Volume 20, Issue 2, June 1997
Donald Wayne Viney
This paper presents, explains, and addresses the pedagogical utility of the “Wachter crystal,” a three-dimensional representation of basic principles of logic designed and created by Thomas Wachter in 1992. The author first discusses a way of understanding relations of logical inference which groups propositions possessing identical truth tables into the same class (that is, a way of conceptualizing rules for replacement). Next, the author presents and explains a 16 x 16 matrix, the most basic figure for representing the inferential relations between the classes of propositional logic. Such a matrix easily maps reflexivity, asymmetry, and transitivity in relations of implication. Moreover, since the relations and properties it illustrates can also be illustrated by a lattice, one can construct a three-dimensional model to represent them. The Wachter crystal, which resembles chemists’ models of molecules, illustrates the same principles as the matrix while foregrounding the commonly-neglected difference between Philonian conditionals (which belong to the object language of propositional logic) and implication (which belong to the metalanguage). In addition to being a perspicuous and aesthetically engaging way to represent basic principles of propositional logic, the Wachter crystal is a bridge to more advanced logical concepts such as modal logic, sentence connectives, and predicates of sentences.