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In this article, we demonstrate that certain Simple Sentence elements that we refer to as “Commutative Elements” form the building blocks of all Sentences. The Commutative Elements that are required to form the subject and predicate of the ten types of Simple Sentences correspond to the ten “Categories” of Aristotle, i.e., Substance, Time, Location, Quantity, Quality, Posture, Possession, Relative, Action, and Passion. In addition, there are some optional Commutative Elements that we may use to supplement the ten Categories, namely Reason and Manner. The Commutative Elements have the feature or Attribute of being movable within the Sentence. This in turn allows us to apply the mathematics of Boolean Structures, including Boolean Algebras, Boolean Rings, Boolean Relations, Boolean Lattices, Boolean Topologies, and Boolean Graphs. The analysis culminates in a proposed Sentence Mapping Theorem that we may use to map language to a precise mathematical structure. We can use the mathematics of Boolean Structures as a foundation for developing software applications, including contract generation software, legal reasoning software, document comparison software, and contract portfolio reporting software, all using mathematics and logic at the Commutative Element level.
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