(This gem is pulled up from the vasty deeps of the ComBox to where it may shine in a more fitting setting. Minor editing, bolding, and comments in blue by BV.)
1). Let us say that a *real* contradiction is a sentence which comes out false according to every possible model (M): i.e., M = language-plus-domain-plus-interpretation, where an ‘interpretation’ is a complete and systematic assignment of extensions to the non-logical terms of the language (L). We assume that L is a well developed natural language such as English and we have a sufficiently rich domain that includes whatever entities are required to implement an interpretation that will suffice for theological purposes.
1.1) Note: We are assuming throughout classical logic in two sense: (a) the logical constants are interpreted classically; (b) there are no *real* true contradictions.
1.2) Sentence S is a *real* contradiction just in case there is no *normal model* M in which it comes out true. A normal model in this context is one which features an interpretation that assigns extensions to the non-logical terms in the usual way prior to resolving any potential ambiguities. On a realist conception of truth, S [if contradictory] has no truth-maker (T-maker) in any normal model or possible world.
2) Let us now define at least one sense of an *apparent contradiction* in model theoretic terms. Let S be a sentence expressible in L and suppose S comes out false in every normal model M. S appears to be a contradiction. Is it really a contradiction? Prof. Anderson maintains that there are sentences which are contradictory in every normal model, but are non-contradictory in some other models of L. How can that be? [Shouldn't Peter have 'false' for contradictory and 'true' for non-contradictory in the preceding sentence? After all, in (1) we are told in effect that contradictoriness is falsehood in every model, which implies that noncontradictoriness is truth in some model. 'Contradictory in every model' is a pleonastic expression.]