Language Lessons from Law in “Reeling and Writhing … and the different branches of Arithmetic”


Michael Heather, Law School, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST,

Nick Rossiter, Informatics, Northumbria University,




Creating, communicating, storing and reasoning in Law all require sophisticated use of language. The formal nature of legal language was recognised not only in the categories of Aristotle, originator of symbolic logic, and even more so by Leibnitz, the pioneer of modern mathematical logic. The complexity of syntax, semantics, pragmatics, rhetoric, teleological and deontic norms, etc embedded only in combinations of words by the Roman jurisconsults led the young Leibnitz to an early appreciation of combinatorial power in his ars combinatoria and even to his inventing the differential calculus.


However his belief that legal disputes can be resolved amicably, objectively and irrefutably by calculemus has not had the success of his differential calculus. No such calculus has yet been found for resolving legal disputes other than by the use of natural language or warfare. Direct application of arithmetic applied to the vision of Leibnitz has been extensively and perhaps exhaustively investigated by workers like Sánchez-Mazas [2] who in a process he called arithmetisation has explored the possibility of assigning to every norm a number rather after the manner of Gödel. Probably for similar reasons that Gödel shows in his famous theorems there are undecidable gaps in every axiomatic number theory.


The recent mathematics of topos theory has been able to fill in some of the gaps, as it appears that topos logic is the world of natural language and of law language. These advances have yet to work their way through the theory. Nevertheless these advances in mathematics of the last fifty years may give us a glimpse of what Leibnitz could have seen in the connection between numbers and strings of characters which embed law. The post-modern theory of categories represented formally in diagrams of arrows has unearthed the existence of fundamental principles like limits and co-limits. These can be described by the Dolittle diagram which subsumes the pullback and the pushout [1]:








Both B and A are subcategories of an ambient context category C. If B is the category of language in some textual form the characters can be strung together in subcategories like the usual natural language registers. The action of writing by an author A who is really a subcategory of the category of persons creates a language text A ´ B by the pullback of B over A. The text A ´ B is then the limit of that person A (for example the objects of A's thoughts on a particular topic) with the appropriate objects of the category B. It is this combinatorial synthesis recognised by Leibnitz that gives text its contents whether in law or in any other subject. Writing is then an operation of the pullback functor. On the other hand given an existing text A ´ B the pushout to a reader A through the language B projects onto A (objects of A's understanding) and onto B as components of language. This gives the analysis into separate categories A + B.


[1] Heather, M A, & Rossiter, B N, Object Awareness in Multimedia Documents, Third International Workshop on Principles of Document Processing (PODP'96 Palo Alto), ed. Nicholas, C, & Wood, D, Lecture Notes in Computer Science 1293 Springer-Verlag 59-86 (1997)).


[2] Sánchez-Mazas, Miguel, Théories Sylogistiques et Déontiques Analysées comme Structures Algébriques, Theoria Revista de Theoria Historia y Fundamentos de la Ciencia, Secunda Epoca, V 12-13, 193-222 (1990).