A theorem in 3-valued model theory with connections to number theory, type theory, and relevant logic

Abstract

Given classical (2 valued) structures {Mathematical expression} and {Mathematical expression} and a homomorphism h of {Mathematical expression} onto {Mathematical expression}, it is shown how to construct a (non-degenerate) "3-valued counterpart" {Mathematical expression} of {Mathematical expression}. Classical sentences that are true in {Mathematical expression} are non-false in {Mathematical expression}. Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). © 1979 Polish Academy of Sciences.