## B entails that a conjunction of determinate truths is determinate

October 26, 2010

I know it’s been quiet for a while around here. I have finally finished a paper on higher order vagueness which has been  a long time coming, and since I expect it to be in review for quite a while longer I decided to put it online. (Note: I’ll come back to the title of this post in a bit, after I’ve filled in some of the background.)

The paper is concerned with a number of arguments that purport to show that it is always a precise matter whether something is determinate at every finite order. This would entail, for example, that it was always a precise matter whether someone was determinately a child at every order, and thus, presumably, that this is also a knowable matter. But it seems just as bad to be able to know things like “I stopped being a determinate child at every order after 123098309851248 nanoseconds from my birth” as to know the corresponding kinds of things about being a child.

What could the premisses be that give such a paradoxical conclusion? One of the principles, distributivity, says that a (possibly infinite) conjunction of determinate truths is determinate, the other, B, says $p \rightarrow \Delta\neg\Delta\neg p$. If $\Delta^* p$ is the conjunction of $p, \Delta p, \Delta\Delta p,$ and so on, distributivity easily gives us (1) $\Delta^*p \rightarrow\Delta\Delta^* p$. Given a logic of K for determinacy we quickly get $\Delta\neg\Delta\Delta^*p \rightarrow\Delta\neg\Delta^* p$, which combined with $\neg\Delta^* p\rightarrow \Delta\neg\Delta\Delta^* p$ (an instance of B) gives (2) $\neg\Delta^* p\rightarrow\Delta\neg\Delta^* p$. Excluded middle and (1) and (2) gives us $\Delta\Delta^* p \vee \Delta\neg\Delta^* p$, which is the bad conclusion.

In the paper I argue that B is the culprit.* The main moving part in Field’s solution to this problem, by contrast, is the rejection of distributivity. I think I finally have a conclusive argument that it is B that is responsible, and that is that B actually *entails* distributivity! In other words, no matter how you block the paradox you’ve got to deny B.

I think this is quite surprising and the argument is quite cute, so I’ve written it up in a note. I’ve put it in a pdf rather than post it up here, but it’s only two pages and the argument is actually only a few lines. Comments would be very welcome.

* Actually a whole chain of principles weaker than B can cause problems, the weakest which I consider being $\Delta(p\rightarrow\Delta p)\rightarrow(\neg p \rightarrow \Delta\neg p)$, which corresponds to the frame condition: if x can see y, there is a finite chain of steps from y back to x each step of which x can see.

## Interpreting the third truth value in Kripke’s theory of truth

March 28, 2010

Notoriously, there are many different theories of untyped truth which use Kripke’s fixed point construction in one way or another as their mathematical basis. The core result is that one can assign every sentence of a semantically closed language one of three truth values in a way that $\phi$ and $Tr(\ulcorner\phi\urcorner)$ receive the same value.

However, how one interprets these values, how they relate to valid reasoning and how they relate to assertability is left open. There are classical interpretations in which assertability goes by truth in the classical model which assigns Tr the positive extension of the fixed point, and consequence is classical (Feferman’s theory KF.) There are paraconsistent interpretations in which the middle value is thought of as “true and false”, and assertability and validity go by truth and preservation of truth. There’s also the paracomplete theory where the middle value is understood as neither true nor false and assertability and validity defined as in the paraconsistent case. Finally, you can mix these views as Tim Maudlin does – for Maudlin assertability is classical but validity is the same as the paracomplete interpretation.

In this post I want to think a bit more about the paracomplete interpretations of the third truth value. A popular view, which originated from Kripke himself, is that the third truth value is not really a truth value at all. For a sentenc to have that value is simply for the sentence to be ‘undefined’ (I’ll use ‘truth status’ instead of ‘truth value’ from now on.) Undefined sentences don’t even express a proposition – something bad happens before we can even get to the stage of assigning a truth value. It simply doesn’t make sense to ask what the world would have to be like for a sentence to ‘halfly’ hold.

This view seems to a have a number of problems. The most damning, I think, is the theory’s inability to state this explanation of the third truth status. For example, we can state what it is to fail to express a proposition in the language containing the truth predicate: a sentence has truth value 1 if it’s true, has truth value 0 if it’s negation is true, and it has truth status 1/2, i.e. doesn’t express a proposition, if neither it nor its negation is true.

