As I mentioned in the last post, it fell to me to introduce the last two chapters in Part III of Field — namely, Ch. 17 in which he rounds out his key technical construction, and Ch. 18, ‘What Has Been Done’. And having got to the end of Field’s core presentation of his story, we are going to call it a day. Since it was the last meeting, I took my cue from Ch. 18 and offered some very quickly written reflections on what has (or rather hasn’t) been achieved. Here they are. Comments more than usually welcome: what am I missing?

Robbie WilliamsJust on Andrew’s point about what’s going to be given up—-you mention that conditional proof fails (your 2). But something much weaker than the version of conditional proof with side premises you formulate fails—conditional proof without side premises. And that’s pretty inevitable, if we have the truth-rules and modus ponens, both usable within subproofs, in a Field-like setting. For given only conditional proof without sidepremises, we can run the Curry paradox. Start by supposing T(C), where C is the Curry sentence. By T-rule, we get T(C)–>falsum. By MP, we have falsum. By conditional proof, we have T(C)–>falsum on no assumptions. And then by the opposite T-rule, we have T(C). By modus ponens again, we have falsum on no assumptions.

Of course, this just makes the weirdness of the conditional that much stronger (especially because the failure of conditional proof with side premises is very familiar in conditional logics—this is far weirder).

Andrew Bacon“A final thing to note, by way of fallback, is that the conservativeness result itself gives a certain kind of restricted “truth in, truth out” result. We’re not going to generate any falsehoods in the non-conditional, non-truth fragment of the language we couldn’t have reached anyway. So the *worst* that can happen following Con2’s guidance, is that we say something untrue in the enriched vocabulary—but that untruth would then be quarantined, and is never going to tell you bad things about the categorical distribution of medium sized dry goods (or subatomic particles, or whatever).”

I don’t think this is quite right. Field never proves a true conservativeness result. For example, if there’s only 1 inaccessible the best Field can show is that the T-schema plus “there are no inaccessibles” is consistent. So (a) this doesn’t guarantee the T-schema plus true sentences is consistent (although it’s plausible that it is) and (b) there’s certainly no guarantee of conservativeness since his result only applies to theories containing false sentences.

Andrew BaconAlthough I guess the first inaccessible doesn’t really count as a a medium sized dry good :-p

Robbie WilliamsAh, yes, I’m relying on memory here… interesting if this fails. Just to clarify: are you saying that Field’s favoured theory, including the conditional, hasn’t been proved to be “conservative over omega-models” as per the Kripke construction, or is this a comment on how to read such conservativeness results?

Andrew BaconSorry I wasn’t clear. I was thinking of some kind of absolute conservativeness: that you can’t prove (in the absolute sense) from truths any Tr-free thing from the T-schema that you couldn’t prove without it. The point, in my example, was that you have the conservativeness of the T-schema over a false theory (one that says there are no inaccessibles) but not conservativeness in the absolute sense.

Of course Field shows more than conservativeness over omega models: he shows conservativeness over standard models of ZFC. It’s unclear to me at the moment how you’d patch that. Assuming more inaccessibles doesn’t really help, and appealing to a second order reflection principle seems to be equally problematic as we would then have second order truths to account for in our conservativeness claim.

Robbie WilliamsOk, great. But isn’t it more of a dilemma: either principles you appeal to in the proof of conservativeness (e.g. there exists an inaccessible cardinal) are false; or else there’s a truth (e.g. there exists an inaccessible cardinal) that you haven’t shown the rest of the stuff to be consistent with. If you thought set theory was itself false-but-reliable, you’re already on the falsity horn. Whether there’s a stable position around here, I don’t know…

Andrew BaconI can’t see why this dilemma is forced on us. At least, I don’t think there’s a general reason an absolute conservativeness proof must accept one of those horns. (Toy example: let PA+ be be the language of arithmetic with a primitive predicate and an axiom stating it applies to everything. I’m pretty sure PA proves “for all x in the language of PA, Bew_PA+(x) -> Bew_PA(x)”.)

Robbie WilliamsHi Sam,

I think one issue here is what we’re thinking of as “the logic”. But one preliminary issue: Field has the stuff about validity inevitably diverging from necessary truth preservation—e.g. instances of modus ponens are valid, but saying they preserve truth means endorsing a Curry sentence. That makes formulating the challenge in terms of truth in/truth out quite delicate. But I take it that by Field’s lights, the question is whether model theoretic logic is “genuinely sound” (i.e. whether model theoretic validity is guaranteed not to outstrip genuine validity).

Let’s call the conditional with a logic specified by some (intuitively sound) set of proof rules Con1, and the conditional with the logic given by the model theory Con 2. (By “the logic” here I just mean the set of sequences of premises/conclusions that is fixed in the respective ways—e.g. by chaining together proof rules; or by the absence of countermodels). The two parts of the Kreisel argument given above would give us that Con1 is a sublogic of Con2.

