Logical hexagon


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

Summary of relationships

The traditional square of opposition demonstrates two sets of contradictories A and O, and E and I, two contraries A and E, and two subcontraries I and O according to Aristotle’s definitions. However, the logical hexagon provides that U and Y are also contradictory.

Interpretations

The logical hexagon may be interpreted in various ways, including as a model of traditional logic, quantifications, modal logic, order theory, or paraconsistent logic.
For instance, the statement A may be interpreted as "Whatever x may be, if x is a man, then x is white."
→ W)
The statement E may be interpreted as "Whatever x may be, if x is a man, then x is non-white."
→ ~W)
The statement I may be interpreted as "There exists at least one x that is both a man and white."
& W)
The statement O may be interpreted as "There exists at least one x that is both a man and non-white."
& ~W)
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."
& W) & & ~W)
The statement U may be interpreted as "One of two things, either 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."
→ W) w → ~W)

Modal logic

The logical hexagon may be interpreted as a model of modal logic such that
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.