In particular, we have the resources to say that the liar sentence does not express a proposition: $\neg Tr(\ulcorner\phi\urcorner)\wedge\neg Tr(\ulcorner\neg\phi\urcorner)$. However, since both conjuncts of this sentence don’t express propositions, the whole sentence,  the sentence ‘the liar does not express a proposition’, does not itself express a proposition either! Furthermore, the sentence immediately before this one doesn’t express a proposition either (and neither does this one.) It is never possible to say a sentence doesn’t express a proposition unless you’ve either failed to express a proposition, or you’ve expressed a false proposition. What’s more, we can’t state the fixed point property: we can’t say that the liar sentence has the same truth status as the sentence that says the liar is true since that won’t express a proposition either: the instance of the T-schema for the liar sentence fails to express a proposition.

The ‘no proposition’ interpretation of the third truth value is inexpressible: if you try to describe the view you fail to express anything.

Another interpretation rejects the third value altogether. This interpretation is described in Fields book, but I think it originates with Parsons. The model for assertion and denial is this: assert just the things that get value 1 in the fixed point construction and reject the rest. Thus the sentences  “some sentences are neither true nor false”, “some sentences do not express a proposition” should be rejected as they come out with value 1/2 in the minimal fixed point. As Field points out, though, this view is also expressively limited – you don’t have the resources to say what’s wrong with the liar sentence. Unlike in the previous case where you did have those resources, but you always failed to express anything with them, in this case being neither true nor false is not what’s wrong with the liar since we reject that the liar is neither true nor false. (Although Field points out that while you can classify problematic sentences in terms of rejection, you can’t classify contingent liars where you’d need to say things like ‘if such and such were the case, then s would be problematic’ since this requires an embeddable operator of some sort.)

I want to suggest a third interpretation. The basic idea is that, unlike the second interpretation, there is a sense in which we can communicate that there is a third truth status, and unlike the first, 1/2 is a truth value, in the sense that sentences with that status express propositions and those propositions “1/2-obtain” – if the world is in this state I’ll say the proposition obtails.

In particular, there are three ways the world can be with respect to a proposition: things can be such that the proposition obtains, such it fails, and such that it obtails.

What happens if you find out a sentence has truth status 1/2 (i.e. you find out it expresses a proposition that obtails)? Should you refrain from adopting any doxastic attitude, say, by remaining agnostic? I claim not – agnosticism comes about when you’re unsure about the truthvalue of a sentence, but in this case you know the truth value. However it is clear you should neither accept nor reject it either – these are reserved for propositions that obtain and fail respectively. It seems most natural on this view to introduce a third doxastic attitude: I’ll call it receptance. When you find out a sentence has truth value 1 you accept, when you find out is has value 0 you reject and when you find out it has value 1/2 you recept. If haven’t found out the truth value yet you should withold all three doxastic attitudes and remain agnostic.

How do you communicate to someone that that the liar has value 1/2? Given that the sentences which says the liar has value 1/2 also has value 1/2, you should not assert that the liar has value 1/2. You assert things in the hopes that your audience will accept them, and this clearly not what you want if the thing you want to communicate has value 1/2. Similarly you deny things in the hope that your audience will reject them. Thus this view calls for a completely new kind of speech act, which I’ll call “absertion”, that is distinct from the speech acts of assertion and denial. In a bivalent setting the goal of communication is to make your audience accept true things and reject false things, and once you’ve achieved that your job is done. However, in the trivalent setting there is more to the picture: you also want your audience to recept things that have value 1/2, which can’t be achieved by asserting them or denying them. The purpose of communication is to induce *correct* doxastic state in your audience, where a doxastic state of acceptance, rejection or receptance in s is correct iff s has value 1, 0 or 1/2 respectively. If you instead absert sentences like the liar, and your audience believes you’re being cooperative, they will adopt the correct doxastic attitude of reception.

This, I claim, all follows quite naturally from our reading of 1/2 as a third truth value. The important question is: how does this help us with the expressive problems encountered earlier? The idea is that in this setting we can *correctly* communicate our theory of truth using the speech acts of assertion, denial and absertion, and we can have correct beliefs about the world by also recepting some sentences as well as accepting and rejecting others. The problem with the earlier interpretations was that we could not correctly communicate the idea that the liar has value 1/2 because it was taken for granted that to correctly communicate this to someone involved making them accept it. On this interpretation, however, to correctly express the view requires only that you absert the sentences which have value 1/2. Of course any sentence that says of another sentence that it has value 1/2 has value 1/2 itself, so you must absert, not assert, those too. But this is all to be expected when the obective of expressing your theory is to communicate it correctly, and that communicating correctly involves more that just asserting truthfully.

