J Robert G Williams

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:

1. R is consistent (in the intuitive sense)
2. So: R is consistent (in the model-theoretic sense). [By a squeezing argument]
3. So: R+T is consistent (in the model-theoretic sense). [By the Kripkean construction]
4. 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.

Kreisel gave a famous and elegant argument for why we should be interested in model-theoretic validity. But I’m not sure who can use it.

Some background. Let’s suppose we can speak unproblematically about absolutely all the sets. If so, then there’s something strange about model theoretic definitions of validity. The condition for an argument to be model-theoretically valid it needs to such that, relative to any interpretation, if the premises are true then the conclusion is true. It’s natural to think that one way or another, the reason to be interested in such a property of arguments is that if an argument is valid in this sense, then it preserves truth. And one can see why this would be—if it is truth-preserving on every interpretation, then in particular it should be truth-preserving on the correct interpretation, but that’s just to say that it guarantees that whenever the premises are true, the conclusion is so too.

Lots of issues about the intuitive line of thought arise when you start to take the semantic paradoxes seriously. But the one I’m interested in here is a puzzle about how to think about it when the object-language in question is (on the intended interpretation) talking about absolutely all the sets. The problem is that when we spell out the formal details of the model-theoretic definition of validity, we appeal to “truth on an interpretation” in a very precise sense—and one of the usual conditions on that is that the domain of quantification is a set. But notoriously there is no set of all sets, and so the “intended interpretation” of discourse about absolutely all sets isn’t something we find in the space of interpretations relative to which the model-theoretic definition of validity for that language is defined. But then the idea that actual truth is a special case of truth-on-an-interpretation is well and truly broken, and without that, it’s sort of obscure what significance the model-theoretic characterization has.

Now, Kreisel suggested the following way around this (I’m following Hartry Field’s presentation here). First of all, distinguish between (i) model theoretic validity, defined as above as preservation of truth-on-set-sized-interpretations (call that MT-validity); and (ii) intuitive validity (call that I-validity)—expressing some property of arguments that has philosophical significance to us. Also suppose that we have available a derivability relation.

Now we argue:

(1) [Intuitive soundness] If q is derivable from P, then the argument from P to q is I-valid.

(2) [Countermodels] If the argument from P to q is not MT-valid, then the argument from P to q is not I-valid.

(3) [Completeness] If the argument from P to q is MT-valid, then q is derivable from P.

From (1)-(3) it follows that an argument is MT-valid iff it is I-valid.

Now (1) seems like a decent constraint on the choice of a deductive system. Friends of classical logic will just be saying that whatever that philosophically significant sense of validity is that I-valid expresses, classical syntactic consequences (e.g. from A&B to A, from ~~A to A) should turn out I-valid. Of course, non-classicists will disagree with the classicist over the I-validity of classical rules—but they will typically have a different syntactic relation and it should be that with which we’re running the squeezing argument, at least in the general case. Let’s spot ourselves this.

(3) is the technical “completeness” theorem for a given syntactic consequence relation and model-theory. Often we have this. Sometimes we don’t—for example, for second order languages where the second order quantifiers are logically constrained to be “well-behaved”, there are arguments which are MT-valid but not derivable in the standard ways. But e.g. classical logic does have this result.

Finally, we have (2). Now, what this says is that if an argument has a set-sized interpretation relative to which the premises are true and the conclusion false, then it’s not I-valid.

Now this premise strikes me as delicate. Here’s why for the case of classical set theory we started with, it seems initially compelling to me. I’m still thinking of I-validity as a matter of guaranteed truth-preservation—i.e. truth-preservation no matter what the (non-logical) words involved mean. And I look at a given set-sized model and think—well, even though I was actually speaking in an unrestricted language, I could very well have been speaking in a language where my quantifiers were restricted. And what the set-sized countermodel shows is that on that interpretation of what my words mean, the argument wouldn’t be truth-preserving. So it can’t be I-valid.

However, suppose you adopt the stance where I-validity isn’t to be understood as “truth-preservation no matter what the words mean”—and for example, Field argues that the hypothesis that the two are biconditionally related is refutable. Why then should you think that the presence of countermodels have anything to do with I-invalidity? I just don’t get why I should see this as intuitively obvious (once I’ve set aside the usual truth-preservation understanding of I-validity), nor do I see what an argument for it would be. I’d welcome enlightenment/references!

We’ve been talking so far about the case of classical set theory. But I reckon the point surfaces with respect to other settings.

For example, Field favours a nonclassical logic (an extension of the strong Kleene logic) for dealing with the paradoxes. His methodology is to describe the logic model-theoretically. So what he gives us is a definition of MT-validity for a language containing a transparent truth-predicate. But of course, it’d be nice if we could explain why we’re interested in MT-validity so-characterized, and one attractive route is to give something like a Kreisel squeezing argument.

