The logic of isomorphism and its uses

We present a class of first-order modal logics, called transformational logics, which are designed for working with sentences that hold up to a certain type of transformation. An inference system is given, and com- pleteness for the basic transformational logic HOS is proved. In order to capture ‘up to isomorphism’, we express a very weak version of higher category theory in terms of first-order models, which makes tranforma- tional logics applicable to category theory. A category-theoretical concept of isomorphism is used to arrive at a modal operator nisoφ expressing ‘up to isomorphism, φ’, which is such that category equivalence comes out as literally isomorphism up to isomorphism.
In the final part of the paper, we explore the possibility of using trans- formational logics to define weak higher categories. We end with two informal comparisons: one between HOS and counterpart semantics, and one between isomorphism logic, as a transformational logic, and Homo- topy Type Theory.