Assertion in this theory behaves much like it does in the paracomplete theory that Field describes, however some of the things Field suggests we should reject we should absert instead (such as the liar.) To get the idea, let me absert some rules concerning absertion:

• You can absert the liar, and you can absert that the liar has value 1/2.
• You can absert that every sentence has value 1, 0 or 1/2.
• You ought to absert any instance of a classical law.
• Permissable absertion is not closed under modus ponens.
• If you can permissibly absert p, you can permissibly absert that you can permissibly absert p.
• If you can absert p, then you can’t assert or deny p.
• None of these rules are assertable or deniable.

(One other contrast between this view and the no-proposition view is that it sits naturally with a more truth functionally expressive logic. The no-proposition view is often motivated by the motivation for the Kleene truth functions: a three valued function that behaves like a particular two valued truth function on two valued inputs, and has value 1/2 when the corresponding two valued function could have had both 1 or 0 depending on how one replaced 1/2 in the three valued input with 1 or 0. $\neg, \vee$ is expressively adequate with respect to Kleene truth functions defined as before. However, Kripke’s construction works with any monotonic truth function (monotonic in the ordering that puts 1/2 and the bottom and 1 and 0 above it but incomparable to each other) and $\neg, \vee$ are not expressively complete w.r.t the monotonic truth functions. There are monotonic truth functions that aren’t Kleene truth functions, such as “squadge”, that puts 1/2 everywhere that Kleene conjunction and disjunction disagree, and puts the value they agree on elsewhere. Squadge, negation and disjunction are expressively complete w.r.t monotonic truth functions.)

## Truth as an operator and as a predicate

November 5, 2009

Suppose we add to the propositional calculus a new unary operator, T, whose truth table is just the trivial one that leaves the truth value of its operand untouched. By adding

• $(Tp \leftrightarrow p)$

to a standard axiomatization of the propositional calculus we completely fix the meaning of T. Moreover this is a consistent classical account of truth that gives us a kind of unrestricted “T-schema” for the truth operator.

On the face of it, then, it seems that if we treat truth as an operator operating on sentences rather than a predicate applying to names of sentences we somehow avoid the semantic paradoxes. But this seems almost like magic: both ways of talking about truth supposed to be expressing the same property – how could a grammatical difference in their formulation be the true source of the paradox?

My gut feeling is that there isn’t anything particularly deep about the consistency of the operator theory of truth: it just boils down to an accidental grammatical fact about the kinds of languages we usually speak. The grammatical fact is this. One can have syntactically simple expressions of type e but not of type t. Without the type theory jargon this just means we can have names that can be the argument of a predicate but not “names” that can be the argument of an operator. Call these latter kind of expressions “name*s”. If $p$ is a name* then $\neg p$ is grammatically well formed and is evaluated as the same as $\neg \phi$ where $\phi$ is whatever sentence p refers* to. If pick $p$ so that it refers* to “$\neg p$” then we are in just the same predicament we were in the case where we were considering names and treating truth like a predicate. One could simply pick a constant and stipulate that it refers to the sentence “~Tr(c)”.

We could make this a little more precise. By restricting our attention to languages without name*s we’re remaining silent about propositions that we could have expressed if we removed the restriction. Indeed, there is a natural translation between operator talk (in the propositional language with truth described at the beginning) and predicate talk. So, on the looks of it, it seems we could make exactly the same move in the predicate case: accept only sentences that are translations of sentences we accept. The natural translation I’m referring to is this:

• $p^* \mapsto p$
• $(\phi \wedge \psi)^* \mapsto (\phi^*\wedge\psi^*)$
• $(\neg \phi)^* \mapsto \neg \phi^*$
• $(T\phi)^* \mapsto Tr(\ulcorner\phi^*\urcorner)$

Here’s a neat little fact which is quite easy to prove. Let M be a model of the propositional calculus (a truth value assignment.)

Theorem. $\phi$ is the translation a true formula in M if and only if $\phi$ appears in Kripke’s minimal fixedpoint construction using the weak Kleene valuation with ground model M.

Note that, because we don’t have quantifiers, the construction tapers out at $\omega$ so we can prove the right-left direction by induction over the finite initial stages of the construction. Left-right is an induction over formula complexity.

