Logic is about the (4!/2!x2!= 6) dual combinations (“oppositions”) of possibility (“some”, Particular Affirmative), necessity (“all”, Universal Affirmative), not-possibility (“not some”, Particular Negative) and not-necessity (“not all”, Universal Negative).

Ancient logic is used in Alchemy. Modern logic is called Geometric Logic and is a part of Modal Logic. Modal Logic works with the notion that propositions can be mapped to sets of possible worlds.

The idea of possible worlds is attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God and argued that our actually created world must be the best of all possible worlds. Possible worlds play an important role in the explanation of Quantum Mechanics.

Ancient Logic and Modern Logic are based on the Square of Opposition of Aristotle. This square was discussed for more than 2000 years.

Recently scientists detected what was wrong with the Square. It misses an extra dimension (“contingency”). This transforms the Square into a Cube.

I will explain Alchemy, Ancient- and Modern Logic and show that Alchemy and Geometric Logic share the same geometry, the Hexagon (the Seal of Solomon), which is a 2D-mapping of the Cube of Space of the Sefer Yetsirah.

**About Alchemy**

**When thou hast made the quadrangle round, **

**Then is all the secret found. **

**George Ripley (d. 1490).**

One of the most complicated ancient architectures is the architecture behind Alchemy. The architecture of Alchemy is full of strange concepts and allegories.

Alchemy comes from the Arabic *al-khimia *meaning*, *the art of *transformation*.

Alchemy is an ancient science that was practiced all over the world. The most famous teacher was called Hermes Trismegistus in Greece and Thoth in Egypt. The old science of Hermes came back in the Renaissance.

The essence of Alchemy is called REBIS (Res Bina), Double Matter. REBIS is the end product of the alchemical *magnum opus* or Great Work.

The Great Work is the chemical and personal quest to create Double Matter, the Philosopher’s Stone. The stone is the agent of chemical transmutation, and the key ingredient in the creation of the elixir of life, said to heal all diseases, induce longevity and even immortality.

In current physics transmutation is proved at room temperature in a proces called Cold Fusion.

The aim of the Great Work is a fusion with the One (the Blazing Star, Saturn, Point) that is divided in the Two (Sun & Moon, Male & Female, Line) and the Four Planets (Square). The Great Work moves from Four to Two to One (or from one to two to four) in a Spiraling Spiral Motion.

The One, to which the elements must be reduced, is a little circle in the center of the squared figure (the cross). The cross is the mediator, making peace between the opposition of the planets.

The winged dragon represents First Matter (Quintessence, The Fifth Element) and suggests ascension, a merging of matter (body) and spirit (Creativity). Creativity (Enthusiasm, Spirit, Inspiration, Imagination) is the engine behind the Spiral Wheel.

The First Matter is the primordial chaos comparable to what we now call the vacuum, the state of lowest energy. This state contains all possibilities.

It can be looked on as an unorganized state of energy that is the same for all substances and exists in an invisible state between energy and matter.

The First Matter is the One, the Point in the Middle of the Circle. In current physics a point is called a singularity or a black hole.

Alchemy is about re-combination summarized in the Latin expression **S*** olve (*break down in separate elements

*)*coming back together (coagulating) in a new, higher form

**et Coagula**(*)*.

Earth (Body) becomes Water (Spirit) by dissolving it in some solvent. Water becomes Air (a moist vapor) by boiling, which further heating turns into Fire (a dry vapor).

Finally Fire becomes Earth by allowing the vapors to condense on a solid material (a stone).

The circulation may repeat if it is done in a reflux condenser, such as the Pelican or *kerotakis*.

According to the early alchemists, the four elements—fire, water, air, and earth—come into existence via the combination of specific qualities, recognized as hot, cold, wet, and dry, being impressed on to the Prime Matter. For example, when hot and dry are impressed on the Prime Matter, we have fire.

(Hot and Cold) and (Wet and Dry) are oppositions. Their fusion is the empty set.

(Hot or Cold) and (Wet or Dry) can be combined -> (Hot, Wet), (Hot, Dry), etc.

The Cold allows substances to get together (Commune, Love). The Hot is the power of separation (not-Commune, Passion).

Wet things tend to be Flexible (Fluent, Movement, Self, Not-Agency) whereas Dry things are fixed and structured (Agency, Resistance, not-Movement).

