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## Propositions

In Bayesian probability theory, probabilities are defined for propositions which follow the laws of Boolean algebra [11,132] and can be either true or false, i.e., propositions which satisfy the laws of ordinary logic. We shall denote the propositions by capital letters. Three basic operations are defined in Boolean algebra: conjunction, disjunction and negation. We shall denote ``A and B'' by AB, ``A or B'' by A+B and ``not A'' by .

Natural language involves propositions whose truth value is ambiguous, ``sky is blue'' for example. The definitions of words sky and blue are more or less ambiguous and therefore it is possible to think that the truth value of ``sky is blue'' is neither completely true nor completely false but something in between. Fuzzy logic [136,133] tries to capture the ambiguity of propositions in natural language, but we shall consider only unambiguously defined propositions.

Harri Valpola
2000-10-31