AIM Framework glossary
Non-Commutativity
Canonical definition
Non-Commutativity is the property of Belief updating whereby the order in which Evidence is evaluated affects the posterior reached, so that two Individuals receiving the same Evidence in different orders arrive at different Belief states. The architectural source of Non-Commutativity is Common-Currency Integration under Source Opacity: each evaluation projects an already source-stripped Scalar onto a reference frame fixed by the current context, and two reference frames that are not aligned yield different results in different orders. Non-Commutativity lies not in the calculus, since strict conditionalisation on fixed likelihood ratios commutes, but in the state-dependence of the evidential force an observation carries against the Belief state it meets. Non-Commutativity holds within a single Individual, is reinforced by memory-recall self-coupling, and is amplified but not created by interpersonal coupling; the network-scale restatement of Non-Commutativity is Non-Commutativity Propagation (N3).
Explanation
The same two pieces of news, heard in opposite orders, leave people believing different things, and neither person has made an arithmetic mistake. The first piece changes where the person stands, and the second is weighed from that new position, so it carries a different weight from the one it would have carried first. Because the mind blends its motives into one signal before any of this happens, it cannot undo the blending and start again from neutral. The route by which evidence arrives is therefore part of what a belief is.
Epistemic status
Framework-Specific Construct. The order effects it names are an established empirical pattern (Chapter 14, Section 14.1); their architectural source is derived.
Derivation
Chapter 14, Section 14.3.3; the location of the order effect in the state-dependence of evidential force at Chapter 14, Section 14.2.2.
Also in the entry
Distinction from the Ignition-Release Asymmetry. Both make a present state depend on the path taken. The Ignition-Release Asymmetry is hysteresis of the fast valuation system: two resting states under the same forcing, separated by an ignition boundary and a release boundary that do not coincide. Non-Commutativity carries no threshold and no second resting state; it is the order-dependence of Belief updating, present at every update.