When the qualities are changed, the elements themselves are changed. Adding Water to Fire substituting wet for dry; hot and dry becomes hot and wet: steam, or Air.As you can see in de picture below Hot & Cold and Wet and Dry are Binary Opposites (“lines”) that are voided by the central cross, the point of the Empty Set, The Void (A and not-A -> Empty).

Hot & Dry = Fire, Hot & Wet = Air, Cold & Wet = Water and Cold & Dry = Earth.

There are four types of Trinities the Passion-trinity (Dry, Wet, Warm), the Structure-trinity (Dry, Wet, Cold), the Resistance-trinity (Warm, Cold, Dry) and the Movement-trinity (Warm, Cold, Wet).

The Square rotates. The wheel of the Square is driven by the four qualities. Wet on the rising side, Hot on the top, Dry on the descending side, and Cold on the bottom.

The start is powerless (Cold). By becoming more flexible (Wet) success and power comes (Heat). At the top Rigidity (Dry) undermines the system and it falls back to a powerless state.

The four Elements all share the Triangle of the Holy Trinity. Every Trinity is seen from another perspective.

The two triangels, The Up-triangel and the Down-Triangel, are Opposites that are created out of two triangels that contain two Opposites that are fused. Humans are opposites in opposition.

Humans are part of the Bilateria, animals with bilateral symmetry.

Humans can be described as a fusion of two mirrored bilateral triangles, the Up-triangel related to the Top of the Body (Mind) and the Bottom-triangel of the lower Body both connected by the Heart. The two triangels and the separate triangels have to be balanced.

The principle of Balancing the two Triangles of Top & Bottom of the Body is represented by the Egyptian Goddess Ma’at who weights the Heart (the connector of Up and Down) of the Dead with her Feather (Nothing).

As you can see the Square of the Cycle of the Four Elements is visible in the picture below. The total picture shows a Hexagram with a Center (The Cross).

The points that are missing in the Square are the top- and the bottom of the two triangels which are the Whole (Up, Heaven) and the not-Whole, Emptiness, the Bottom (Hell).

A hexagon is a 2D-projection of a Cube.

The hexagon of Solomon can be transformed into the Cube of Space of the Sefer Yetsirah.

The Cube of Space can be transformed into the Tree of Life.

The Tree of Life also called the tree of knowledge or world tree connects heaven and underworld and all forms of creation and is portrayed in various religions and philosophies all over the world.

The Tree of Life shows the levels/scales (three) and possible paths of the Opus Magnum, the alchemical journey.

The Spiral of Alchemy is about the Chemical Transformation of the Body by the Conceptual Transformation of the Mind.

**About The Square of Opposition**

People have always believed in the fundamental character of binary oppositions like Hot & Cold and Wet & Dry are used in Alchemy.

In this part we move to the Field of Logic. In this case Hot/Cold and Wet/Dry transform into Necessary /Not-Necessary and Possible/not-Possible.

Oppositions can be divided in:

**Digital**oppositions, contradictories, contain mutually exclusive terms (Gender (Male/Female)).**Analogue**oppositions, antonyms or*contraries*: contain terms that are ordered on the same dimension (Temperature (Minimum->Cold/maximum->Hot)).

The doctrine of the square of opposition originated with Aristotle in the fourth century BC and has occurred in logic texts ever since.

The logical square is the result of two questions: Can two things be false together? Can two things be true together?

This gives 4 possibilities: no-no: contradiction (A/O, E/I); no-yes: subcontrariety (I/O); yes-no: contrariety (A/E); yes-yes: subalternation (A/I, E/O) better known as implication (A->B).

The four corners of the square represent the four basic forms of propositions recognized in classical logic:

**A**propositions, or universal affirmatives take the form: All S are P.**E**propositions, or universal negations take the form: No S are P.**I**propositions, or particular affirmatives take the form: Some S are P.**O**propositions, or particular negations take the form: Some S are not P.

The square of opposition was debated for many reasons for more than two thousand years.

One of the discussions is about the difference between possible and necessary. In the so called Master Argument Diodorus proved that the future is as certain and defined as the past. The essence of logic (necessaty) implies the non-existence of freedom (possibility). X is possible if and only if X is necessary.

The term ‘possible’, in Aristotle’s view, is ambiguous. It has two senses known as one-sided possibility and two-sided possibility (or contingency). Being two-sided possible means being neither impossible nor necessary, and being one-sided possible simply means being not impossible.

