Archive for the ‘Philosophy of Language’ Category


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!)]



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.


Vagueness and uncertainty

June 17, 2009

My BPhil thesis is finally finished so I thought I’d post it here for anyone who’s interested.


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.


The Sorites paradox and non-standard models of arithmetic

December 16, 2008

A standard Sorites paradox might run as follows:

  • 1 is small.
  • For every n, if n is small then n+1 is small.
  • There are non-small numbers.

On the face of it, these three principles are inconsistent, since the first two premisses entail that every number is small by the principle of induction. As far as I know, there is no theory of vagueness that gives us that these three sentences are true (and none of them false.) Nonetheless, it would be desirable if these sentences could be satisfied.

The principle of induction seems to do fine when we are dealing with precise concepts. Thus the induction schema for PA is fine, since it only says that it holds for properties definable in arithmetical vocabulary – all of which is precise. However, if we read the induction schema as open ended, that is, to hold even if we were to extend the language with new vocabulary, it is false. For it fails when we introduce into the language vague predicates.

The induction schema is usually proved by appealing to the fact that the naturals are well-ordered: every subset of the naturals has a least element. If the induction schema is going to fail if we allow vague sets, so should the well ordering principle. And that seems right: the set of large numbers doesn’t appear to have a least element – there is no first large number. So we have:

  • The set of large numbers has no smallest member.

Again no theory I know of delivers this verdict. The best we get is with non classical logics, where it is at best vague whether there exists a least element of the set of large numbers.

Finally, I think we should also hold the following:

  • For any particular number, n, you cannot assert that n is large.

That is, to assert of a given number, n, that it is large is to invite the Sorites paradox. You may assert that there exist large numbers, its just you can’t say exactly which they are. To assert that n is large, is to commit yourself to an inconsistency by standard Sorites reasoning, from n-1 true conditionals and the fact that 0 is not large.

The proposal I want to consider verifies all three of the bulletted points above. As it turns out, given a background of PA, the initial trio isn’t inconsistent after all. It’s merely \omegainconsistent (given we’re not assuming open ended induction.) But this doesn’t strike me as a bad thing in the context of vagueness, since after all, you can go through each of the natural numbers and convince me its not large by Sorites reasoning, but that shouldn’t shake my belief that there are large numbers.

\omega-inconsistent theories are formally consistent with the PA axioms, and thus have models by Gödel’s completeness theorem. These are called non-standard models of arithmetic. They basically have all the sets of naturals the ordinary natural numbers have, except they admit more subsets of the naturals – they admit vague sets of natural numbers as well as the old precise sets. Intuitively this is right – when we only had precise sets we got into all sorts of trouble. We couldn’t even talk about the set of large numbers because it didn’t exist; it was a vague set.

What is interesting is that some of these new sets of natural numbers don’t have smallest members. In fact, the set of all non-standard elements is one of these sets, but there are many others. So my suggestion here is that the set of large numbers is one of these non-standard sets of naturals.

Finally, we don’t want to be able to assert that n is large, for any given n, since that would lead us to true contradiction (via a long series of conditionals.) The idea is we may assert that there are large numbers out there, but we just cannot say which ones. On first glance this might seem incoherent, however, it is just another case of \omega-inconsistency. \{\neg Ln \mid n a numeral \} \cup \{\exists x Lx\} is formally consistent. For example, this is satisfied in any non-standard model of PA with L interpreted as the set of non-standard elements.

How to make sense of all this? Well, the first thing to bear in mind is that the non-standard models of arithmetic are not to be taken too seriously. They show that the view in question is consistent, and are also a good guide to seeing what sentences are in fact true. For example in a non-standard model the second order universally quantified induction axiom is false, since the second order quantifiers range over vague sets, however the induction schema is true, provided it only allows instances of properties definable in the language of arithmetic (this is how the schema is usually stated) since those instances define only precise sets. We should not think of the non-standard models as accurate guides to reality, however, since they are constructed from purely precise sets, of the kind ZFC deals with. For example, the set of non-standard elements is a precise set being used to model a vague set. Furthermore, the non-standard models are described as having an initial segment which are the “real” natural numbers, and then a block of non-standard naturals coming after them. The intended model of our theory shouldn’t have these extra elements, it should have the same numbers, just with more sets of numbers, vague and precise ones.

Another question is, which non-standard model makes the right (second order) sentences true? Since there are only countably many naturals, we can add a second order sentence stating this to our theory (we’d have to check it still means the same thing once the quantifiers range over vague sets as well.) This would force the model to be countable. Call the first order sentences true in the standard model plus the second order sentence saying the universe is countable, plus the statements: (i) 0 is small, (ii) for every n, if in is small, n+1 is small and (iii) there are non small numbers, T. T is still consistent (by the Lowenheim-Skolem theorem), and I think this will uniquely pick out our model as \mathbb{N} + \mathbb{Q} by a result from Skolem (I can’t quite remember the result right now, but maybe someone can correct me if its wrong.) This only gives us the interpretation for the second order quantifiers and the arithmetic vocabulary, obviously it won’t tell us how to interpret the vague vocabulary.


Higher Order Vagueness and Sharp Boundaries

September 1, 2008

One of the driving intuitions that motivates the rejection of bivalence, in the context of vagueness, is the intuition that there are no sharp cut off points. There is no number of nanoseconds such that anything living that long is young, but anything older would cease to be young (a supervaluationist will have to qualify this further, but something similar can be said for them too.) The thought is that the meaning determining factors, such as the way we use language, simply cannot determine such precise boundaries. Presumably there are many different precise interpretation of language that are compatible with our usage, and the other relevant factors.

