## Can we mathematically represent the unconscious activity of human brain?

Modeling human decisions under uncertainty is one of the issues that became crucial in AI in the recent years. Mathematical models for decision making under risk, such as those developed in the Expected Utility Theory, provide optimal decision. Nevertheless experiments and studies show that these rational decision models are not always able to describe the typical human approach to making decisions.

In the Prospect Theory human reasoning and human decision making is investigated by different experiments which evidence biases of human intuition. Experimental methods lead to describe the dual process of the brain activity as regulated by two different ways of thinking namely fast and slow thinking denoted also as System 1 and System 2, respectively. System 1 regulates intuitive, involuntary, unconscious and effortless activities while System 2 is the conscious part of the brain in charge of logical reasoning .

Also according to Matte Blanco conscious and unconscious activities are two different modes of being; he draws a distinction between logical conscious/asymmetrical thought – structured on the categories of time and space and ruled by Aristotle’s principle of non-contradiction and unconscious/symmetrical thought – based upon the principles of symmetry and generalization. Both types of thoughts are supposed to combine in different human thinking experiences since they yield a bi-logic asset. Emotions are ways to reach and decode the unconscious since they function at the same way.

A model of coherent upper and lower conditional previsions based on Hausdorff outer and inner measures is proposed to describe this dual aspect of human brain’s activity. Coherent upper and lower conditional previsions are non-linear functions satisfying the axioms of coherence and coherent lower and upper conditional probabilities are obtained only indicator functions are considered.

Many properties of coherent lower conditional previsions can be obtained by the conjugate coherent upper conditional previsions but the two non-linear functionals represent different binary relations between random variables; in fact preference orderings represented by the coherent lower previsions satisfy the antisymmetric property which is not satisfied by the binary relation represented by their conjugate coherent upper conditional previsions . Coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures are proposed to represent respectively a partial strict order and a complete equivalence relation between random variables. The two binary relations can describe the activity of the conscious human thought ruled by the antisymmetric property and of the unconscious human thought which is governed by the symmetric principle and the generalization principle according to the theory developed by Matte Blanco.