It is well known that the set of 16 connectives ('and,' 'or,' etc.) of two-variable Boolean logic and the set of 16 subsets of an four-element set (a 4-set) have the same structure-- a Boolean lattice. It is also well known that the group of Boolean-lattice automorphisms, or symmetries, of the 16 connectives is isomorphic to the group of 24 permutations of the 4-set. Nevertheless, some authors* have proposed additional automorphisms of the 16 connectives, based on symmetries of geometric figures (such as a Hasse diagram of the Boolean lattice) labeled by the connectives. Such proposed automorphisms may be viewed in a larger context-- that of the 322,560 automorphisms of the four-dimensional affine space A over the two-element Galois field GF(2). The phrase 'geometry of logic' has been used by, among others, David Miller of the department of philosophy at the University of Warwick. The following is from a talk by Miller: As the above figures suggest, a rather different 'utterly literal' approach might involve, not a metric or pseudometric space, but instead the finite geometry A. In summary, geometrically-defined permutations acting on the 16 Boolean connectives have seemed, to some,* a nontrivial topic for study. The 4x4 space A provides a new** setting for such investigations, a setting that accommodates many more automorphisms than have previously been considered in what has been called 'the geometry of logic.' * That mathematically unlearned authors such as Jean Piaget (see Jean-Blaise Grize's discussion of Piaget, 'Operatory Logic,' pp. 77-86 in Inhelder, De Caprona, and Cornu-Wells (eds.), Piaget Today, Psychology Press (UK), 1987) and Shea Zellweger (see his 'Untapped Potential in Peirce's Iconic Notation for the Sixteen Binary Connectives', pp. 334–386 in Houser, Roberts, Van Evra (eds.), Studies in the Logic of Charles Sanders Peirce, Indiana University Press, Bloomington, IN, 1997) have discussed variants of the geometric-figure approach to Boolean automorphisms does not necessarily mean that it is unworthy of attention. For an approach more sophisticated than those of Piaget and Zellweger, see, for instance, Achille C. Varzi's 'The Geometry of Negation' (with Massimo Warglien), Journal of Applied Non-Classical Logics, 13 (2003), 9-19. (Thanks to an anonymous Usenet author for this reference.) A search for references to Zellweger leads to a detailed description of a group of 16 'logical automorphisms' of the 16 binary connectives in the paper 'Simetria y Logica: La notacion de Peirce para los 16 conectivos binarios,' by Mireya Garcia, Jhon Fredy Gomez, and Arnold Oostra. (Published in the Memorias del XII Encuentro de Geometria y sus Aplicaciones, Universidad Pedagogica Nacional, Bogota, June 2001, on the Web at http://www.unav.es/gep/Articulos/SimetriaYLogica.pdf.) The authors do not identify this group as isomorphic to a subgroup of the affine group of A, this can serve as an exercise. The Oostra paper also suggests an introductory exercise for those unfamiliar with Boolean lattices: determine whether the authors' order-16 group includes all transformations that might reasonably be called 'logical automorphisms' of the 16 binary connectives. Update of Sept. 27, 2007: The above association of the affine 4-space A with the 16 Boolean connectives is not new. See, for instance, Carol von der Lin on Projective Structures (unsigned, undated, web page, parts of which are from Nov. 2003 or before). The resulting association of the Boolean connectives with group actions on the 4x4 square may, however, be new. For a much more sophisticated approach to geometry and logic, see Steven Vickers on geometric logic. Source.


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