The intuition extends further. Surely there is no sharp cut off point between being young, and being borderline young (and between being borderline and being not young.) There are borderline bordeline cases. And similarly there shouldn’t be sharp cut off points between youngness, and borderline borderline youngness etc… Thus there should be all kinds of orders of vagueness – at each level we escape sharp cut off points by positing higher levels of vagueness.

This line of thought is initially attractive, but it has its limitations. Surely there must be sharp cut off points between being truly young and failing to be truly young – where failing to be young includes being borderline young, or borderline borderline young, and so on. Basically, failing to be true involves anything less than full truth.

Talk of ‘full truth’ has to be taken with a pinch of salt. We have moved from talking about precise boundaries in the object language, to metatheoretical talk about truth values. This assumes that those who reject sharp boundaries identify the intended models with the kinds of many truth valued models used to characterise validity (my thinking on this was cleared up a lot by these helpful two posts by Robbie Williams.) Timothy Williamson offers this neat argument that we’ll be committed to sharp boundaries either way, and it can be couched purely in the object language. Suppose we have an operator, \Delta p, which says that p is determinately true. In the presence of higher order vagueness, we may have indeterminacy about whether p is determinate, and we may have indeterminacy about whether p is not determinate. I.e. \Delta fails to be governed by the S4 and S5 axioms respectively. However, we can introduce the following operator, which is supposed to represent our notion of being absolutely determinately true as the following infinite conjunction:

  • \Delta^\omega p := p \wedge \Delta p \wedge \Delta\Delta p \wedge \ldots

We assume one fact about \Delta. Namely: that an arbitrary conjunction of determinate truths is also determinate (we actually only need the claim for countable conjunctions.)

  • \Delta p_1 \wedge \Delta p_2 \wedge \ldots \equiv \Delta (p_1 \wedge p_2 \wedge \ldots )

From this we can deduce that \Delta^\omega p obeys S5 S4 [edit: only if you assume that \Delta obeys the B princple do we get S5. Thanks to Brian for correcting this] (its exactly the same way you show that common knowledge obeys S4, if you know that proof.) If \Delta^\omega p holds, then we have \Delta p \wedge \Delta\Delta p \wedge \Delta\Delta\Delta p \wedge \ldots by conjunction elimination, and by the definition of \Delta^\omega. By the assumed fact, this is equivalent to \Delta(p \wedge \Delta p \wedge \ldots ) which by definition is \Delta(\Delta^\omega) p. We then just iterate this, to get each finite iteration \Delta^n\Delta^\omega p and collect them together using an infinitary conjunction introduction rule to got \Delta^\omega\Delta^\omega p.

But what of the assumption that a conjunction of determinate truths is determinate? It seems pretty uncontroversial. In the finite case its obvious. If A and B are determinate, how can (A ^ B) fail to be? Where would the vagueness come from? Not A and B, by stipulation, and presumably not from ^ – it’s a logical constant after all. The infinitary case seems equally sound. However – the assumption is not completely beyond doubt. For example in some recent papers Field has to reject the principle. On his model theory, p is definitely true if a sequence of three valued interpretation eventually converges to 1 – after some point it is 1’s all the way. Now its possible for each conjunct, of an infinite conjunction, to all converge on 1, but that there is no point along the sequence such that all of them are 1 from then on.

I think we can actually do away with Williamsons’ assumption – all we need is factivity for \Delta.

  • \Delta p \rightarrow p

To state the idea we’re going to have to make some standard assumptions about the structure of the truth values sentences can take. We’ll then look at how to eliminate reference to truth values all together, so we do not beg the question against those who just take the many valued semantics as a way to characterise validity and consequence relations. The assumption is that the set of truth values is a complete lattice, and that conjunction, disjunction, negation, etc… all take the obvious interpretations: meet, join, complement and so on. They form a lattice because conjunction, negation etc… must retain their characteristic features, and the lattice is complete because we can take arbitrary conjunctions and disjunctions. As far as I know, this assumption is true of all the major contenders: the three valued logic of Tye, continuum valued logic of Lukasiewicz’, supervaluationism (where elements in the lattice are just sets of precisifications), intuitionism and, of course, bivalent theories are all complete lattices.

In this framework the interpretation of \Delta will just be a function, \delta, from the lattice to itself and the sharp boundaries hypothesis corresponds to obtaining a fixpoint of this function for each element by iterating \delta. There are a variety of fixpoint theorems available for continuous functions (these are obtained by iterating \omega many times – indeed the infinitary distributivity of \Delta in the Williamson proof is just the assumption of continuity for \delta.) However, it is also true that every monotonic function will have fixpoint that is obtained by iterating – the only difference is that we will have to iterate for longer than omega. (Note that our assumption of factivity for \Delta ensures monotonicity of \delta – i.e. that \delta (x) \sqsubseteq x.) Define \delta^\alpha for ordinals alpha recursively as follows:

  • \delta^0(a) := a
  • \delta^{\alpha + 1}(a) := \delta(\delta^\alpha(a))
  • \delta^\gamma (a) : = \sqcap_{\alpha < \gamma}\delta^\alpha(a)

(We can define \Delta^\alpha analogously if we assume a language which allows for arbitrary conjunctions.) Fix a lattice, B, and let \kappa := sup\{ \alpha \mid \mbox{there is an } \alpha \mbox{ length chain in B}\} where a chain is simply a subset of B that is linearly ordered. It is easy to see that \delta(\delta^\kappa(a)) = \delta^\kappa(a) for every a in B, and thus that \delta^\kappa(\delta^\kappa(a)) = \delta^\kappa(a). This ensures the S4 axiom for \Delta^\kappa

  • \Delta^\kappa p \rightarrow \Delta^\kappa\Delta^\kappa p

We can show the S5 axiom too.

