From the Square to Octahedra
Colwyn Williamson (Notre Dame J. Formal Log. 13:497–500, 1972) develops a comparison between propositional and syllogistic logic. He outlines an interpretation of the traditional square of opposition in terms of propositional logic, that is, the statements corresponding to the corners of the traditional square can be represented with propositional logic operators. His goal is to present a twofold square that preserves the truth conditions of the relationships between the formulas, and define other set of formulas that complete the traditional square to outline an octagon of opposition. We present two octahedra inspired in these squares. The octahedra hold the relations of the traditional square of opposition and also keep (and with some restrictions, extend) the equipollence and immediate inference rules.
KeywordsHexagon Octagon Propositional logic Square of opposition Syllogistic
Mathematics Subject Classification (2000)Primary 03B05; Secondary 03B22 03B35 03B10
Many thanks to Yessica Ramos for their detailed revisions to previous versions of this paper, and I especially want to thank the two jurors who reviewed earlier versions of this work. Without his detailed comments would have been impossible to reach the conclusions presented here.