On the symmetry of logic

Embedding logical operations in non-dissipative physical processes requires the use of reversible logic. Following Feynman's (1986) approach, the sixteen distinct truth tables of classical logic are shown to be contained in the 8! reversible logic operations covered by the symmetric group S₈, which permute the eight values of three logical variables. Small subgroups of S₈ are shown to cover, respectively, reversible logic, reversible switching and reversible arithmetic. A new universal primitive is found which generates a covering group of reversible logic. It is shown that the octahedral group in four dimensions covers both reversible logic and switching and, hence, that the orthogonal group O(4) provides a covering group for quantum gates.