Note that trivially, every (weakly) monotonic function on a lattice has a fixpoint: the top/bottom element. The crucial point of the above is that we are able to define the fixpoint in the language. Secondly, note also that \kappa depends on the size of the lattice in question – which allows us to calculate \kappa for most of the popular theories in the literature. For bivalent and n-valent logics its finite. For continuum valued fuzzy logics the longest chain is just 2^{\aleph_0}. Supervaluationism is a bit harder. Precisifications are just functions from a countable set of predicates to sets of objects. Lewis estimates the number of objects is at most \beth_2, making the number of extensions \beth_3. Obviously \beth_3^{\aleph_0} = \beth_3 so the longest chains are at most \beth_4 (and I’m pretty sure you can always get chains that long.)

All the theories, except possibly supervaluationism, define validity with respect to a fixed lattice. Things get a bit more complicated if you define validity with respect to a class of lattices. Since the class might have arbitrarily large  lattices, there’s no guarantee we can find a big enough \kappa. But that said, I think we could instead introduce a new operator, \Delta^* p := \forall \alpha \in On \Delta^\alpha p which would do the trick instead (I know, I’m being careless here with notation – but I’m pretty sure it could be made rigorous.)

Final note: what if you take the role of the many valued lattice models to be a way to characterise validity rather than as a semantics for the language? For example – couldn’t you adopt a continuum valued logic, but withhold identifying the values in [0,1] with truth values. The worry still remains however. Let c := 2^{\aleph_0}. Since \Delta^c p \rightarrow \Delta^c\Delta^c p is true in all continuum valued models, it is valid. Valid gives us true, so we end up with sharp boundaries in the intended model – even though the intended model needn’t look anything like the continuum valued models.

Apologies: if you read this just after I posted it. Throughout I wrote ‘Boolean algebra’ where I meant ‘lattice’ :-s. I have corrected this now.


Counterexamples to Modus Ponens?

August 19, 2008

Moritz has a very interesting post on McGee’s counterexample to modus ponens. One thing that came up in the comments was how things looked on the restrictor analysis of conditionals. On this view ‘if’ is not to be thought of syntactically as a connective, but rather as a device for restricting modals. The basic idea is, given a modal, O, and antecedent and consequent, p and q, ‘if’ allows us to restrict the domain of O to p-worlds. Roughly, ‘if p, Oq’ means that q is true in all the best O-accessible p-worlds. In this note I want to argue that the restrictor view will have to admit exceptions to modus ponens even for simple (non nested) conditionals.

Moritz points out that, since we are no longer dealing with a connective, it is not clear what counts as an instance of modus ponens. One reaction, then, might be to simply say that there are no instances of modus ponens for the English conditional; the rule is only applicable to connectives, so we should reserve the rule for formal connectives that behave in the right way syntactically. I think this is a little extreme – after all, modus ponens presumably originated from a rule of inference for natural language conditionals, and we seem to be able to recognise instances of it when we them in natural language reasoning.

On the syntactic understanding of modus ponens for connectives the rule is something like the following: from p and p \rhd q infer q (where \rhd is syntactically a connective.) Thus we can say, for example, that modus ponens holds for the intuitionist conditional, the Stalnaker conditional, even for classical conjunction, but not for classical disjunction, the reverse conditional (\leftarrow) and so on. But even here we’re not home and dry. We still need to be able to distinguish the antecedent from the consequent syntactically. Does modus ponens hold for the connective \frac{p}{q} defined to have the same truth conditions as p \rightarrow q? Depending on which counts as the antecedent, we get the material conditional or the material reverse conditional – MP holds for the former but not latter.

In short, I think an explicit syntactic characterisation of MP is not the correct way to go. For now I think we can just settle for a rule of thumb: if we can recognise it as in instance of modus ponens in English, then its underlying logical form forms the basis for an admissible syntactic characterisation for modus ponens. This allows us to characterise modus ponens even for syntactic items that aren’t connectives. Indeed, if English conditionals really do have the logical form the restrictor analysis predicts, then the following schemas are plausible syntactic characterisations of MP. (Notation: if \Box is an operator, \mbox{[if }p: \Box ] is the new operator formed by restricting by p.)

  1. From p and \mbox{[if }p: \Box ]q, infer q
  2. From p and \mbox{[if }p: \Box ]q, infer \Box q

Now I’m not going to defend the claim that both of these form genuine rules that capture modus ponens for the English conditional. Rather, all I want is that there are no other candidate rules to fill this role which are remotely plausible. This can be argued for by brute force. We have no difficulty identifying the antecedent and the conditional, thus premisses are definitely as represented as above. Combinatorially, there are only four candidates for the conclusion that we can construct from the available items: p, q, \Box p, and \Box q. Clearly we can eliminate p and \Box p.

Let us start with (1). Since the type of ‘if’ is a relation between two propositions and an operator, (1) is only valid if no matter how we reinterpret p, q and \Box the conclusion is true whenever the premises are. A related question, which will determine the answer above one, is to ask for what interpretations of \Box is the above valid (i.e. fixing the interpretation of \Box and varying only p and q.) It turns out (1) is valid relative to an interpretation of \Box iff it has a reflexive accessibility relation. Let’s consider just the direction of the biconditional that invalidates (1): it is perfectly possible for p to be true at a world w, and for q to be true at all the accessible p-worlds, and q false at w so long as w isn’t among the accessible p-worlds. Since w is a p-world, it must be inaccessible – i.e. this happens when the accessibility relation isn’t reflexive.

To illustrate, consider an example where the restricted modal isn’t reflexive. The easiest cases to consider are doxastic and deontic modals.

  • John’s a murderer
  • If John’s a murderer, he ought to be in jail
  • Therefore, John is in jail

This is a counterexample to (1), there is not enough evidence to put John away. You might object – surely this was not a plausible characterisation of modus ponens in the first place. What we should rather conclude from the first two premises is that John ought to be in jail. This is essentially an appeal to (2).

