Aristotelian Fragments and Subdiagrams for the Boolean Algebra $\mathbb{B}$₅

On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra ${\mathbb{B}}_5$, 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 ${\mathbb{B}}_5$, which are the result of moving from ${\mathbb{B}}_5$ to ${\mathbb{B}}_4$ 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 ${\mathbb{B}}_4$ to ${\mathbb{B}}_3$.