We are now going to permit some constraints on this assignment. While we may not consider the existence of such abstract entities logically necessary, it seems also inappropriate to consider the existence contingent, since not empirical observation can provide evidence for or against their existence. The idea is that the semantics of the names in the language is determined by defining a subset of the interpretations of the language which are consistent with the intended meaning of the names. This feature is only intended for abstract entities, and is subject to the constraint of consistency, which requires that there must be at least one standard interpretation. The interpretation of concrete references may not be constrained, but there is some difficulty in clarifying this distinction.
Mathematics now becomes necessary.
There remains some awkwardness in relation to concrete language and ontology which may be fatal for this account of possible world. We get these by adopting some convenient but otherwise arbitrary abstract ontology, which we use for building abstract models of the real world. Models of the real world require no new concrete ontology but simply the introduction of new constants which are denote abstract surrogates for the real world. My own preference is for a notion of possible world which I would guess is pretty much the kind of thing a mathematical physicist would go for. First let me observe that I don't see any way we can get a handle on the real world other than through various kinds of models.