# Quantum Cognition

Quantum Cognition is the general name given to an approach to constructing cognitive models based on the mathematics of quantum probability theory, rather than the more familiar classical probability theory.

Classical probability theory is the most familiar way of formalising uncertainty, but there are technically an infinite number of possible frameworks that can be used to assign ‘probabilities’ to a set of events. What makes a particular probability framework useful or not is whether the structural properties of the framework match the properties of the set of events we want to assign probabilities to.

Classical probability theory is useful when the events in question can be described by the structure of set theory, so that we can define unions and intersections of events in the normal way. For example, we can think of the set of days in which it rains [latex]S(R)[/latex] and the set of days on which is is windy [latex]S(W)[/latex] to be proper subsets of the set of all days in a given year, and then the set of days on which it both rains and is windy [latex]S(R\wedge W)[/latex] is given by the intersection [latex]S(R)\cap S(W)[/latex].

However even if the set of days does have this structure, it is less obvious that our mental representation has the same structure. Some evidence for this comes from well known effects in decision making, such as order effects and the conjunction fallacy. In these examples people seem to give judgments at odds with the predictions of classical probability theory, for example in the conjunction fallacy people judge [latex]p(A\wedge B)>p(A)[/latex], which implies [latex]S(A)\cap S(B) \subset S(A)[/latex] which is impossible. In this case we can build a quantum model which can reproduce the conjunction fallacy, see this page.

Quantum probability theory is based not on the algebra of sets, but on the algebra of subspaces of a vector space. One can define analogs of the intersection and union operations, but they behave rather differently from their set theoretic counterparts. One notable example is that they are not distributive, which means that for two events A and B, [latex]A\wedge(B\vee \neg B)\neq (A\wedge B)\vee(A\wedge \neg B)[/latex], which has the consequence that in quantum probability theory, [latex]p(A)\neq p(A\wedge B)+p(A\wedge \neg B)[/latex].

Quantum probability theory has many features, such as order effects, context effects and constructive judgments, that seem to align well with what we know about real human decision making. Note however that ultimately whether quantum or classical cognitive models best describe human reasoning is an empirical question and probably also a very domain specific one. Initial research involving quantum models tended to focus mainly on explaining results previously seen as paradoxical from the point of view of classical probability theory, and there have been a number of successes in this area. More recently, the focus has switched to some extent to testing new predictions arising from quantum models, and designing better tests of quantum vs classical decision theories.

These pages are designed to explain a number of results in the quantum cognition literature, with the philosophy that playing with models yourself is a more useful learning aide than simply reading about them. Some of these pages require you to have the Wolfram CDF player installed, which is a free program that lets you interact with some of the models.

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