Unfortunately (2) admits counterexamples as well. Suppose there are accessible ~p worlds, where q is false, but q is true at all the accessible p worlds, and that p is true at the actual world. Certainly p is true, and \mbox{[if }p: \Box ]q is true because q is true in all the accessible p-worlds. However, \Box q is false, since there are accessible p-worlds where q is false.

Examples are difficult to conjure up, since modals are often sensitive to the conversational background. Whenever I assert p, I essentially restrict the \Box-accessible worlds to p-worlds. Thus asserting p and then \mbox{[if }p: \Box ]q invariably places us in a context according to which \Box q is true. Let us try anyway. Suppose that for all we know, Jones isn’t in his office. So: it is not the case that Jones must be in his office. Be that as it may, if it is 3.00pm, Jones must be in his office, because we know his lunch break ends at 2.00pm. Suppose further that unbeknownst to us, it is be 3.00pm. That sounds like a consistent story right? But now if I assert them in the following order:

  • It’s 3.00pm
  • If it’s 3.00pm, then Jones must be in his office
  • It’s not the case that Jones must be in his office

the trio above sounds inconsistent. But strictly speaking they’re all true, provided we keep the same context throughout.

What is interesting is that the McGee counterexamples, on a plausible syntactic analysis, form more convincing counterexamples to modus ponens on characterisation (2). For on a plausible syntactic analysis of nested conditionals, ‘if p, if q, then must r‘ the inside conditional ‘if q, then must r‘ has the form s = \mbox{[if }q: \Box ]r. Since s has the form operator:proposition, ‘if p, s’ is naturally read with ‘if p’ restricting the compound operator, rather than some covert modal. Thus the embedded conditional gets the form \mbox{[if }p: \mbox{[if }q: \Box ]]r. Essentially we are doubly restricting \Box. The resulting McGee example is not an instance of modus ponens on characterisation (1). However it does represent a failure for the rule (2), where the modal is the compound operator \mbox{[if }q: \Box ].


Are we living in a possible world?

May 30, 2008

Assume the following kind of picture. You have possible worlds – maximally specific ways the world can be. Worlds can overlap – one and the same individual may be a part of many different worlds. But suppose also that worlds can overlap in a more extreme way, worlds can share large chunks of history, the same state of affairs, or possible situation might be a part of or exist in many different worlds.

Now ask the question: which world do I belong to? Which of these worlds is mine? Since I exist in many worlds there is no unique answer. But intuitively there should be: ‘the actual world’. My impression, (but that’s all it is) is that people generally think that this is not a particularly deep question – the actual world can just be picked out indexically – ‘the world I belong to’, or ‘this one here’. But as we noted, ‘the world I belong to’ won’t do, I belong to many, and if you follow me in thinking situations are parts of multiple worlds, ‘this one here’ won’t do either. The most I can do is demonstratively pick out the situation I’m currently in – I can specify the situation in so far as I can point to or describe it.

Picking it out indexically is not the way to go. Presumably you will say that the actual world is somehow metaphysically privileged – my world is the metaphysically privileged one. But then there is a puzzle about how non-actual English speakers can say things like ‘We live in the actual world’ – that’s supposed to be true relative to their context of utterence. But if ‘actual’ isn’t working indexically there, it can’t be true. Even if, for some mysterious reason, it is working indexically for them and not for us, we want to know what something with the same linguistic meaning as our utterence would have said in that situation.

What I’m interested in is the idea that we’re not actually in any possible world – that we’re living in some less than maximal possible situation. Maybe there is some metaphysical fact about which situation that is, or maybe the actual situation depends on my context of uttence – I shall leave it open. I just want to see how things pan out.

Here’s one candidate: the actual situation is just everything that’s happened so far. This situation is part of many different possible worlds – it’s part of a possible world in which I get a haircut tomorrow, and a possible world in which I don’t. There is no matter of fact, yet, whether I get a haircut tomorrow, because actuality just doesn’t say anything about it, its incomplete. (Although we could supervaluate and say I either do or I don’t, because in every world which contains actuality, I do or I don’t.) It’s an submaximal situation in so far as the future is genuinely indeterminate.

Here’s another one: maybe the actual situation is incomplete in that it doesn’t make true the following kinds of proposition: the boundaries of mount Everest are thus and so, this droplet is a part of this cloud, and so on. Maybe the world is inherently vague. If we go for the first option, the actual situation depends on context, we simply cannot demonstratively pick out one world where Everest has such and such borders, over the other one where it doesn’t.

Lastly – the actual situation doesn’t include facts like: this quark is located here with these properties. If the actual situation is indeterminate in the way that quantum mechanics says it is, then we might even be able to argue that there is no maximally specific world containing this one. Some situations can’t be extended to a maximally specific world because the physical laws of the situation prohibit it.

Ok, so those are my thoughts so far (a bit brief I know.) It all sounds a bit radical, but I haven’t been able to think up a nice solution to the motivating worry. (It’s not clear that going submaximal in the way I described helps either.) Does anyone know any literature that might be relavent?


The Vagueness of “Vague”

May 23, 2008

Back over here I was trying to argue that the determinately operator, \Delta, could not be a logical constant because it is vague. In the course of my pitch for this I had to appeal to an argument due to Sorensen for the vagueness of “vague”. Define:

  • x is k-small iff x < k or x is small

Since being 0-small is equivalent to being small, 0-small is vague. Since n := Rayo’s number isn’t small, being n-small is equivalent to being less than n, which is precise, so n-small isn’t vague. Since the predicates ‘k-small’ for k < n form a sorites of “is vague”, “is vague” must be vague.

