The model-theoretic definition of logical consequence presupposes a division of the expressions of a language into logical and non-logical. The idea that invariance of denotations captures an important aspect of the formality of logic has long been taken to supply at least a necessary criterion for the logicality of an expression. This idea, however, seems to rely on classical presuppositions, such as bivalence and a (classical) set-theoretic background theory. This is problematic since the model-theoretic definition finds application in a host of non-classical logics. Yet, in how far is its usage in these contexts justified if it crucially depends on classical assumptions?

In this talk, we investigate the nature, role and justification of invariance criteria in non-classical settings. We examine whether commonly used criteria continue to provide reliable classifications of logical expressions in these frameworks and study how they might be modified or supplemented in order to provide well-motivated and adequate demarcations of non-classical constants. A core contention of our proposal is that the type of invariance appropriate for a given logic is not independent of its characteristic (non-classical) features. We demonstrate how this idea can be spelled out for a selection of non-classical logics.