Necessity, a Leibnizian Thesis, and a Dialogical Semantics

In this paper, an interpretation of "necessity", inspired by a Leibnizian idea and based on the method of dialogical logic, is introduced. The semantic rules corresponding to such an account of necessity are developed, and then some peculiarities, and some potential advantages, of the introduced dialogical explanation, in comparison with the customary explanation offered by the possible worlds semantics, are briefly discussed

Apolarity for determinants and permanents of generic matrices

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the Hafnian of a generic symmetric matrix are each generated in degree two. In each case we specify the generators and a Gr\"{o}bner basis of the apolar ideal. As a consequence, using a result of K. Ranestad and F. O. Schreyer we give lower bounds to the cactus rank and rank of each of these invariants. We compare these bounds with those obtained by J. Landsberg and Z. Teitler