Information Gain from Inference

An argument is made for a double implicational associative information frame, 〈S, , , →, ←〉 being a relevant structure insofar as the specification of the information gain from inference (IGfI) is concerned. A strong informational pluralism is adopted throughout. The rejection of the structural rules K, K′, W, WI, C and CI is justified on the basis of the identification, frequency, and order of formulae as being a necessary condition on the tracking of the information gain resulting from inferential procedures. The retention of the structural rule B is justified on the basis of inference being a sequential process. The information structure in question, I, is identified with its corresponding algebraic structure, a partially ordered residuated semigroup, (S, ≤, , →, ←). A possible philosophical basis for extending I from a semigroup structure to a monoidal structure in the context of IGfI is then explored.