Where does transitivity collide?

In decision theory and economics, transitivity is a consistency requirement on preferences: if an agent prefers A to B and B to C, then to count as rational the agent must also prefer A to C. Von Neumann and Morgenstern's 1944 Theory of Games and Economic Behavior placed this among the axioms a preference ordering must satisfy for choices to be represented by a utility function, making transitivity a cornerstone of the theory of rational choice. The referent is the internal coherence of one agent's ranking over options, tested by hunting for preference cycles such as A over B, B over C, yet C over A. It is the ordering property applied to preference, distinct from a child's inference about quantities, a verb's grammar, or the closing of triangles in a network.[1]

Unit

Whether a preference relation admits no cycles, so that A over B and B over C imply A over C

Recovery means

Not applicable

Measured by

Tested by presenting binary choices and checking for intransitive preference cycles

In developmental psychology, transitivity, or transitive inference, is the cognitive ability to derive an unstated relation from known ones: told that stick A is longer than B and B longer than C, a child concludes that A is longer than C without comparing them directly. Bryant and Trabasso's 1971 Transitive inferences and memory in young children showed that when the premises are well memorised, children far younger than Piaget had claimed can make the inference, relocating the difficulty from logic to memory. The referent is a reasoning skill in an individual mind, measured by accuracy on inferred pairs after training on adjacent ones. It is a mental accomplishment, not an axiom on preferences, a property of verbs, or a network statistic.[2]

Unit

Accuracy in inferring an unstated ordered relation from learned adjacent relations

Recovery means

Not applicable

Measured by

Assessed by training adjacent pairs such as A over B and B over C, then testing inference on the non-adjacent pair A over C

In linguistics, transitivity is the property of a clause that carries an action from an agent over to an affected object, as in she closed the door, in contrast to intransitive clauses that take no such object. Hopper and Thompson's 1980 Transitivity in grammar and discourse reconceived it not as a yes-or-no trait of a verb but as a graded cluster of features, including how many participants there are, whether the action is completed, and how affected the object is, all tracking how central a clause is to the ongoing discourse. The referent is the argument structure and semantic force of a clause, read off from its morphology and syntax. It is a grammatical notion, sharing only the word with a preference axiom, a reasoning skill, or a graph's clustering.[3]

Unit

The degree to which a clause transfers an action from an agent to an affected object

Recovery means

Not applicable

Measured by

Rated across the clause's participant count, aspect, affectedness of the object, and related component features

In network science, transitivity is the tendency of a network's ties to close into triangles: if A is linked to B and B to C, then A and C are unusually likely to be linked as well, the friend-of-a-friend-is-a-friend pattern. Watts and Strogatz's 1998 Collective dynamics of 'small-world' networks quantified this with the clustering coefficient and showed that many real networks combine high transitivity with short path lengths. The referent is a statistical property of a whole graph's triads, measured as the fraction of connected triples that are closed. It is a structural regularity of networks, not a rule of rational preference, a reasoning ability, or the grammar of a clause.[4]

Unit

The clustering coefficient, the fraction of connected triples of nodes that form closed triangles

Recovery means

Not applicable

Measured by

Computed as the ratio of closed triangles to connected triples of nodes across the network