In the comments Robbie pressed me on this last step. It seems the existence of a Sorites isn’t always sufficient for a predicate to be vague. Especially when the terms in the sorites sequence are themselves vague (as in this case.) Anyway, I thought I’d repost some of the discussion here for a wider audience (and also give me a chance to get clearer on what’s going on.) Robbies example was as follows:

“Suppose we have a paradigm electron, Sparky, and a paradigm non-electron, Robbie. Let “item number n” be introduced as the x satisfying the following reference-fixing description: either x is Sparky, and n is small; or x is Robbie, and n is not small. Now the first few terms (where x is determinately small) will determinately refer to Sparky, and the last few (where x is determinately not small) will determinately refer to Robbie. But “item number n” for borderline-small n will be indeterminate in reference between the two.

Now consider the collection of claims: (0) “item 0 is an electron”; (1) “item 1 is an electron”…. (n) “item n is an electron”. This’ll display the characteristics of a forced-march sorites, I guess, and we could turn it into a sorites paradox without too much trouble. Does it show “electron” is vague? Not at all—intuitively the reason we get a sorites is because of the vagueness in the referential terms.”

This left me a bit worried. Not just because I couldn’t establish the conclusion I wanted, but because it seems we don’t have any sufficient condition under which we can conclude that any expression is vague.

That said, I don’t think Sorites are completely useless in testing whether predicates are vague. Since the counterexamples only work when the terms in the Sorites sequence are vague, the existence of a Sorites for F should show at least the following disjunctive claim:

  • Either (i) the terms in the Sorites are vague, or (ii) F is vague.

And I think we can get a more useful version for testing for vagueness from this, namely:

  • If there is a Sorites for F, and all the terms in the Sorites sequence are precise, the F is vague.

With that in place, I was wondering if I could run a better argument for the vagueness of “vague”.

Let “item number k” refer (determinately) to the predicate “is k-small”. Now we can run a Sorites as follows: (i) item number 0 is vague (ii) if item number k is vague, so is item number k+1, (iii) item number n isn’t vague, where n := Rayo’s number.

The idea is that we have reformulated Sorensens argument so that the terms of the Sorites sequence are completely precise. With our new principle for testing vagueness in place, this should be sufficient to establish its vagueness.

Two things to note. Nothing depends on whether you think it is predicates like “is small” that are vague, or the semantic values of these things that are vague. On the first view a predicates reference is indeterminate between many precise extensions, on the second predicates have exactly one reference, a vague semantic value (a function from precisifications to extensions.) The argument could be run either way as far as I can see.

Secondly, if we can succeed in determinately referring to things at all, the names “item number k”, for each k, must be among those cases. Now I don’t have much of an argument for this yet, but I just can’t see how indeterminacy could enter into the picture there at all. We are talking about abstract entities (strings of letters, or semantic values dependingly) and it seems to me that if we can refer to these kinds of things at all we can refer to them precisely.

Anyway, that’s that – I’m still not 100% sure this will do the job. One worry is that if we go for the second option above (it’s semantic values, not syntactic entities, that are vague) then it isn’t completely clear that determinate reference has been achieved. For suppose the semantic values of predicates are functions from admissible precisifications to extensions. Then, in the presence of higher order vagueness, it will be vague which precisifications are admissible – and thus it will be vague what the domain of these functions are. In other words, it might still be vague what the semantic value of “is k-small” is, and my “item number k” trick won’t do the job. (It’s interesting, I think, that this problem doesn’t arise for the view that it is predicates that are vague.)


Presuppositions and Modal Operators

May 13, 2008

Ok, so I don’t know anything about presupposition failure, so I’m going to keep this short and simple and hope someone can set me straight.

Suppose the Strawsonian view of definite descriptions. Now read the following giving the description narrower scope than necessity.

  1. Necessarily the Queen of England is a queen.

I want to know what happens to this sentence on the Strawsonian view. My first thought is that 1. is truthvalueless because there are worlds in which the embedded sentence is truthvalueless because there is no queen of England (the intuition is that 1. is like the conjunction “at w, the Queen of England is a queen and at w’ the Queen of England is a queen and …”, and a conjunction is truthvalueless if one of its conjuncts is.)

But this doesn’t sound right to me at all. 1. doesn’t seem like a presupposition failure – I’ve just ascribed necessity to a perfectly well behaved proposition (all of its parts exist.) I admit, it may be controversial whether 1) is true or false (depending on whether we consider worlds where England has no queen), but to my mind it certainly isn’t a presupposition failure – it only represents a possible presupposition failure.

Maybe you could say that p is necessary iff its true or truthvalueless in every world. But then

  1. Necessarily the Queen of England exists

comes out true. This is bad, especially on it’s narrow scope reading (so bad independently of your views on fixed/variable domain Kripke semantics.)

I considered a couple of other ways of treating 1. but they don’t seem to work either, so I think I’ll leave it there. Can anyone tell me how this is supposed to work?


Is ‘determinately’ a logical constant?

April 28, 2008

I’ve had a very interesting discussion about the logical status of the determinately operator over at Theories n’ Things recently. One counter (due to Tim Williamson I think) to the claim that supervaluationism preserves classical logic is the fact that bad things happen when a \Delta (‘determinately’) operator is added to the language. For example, if validity is thought of as preservation of supertruth, then the deduction theorem fails.

  • If p\models \Delta p then \models (p \rightarrow \Delta p)

Any model in which p is supertrue is one in which \Delta p is supertrue. However, take a model in which p is true on some but not all precisifications. Let v be a precisification on which p is true. Since there are precisifications on which which p is false, \Delta p is false on all precisifications, in particular v. So the conditional is false on v, and is thus not supertrue.

