> ## Documentation Index > Fetch the complete documentation index at: https://agentconcepts.io/llms.txt > Use this file to discover all available pages before exploring further. # transitivity > A relation's chaining-inference license: when R(A,B) and R(B,C) hold, R(A,C) follows. The concept is the license and its failure modes, not any one relation. mathematics psychology # Transitivity ## 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. ## 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. ## 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. ## Structure Internal structure of transitivity: a table of its component slots and the concepts that fill them. 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. ## Relationships Relationship neighborhood of transitivity: a graph of the concepts it connects to and the concepts it is a part of. * [root-cause-analysis](/concepts/root-cause-analysis) — chasing a cause down a chain presumes causation is transitive; where it isn't, the descent over-reaches. * [gradient](/concepts/gradient) — a coherent single-direction gradient presumes a transitively-chainable "more than"; without transitivity the local steps don't compose into a global direction. ## Examples Rational-choice theory takes transitivity of preference as an axiom: if a chooser prefers A to B and B to C, they should prefer A to C. Tversky showed experimentally that real people systematically violate this. When alternatives vary on several attributes — gambles trading off probability against payoff, job candidates trading off intelligence against experience — subjects fell into preference *cycles*, choosing A over B, B over C, and then C over A. The mechanism was a lexicographic semiorder: along each adjacent pair the chooser ignored a small difference on the dominant attribute and decided on the lesser one, but those ignored small differences accumulated around the cycle until the endpoints reversed. **Inference**: This is the load-bearing demonstration that transitivity is a property a relation may *lack*. "Preferred-to" is exactly the kind of relation that looks chainable and isn't — the same trap as similarity chains, where adjacent pairs share a property the endpoints do not. Any system that models a chooser as if preference transitivity held (and then chains pairwise comparisons into a global ranking) silently breaks wherever the small-difference-ignoring strategy is in play. "Ancestor of" is the textbook example of a transitive relation: if X is an ancestor of Y and Y is an ancestor of Z, then X is an ancestor of Z. What makes it instructive is its relationship to "parent of," which is *not* transitive — X being the parent of Y and Y being the parent of Z makes X the grandparent of Z, not the parent. "Ancestor of" is precisely the transitive closure of "parent of": the smallest transitive relation that contains it, obtained by chaining parent-links as far as they go. **Inference**: This is the clean positive case of the license — the relation genuinely chains, so reachability across a family tree is decidable from the direct parent edges alone. It also names the general move: a non-transitive base relation (parent-of, depends-directly-on, links-to) has a transitive closure (ancestor-of, depends-on, reachable-from) that *is* chainable. The license isn't a property you assume; it's something you construct by closing the base relation under composition.