If the rule is to simply reject all sentences which aren’t translations of an operator sentence then it appears that the neat classical operator view is really just the well known non-classical view based on the weak Kleene valuation scheme. It is well known that the latter only appears to be classical when we restrict attention to grounded formulae; it seems the appearance is just as shallow for the former view.

Incidentally, note that there’s no natural way to extend this result to languages with quantifiers. This is because there’s no “natural” translation between the propositional calculus with propositional quantifiers and a quantified language with the truth predicate capable of talking about its own syntax.

## Rigid Designation

October 23, 2009

Imagine the following set up. There are two tribes, A and B, who up until now have never met. It turns out that tribe A speaks English as we speak it now. However, tribe B speaks English* – a language much like English except it doesn’t contain the names “Aristotle” or “Plato”, and contains two new names, “Fred” and “Ned”.

Suppose now that these two tribes eventually meet and learn each others language. In particular tribe A and B come to agree that the following holds in the new expanded language: (1) necessarily, if Socrates was a philosopher, Fred was Aristotle and Ned was Plato, and (2) necessarily, if Socrates was never a philosopher, Fred was Plato and Ned was Aristotle.

Now we introduce to both tribes some philosophical vocabulary: we tell them what a possible world is, what it means for a name to designate something at a possible world. Both tribes think they understand the new vocabulary. We tell them a rigid designator is a term that designates the some object at every possible world.

Before meeting tribe B, tribe A will presumably agree with Kripke in saying that “Aristotle” and “Plato” are rigid designators, and after learning tribe B’s language will say that “Fred” and “Ned” are non-rigid (accidental) designators.

However tribe B will, presumably, say exactly the opposite. They’ll say that “Aristotle” is a weird and gruesome name that designates Fred in some worlds and Ned in others. Indeed whether “Aristotle” denotes Fred or Ned depends on whether Socrates is a philosopher or not, and, hence, tribe A are speaking a strange and unnatural language.

Who is speaking the most natural language is not the important question. My question is rather, how do we make sense of the notion of ‘rigid designation’ without having to assume English is privileged in some way over English*. And I’m beginning to think we can’t.

The reason, I think, is that the notion of rigid designation (and, incidentally, lots of other things philosophers of modality talk about) cannot be made sense of in the simple modal language of necessity and possibility – the language we start off with before we introduce possible worlds talk. However the answer to whether or not a name is a rigid designator makes no difference to our original language. For any set of true sentences in the simple modal language involving the name “Aristotle” I can produce you two possible worlds models that makes those sentences true: one that makes “Aristotle” denote the same individual in every world and the other which doesn’t.* If this is the case, how is the question of whether a name is a rigid designator ever substantive? Why do we need this distinction? (Note: Kripke’s arguments against descriptivism do not require the distinction. They can be formulated in pure necessity possibility talk.)

To put it another way, by extending our language to possible world/Kripke model talk we are able to postulate nonsense questions: Questions that didn’t exist in our original language but do in the extended language with the new technical vocabulary. An extreme example of such a question: is the denotation function a set of Kuratowski or Hausdorff ordered pairs? These are two different, but formally irrelevant, ways of constructing functions from sets. The question has a definite answer, depending on how we construct the model, but it is clearly an artifact of our model and corresponds to nothing in reality.

Another question which is well formed and has a definite answer in Kripke model talk: does the name ‘a’ denote the same object in w as in w’. There seems to be no way to ask this question in the original modal language. We can talk about ‘Fred’ necessarily denoting Fred, but we can’t make the interworld identity comparison. And as we’ve seen, it doesn’t make any difference to the basic modal language how we answer this question in the extended language.

[* These models will interpret names from a set of functions, S, from worlds to individuals at that world and quantification will also be cashed out in terms of the members of S. We may place the following constraint on S to get something equivalent to a Kripke model: for $f, g \in S$, if f(w) = g(w) for some w then f=g.

One might want to remove this constraint to model the language A and B speak once they’ve learned each others language. They will say things like: Fred is Aristotle, but they might have been different. (And if they accept existential generalization they’ll also say there are things which are identical but might not have been!)]

## Precisifications

August 19, 2009

I’ve been wondering just how much content there is to the claim that vagueness is truth on some but not all acceptable ways of making the language precise. It is well known that both epistemicists and supervaluationists accept this, so the claim is clearly not substantive enough to distinguish between *these* views. But does it even commit us to classical logic? Does it rule out *any* theory of vagueness.