I don’t particularly like the definition of validity used here. If you think that truth is fundamentally a property of propositions, and you think vague sentences express multiple propositions then the idea that validity preserves truth can be fleshed out as valid arguments preserve truth at a precisification (we’re starting off with a relation, \models, between propositions, and we’re raising it to a relation between sentences which are ambiguous.)

But of course, even if we go for validity-as-preservation-of-supertruth, the argument is still not straightforward, for it depends on whether we’re keeping the interpretation of \Delta fixed across models. If we’re not, then the deduction theorem holds again – interpret \Delta as negation and the antecedent fails. Robbie pointed me to a nice paper where he made essentially this point.

So why shouldn’t we treat \Delta as a logical constant? Well one test is to see if it is invariant under arbitrary permutation of the domain. It’s not clear how to apply this to intensional operators, but there are similar invariance tests you can apply to the accessibility relation. As far as I can tell \Delta fails (but I still need to check up on what the MacFarlane tests are here.)

Another reason to think that \Delta isn’t logical is as follows: the logical constants cannot be vague. Why is this? Well the supervaluationist will presumably say logical truths must be true at all precisifications (and all worlds etc…) But it is independently plausible (and also widely believed, see e.g. the Lewis/Sider argument for unrestricted composition.)

I want to argue that in a setting which countenances higher order vagueness, \Delta is vague, and thus cannot be a logical constant. Unfortunately the standard tests for vagueness are difficult to apply. If our term is of the type of a predicate, <e, t>, then we can just see if it is susceptible to a sorites paradox (this isn’t perfect since it doesn’t give a positive test for when a predicate isn’t vague, i.e. precise, but its good enough for most purposes.) Then we can recursively apply the test to other types as follows: assume we can test for vagueness for the lower types. Combine the higher type term we want to test with precise terms (we can test these for precision by inductive hypothesis.) Finally test the result (of a lower type) for vagueness.

But in our case, if \Delta p is vague, then there is a good case for saying that p wasn’t precise in the first place. If a term is precise, this fact should itself be precise. Not only is it determinately true that everything is self identical, but it’s determinately determinately true, and so on. But in the context of higher order vagueness, this is exactly the principle we want to fail.

It’s not looking so good for the idea that \Delta is vague. But nonetheless the existence of higher order vagueness (in p, say – so \nabla \Delta p) strikes me as compelling evidence that there is vagueness located in \Delta rather than in p. I was hoping to run a more precise version of this argument by constructing a sorites for \Delta directly. Of course its not that simple since it’s an operator and not a predicate.

Assume the following

  • The operation of lambda abstraction is a precise operation. (I also think its an interesting case of a logical operation – it does not fit into the type heirarchy.)
  • Existential quantification is precise.

Sider has a cute argument for the second claim: suppose \forall_1 and \forall_2 are different precisifications of universal quantifiers. Then they must have different extensions. That is, there must be an object, x, which one ranges over but the other doesn’t. But then one of them would not be an admissible precisification of the universal quantifier since it missed an object out.

Going back to our test for vagueness: if composing \Delta with precise operations yields a vague term, then \Delta is vague. Thus, if

  • \lambda F \exists x \nabla Fx

is vague, then \Delta must be vague (in what follows, abbreviate to Vague(F).) Note that this predicate is true of a predicate F iff it has a borderline case. But it turns out this predicate is vague, by an argument due to Sorensen. Define:

  • x is k-small iff x < k or x is small

Since being 0-small is equivalent to being small, 0-small satisfies Vague(F) since it has a borderline cases. Since n := Rayo’s number isn’t small, being n-small is equivalent to being less than n, which is precise, so n-small does not satisfy Vague(F). Since the predicates ‘k-small’ for k < n form a sorites of Vague(F), Vague(F) must be vague. And hence, by our assumption, \Delta must be vague.


A Game Theoretic Semantics for Vagueness

April 6, 2008

Ok, so there’s a risk I’m going to alienate my readers with all these wacky theories of vagueness, but here’s another one if you’re keeping track. I was thinking of trying to capture the (possibly) Fregean idea that the sense of an expression is a method for determining what the referent of the expression is. For vague expressions this method may be non-deterministic – on some ways of carrying out the method you arrive at one referent, on others, other referents. Like supervaluationism, this position views vagueness as a kind of semantic underdeterminacy, but, I shall argue, gives us a very different logic.

I’ve been considering two different ways of representing a ‘method for determining the referent’ formally: 1) to think of senses as computer programs of some sort, 2) to think of them as a game between two players. I’m going to be considering the second option here. First, let’s look at the game semantics for first order logic for those not familiar with it. Given a model, M, and an assignment of variables, v, we can define an assortment of games between two players G_M(\ulcorner \phi \urcorner , v) as follows.

  • G_M(\ulcorner \phi \vee \psi \urcorner , v): the verifier chooses between \phi and \psi then the game continues with G_M(\ulcorner \chi \urcorner , v) where \chi is the chosen formula.
  • G_M(\ulcorner \phi \wedge \psi \urcorner , v): the falsifier chooses between \phi and \psi then the game continues with G_M(\ulcorner \chi \urcorner , v) where \chi is the chosen formula.
  • G_M(\ulcorner \neg \phi \urcorner , v): the verifier and falsifier swap roles and the game continues with G_M(\ulcorner \phi \urcorner , v)
  • G_M(\ulcorner \exists x \phi \urcorner , v): the verifier chooses an assignment v^\prime that differs from v at most in its assignment to x, and the game continues with G_M(\ulcorner \phi \urcorner , v^\prime).
  • G_M(\ulcorner \forall x \phi \urcorner , v): the falsifier chooses an assignment v^\prime that differs from v at most in its assignment to x, and the game continues with G_M(\ulcorner \phi \urcorner , v^\prime).
  • G_M(\ulcorner P^n_i(x_1, \ldots , x_n) \urcorner , v): if \langle v(x_1), \ldots , v(x_n) \rangle \in P^M then the player playing the role of verifier wins. Otherwise the falsifier wins.

