Relativizations of the Principle of Identity

We discuss some logico-mathematical systems which deviate from classical logic and mathematics with respect to the concept of identity. In the first part of the paper we present very general formulations of the principle of identity and show how they can be ‘relativized’ to objects and to properties. Then, as an application, we study the particular cases of physics (the transgression of the principle of identity by quantum objects) and logic (some logics in which the principle of replacement is not valid are presented). In the last part of the paper, we discuss the alphabar logics, that is, those logical systems which violate a formulation of one of the most fundamental versions of the principle of identity; in these logics, there are formulas which are not deducible from themselves.