One question is: what’s the logic which we’re interested in a safety result for? In particular, is it Con1 or Con2? By the subset result for Con1 and the conservativeness result for Con2, neither will engender paradox. But do we have something like soundness for either? If not, isn’t it bad to be relying on them in inference?

Now, for Con1, the presupposition is that we do have soundness—that we can recognize that any sequent in Con1 is genuinely valid. But without a completeness result that’d tell us that Con2 and Con1 coincide, we don’t yet have any reason to think that Con2 is intuitively sound—which I take it is Peter’s point. This is problematic if we’re going to help ourselves to anything in Con2; but no problem if we restrict ourselves to using Con1. And if, using Con1 only, we can show that we have whatever expressive, practical or theoretical ends we wanted the conditional for in the first place, this looks like an achievement. (Of course, if we spot an sequent that’s part of Con2 and not part of Con1, but is intuitively sound, we can of course chuck it in, getting a new logic Con1*, and we can repeat the same story).

Where would this leave us? Well, we wouldn’t have a definitive characterization of the “real” logic of the language that includes truth/conditional—either proof theoretically or model theoretically. What we might have are lower and upper bounds for the logic. One position to take at this point is that there’s a fact of the matter what the real logic is, and we’re just ignorant of it. Is that a problem? Sure, it’d be nice to know exactly what the logic is, but we already know a lot—that it’ll be as powerful as Con1; and that (in virtue of being a sublogic of Con2) it’s not paradox engendering. Why shouldn’t we be satisfied with this? (Another attitude to take is that there’s no fact of the matter which inferences are genuinely valid—something that Peter suggested as a default reasonable position for Field to take).

A final thing to note, by way of fallback, is that the conservativeness result itself gives a certain kind of restricted “truth in, truth out” result. We’re not going to generate any falsehoods in the non-conditional, non-truth fragment of the language we couldn’t have reached anyway. So the *worst* that can happen following Con2’s guidance, is that we say something untrue in the enriched vocabulary—but that untruth would then be quarantined, and is never going to tell you bad things about the categorical distribution of medium sized dry goods (or subatomic particles, or whatever). There’s various fallback positions one could adopt where this alone would be enough to justify reliance on Con2, whether or not it’s sound. (Compare: the inference from “F and G are equinumerous” to “the number of Fs = the number of Gs” takes you from truth to falsity on Field’s 1980 view of maths. But we’re justified in reasoning this way, says Field, in virtue of a conservativeness result for arithmetic over nominalized physics. I’m not saying that this is his 2008 view on conditionals, but if we were forced back to it, it wouldn’t be an unfamiliar situation).

Sam RobertsHi Rrobbie,

Thanks for that, that was really helpful.

I think that the point about quarantine might be understated. Of course, if we just want to take account of truth, those sentences which we might come to accept though Con 2 which are untrue could be few and far between. I worry, however, whether this will be the case if we want to take account of vagueness, which is ubiquitous. In particular we might have the rule ‘A implies DA’ in a Con 1, but in every model D lowers the value of A. So there is the worry that we could reason in the logic from something true, though slightly vague, to something untrue relatively easily.

Thanks again.

Sam

Andrew BaconHi Peter,

I guess you could accuse anyone who changes their logic of changing the subject. But I assume you think that principles (a)-(d) are particularly important in this respect? Giving up on (d) (and thus the closely related (b)) are pretty much essential to any non-classical solution, since they’d give rise to Curry’s paradox (with modus ponens.) I think resting your case on those principles will ultimately end in one of those hard to resolve arguments where the trade off is between giving up the T-schema or giving up on the logic of the conditional. Either side can argue that the other is changing the subject. (It’s interesting that (a) fails in Fields logic. I think (a) plays an important part in the omega-inconsistency of Lukasiewicz logic. Either way I don’t have many intuitions about it.) By the way he’s got a pretty good motivation for rejecting induction based on the Sorites and Berry paradoxes. In fact, rejecting induction seems to be the central moving part in the non-classical response to the Sorites: that you can reject the existence of a first large number contradicting the least number principle.