We can then say that a formula \phi is true (in M, on v) if the player who starts off playing the verifier has a winning strategy for G_M(\ulcorner \phi \urcorner , v), and say its false if the falsifier has a winning strategy for this game. To extend to a simple system with vagueness, we can say \phi is supertrue (superfalse) if the verifier (falsifier) has a winning strategy for the game that starts with the falsifier (verifier) picking a precisification and continues as G_M(\ulcorner \phi \urcorner , v). This is equivalent to standard supervaluationist semantics.

Note: there is a related way to do things which gives different results. If you allow games of imperfect information then not every game is determined, and you can get violations of LEM. This is relevant to vagueness. Say that a sentence \phi is true (false) if the verifier (falsifier) has a winning strategy for the following game:

  • A precisification is chosen at random without the verifier or falsifier knowing which. The game then continues with G_M(\ulcorner \phi \urcorner , v).

In this case we get different results, for example (p \vee \neg p) is neither true nor false, when p is borderline (true on some but not all precisifications.) In fact, this version will be equivalent to the strong Kleene 3-valued logic with the neutral value holding when neither the verifier or the falsifier have a winning strategy.

But the view I’m interested isn’t this one. I want to identify the sense of an atomic formula with a game G_M(P(x_1, \ldots , x_n), v), in this case given by the model, M, which may or may not be determined. The idea is that vague expressions correspond to games in which neither player has a winning strategy, because the method for determining the truth value does not always land you with the same result. The method is unreliable, inaccurate or ill defined. This reflects the idea that if someone asks you to determine whether borderline balding Billy is bald there is no well defined procedure, or way to go about doing this.

Adopting the rules above except for the atomic case, which we replace with the game supplied by the model, M, we again get a non-classical logic. In this case (I think) we get the weak Kleene 3-valued logic, which is interesting, because as far as I know, no-one takes this as the logic of vagueness.


Vagueness and Boundaries

April 2, 2008

Lately I’ve been trying to make sense of the idea that vague concepts are boundaryless. It’s very hard to make sense of this when the semantic value for a predicate is a set, fuzzy or sharp, because in both cases there is always a sharp cut off point between being in the set (to some degree or other), and not being in it.

I’ve been thinking that maybe we shouldn’t be thinking about things set theoretically, but topologically – after all topology is the mathematical study of boundaries. Anyway, I’m not sure if this will work, but I’m going to throw it out there. I certainly haven’t thought it through!

Let’s start off with a metric space \langle S, c(x, y) \rangle. S is our set of states – for now we will work in a supervaluationist framework and take them to be precisifications of the language. c(x, y) is a closeness metric telling how similar (measured in real numbers) two states are to one another. Now let \mathcal{O} be the regular open set lattice from the standard ball topology over S. Elements of \mathcal{O} will serve as the denotata of sentences of our language. They are desirable for two reasons. Firstly they are necessarily ‘blurry’, that is, they are regions of our precisification space, as opposed to ‘points’. Regions represent a range of precisifications, whereas a point would represent a maximally specific way in which language is completely sharpened (we assume for now that our language contains only vague predicates.) Secondly, there are no boundaries between these regions. For example there is no boundary between the region corresponding to x being red, and its complement in \mathcal{O}, the region corresponding to x not being red. We want to capture the idea that there is no last red thing and no first non-red thing (across a rainbow for example.) Although we constructed these regions from maximally specific precisifications, the idea is to take the regions as primitive.

We shall give the semantics as follows. Define \sigma := Int \circ Cl, the composition of the interior and the closure operation. Let D be the domain of discourse.

  • \mbox{ }[ \![ P^n_i]\!] : D^n \rightarrow \mathcal{O}
  • \mbox{ }[ \! [ \neg \phi ] \! ] := \sigma (S \setminus [ \! [\phi ] \! ])
  • \mbox{ }[ \! [ \phi \wedge\psi]\! ] := [\! [\phi ]\! ] \cap [\! [ \psi ]\! ]
  • \mbox{ }[ \! [ \phi \vee \psi]\! ] := \sigma([\! [\phi ]\! ] \cup [\! [ \psi ]\! ] )
  • \mbox{ }[ \! [ \forall x\phi]\! ] := \sigma(\bigcap_{x \in D} [\! [\phi(x) ]\! ])
  • \mbox{ }[ \! [ \exists x\phi]\! ] := \sigma(\bigcup_{x \in D} [\! [\phi(x) ]\! ])

The logic will be classical since \mathcal{O} is a complete Boolean algebra under the operations given above. It will, however, become non-classical if we add in precise predicates (whose semantic values can take non regular open sets.) I don’t know how this this particular semantics will pan out in the long run though. For example, with predicates like “red” it’s ok to have no boundaries between the red and not red, but with vague discreet predicates, like “small number” it looks like you might end up with there being no small numbers. Anyway, I was hoping that something in this spirit might be able to put the ‘no boundaries’ conception of vague predicates on a firmer footing, even if it doesn’t ultimately work. Any thoughts would be welcome…


Properties, Worlds and Propositions

March 28, 2008

Two different properties can have the same extension – examples should be familiar, for example the property has a kidney is different from the property has a heart, yet they have the same extension. The fact that they are distinct is usually established by considering their modal properties. For example, having a heart could have had an instant that didn’t instantiate having a kidney and vice versa. This is certainly a sufficient condition for properties to be distinct, but not a necessary one, again examples should be familiar: being triangular and being trilateral are standard examples, supposedly different, yet necessarily coextensive.

