mathematics psychology

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Transitivity

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Description

Transitivity is the property that lets you chain a relation: if a relation R holds between A and B, and between B and C, then it holds between A and C — and, by repetition, across the whole transitive closure. The catalog entry is the license itself, not any particular relation that happens to carry it. Some relations are transitive (≤, ancestor-of, implies, is-a-prerequisite-of); some are flatly not (beats, is-friends-with, is-similar-to); and the entire diagnostic value is knowing which, because the cost of assuming a license that isn’t there is a chain of false inferences. The diagnostic question — “if R(A,B) and R(B,C), does R(A,C) actually follow, or am I assuming it?” — is what the concept fires on. When the answer is yes, transitivity is what makes long-range deduction cheap (you don’t re-derive every pair; you compose). When the answer is no, every chaining step is a category error waiting to compound.

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Triggers

User-initiated: User reasons across a chain — “A depends on B, B depends on C, so A depends on C”, “she’s an ancestor of him, he’s an ancestor of them” — or proposes that a property “carries over” or “propagates along” a sequence of links. Vocabulary cues: “transitive,” “follows from,” “by extension,” “so therefore,” “chain.” Agent-initiated: Agent notices a multi-step inference resting on an unexamined chaining assumption — a “beats” or “prefers” or “is similar to” relation being treated as if endpoints inherit the property of adjacent links. Candidate inference: “is this relation actually transitive, or is this a preference-cycle / sorites trap?” Situation-shape signals: Dependency resolution, prerequisite graphs, syllogistic deduction, kinship/ancestry queries, ranking-from-pairwise-comparisons. Anywhere an inference jumps from adjacent pairs to a non-adjacent conclusion.

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Exclusions

• Intransitive / cyclic dominance — rock-paper-scissors, Condorcet cycles, non-transitive dice, unbalanced “beats” relations. A>B and B>C do not license A>C; the license must be earned per relation, not assumed. The load-bearing failure mode.
• Similarity chains — “close to” / “looks like” accumulate small differences (sorites); adjacent pairs are similar but endpoints aren’t. Treating similarity as transitive is the classic trap.
• Single links with no chain — one R(A,B) with no R(B,C) has nothing to derive. Transitivity is a property of a relation, not of a lone edge.
• Empirically intransitive preferences — real human choice over multi-attribute options cycles (Tversky 1969). Assuming preference transitivity silently breaks where small attribute differences get ignored.

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Structure

A relation R, a chain of adjacent pairs (R(A,B), R(B,C)), and the licensed non-adjacent inference R(A,C). The concept is which relations carry the license: order relations and “implies” and “ancestor-of” do; “beats” and “is-similar-to” and observed preference often don’t.

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Relationships

• root-cause-analysis — chasing a cause down a chain presumes causation is transitive; where it isn’t, the descent over-reaches.
• gradient — a coherent single-direction gradient presumes a transitively-chainable “more than”; without transitivity the local steps don’t compose into a global direction.

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Examples