Suppose you have some theory R, formulated in that fragment of English that is free of semantic vocabulary. The theory, we can assume, is at least “effectively” classical—e.g. we can assume excluded middle and so forth for each predicate that it uses. Now think of total theory—which includes not just this theory but also, e.g. a theory of truth.

It would be nice if truth in this widest theory could work “transparently”—so that we could treat “p” and “T(p)” as intersubstitutable at least in all extensional contexts. To get that, something has to go. E.g. the logic for the wider language might have to be non-classical, to avoid the Liar paradox.

One question is whether weakening logic is enough to avoid problems. For all we’ve said so far, it might be that one can have a transparent truth-predicate—but only if one’s non-semantic theories are set up just right. In the case at hand, the worry is that R cannot be consistently embedded within a total theory that includes a transparent truth predicate. Maybe in order to ensure consistency of total theory, we’d have to play around with what we say in the non-semantic fragment. It’d be really interesting if we could get a guarantee that we never need to do that. And this is one thing that Kripke’s fixed point construction seems to give us.

Think of Kripke’s techniques as a “black box”, which takes as input classical models of the semantics-free portion of our language, and outputs non-classical models of language as a whole—and in such a way as to make “p” and “Tp” always coincide in semantic value. Crucially, the non-classical model coincides with the classical model taken as input when it comes to the semantics-free fragment. So if “S” is in the semantics-free language and is true-on-input-model, then it will be true-on-the-output model.

This result seems clearly relevant to the question of whether we disrupt theories like R by embedding them within a total theory incorporating transparent truth. The most obvious thought is to let M be the intended (classical) model of our base language—and then view the Kripke construction as outputting a candidate to be the intended interpretation of total language. And the result just given tells us that if R is true relative to M, it remains true relative to the outputted Kripkean (non-classical model).

But this is a contentious characterization. For example, if our semantics-free language contains absolutely unrestricted quantifiers, there won’t be a (traditional) model that can serve as the “intended interpretation”. For (traditional) models assign sets as the range of quantifiers, and no set contains absolutely everything—in particular no set can contain all sets. And even if somehow we could finesse this (e.g. if we could argue that our quantifiers can never be absolutely unrestricted), it’s not clear that we should be identifying true-on-the-output-model with truth, which is crucial to the above suggested moral.

Field suggests we take a different moral from the Kripkean construction. Focus on the question of whether theories like R (which ex hypothesi are consistent taken alone), might turn out to be inconsistent in the light of total theory—in particular, might turn out to be inconsistent once we’ve got a transparent truth predicate in our language. He argues that the Kripkean construction gives us this.

Here’s the argument. Suppose that R is classically consistent. We want to know whether R+T is consistent, where R+T is what you get from R when you add in a transparent truth-predicate. The consistency of R means that there’s a classical model on which it is true. Input that into Kripke’s black box. And what we get out the other end is a (non-classical) model of R+T. And the existence of such a model (whether or not it’s an “intended one”) means that R+T is consistent.

Field explicitly mentions one worry about this–that it might equivocate over “consistent”. If consistent just means “has a model (of such-and-such a kind)” then the argument goes through as it stands. But in the present setting it’s not obvious what all this talk of models is doing for us. After all, we’re not supposed to be assuming that one among the models is the “intended” one. In fact, we’re supposed to be up for the thesis that the very notion of “intended interpretation” should be ditched, in which case there’d be no space even for viewing the various models as possibly, though not actually, intended interpretations.

This is the very point at which Kreisel’s squeezing argument is supposed to help us. For it forges a link between intuitive consistency, and the model-theoretic constructions. So we could reconstruct the above line of thought in the following steps:

- R is consistent (in the intuitive sense)
- So: R is consistent (in the model-theoretic sense). [By a squeezing argument]
- So: R+T is consistent (in the model-theoretic sense). [By the Kripkean construction]
- So: R+T is consistent (in the intuitive sense). [By the squeezing argument again]

Now, I’m prepared to think that the squeezing argument works to bridge the gap between (1) and (2). For here we’re working within the classical fragment of English, and I see the appeal of the premises of the squeezing argument in that setting (actually, for this move we don’t really need the premise I’m most concerned with—just the completeness result and intuitive soundness suffice).

But the move from (3) to (4) is the one that I find dodgy. For this directly requires the principle that if there is a formal (3-valued) countermodel to a given argument, then that argument is invalid (in the intuitive sense). And that is exactly the point over which I voiced scepticism in the previous post. Why should the recognition that there’s an assignment of values to R+T on which an inference isn’t value-1 preserving suggest that the argument from R+T to absurdity is invalid? Without illegitimately sneaking in some thoughts about what value-1 represents (e.g. truth, or determinate truth) I can’t even begin to get a handle on this question.

In the previous post I sketched a fallback option (and it really was only a sketch). I suggested that you might run a squeezing argument for Kleene logic using probabilistic semantics, rather than 3-valued ones, since we do have a sense of what a probabilistic assignment represents, and why failure to preserve probability might be an indicator of intuitive invalidity. Now maybe if this were successful, we could bridge the gap—but in a very indirect way. One would argue from the existence of a 3-valued model, via completeness, to the non-existence of a derivation of absurdity from R+T. And then, by a second completeness result, one would argue that there had to exist a probabilistic model for R+T. Finally, one would appeal to the general thought that such probabilistic models secured consistency (in the intuitive sense).

To summarize. The Kripkean constructions obviously secure a technical conservativeness result. As Field mentions, we should be careful to distinguish this from a true conservativeness result: the result that no inconsistency can arise from adding transparent truth to a classically consistent base theory. But whether the technical result we can prove gives us reason (via a Kreisel-like argument) to believe the true conservativeness result turns on exactly the issue of whether a 3-valued countermodel to an argument gives us reason to think that that argument is intuitively invalid. And it’s not obvious at all where that last part is coming from—so for me, for now, it remains open whether the Kripkean constructions give us reason to believe true conservativeness.