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A Quantitative Theory of Cognition with Applications
1  University of Copenhagen, Department of Mathematical Sciences

Abstract: An abstract non-probabilistic model of "situations" from the "world" with a focus on the interplay between "Nature" and "Observer" is presented and some applications, probabilistic and non-probabilistic, discussed. Nature is the holder of "truth". Observer seeks the truth but is restricted to considerations of, especially, "belief", "action" and "control". Inference is based on these elements. To simplify, our model identifies action and control and derives these concepts from belief - recall that "belief is a tendency to act" (Good 1952).  "Knowledge" is mainly thought of as "perception", the way situations from the world are presented to Observer. "Interaction" connects truth and belief with knowledge. If interaction leads to undisrupted truth, you are in the "classical world". Adding probabilistic elements, you are led to elements of Shannon theory. If mixtures of truth and belief  represent the rules of the world, you are led to Tsallis entropy instead.The quantitative basis for the theory is the view that "knowledge is obtained at a cost" and the associated modelling by a "proper effort function" (inspired by the use in statistics of proper scoring rules). Specific situations depend on "preparations". Certain preparations defined by reference to the effort function is suggested to represent what can be known, the "knowable". Other philosophically inclined considerations serve to provide natural interpretations. This includes the introduction of game theoretical thinking in the interplay between Nature and Observer. "Entropy", "redundancy" and the related notion of "divergence" make sense in the abstract theory. Other notions from Information Theory also appear, e.g. the "Pythagorean (in)equalities" as known from information theoretical inference. But applications are not only to information theory. For example the (in)equalities just pointed to also lead to the classical geometric results bearing Pythagoras' name.  Among applications we point to notions of Bregman divergence and elaborations which are presently considerd for a relation to classical duality theory.
Keywords: Truth; belief; knowledge; entropy; divergence; proper effort function; fundamental inequality; core; Tsallis entropy; Bregman divergence