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

Given classical (2 valued) structures $$\mathfrak{A}$$ and $$\mathfrak{A}'$$ and a homomorphism h of $$\mathfrak{A}$$ onto $$\mathfrak{A}'$$ , it is shown how to construct a (non-degenerate) “3-valued counterpart” $$3\mathfrak{A}'$$ of $$\mathfrak{A}'$$ . Classical sentences that are true in $$\mathfrak{A}$$ are non-false in $$3\mathfrak{A}'$$ . 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).