One really helpful distinction that Etchemendy makes in his “Concept of Logical Consequence” is between representational and interpretational semantics. Representational semantics, in a model, give ways the world might be given what the words/etc. mean. Thus, a given truth table for a lot of propositions will represent the myriad of possibilities the world could have been depending on whether one or another proposition was true or false. Interpretational semantics are the more standard model theoretic kind. They vary the meanings of terms and give us what would be true given that the words have the interpretation in question. Very roughly, the prior varies worlds and maintains meaning, while the latter varies meaning and maintains worlds.

This gets deployed by Etchemendy when looking at cases where the two conceptions come apart. Representational semantics supports counterfactuals involving differing ways things may be. Would a sentence A&B be true in a world where A held but not B? No. More concretely, would “snow is white” be true in a world where snow is black? No, again. Therefore, it isn’t necessary that snow is white. Interpretational semantics tells us that “snow is white” would be true if ‘snow’ meant tar and ‘is white’ meant ‘is black’. Again, there are clearly interpretations on which this sentence is false.

Representational semantics is what connects to our ideas about necessity. If something is true in all the different situations represented, if the things under consideration are changing, then it is necessary. Interpretational semantics tells us when things are true in virtue of meaning, i.e. no matter what meaning is assigned to the parts of a sentence (holding to type restrictions and other things), it comes out true. We can’t get necessity out of this though, cause it involves only meanings with regard to the one domain. Necessity deals with possible worlds, and so we need those in the picture to properly claim that something is necessary. It is pretty easy to slide between the two ideas, especially since they are extensionally equivalent in some cases. The urge is even stronger if we think of models as being worlds, but evaluating sentences using interpretational semantics. We get to have our necessity cake and eat it too.