What is "trichotomy"

Trichotomy \Tri*chot"o*my\, n. [Gr. tri`cha threefold, in three parts + te`mnein to cut or divide: cf. F. trichotomie.] Division into three parts.

n. 1 division or separation into three groups or pieces. 2 (context algebra English) the following property of an order relation (e.g., "less than"): for any two elements (of a given algebraic structure) there are exactly three possibilities: either the first element is less than the second one, the second element is less than the first one, or the two elements are equal.

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero. More generally, trichotomy is the property of an order relation  <  on a set X that for any x and y, exactly one of the following holds: x < y, x = y, or x > y.

In mathematical notation, this is

∀x, y ∈ X ((x < y  ∧  ¬(y < x)  ∧  ¬(x = y) ) ∨  (¬(x < y)  ∧  y < x  ∧  ¬(x = y) ) ∨  (¬(x < y)  ∧  ¬(y < x)  ∧  x = y  )) .

Assuming that the ordering is irreflexive and transitive, this can be simplified to

∀x ∈ X ∀y ∈ X ((x < y)  ∨  (y < x)  ∨  (x = y)) .

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers. The law does not hold in general in intuitionistic logic.

In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).

More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds. If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example, in the case of three element set {a,b,c} the relation R given by aRb, aRc, bRc is a strict total order, while the relation R given by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.

In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as more foundational than the law of total order.

A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it is trivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

A trichotomy is a splitting into three parts, and, apart from its normal literal meaning, can refer to:

• Trichotomy (mathematics), in the mathematical field of order theory
• Trichotomy theorem in finite group theory
• Trichotomy (philosophy), for the theological idea that man has a threefold nature; also for a three-way classificatory division, especially when done according to a pattern
• In taxonomy, a trichotomy is speciation of three groups from a common ancestor, where it is unclear or unknown in what chronological order the three groups split.

A trichotomy is a three-way classificatory division. Some philosophers pursued trichotomies.

Important trichotomies discussed by Aquinas include the causal principles (agent, patient, act), the potencies for the intellect (imagination, cogitative power, and memory and reminiscence), and the acts of the intellect (concept, judgment, reasoning), with all of those rooted in Aristotle; also the transcendentals of being (unity, truth, goodness) the requisites of the beautiful (wholeness, harmony, radiance).

Kant expounded a table of judgments involving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) Relation, (4) Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories. Kant also adapted the Thomistic acts of intellect in his trichotomy of higher cognition—(a) understanding, (b) judgment, (c) reason—which he correlated with his adaptation in the soul's capacities—(a) cognitive faculties, (b) feeling of pleasure or displeasure, and (c) faculty of desire—of Tetens's trichotomy of feeling, understanding, will.

Hegel held that a thing's or idea's internal contradiction leads in a dialectical process to a new synthesis that makes better sense of the contradiction. The process is sometimes described as thesis, antithesis, synthesis. It is instanced across a pattern of trichotomies (e.g. being-nothingness-becoming, immediate-mediate-concrete, abstract-negative-concrete); such trichotomies are not just three-way classificatory divisions; they involve trios of elements functionally interrelated in a process. They are often called triads (but 'triad' does not have that as a fixed sense in philosophy generally).

Charles Sanders Peirce built his philosophy on trichotomies and triadic relations and processes, and framed the " Reduction Thesis" that every predicate is essentially either monadic (quality), dyadic (relation of reaction or resistance), or triadic (representational relation), and never genuinely and irreducibly tetradic or larger.