April 23, 2009

I’ve been casually reading Field’s “Saving Truth from Paradox” for some time now. I think it’s a fantastic book, and I highly recommend it to anyone interested in the philosophy of logic, truth or vagueness.

I’ve just read Ch. 21 where he discusses a paradox presented in Restall 2006. The discussion was very enlightening for me, since I had often thought this paradox to be fatal to non-classical solutions to the liar. But although Fields discussion convinced me Restall’s argument wasn’t as watertight as I thought it was, I was still left a bit uneasy. (I think there is something wrong with Restall’s argument that Field doesn’t consider, but I’ll come to that.)

Before I continue, I should state the paradox. The problem is that if one has a strong negation in the language, $\neg$, one can generate a paradoxical liar sentence which says of itself that it’s strongly not true. Strong negation has the following properties which ensures that that last sentence is inconsistent:

1. $p, \neg p \models \bot$
2. If $\Gamma , p \models \bot$ then $\Gamma \models \neg p$

Roughly, the strong negation of p is the weakest proposition inconsistent with p – the first condition guarantees that it’s inconsistent with p, the second that it’s the weakest such proposition. It’s not too hard to see why having such a connective will cause havoc.

Restall’s insight (which was originally made to motivate a “strong” conditional, but it amounts to the same thing) was that one can get such a proposition by brute force: the weakest proposition inconsistent with p is equivalent to the disjunction of all propositions inconsistent with p. Thus, introducing infinitary disjunction into the language, we may just “define” $\neg p$ to be $\bigvee \{q \mid p \wedge q \models \bot \}$. Each disjunct is inconsistent with p so the whole disjunction must be inconsistent with p, giving us the first condition. If q is inconsistent with p, then q is one of the disjuncts in $\neg p$ so q entails $\neg p$, giving us (more or less) the second condition.

An initial problem Field points out is that this definition is horribly impredicative – $\neg p$ is inconsistent with p, so $\neg p$ must be one of it’s own disjuncts. Field complains that such non-well founded sentences give rise to paradoxes even without the truth predicate, for example, the sentence that is it’s own negation. (I personally don’t find these kinds of languages too bad, but maybe that’s best left for another post.) This problem is overcome since you can run a variant of the argument by only disjoining atomic formulae so long as you have a truth predicate.

The second point, Field’s supposed rebuttal of the argument, is that to specify a disjunction by a condition, F say, on the disjuncts, you must first show F isn’t vague or indeterminate, or else you’ll end up with sentences such that it is vague/indeterminate what their components are. Allowing such sentences means they can enter into vague/indeterminate relations of validity – for example, it is vague whether a sentence such that it is vague whether it has “snow is white” as a conjunct entails “snow is white”. But the property F, in this case, is the property of entailing a contradiction if conjoined with p. Thus to assess whether F is vague/indeterminate or not, we must ask if entailment can ever be vague. But to do this we must determine whether there are sentences in the language such that it is indeterminate what their components are. Since the language contains the disjunction of the F’s, this requires us to determine whether F is vague – so we have gone in a circle.

Clearly something weird is going on. That said, I don’t quite see how this observation refutes the argument. It’s perfectly consistent with what’s been said above that entailment for the expanded language with infinitary disjunction is precise, that there is a precise disjunction of the things inconsistent with p, and that Restall’s argument goes through unproblematically. It’s also consistent that there *are* vague cases of entailment – but that the two conditions for strong negation above determinately obtain (there are some subtle issues that must be decided here, e.g., is “p and q” determinately distinct from the sentence that has p as its first conjunct, but only has q as its second conjunct indeterminately.)

Even so, I think there are a couple of problems with Restall’s argument. The first is a minor problem. To define the relevant disjunction, we must talk about the property of “entailing a contradiction if conjoined with p”. But to do this we are treating “entails” like it was a connective in the language. However, one of Fields crucial insights is that “A entails B” is not an assertion of some kind of implication holding between A and B, but rather the conditional assertion of A on B. “entails” cannot be thought of like a connective. For one thing, connectives are embeddable, whereas it doesn’t make much sense to talk of embedded conditional assertions. Secondly, a point which I don’t think Field makes explicit, is that it is crucial that “entails” doesn’t work like an embeddable connective, otherwise one could run a form of Curry’s paradox using entailment instead of the conditional.

This not supposed to be a knockdown problem. After all, so what if you can’t *define* strong negation, there is, nonetheless, this disjunction whose disjuncts are just those propositions inconistent with p. We may not be able to define it or refer to it, but God knows which one it is all the same.

