Boolean Semantics and Categorial Polyvalency

Boolean semantics is a version of formal semantics in which models are supposed to have algebraic, preferably Boolean, structure (Keenan and Faltz 1985). Thus for any category C there is a corresponding denotational Boolean algebra DC of possible denotations of expressions of category C. Furthermore, given that most categories are functionally related (in principle all ”major” categories are Boolean), the corresponding denotational algebras are not independent of each other. In particular the algebra DA/B has as elements functions from DB to DA. Given that functions interpreting functional expressions may satisfy various ”universal” constraints, we will usually consider just some sub-algebras of the algebra of all functions from DB to DA. An important feature of denotational algebrasDC is that they are atomic. This means that any element different from the zero of the algebra contains an atom. An atom is an element which contains only itself and the zero element. Atoms of the algebra DA/B are determined by atoms and/or elements of the resulting algebraDA. Obviously the binary algebra {0, 1} has only one atom. This is the element corresponding to truth. Thus atoms of denotational algebras of functional expressions are eventually defined by truth. Let me illustrate this point by the atomicity of DA/B in the case when there are no constraints on functions from DA on DB . In this case atomicity of DA/B is inherited from the atomicity of DA. More precisely atoms of DA/B are determined by atoms of DA in the following way (Zuber 2001):