I’m not so pessimistic as you and Robbie are about developing a non-classical model theory. I’ve worked out some of the basics in the propositional case here if you’re interested (which applies to a range of logics including Fields): http://users.ox.ac.uk/~lady1900/papers/Non-classical%20metatheories.pdf

Regarding the conceptual motivation behind Field’s construction I think that it isn’t supposed to be anything more than a consistency proof. Kripke’s ungroundedness intuition has to be taken with a pinch of salt anyway: the notion of ungroundedness is just as riddled with paradoxes as truth is (for example Mirimanoff’s paradox.) The conceptual motivation behind that model theoretic construction as applied to English is ultimately inexpressible as it essentially requires a richer metalanguage. Field’s determinacy operator is really what is supposed to be expressing what’s wrong with the liar and related sentences. Unlike ungroundedness talk, this operator is in the object language and can be applied to itself. He doesn’t give us an explicit analysis of his operator but he does give us a fix on its conceptual role. (In Lukasiewicz you can analyse “it’s indeterminate whether p” as “p iff ~p” which is nice since intuitively the liar is true if and only if it isn’t. Field doesn’t quite have this but it’s in the general ball park.)

Sam RobertsHi Robbie,

Just a quick question on the last point you make.

Of course, if we have a relative consistency proof for the logic at large, then any subset of it wont lead to paradox. But I took the more interesting point, which I think Peter is making, to be that this does not guarantee us that we will only come to accept truths (simpliciter) using the logic. Isn’t it a fair requirement to have some reason to think truths in truths out?

Any help would be great.

Regards

Sam

Robbie WilliamsHi Peter,

I think I’ve got a more positive take on what’s achieved than you do; though I had several of the worries you mention too. Here’s my take on a couple of the things you mention.

For starters, I pretty much set aside the stuff about appeals to a non-classical model theory. It’d be nice if someone could do that; but without something more than promissory notes, I think it’s better to keep faith with the explicitly instrumentalist treatment of model theory that he seems to endorse for most of the book.

Re the alleged double-standards vis a vis Kripke. I think it’s a really good question why we (/Field) should want more than the intersubstitutable truth predicate in Kleene logic that Kripke gave us. One thought backing this up is this: liars (or contingent liars) are peripheral to our interests most of the time. And given relevant instances of excluded middle, the Kleene “material conditional” collapses to the standard one, so we have something that’s well-behaved whenever we can presuppose excluded middle for the sentences in question. Sure, things will break down when we hit something ungrounded, and maybe in special circumstances we might not quite say what we wanted to say in truth-attributions—but why care?

I think there are two reasons given in the book for wanting more. The first is based on the claim that we need to retain fragments of a “theory of truth”—-claims about how truth in logically complex things relates to truth of their simpler parts, and so on. Maybe this is supposed to help (or be required) if the truth predicate is going to play the expressive role that the strong disquotationalist wants it to.

Here I’ve got similar worries to you: I never really understood *how much* we wanted on this front—-e.g. what would count as “success” (perhaps because I didn’t really see what motivated it in the first place). So I find this motivation quite confusing.

The far cleaner motivation, in my view, is the connection Field sees between the semantic paradoxes and vagueness. It might be ok to live with a conditional that’s well-behaved except in non-classical cases, if those non-classical cases were peripheral to everyday concerns. But suppose you thought that excluded middle failed for borderline bald Harry. Now, suddenly, we’re faced with non-classical behaviour all the time. And this is problematic even when we’re not talking about truth—all sorts of platitudinous sounding claims like “If a man with n hairs is bald, a man with less than n hairs is bald” should be rejected if the conditional is Kleene-like. So—assuming we’re after a unified theory—-the motivation for a conditional that’s better behaved than the Kleene one is very strong. Notice that nowhere in this way of setting up the problem do we need to appeal to the desire to keep Tarskian biconditionals—they’re a nice bonus we get automatically from (a) the idea that a minimal conditional of reasonableness of a conditional for use in ordinary contexts is that it satisfy all instances of identity (A–>A); (b) full intersubstitutability for truth.

(Incidentally, the way the logic of “definitely” flows from the conditional in Field’s treatment of vagueness seems ultra-elegant to me. Most theories of vagueness—aside from epistemicism—resort to something that looks like monster-barring to make room for nontrivial higher order vagueness. On Field’s theory, an attractive logic of definiteness falls out of the logic of the conditional.)

Regarding the insouciance over completeness. Here’s a quick thought: Suppose we have some syntactical proof system including a Fieldian conditional—and suppose we’re happy with it (it allows us to formulate a decent truth theory, express whatever we need to express, formalize good reasoning in the presence of vagueness, etc). Now the question arises: is this proof theory going to generate paradoxes? If we persuade ourselves that every syntactically valid inference is genuinely valid (one piece of the Kreisel argument) and every model-theoretic counterexample is a genuine counterexample (another piece of the Kreisel argument) then showing that the conditional as defined by model theory isn’t paradox-engendering will suffice to show that we’ll get no paradox out of the rules we have in the proof theory. In that way, core elements of the Kreisel argument can be used to give a “safety result” for the proof theory, even while we’re neutral about completeness. (Maybe this is what you call the “existence proof” reading of his enterprise.)