eJournal: uffmm.org, ISSN 2567-6458
Author: Gerd Doeben-Henisch
Some corrections: 28.Dec.2018
As mentioned in a preceding post the AAI paradigm has to be reconsidered in the light of the quantum logic (QL) paradigm. Here some first concepts which have to be considered (see for the following text chapter two of the book: Jerome R. Busemeyer and Peter D. Bruza, Quantum Models of Cognition and Decision, Cambridge University Press, Cambridge (UK), 2012).
The paradigms of ‘quantum logic’ as well as ‘quantum probability theory’ arose in the field of physics, but as it became clear later these formalisms can be applied to other domains than physics too.
The basic application domain is a appears as a paradigm – real or virtual – in which one can distinguish ‘events‘ which can ‘occur‘ along a time-line as part of a bigger state. The ‘frequency‘ of the occurrences of the different events can be ‘counted’ as a function of the presupposed time-line. The frequency can be represented by a ‘number‘. The frequency can be a ‘total frequency’ for the ‘whole time-line’ or a ‘relative frequency’ with regard to some part of a ‘partition of the time-line’. Having relative frequencies these can possibly ‘change‘ from part to part.
The basic application domain can be mapped into a formalism which ‘explains’ the ‘probability’ of the occurrences of the events in the application domain. Such a formalism is an ‘abstraction’ or an ‘idealization‘ of a certain type of an application domain.
The two main types of formalisms dealt with in the mentioned book of Busemeyer and Bruza (2012) are called ‘classical probability theory’ and ‘quantum probability theory’.
The classical theory of probability (CTP)has been formalized as a theory in the book by A.N. Kolmogorov, Foundations of the Theory of Probability. Chelsea Publ. Company, New York, 2nd edition, 1956 (originally published in German 1933). The quantum logic version of the theory of probability (QLTP) has been formalized as a theory in the book John von Neumann, Mathematische Grundlagen der Quantenmechanik, published by Julius Springer, Berlin, 1932 (a later English version has been published 1956).
In the CTP the possible elementary events are members of a set E which is mapped into the set of positive real numbers R+. The probability of an event A is written as P(A)=r (with r in R*). The probability of the whole set E is assumed as P(E) = 1. The relationship between the formal theory CTP and the application domain is given by a mapping of the abstract concept of probability P(A) to the relation between the number of repetitions of some mechanism of event-generation n and the number of occurrences m of a certain event A written as n/m. If the number of repetitions is ‘big enough’, then – according to Kolmogorov — the relation ‘n/m’ will differ only slightly from the theoretical probability P(A) (cf. Kolmogorov (1956):p.4)
The expression ‘mechanism of event-generation‘ is very specific; in general we have a sequence of states along a time-line and some specific event A can occur in one of these states or not. If event A occurs then the number m of occurrences m is incremented while the number of repetitions n corresponds to the number of time points which are associated with a state of a possible occurrence counted since a time point declared as a ‘starting point‘ for the observation. Because time points in an application domain are related to machines called ‘clocks‘ the ‘duration‘ of a state is related to the ‘partition’ of a time unit like ‘second [s]’ realized by the used clock. Thus depending from the used clock can the number of repetitions become very large. Compared to the human perception can this clock-based number of repetitions be ‘misleading’ because a human observer has seen perhaps only two occurrences of the event A while the clock measured some number n* far beyond two. This short remark reveals that the relationship between an abstract term of ‘probability’ and an application domain is far from trivial. Basically it is completely unclear what theoretical probability means in the empirical world without an elaborated description of the relationship between the formal theory and the sequences of events in the real world.