 The relational semantics of normal modal logics, however, does come with a commitment to a substantial property of negation understood as impossibility or as unnecessity. The analysis of negation as a normal impossibility operator can be traced back to work by Birkhoff and von Neumann (1936) and Goldblatt (1974) on negation in quantum logic. It has been developed by Vakarelov (1977, 1989b) and Došen (1984, 1986, 1999) and has been further investigated in the algebraic setting of Michael Dunn’s gaggle theory (see Bimbó and Dunn 2008) by Dunn (1993, 1996, 1999) and Dunn and Zhou (2005). A relational model (or Kripke model) is a structure \(\cal M\) \(=\) \((W, R, v)\), where \(W\) is a non-empty set of information states, \(R\) is a two-place “accessibility” relation on \(W\), and \(v\) is a valuation function. Dunn denotes the accessibility relation by \(\bot\) (pronounced “perp”) and regards it as a relation of incompatibility or orthogonality between states. Negation as impossibility, denoted by \(\osim \), is then semantically defined by postulating that \(\osim A\) is true at a state \(w\) in model \(\cal M\) iff \(w\) is incompatible with all states \(u\) (from \(W\)) at which \(A\) is true: \({\cal M}, w \models \osim A\) iff (for every \(u\): \({\cal M}, u \models A \mbox{ implies } w \bot u).\) Alternatively, the relation \(R\) may be understood as a relation of compatibility between states, denoted by \(C\). \({\cal M}, w \models \osim A\) is then defined by requiring that for every \(u\): \(wCu \mbox{ implies } {\cal M}, u \not \models A.\) Negation as unnecessity, denoted by \(\neg\), is accordingly defined by the following clause: \({\cal M}, w \models \neg A\) iff (there exists \(u\) with \(wCu \mbox{ and } {\cal M}, u \not \models A)\).