If one allows quantification over non-classical interpretations it seems clear that this doesn’t impose much of a constraint. For example, if we include among our admissible interpretations Heyting algebras, or Lukasiewicz valuations, or what have you, it seems clear that we needn’t (determinately) have a classical logic. Similar points apply if one allowed non-classically described interpretations; interpretations that perhaps use bivalent classical semantics, but are constructed from sets for which membership may disobey excluded middle (e.g., the set of red things.)

In both cases we needn’t get classical logic. But this observation seems trite; and besides they’re not really ‘ways of making the language precise’. A precise non-bivalent interpretation is presumably one in which every atomic sentence letter receives value 1 or 0, thus making it coincide with a classical bivalent interpretation – and presumably no vaguely described precisification is a way of making the language completely precise either.

So a way of sharpening the claim I’m interested in goes as follows: vagueness is truth on some but not all admissible ways of making the language precise, where ‘a way of making the language precise’ (for a first-order language) is a Tarskian model constructed from crisp sets. A set X is crisp iff $\forall x(\Delta x\in X \vee \Delta x \not\in X)$. This presumably entails $\forall x(x\in X \vee x\not\in X)$ which is what crispness amounts to for a non-classical logician. An admissible precisification is defined as follows

• v is correct iff the schema $v \models \ulcorner \phi \urcorner \leftrightarrow \phi$ holds.
• v is admissible iff it’s not determinately incorrect.

Intuitively, being correct means getting everything right – v is correct when truth-according-to-v obeys the T-schema. Being admissible means not getting anything determinately wrong – i.e., not being determinately incorrect. Clearly this is a constraint on a theory of vagueness, not an account. If it were an account of vagueness it would be patently circular as both ‘crisp’ and ‘admissible’ were defined in terms of ‘vague’.

Now that I’ve sharpened the claim, my question is: just how much of a constraint is this? As we noted, this seems to be something that every classicist can (and probably should) hold, whether they read $\nabla$ as a kind of ignorance, semantic indecision, ontic indeterminacy, truth value gap, context sensitivity or as playing a particular normative role with respect to your credences, to name a few. Traditional accounts of supervaluationism don’t really say much about how we should read $\nabla$, so the claim that vagueness is truth on some but not all admissible precisifications doesn’t say very much at all.

But what is even worse is that even non-classical logicians have to endorse this claim. I’ll show this is so for the Lukasiewicz semantics but I’m pretty sure it will generalise to any sensible logic you’d care to devise. [Actually, for a technical reason, you have to show it’s true for Lukasiewicz logic with rational constants. This is no big loss, since it’s quite plausible that for every rational in [0,1] some sentence of English has that truth value: e.g. the sentences “x is red” for x ranging over shades in the spectrum between orange and red would do.]

Supposing that $\Delta \phi \leftrightarrow \forall v(admissible(v) \rightarrow v \models \ulcorner \phi \urcorner)$ has semantic value 1, you can show, with a bit of calculation, that this requires that $\delta \|\phi\| = inf_{v\not\models \phi}(\delta(1-inf_{v \models \psi}\|\psi\|))$, where $\delta$ is the function interpreting $\Delta$. Assuming that $\delta$ is continuous this simplifies to: $\delta\|\phi\| = \delta(1-sup_{v\not\models \phi}inf_{v\models \psi}\|\psi\|)$. Now since no matter what v is, so long as $v \not\models \phi$, we’re going to get that $inf_{v \models \psi}\|\psi\| \leq \|\neg\phi\|$, since v is classical (i.e. $v \models \neg\phi$.) But since we added all those rational constants the supremum of all these infs is going to be $\|\neg\phi\|$ itself. So $\|\phi\| = 1-sup_{v\not\models \phi}inf_{v\models \psi}\|\psi\|$ no matter what.

So if one assumes that $\delta$ is continuous it follows that determinacy is truth in every admissible precisification (and that vagueness is truth in some but not all admissible precisifications.) The claim that $\delta$ should be continuous amounts to the claim that a conjunction of determinate truths is determinate, which as I’ve argued before, cannot be denied unless one either denies that infinitary conjunction is precise or that vagueness is hereditary.

## Truth Functionality

May 4, 2009

I’ve been thinking a lot about giving intended models to non-classical logics recently, and this has got me very muddled about truth functionality.

Truth functionality seems like such a simple notion. An n-ary connective, $\oplus$, is truth functional just in case the truth value of $\oplus(p_1, \ldots, p_n)$ depends only on the truth values of $p_1, \ldots, p_n$.

