Star and Perp: Two Treatments of Negation
Jon Michael Dunn · Philosophical Perspectives · 1993
The first uses a unary operation * (star) on some underlying set of states (worlds, situations, set-ups, cases, whatever). The second definition uses a binary relation I (perp) on such an underlying set.2 It is the purpose of this paper to show that there is a close connection between these two apparently different treatments. The definition (_-*) is perhaps most famous from the Routley-Meyer semantics for relevance logic (see e.g., Routley and Routley (1972), Routley and Meyer (1973)), though its mathematical essence can be traced to the BialynickiBirula and Rasiowa (1957) representation of De Morgan lattices (cf. Dunn (1966, 1967, 1986).3 The definition (-_I) is perhaps most famous from the Goldblatt (1974) semantics for orthologic, though its most familiar current use is in the Girard (1987) semantics for linear logic. It too has a more ancient history, going back to Birkhoff (1941) in his example of a Galois connection as determined by a polarity, defined using an arbitrary binary relation. This in turn generalizes the orthogonality operator on closed subspaces of a Hilbert space. K. DoAn (1986) should also be recommended for a treatment of various negations in the neighborhood of the intuitionistic one, but with semantics done in the perp style.4 My main interest in the relationship between the two treatments of negation is motivated by the fact that the perp definition is the one that falls right out of the general gaggle theoretic considerations of Dunn (1990) about how to define semantical conditions for n-placed logical operators using n + 1-placed accessibility relations,5 and yet it is often convenient to understand the De