The minimal modal logic: a cautionary tale about primitives and definitions.
Marilyn Milberger · Notre Dame Journal of Formal Logic · 1978
On page 254 of [2], Hughes and Cresswell list several weak adequacy criteria for propositional modal logics.One is that a) ~D~/> =Op and b) ~O~/>Ξ Up be theses.On page 255 they say that the weakest logic satisfying their conditions is the SO.5 of [3].A formulation of this logic is given in which the square is the only primitive modal operator, and they add that "the other modal operators are defined so as to satisfy" the condition given and similar requirements for strict implication and strict coimplication.As for the sense of necessity represented by the SO.5 square, they quote Lemmon, in [4], as proposing that it be read "it is tautologous by truth tables that...."It would seem that, as they present SO.5,*This paper was written as a result of conversations with Jay Hartman and correspondence with Robert Purdy.