwise, 2018).

Given the higher dimensional implications of the cubic metaphor which inspired this exploration, Kauffman's diagrammatic application of the notaion of Laws of Form are potentially of the greatest relevance to the possibility of transcendent modes of discourse.

Application of the notation of the Laws of Form to logical connectives
16 Binary Boolean Connectives	Planar graph of the rhombic dodecahedron	Logical Garnet (Zellweger)

Reproduced from Louis H Kauffman (Laws of Form: an exploration in mathematics and foundations)