But cashing out what “depends” means here is harder than it sounds. Consider, for example, the following (familiar) connectives.

• $|\Box p| = T$ iff, necessarily, $|p| = T$.
• $|p \vee q| = T$ iff $|p| = T$ or $|q| = T$.

Why, in the second example but not the first, does the truth value of $\Box p$ depend on the truth value of p? They’ve both been given in terms of the truth value of p. It would be correct, but circular, to say that the truth value of $\Box p$ doesn’t depend on the truth value of p, because it’s truth value isn’t definable from the truth value of p using only truth functional vocabulary in the metalanguage. But clearly this isn’t helpful – for we want to know what counts as truth functional vocabulary whether in the metalanguage or anywhere. For example, what distinguishes the first from the second example. To say that $\vee$ is truth functional and $\Box$ isn’t because “or” is truth functional and “necessarily” isn’t, is totally unhelpful.

Usually the circularity is better hidden than this. For example, you can talk about “assignments” of truth values to sentence letters, and say that if two assignments agree on the truth values of $p_1, \ldots, p_n$ then they’ll agree on $\oplus(p_1, \ldots, p_n)$. But what are “assignments” and what is “agreement”? One could simply stipulate that assignments are functions in extension (sets of ordered pairs) and that f and g agree on some sentences if f(p)=g(p) for each such sentence p.

But there must be more restrictions that this: presumably the assignment that assigns p and q F and $p \vee q$ T is not an acceptable assignment. There are assignments which give the same truth values to p and q, but different truth values to $p \vee q$, making disjunction non truth functional. Thus we must restrict ourselves to acceptable assignments; assignments which preserve truth functionality of the truth functional connectives.

Secondly, there needs to be enough assigments. The talk of assignments is only ok if there is an assignment corresponding to the intended assignment of truth values to English sentences. I beleive that it’s vague whether p, just in case it’s vague whether “p” is true (this follows from the assertion that the T-schema is determinate.) Thus if there’s vagueness in our langauge, we had better admit assignments such that it can be vague whether f(p)=T. Thus the restriction to precise assignments is not in general OK. Similarly, if you think the T-schema is necessary, the restriction of assignments to functions in extension is not innocent either – e.g., if p is true but not necessary, we need an assignment such that f(p)=T and that possibly f(p)=F.

Let me take an example where I think it really matters. A non-classical logician, for concreteness take a proponent of Lukasiewicz logic, will typically think there are more truth functional connectives (of a given arity) than the classical logician. For example, our Lukasiewicz logician thinks that the conditional is not definable from negation and disjunction. (NOTE: I do not mean truth functional on the continuum of truth values [0, 1] – I mean on {T, F} in a metalanguage where it can be vague that f(p)=T.)) “How can this be?” you ask, surely we can just count the truth tables: there are $2^{2^n}$ truth functional n-ary connectives.

To see why it’s not so simple consider a simple example. We want to calculate the truth table of $p \rightarrow q$.

• $p \rightarrow q$: reads T just in case the second column reads T, if the the first column does.
• $p \vee q$: reads T just in case the first or the second column reads T.
• $\neg p$: reads T if the first column doesn’t read T.

The classical logician claims that the truth table for $p \rightarrow q$ should be the same as the truth table for $\neg p \vee q$. This is because she accepts the equivelance between the “the first column is T if the second is” and “the second column is T or the first isn’t” in the metalanguage. However the non-classical logician denies this – the truth values will differ in cases where it is vague what truth value the first and second columns read. For example, if it is vague whether both columns read T, but the second reads T if the second does (suppose the second column reads T iff 87 is small, and the second column reads T iff 88 is small), then the column for $\rightarrow$ will determinately read T. But the statement that $\neg p \vee q$ reads T will be equivalent to an instance of excluded middle in the metalanguage which fails. So it will be vague in that case whether it reads T.

The case that $\rightarrow$ is truth functional for this non-classical logician seems to me pretty compelling. But why, then, can we not make exactly the same case for the truth functionality of $\Box p$? I see almost no disanalogy in the reasoning. Suppose I deny that negation and the truth operator are the only unary truth functional connectives, I claim $\Box p$ is a further one. However, the only cases where negation and the truth operator come apart from necessity is when it is contingent what the first column of the truth table reads.

I expect there is some way of unentangling all of this, but I think, at least, that the standard explanations of truth functionality fail to do this.

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.