The real problem, I think, is the following. How are we construing $\neg p$? Is it a new connective in the language, stipulated to mean the same as “the disjunction of those things inconsistent with p”? If it is, how do we know it is a logical connective? (If $\neg$ weren’t logical neither (1) nor (2) would hold, since there would be no logical principles governing it.) Field objects to a similar argument from Wright, because “inconsistent with p” is not logical. Inconsistency is not logical: for a start it can only be had by sentences, so it is not topic neutral.

The way of construing $\neg p$ that makes it different from Wright’s argument, and allegedy problematic, is to construe $\neg p$ as schematic for a large disjunction. The symbol $\neg$ does not actually belong to the language at all – writing $\neg p$ is just a metalinguistic shorthand for a very long disjunction, a disjunction that will change, depending in each case, on p. Treating it as such guarantees that (1) and (2) hold, since when they are expanded out, are just truths about the logic of disjunction and don’t contain $\neg$ at all.

But treating $\neg p$ as schematic for a disjunction means it doesn’t behave like an ordinary connective. For one you can’t quantify into it’s scope. What sentence would $\exists x\neg Fx$ be schematic for? What we want it to mean is that there is some object, a, such that the disjunction of things inconsistent with Fa holds. But there’s no single sentence involved here.

Another crucial shortcoming is that it’s not clear that we can “put a dot” under $\neg$. That is, define a function which takes the Gödel number of p, to the Gödel number of the disjunction of things inconsistent with p. Firstly there might not be enough Gödel numbers to do this (since we have an uncountable language now!) But secondly, how do we know we can code “inconsistent with p” in arithmetic? Fields logic isn’t recursively axiomatizable (Welch, forthcoming) so it seems like we’re not going to be able to code “inconsistent with p” or the strong negation of p – and thus it seems we’re not going to be able to run the Gödel diagonalisation argument. (I was always asleep in Gödel class so maybe someone can check I’m not missing something here.)

So you can’t get a strongly negated liar sentence through Gödel diagonalisation, but what about indexical self reference? “This sentence is strongly not true” is schematic for a sentence not including “strongly not”, but with a large disjunction instead. However, which disjunction is it? We’re in the same pickle we were in when we tried to quantify into the scope of $\neg$. In both cases, the disjunction needed to vary depending on the value of the variable “x” or in this case, the indexical “this”.

I can’t say I’ve gotten to the bottom of this, but it’s no longer clear to me how problematic Restall’s argument is for the non classical logician.

I agree that Field has a way out, as he describes, but I think it comes at a significant cost. My initial statement of the problem wasn’t directed at Field, of course (I’d been working on this before Hartry was working on non-classical approaches to the paradoxes). Rather, the original problem was a dilemma for non-classical logicians of all stripes.

Now, to the detail: “To define the relevant disjunction, we must talk about the property of “entailing a contradiction if conjoined with p”. But to do this we are treating “entails” like it was a connective in the language” — I think that’s wrong. (In my original paper, by the way, I wasn’t defining negation, but rather, the intuitionistic conditional, and coding up Curry’s paradox, but the difference between what I said and Hartry’s re-telling is not material for this discussion.) It’s true that we need to talk of such things as what propositions entail what propositions, but everyone doing logic needs to do that. The point of the selection is to select which propositions to disjoin, and the argument only applies to those whose logic supplies the resources to allow us to make that selection. Given a proposition p, does the class of all propositions inconsistent with p have a disjunction? In some logics, the answer is ‘yes’, in others, the answer is ‘no’.

My main target, of course, is someone like Priest, who uses logics in which propositions are sets of (normal and non-normal) worlds, where disjunction of propositions is the union of those sets. Inconsistency (in the salient sense here) is having an empty intersection with, the relevant classes of propositions just does have a union, and this union is itself a set of worlds — proposition. The algebra of propositions in these models supports the reasoning, whether or not we have a connective in the object language that does the job… But now to say that even though the algebra of propositions is closed under that operation, such a thing _cannot_ be expressed in the object language in pain of paradox? This is exactly the kind of failure of semantic closure that people like Priest have argued against, forever.

That’s the dialectic of the argument as I originally construed it, anyway.

2. Hi Greg,

Thanks a lot for the feedback – that was very helpful!

“I agree that Field has a way out, as he describes, but I think it comes at a significant cost. “

Maybe I was missing something with Fields response then, because it seemed to me consistent with what he said that there are no indeterminate cases of entailment (at least, he didn’t give a positive reason to think there would be, just a reason to think there might be.)

