Aristotelian Fragments and Subdiagrams for the Boolean Algebra B5

Abstract

On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra B5, i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification with fractions and percentages—as in two thirds/66 percent of the children are asleep—and normative quantification—as in not enough/too many children are asleep. On a more theoretical level, we study two distinct Aristotelian subdiagrams in B5, which are the result of moving from B5 to B4 either by collapsing bit positions or by deleting bit positions. These two operations are also argued to shed a new light on earlier results from Logical Geometry, in which the collapsing or deletion of bit positions triggers a shift from B4 to B3.