So when we left off our story, Barwise and Etchemendy were modelling Russellian propositions with Aczelian hypersets, and the Liar Sentence came out false, but this fact wasn't included in "the world," because if it was, that would violate the "coherence condition" preventing a set-theoretic proposition-object and its "dual" from both being present in a model.
They describe this consequence as "counter-intuitive." I'd describe it as absolutely incoherent.
In any case, when they turn to Austinian propositions, one of the big advantages of the shift was supposed to be that this "counter-intuitive" consequence can be avoided. Remember, Austinian propositions include the situations they are about--so, e.g. if you are at a card game and you mistake the player holding the 3 of Hearts for Jill and claim that "Jill has the 3 of Hearts," the proposition comes out false even if by coincidence Jill actually is playing cards across town and she does happen to have the 4 of Hearts.
As such, the proposition being expressed by any given use of the LS is about some specific situation, since all propositions are on the "Austinian" model. How does this help?
Well, as in the Russellian case, the LS always comes out false. We can see why by analogy to the behavior of the Truth-Teller ("This sentence is true.") In situations that don't include any semantic facts, the Truth-Teller is false, since there is nothing to make it true. In situations that include semantic facts, the Truth-Teller is sometimes true and sometimes false, depending on what those semantic facts are...i.e. in a situation that includes the semantic fact that the Truth-Teller of that situation is true, it's true, and in a situation that includes the semantic fact that the Truth-Teller of that situation is false, it's false. Where these "semantic facts" about the Truth-Teller are supposed to come from, since they obviously don't supervene on the non-semantic facts, is a bit of a mystery to me, but whatever. Let's move on.
The Liar behaves in exactly the same way, with one major difference. The relevant semantic facts are always excluded from the situation used to determine its truth value. Why? Well, if we allowed them to included, that would give us the result that each Liar was both true and false, and that would violate the coherence rule. So it's in "the world," but not the part of it used to evaluate the truth-value of the proposition.
So the Liar of a situation that doesn't have any semantic facts in it is false--just false and not true, which is a neat trick--and the semantic fact that "the Liar of Situation 1 is false" is included not in Situation 1 but in Situation 2. Sure, Situation 2 has a Liar of its own, which is made false because there is nothing in Situation 2 to make it true (after all, the semantic fact that the Liar of Situation 1 is false is about the Liar of Situation 1, not this new Liar, so it is irrelevant), and its falsehood is a fact not of Situation 2 but of Situation 3. On and on forever.
Two comments seem to be in order about this picture.
The first is that, at the beginning of the book, Barwise and Etchemendy rejected Tarski's hierarchy-of-artificial-languages solution because it was arbitrary, didn't really solve anything on the intuitive level, etc. I agree with them on that. It is interesting, though, given that, that the situation-theoretic hierarchy they posit seems to be structurally exactly parallel to Tarski's languages model.
Secondly, and more importantly, it seems to me that the one fact being excluded from the situations is the only one that cannot be plausibly excluded from them. It is, in fact, the ONLY thing that belongs in them. If the Liar Sentence really does express a proposition, then the one and only thing its about is its own truth value. That semantic fact is the one and only fact of any kind that belongs in its Austinian "situation," so a Liar that really was 'about' a situation without semantic facts, or even without this one relevant semantic fact, is simply an incoherent non-possibility.
As such, taken as a defense of the Law of Non-Contradiction--and, as an attempt to explain why the Liar does not have its prima facie property of being true if false and false if true, that's what it amounts to, even without a dialetheist who actually asserts that it is both true and false as the conscious target target--it boils down to the following argument.
"There are no counter-examples to the LNC, therefore the Liar is not a counter-example to the LNC."
That's a perfectly valid instance of universal instantiation, but those of us who want to defend the LNC in the context of an actual argument are going to have to do better than that.