Leibniz distinguished between *necessary truths*, which are true for a reason—i.e, their opposite is a contradiction—and *contingent truths*, such as the fact that the president of France is François Hollande. A contingent truth cannot be proved logically or mathematically; it is accidental, or historical.

Modal Logic is about the fusion of necessity and possibility, contingency.

In Modal Logic the propositions are modelled in a logical hexagon where:

**A**is interpreted as necessity: the two propositions must be either simultaneously true or simultaneously false. A is a Proof or Law.**E**is interpreted as impossibility. E is an Observation that contradicts the Proof of A.**I**is interpreted as possibility: the truth of the propositions depends on the system of logic being considered . I is an**Idea**implied by the Proof that Contradicts E.**O**is interpreted as ‘not necessarily’. O contradicts A.**U**is interpreted as non-contingency: neither logically necessary nor logically impossible, its truth or falsity can be established only by sensory observation.**Y**is interpreted as contingency: propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions). Their truth depends on the truth of the facts that are part of the proposition. Y is a possible proven theory.

The Logical Hexagon has the same geometry as the Seal of Solomon. It contains the Square of Opposition (A,E,I,O).

The Logical Hexagon can be transformed just like the Seal of Solomon to the Cube of Space now called The Cube of Opposition.

The Logical Hexagon is used in many fields of Sciences like Musical Theory or Scientific Discovery.

Every opposite can be defined by a string of zero’s and one’s like 1010. An opposite is 0101. This means that there are 2**4/2 = 4×4/2= 16 / 2 = 8 possible opposites. This is called the Octagon of Opposites.

The Octagon contains six reachable and two unreachable points in the Center (0000, 1111), the Empty Set (the Void, the Hole, Contradiction) or its Opposite (The Whole, Tautology).

It is clear that it must be possible to create higher dimensional models based on n-based-logics.

When n becomes very big the geometry will tend to a n-Dim circle in which every point is in opposition to the point at the other side.

The big problem with n-based logics is language. We don’t have the names to articulate the many grades of opposition that are possible.

The other problem is that a negation is not a symmetric operation (A = not-not-A) but a rotation with an angle 360/n.

The n-opposite-geometries are highly related to Simple Non-Abelian Groups (SNAGs). They play an important role in biology.

The last step is to transform the static n-opposite geometries by making n very large (infinite) to a dynamic “opposition field”.

**Examples of Comparable Models**

In current Psychology Hot & Cold are called Communion and Moist & Dry, Agency. Together they create the so called Interpersonal Circumplex.

The four stages of Learning by Jean Piaget.

The mathematical model behind the theory of Piaget is called the Klein Four Group or Identity/Negation/Reciprocity/Correlation-model.

In the Science of Ecology (Panarchy) Agency & Communion are called Connectedness and Potential. The Panarchy model looks like a Mobius Ring.

A model that describes the Four Perspectives on Security:

The Semiotic Square of Greimas:

The Chinese Sheng-Cycle: The Chinese Five Element Model is comparable to the Western Four Element model. It contains as a Fifth Element, the Whole-Part-relationship (Earth, Observer State). Emptiness (Nothing, the Tao) is represented by the recursive Pentangle inside.

**The lesson**

**“ When thou hast made the quadrangle round, then is all the secret found“.**

The secret: It is impossible to move though the center because the paths that go through the center are a contradiction or a tautology (I am what I am, JHWH).

We have to move with the Cycle (making the rectangle round (a circle)), around the Singularity (the Hole of the Whole), going Up and going Down all the time.

The only way to solve this problem is to *join the opposites *by accepting only one thruth-value, by:

- accepting everything (Wu Wei),
- denying everything (Stoics) or
- diminishing the amount of dimensions by moving up in abstraction (Hot & Cold -> Temperature -> Everything is Energy) to the level of zero dimensions, a point.

**LINKS**

About the Geometry of Negation

How to use the Square of Opposition as a Research Tool

About the Cube of Space of Sefer Yetzirah

About Chemical Transformation (Cold Fusion)

About the Ars Generalis Ultima of Ramon Lull (1305)

About the Six Domains of the Polynomic System of Value

Why Innovation is Re-Combination

How to Balance the Seal of Solomon

About the Hexagon of Opposition

Anti-Fragility and the Square of Opposites

An Introduction to Hexagonal Geometry