The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

Possibility Theory

Possibility theory (formulated by Zadeh in 1977, and later expanded by French mathematicians Didier Dubois and Henri Prade) developed as a branch of the theory of fuzzy sets in order to deal with the lexical elasticity of ordinary language (i.e., the fuzziness of words such as "small" and "many"), and other forms of uncertainty which are not probabilistic in nature. The subject of possibility  theory is the possible (not probable) values of a variable.

Possibility theory is both a theory of imprecision (represented by fuzzy sets) and a theory of uncertainty.  The uncertainty of an event is described by a pair of degrees: the degree of possibility of the event and the degree of possibility of the contrary event. The definition can be dually stated in terms of necessity, necessity being the complement to one of possibility.

Its basic axioms are that: 1. the degree of possibility is one for a proposition that is true in any interpretation and is zero for a proposition that is false in any interpretation; 2. the degree of possibility of a disjunction of propositions is the maximum degree of the two. When the degree of necessity of  a proposition is one, the proposition is true. When the degree of possibility of a proposition is zero, the proposition is false. When the degree of necessity is zero, or the degree of possibility is one, nothing is known about the truth of the proposition. 

Possibility Logic has a graded notion of possibility and necessity.

 


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