What would this look like? Well, we’d need to endorse (1)—to pick out a syntactic consequence relation and judge the basic principles to be I-valid. Let’s spot ourselves that. We’d also need (3), the completeness result. That’s tricky. For the strong Kleene logic itself, we have a completeness result relative to a 3-valued semantics. So relative to K3 and the usual 3-valued semantics, we’ve got (3). But Field’s own system adds to the K3 base a strong conditional, and the model theory is far more complex than a 3-valued one. And completeness just might not be available for this system—see p.305 of Field’s book.

But even if we have completeness (suppose we were working directly with K3, rather than Field’s extension) to me the argument seems puzzling. The problem again is with (2). Suppose a given argument, from P to q, has a 3-valued countermodel. What do we make on this? Well, this means there’s some way of assigning semantic values to expressions such that the premises all get value 1, and the conclusion gets value less than 1 (0.5, or 0). But what does that tell us? Well, if we were allowed to identify having-semantic-value-1 with being-true, then we’d forge a connection between countermodels and failure-to-preserve-truth. And so we’d be back to the situation that faced us in set-theory, in that countermodels would display interpretations relative to which truth isn’t preserved. I expressed some worries before about why if I-validity is officially primitive, this should be taken to show that the argument is I-valid. But let’s spot ourselves an answer to that question—we can suppose that even if I-valid is primitive, then failure of truth-preservation on some interpretation is a sufficient condition for failure to be I-valid.

The problem is that in the present case we can’t even get as far as this. For we’re not supposed to be thinking of semantic value 1 as truth, and nor are we supposed to be thinking of the formal models as specifying “meanings” for our words. If we do start thinking in this way, we open ourselves up to a whole heap of nasty questions—e.g. it looks very much like sentences with value 1/2 will be thought of as truth-value gaps, whereas the target was to stablize a transparent notion of truth—a side-effect of which is that we will be able to reduce truth-value gaps to absurdity.

Field suggests a different gloss in some places—think of semantic value 1 as representing determinate truth, semantic value 0 as representing determinate falsity, and semantic value 1/2 as representing indeterminacy. OK: so having a 3-valued countermodel to an argument should be glossed have displaying a case where the premises are all determinately true, and the conclusion is at best indeterminate. But recall that “indeterminacy” here is *not* supposed to be a status incompatible with truth—otherwise we’re back to truth-value gaps—so we’ve not got any reason here to think that we’ve got a failure of truth-preservation. So whereas holding that failure of truth-preservation is a sufficient condition for I-invalidity would be ok to give us (2) for the case of classical set theory, in the non-classical cases we’re thinking about it just isn’t enough to patch the argument. What we need instead is that failure of determinate-truth-preservation is a sufficient condition for I-invalidity. But where is that coming from? And what’s the rationale for it?

Here’s a final thought about how to make progress on these questions. Notice that the Kreisel squeezing argument is totally schematic—we don’t have to pack in anything about the particular model theory or proof theory involved, so long as (1-3) are satisfied. As an illustration, suppose you’re working with some model-theoretically specified consequence relation where there isn’t a nice derivability relation which is complete wrt it—where a derivability relation is nice if it is “practically implementable”–i.e. doesn’t appeal to essentially infinitary rules (like the omega-rule). Well, nothing in the squeezing argument required the derivability relation to be *nice*. Add in whatever infinitary rule you like to beef up the derivability relation until it is complete wrt the model theory, and so long as what you end up with is intuitively sound—i.e. (1) is met—then the Kreisel argument can be run.

A similar point goes if we hold fixed the proof theory and vary the model theory that defines MT-validity. The condition is that we need a model theory that (a) makes the derivability relation complete; and (b) is such that countermodels entail I-invalidity. So long as something plays that role, we’re home and dry. Suppose, for example, you give a probabilistic semantics for classical logic (in the fashion that Field does, via Popper functions, in his 1977 JP paper), and interpret an assignment of probabilities as a possible distribution of partial beliefs over sentences in the language. An argument is MT-valid, on this semantics, just in case if whenever premises are probability 1 (conditionally on anything) then so is the conclusion. Slogan: MT-validity is certainty-preservation. A countermodel is then some representation of partial beliefs whereby one is certain of all the premises, but less than certain of the conclusion. Just as with non-probabilistic semantics, there’ll be a question of whether the presence of a countermodel in this sense is sufficient for I-invalidity—but it doesn’t seem to me that we weaken our case by this shift.

But what seems significant about this move is that, in principle, we might be able to do the same for the non-classical cases. Rather than do a 3-valued semantics and worry about what to make of “semantic value 1 preservation” and its relation to I-validity, one searches for a complete probabilistic semantics. The advantage is that we’ve got a interpretation standing by of what individual assignments of probabilities means (in terms of degrees of belief)—and so I don’t envisage new interpretative problems arising for this choice of semantics, as they did for the 3-valued way of doing things.

[Of course, to carry out the version of the squeezing argument itself, we’ll need to actually have such a semantics—and maybe to keep things clean, we need an axiomization of what a probability function is that doesn’t tacitly appeal to the logic itself (that’s the role that Popper’s axiomatization of conditional probability functions in Field’s 1977 interpretation). I don’t know of such an axiomitization—advice gratefully received.]