First Steps Towards a Logic of Ordered Pairs

Logical proportions are a type of propositional connector that involves four variables, expressed as a formula that encodes the conjunction of two equivalences. These equivalences refer to indicators of similarity or dissimilarity between two ordered pairs of variables, say (a, b) and (c, d). An example of a logical proportion is the analogical proportion, which is of the form: “a is to b as c is to d”. The concept of logical proportion is used here to develop a logic that deals with ordered pairs of (vectors of) variables. In particular, we outline a logic of change which controls how the differences inside ordered pairs can be logically combined and propagated via a consequence relation between pairs of vectors.