Logical hexagon - AbsoluteAstronomy.com

The logical hexagon is a conceptual model

Conceptual model

In the most general sense, a model is anything used in any way to represent anything else. Some models are physical objects, for instance, a toy model which may be assembled, and may even be made to work like the object it represents. They are used to help us know and understand the subject matter...

of the relationships between the truth values of six statement

Statement (logic)

In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...

s. It is an extension of Aristotle

Aristotle

Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

's square of opposition

Square of opposition

In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...

. It was discovered independently by both Augustin Sesmat and Robert Blanché

Robert Blanché

Robert Blanché was an associate professor of philosophy at the University of Toulouse. He wrote many books addressing the philosophy of mathematics.-Works :...

.

This extension consists in introducing two statements Y and U. Y is the conjunction

Logical conjunction

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

of the two traditional particulars I and O. Whereas, U is the disjunction

Logical disjunction

In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

of A and E.

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.

Interpretations of the logical hexagon

The logical hexagon may be interpreted in various ways, including as a model of traditional logic

Term logic

In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...

, quantification

Quantification

Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...

s, modal logic

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

, order theory

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

, or paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

.

For instance, the statement A may be interpreted as "Whatever x may be, if x is a man, then x is white."(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."(M(x) → ~W(x))

The statement I may be interpreted as "There exists at least one x that is both a man and white."(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"(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" ((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."(M(x) → W(x)) ∨ (x)(M(x) → ~W(x))

Modal logic

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
Logical possibility
A logically possible proposition is one that can be asserted without implying a logical contradiction. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true...
O is interpreted as 'not necessarily'
U is interpreted as non-contingency
Y is interpreted as contingency
Contingency (philosophy)
In philosophy and logic, contingency is the status of propositions that are neither true under every possible valuation nor false under every possible valuation . A contingent proposition is neither necessarily true nor necessarily false...

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.

Summary of relationships

Interpretations of the logical hexagon

Modal logic

Further extension

Further reading

See also