“My initial statement of the problem wasn’t directed at Field, of course (I’d been working on this before Hartry was working on non-classical approaches to the paradoxes). Rather, the original problem was a dilemma for non-classical logicians of all stripes.”

Sorry to make it all about Field! Although, I do think that this is a good test case (and obviously, Field thinks it is highly relevant to his project!)

“It’s true that we need to talk of such things as what propositions entail what propositions, but everyone doing logic needs to do that. The point of the selection is to select which propositions to disjoin, and the argument only applies to those whose logic supplies the resources to allow us to make that selection. Given a proposition p, does the class of all propositions inconsistent with p have a disjunction?”

Right. I wasn’t so much objecting to the existence of such a disjunction, but more to our ability, as finite beings, to specify it. The way Field cashes out validity in terms of conditional assertion, rather than the assertion of a conditional, won’t allow us to simply say “that disjunction, whose disjuncts are just those things which when conjoined with p entail a contradiction” (if we cash out argument validity instead as the validity of a conditional, we can say things like this, but this kind of validity won’t do for the argument.) This is inability to specify the disjunction is not entirely to do with it being infinite, but rather to do with the way we need to specify the disjuncts. Adding an “entailment” *connective* generates Curry like paradoxes of it’s own – I take this distinction between conditional assertion and assertion of a conditional to be a nice philosophical motivation for not adding such a connective into the language.

“My main target, of course, is someone like Priest, who uses logics in which propositions are sets of (normal and non-normal) worlds, where disjunction of propositions is the union of those sets. Inconsistency (in the salient sense here) is having an empty intersection with, the relevant classes of propositions just does have a union, and this union is itself a set of worlds — proposition. The algebra of propositions in these models supports the reasoning, whether or not we have a connective in the object language that does the job… But now to say that even though the algebra of propositions is closed under that operation, such a thing _cannot_ be expressed in the object language in pain of paradox? This is exactly the kind of failure of semantic closure that people like Priest have argued against, forever.”

I see. This argument seems closer to the argument you labelled “The Algebra of Propositions” in your paper, rather than the infinitary disjunction argument which could be carried out purely in the object language (no?) The latter argument seems stronger to me, since it applies also to people who deny the existence of a (classical) intended model (such as Field).

3. Here are a couple of things I’m puzzled about the argument in the “Infinitary Disjunction” section – maybe you could help me.

Consider the version of the argument for strong negation rather conditional (call strong negation, Negation, for short.) I take that the existence of such disjunctions guarantees the following:

“For each Negation free formula, $\phi$, there is a Negation free infinitary disjunction equivalent to the Negation of $\phi$.”

What I’d like to know is (a) whether there is a Negation free equivalent of any formula (including formulae with embedded Negations) (b) whether you can code the function taking the Gödel number of a sentence to the Gödel number of it’s Negation, (c) whether the fact above should allow the introduction of a Negation connective if (a) obtains (and if it doesnt?)

4. Somewhat off topic, but since you brought it up… I grant that it makes little sense for “entails” to embed on the conception that “A entails B” is really an assertion about the conditional assertion of B upon A… but what about:

(E) That the king of France is bald entails that France has a king entails that something entails that France has a king.

We could either conclude, with Field’s conception of entailment, that (E) is nonsense. But it doesn’t strike me as nonsense. So I’m inclined to think that “entails” does embed.

5. Hi Colin. Thanks for the comment!

Yes that is worrying. I take it your choice of example was supposed to prevent me from interpreting these as valid assertions of conditionals (rather than valid conditional assertions)?

That said – the problem of Curry’s paradox for an embeddable entailment conditional seems to me quite general. It seems like you have a burden of proof there, at least of defending against Curry style arguments, if you are to claim entailment is embeddable.

6. […] finally, a smattering of logic: Andrew Bacon discusses Restall’s Paradox, Kenny Easwaran endorses a particular kind of probabilistic proof, and I consider a slight […]

7. For what it’s worth, the credit for interpreting “entails” or the turnstile as conditional assertion is really due to Dana Scott in a series of papers from the early 70s.

8. Thanks! Do you have any references?

9. “On Engendering an Illusion of Understanding” http://www.jstor.org/discover/10.2307/2024952?uid=3739832&uid=2&uid=4&uid=3739256&sid=21102000500171

“Completeness and Axiomatizability in many-valued logic”