Logical hexagon

The logical hexagon is a conceptual model of the relationships between the truth values of six statements. It is an extension of Aristotle's square of opposition. It was discovered independently by both Augustin Sesmat and Robert Blanché.^[1]

This extension consists in introducing two statements Y and U. Y is the conjunction of the two traditional particulars I and O. Whereas, U is the disjunction of A and E.

Example

For instance, the statement A may be interpreted as "Whatever x may be, if x is a man, then x is white."

(x)(M(x) → W(x))

The statement E may be interpreted as "Whatever x may be, if x is a man, then x is non-white."

(x)(M(x) → ~W(x))

The statement I may be interpreted as "There exists at least one x that is both a man and white."

(∃x)(M(x) & W(x))

The statement O may be interpreted as "There exists at least one x that is both a man and non-white"

(∃x) M(x) & ~W(x)

The statement Y may be interpreted as "There exists at least one x that is both a man and white and there exists at least one x that is both a man and non-white"

(∃x)(M(x) & W(x)) & (∃x)(M(x) & ~W(x))

The statement U may be interpreted as "Whatever x may be, if x is a man, then x is white or whatever x may be, if x is a man, then x is non-white."

(x)(M(x) → W(x)) v (x)(M(x) → ~W(x))

Summary of relationships

The traditional square of opposition demonstrates two sets of contradictories A and O, and E and I (i.e. they cannot both be true and cannot both be false), two contraries A and E (i.e. they can both be false, but cannot both be true), and two subcontraries I and O (i.e. they can both be true, but cannot both be false) according to Aristotle’s definitions. However, the logical hexagon provides that U and Y are also contradictory.

History

In 1966, Blanché published Structures intellectuelles, wherein he presented a hexagonal figure introducing Y and U, a third pair of contrary and subcontrary logical propositions for a total of six, in addition to those of the traditional square, the contraries A and E, and the subcontraries I and O.

Interpretations of the logical hexagon

The logical hexagon may be interpreted in various ways, including as a model of quantifications, modal logic, order theory, and paraconsistent logic.

Modal logic

For instance, the logical hexagon may be interpreted as a model of modal logic such that

A is interpreted as necessity
E is interpreted as impossibility
I is interpreted as possibility
O is interpreted as 'not necessarily'
U is interpreted as non-contingency
Y is interpreted as contingency

Further extension

It has been proven that both the square and the hexagon, followed by a “logical cube”, belong to a regular series of n-dimensional objects called “logical bi-simplexes of dimension n.” The pattern also goes even beyond this.^[2]

References

^ N-opposition theory logical hexagon
^ Moretti, Pellissier

[1] N-opposition theory logical hexagon

[2] Moretti, Pellissier

[1]

[2]