nontransitive relation

logic

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July 1, 2026

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In logic, a relation is a set of ordered pairs, triples, or more. A relation can be described by its properties, such as symmetry, transitivity, or reflexivity.1

Transitive Relation: If a relation holds between a and b, and also between b and c, it must also hold between a and c. For example, "is greater than" is a transitive relation.21
Intransitive Relation: If a relation holds between a and b, and between b and c, it cannot hold between a and c. An example is "is the daughter of".2
Nontransitive Relation: A relation that may or may not hold between a and c if it holds between a and b, and between b and c. "Loves" is an example of a nontransitive relation.2

Logical relations are fundamental to reasoning, ensuring that discourse is intelligible and coherent.3

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formal logic
Classification of dyadic relations
…intransitive is said to be nontransitive. Thus, ϕ is nontransitive if (∃x)(∃y)(∃z)(ϕxy · ϕyz · ϕxz) · (∃x)(∃y)(∃z)(ϕxy · ϕyz · ∼ϕxz) (example: “is a first cousin of”).
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…intransitive is said to be nontransitive. Thus, ϕ is nontransitive if (∃x)(∃y)(∃z)(ϕxy · ϕyz · ϕxz) · (∃x)(∃y)(∃z)(ϕxy · ϕyz · ∼ϕxz) (example: “is a first cousin of”).
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