As I said in the last post, we’ve fairly recently started working through Field’s* Sa**ving Truth from Paradox* in our reading group.

It fell to me to introduce Ch. 4 on Curry’s Paradox and (infinite valued) Lukasiewicz logics yesterday. I very speedily dashed off some introductory remarks (expecting, rightly, that few there would have actually read either Curry or Lukasiewicz so some scene-setting background might be helpful). For what little they are worth, here are my notes, now very slightly revised (most recently on 29th June).

Andrew Bacon“As I remarked before, in the original three-valued framework it would be better to say that there are still just two values that a proposition can take, truth and falsity: we are simply explicitly marking the (supposed) possibility that a proposition might not (yet) get to have one of those value. Indeterminate is not really an intermediate value, so much as a lack of value: but then what sense does it make to suppose we should interpose more of the same between truth and falsehood? Frankly I have no idea!”

I don’t think this is what he’s up to at all. In fact, I think setting the discussion up in terms of Lukasiewicz’s interpretation of the logic is quite unhelpful given that Field is essentially just interested in building a *logic* which can support full intersubstitutivity.

Nothing is neither true nor false for Field, or else one would have a contradiction via full intersubstitutivity.

“Where on earth has the idea of some fuzziness in true come from?”

Well, he thinks it’s indeterminate whether the liar is true. This is also what Lukasiewicz logic predicts if you define ‘indeterminate’ in the way he suggests (which doesn’t correspond to the value 1/2 in the semantics in the way Lukasiewicz was thinking about it.)

Peter SmithThanks for this! Actually I agree that looking back at Lukasiewicz isn’t much help for grasping what Field is up to! But — given some of the seminar participants hadn’t come across his three-valued logic before — it was worth saying

somethingabout where Lukasiewicz was coming from. And worth noting that the crucial Lukasiewicz rules for evaluating the conditional are not immediately attractive, are in fact not well motivated byhim, and we look in vain for a better motivation in Field.You say “Field is essentially just interested in building a *logic* which can support full intersubstitutivity”. Does that mean he is just trying to find

someapparatus which is vaguely similar to standard, semantically motivated, logics and which happens to do the techie trick? I’d have thought we should want more thanthat, and requiresomeconception of the significance of the many values.As to Field on indeterminacy, fuzziness, etc. let’s just say that the exposition at this early point in the book is … erm … fuzzy and indeterminate.

Andrew Bacon“Does that mean he is just trying to find some apparatus which is vaguely similar to standard, semantically motivated, logics and which happens to do the techie trick? I’d have thought we should want more than that, and require some conception of the significance of the many values.”

I’d say that’s pretty much exactly what he’s doing. The many valued models are, for Field, just to be thought of as a way of characterising the extension of the consequence relation (which is why he’s interested in the squeezing arguments by the way) and so the values shouldn’t be thought of as semantically motivated. After all for Field semantic notions (truth, reference, satisfaction) are disquotational and non-classical, whereas “having semantic value 1 relative to a classically described MV model” is classical (he thinks membership statements obeys LEM) and isn’t disquotational for any model (there’s always, for example, the sentence “this sentence does not have semantic value 1 according to M”).

In the end I think he uses a many valued model to prove the consistency of his preferred theory (although he doesn’t quite put it like that.) From what I took away from the book the philosophically important notions are truth (which doesn’t admit gaps) and determinacy (which does admit gaps, but still isn’t classical).

Peter SmithThanks again for this — very helpful!

“The many valued models are, for Field, just to be thought of as a way of characterising the extension of the consequence relation” So he isn’t doing semantics proper, but just algebra. Fine. But (a) you’d think he’d say that when he himself asks the question about the significance of the continuum valued semantics on p. 86, and (b) you’d think he’d immediately interested in telling us e.g. about a propositional deductive system which is sound and complete with respect to the algebraic models.

But if you are right that “In the end … he uses a many valued model to prove the consistency of his preferred theory” that gives a clean story to tell. It isn’t clearly out there, however, in Ch 4 (and after al,l this comes straight after talking about Kripke’s theory which is semantically driven: Field doesn’t signal any shift of focus …).

Andrew BaconHis favoured interpretation of Kripke’s theory is one that denies there are truth value gaps so I thought it was relatively clear he’s not thinking about it as being completely driven by this view of there being three truth values (by which I mean something like the picture you were ascribing to Lukasiewicz.)

I agree it is hard to find a nice clear statement of the distinction between the two approaches though.

Peter SmithI didn’t mean that Kripke’s theory is driven by the view that there are three truth-values — just that there is a genuinely semantic guiding conception of what is going on in that story (in talking of grounding, and successive rounds of fixing more and more values). I’m missing an analogous guiding semantic conception (at least in this early chapter!) of what might drive us to something like a Lukasiewicz logic. So you may well be right that construing things merely algebraically is the the way to go!

Andrew BaconBy the way, it’s not entirely clear how the second Hajek/Paris/Shepardson result you mentioned relates to Fields project. He explicitly rejects the induction schema once you start inserting vague and indeterminate predicates into it. For, example, he thinks it’s vague whether there’s a last small number, and thus rejects the least number principle (and it’s negation.) So induction fails for “small”. Similarly, one would imagine, it would fail for “true” although this perhaps only comes out clearly when we are considering Berry’s paradox rather than standard liars. (I think, but I can’t remember exactly, that it’s indeterminate whether there’s a first number not definable by a formula of less than 40 symbols.)

Incidentally, the first result you mentioned was shown already by Greg Restall: http://consequently.org/writing/arithluk/.

Peter SmithThanks for this too — point taken.