I think this second point can be made more clearly when we consider properties for which it is quite hard to make sense of the property ‘being had in a world’. For example exists in more than two worlds and exists in more than three worlds. Presumably I have both these properties, but I’m not sure if I have them in any world in particular! Either way, they are two distinct properties. Now consider properties of worlds, for example the property a world has iff donkeys talk in that world, and the one a world has iff pigs fly in that world. Here again, we can’t talk about about these properties coming apart in different worlds because nothing has these properties in a world. But clearly these properties are different – in this case they have different extensions because there are worlds in which donkeys talk, but pigs are earthbound.

Where am I going with this? Well, once we have the following two assumptions we can solve a difficult problem in the philosophy of language. The assumptions are:

  • You can have two distinct yet coextensive properties.
  • Properties aren’t in any sense parasitic on worlds (so we can make sense of worlds having properties.)

The puzzle is to do with propositional attitudes. Often propositions are treated as sets of worlds. But if this is the case then we cannot explain the semantic difference between “Hesperus is Phosphorus” and “Hesperus is Hesperus”, e.g. when they appear in belief reports. Why? Because the set of worlds they each represent are coextensive, and whenever you have two coextensive sets they are identical. Thus they express the same proposition.

My idea was, instead identifying propositions with sets of worlds, to identify propositions with properties of worlds. In this case the property a world has iff Hesperus is Phosphorus in that world can be different from the property a world has iff Hesperus is Hesperus in that world, even though they are coextensive! Do we even need to postulate necessarily coextensive distinct properties to make this work? I think not – it doesn’t make sense to talk about world properties being necessarily coextensive – these two properties aren’t necessarily coextensive because their extensions aren’t world dependent.


Plural Reference

March 16, 2008

Colin has a post over at at inconsistent thoughts which I thought was quite interesting. The idea was to treat plural and singular reference as the same species of reference by simply relaxing the constraint that the reference relation must be functional. In the case of an ordinary singular variable or noun phrase there will be exactly one thing standing in the reference relation to it, but in the case of plurals there may be many.

I like this approach for several reasons. For one thing, it does away with the difference between the object, a, and the singleton ‘plurality’ containing only a. Of course, pluralities aren’t objects in their own right, so the only way to make sense of the difference here is that it is a difference in kind of reference involved. But prima facie, there is no difference in the case of a singular name, and a plural term denoting a “singleton”.

Colin then poses a problem for this kind of view: how does one then account for collective (non-distributive) plural predicates – predicates as found in “the students surrounded the building”. It’s not the case that each student surrounded the building, they did so collectively, so we can’t simply say that “The F’s surrounded the building” iff for each x such that Ref(“the F’s”, x), x surrounded the building. Nor is there any other obvious way to do it along these lines.

Maybe you could take the same line with predicates as with terms (in particular, with collective plural predicates.) So instead of a predicate standing in the reference relation to exactly one set of objects, as per usual in model theory, it could stand in it to many. So we treat the predicate reference relation as non-functional too. We could then say Ref2(“surrounded the building”, P) holds iff P is a set of things that surrounded the building, and generally a plural predicate is true of a term iff the predicate refers to the set of referents of the term. What do people think?

Update: Just an extra detail: nothing relies on the semantic value of a predicate being a set. We could treat predicates as plurally referring to their members. This way Ref2 would take plurality rather than a set as its second argument, making Ref2 an irreducibly plural predicate. I take it that having plural predication in the metalanguage is not problematic though.


Counterparts and Actuality

March 4, 2008

I’ve been reading this paper by Delia Graff Fara for one of the discussion groups I’ve been going to. It’s basically a follow up to the Williamson and (Michael) Fara paper from a couple of years back, highlighting some problems counterpart theory would face if augmented by an actuality operator. I had some general methodological problems with these papers (for example, they would argue that CPT could not provide faithful interpretations of QML formulae – when CPT’s aims are to provide interpretations of English, and further, they claim, to do so more faithfully than QML). But that aside, there was one obvious response Lewis could make which neither paper seemed to consider. I mentioned it in the discussion group, but didn’t get a chance to think it through properly, so I though I might take this opportunity to expand it some more (so apologies in advance for any obvious errors!)

Consider a world of eternally recurring, qualitatively identical epochs (call it w_e.) Now Lewis wants to reconcile two things. He wants to deny a version of haecceitism that states that there can be qualitatively identical possible worlds which differ with respect to what de re possibilities they represent for some individual; while making sense of the intuitive claim that Bob, who, lets say, lives in the 17th epoch, might have lived in the 18th epoch (i.e. where is qualitatively identical twin, boB, lives.) To do this he allows the counterpart relation to hold between individuals that live in the same world. This amounts, as Graff Fara notes, to individuating possibilities more finely than possible worlds. For example, there is one possibility in which Bob lives in the 17th epoch, and one in which he lives in the 18th, yet there is only one possible world involved. In Lewis’s own words: “Possibilities are not always possible worlds. There are possible worlds, sure enough, and there are possibilities, and possible worlds are some of the possibilities.” (PoW, p230)

So why don’t we just interpret the actuality operator as being true in the actual possibility, rather than being true in the actual world? To fix ideas, let us think of a possibility as an ordered pair of a world, w, and a function from individuals in w to individuals in w. For example the first possibility we considered was w_e, with the identity mapping, taking Bob to himself, but when we considered the possibility that Bob might have lived in the 18th epoch we were considering the pair w_e and the mapping that takes Bob to boB, Bob’s 18th epoch twin, (and which is the identity elsewhere.)

Does interpreting the actuality operator like this help? For example, do we get all the inferences we usually get from it? We can show that we do by simply interpreting QML+@ (quantified modal logic augmented with an actuality operator) in terms of possibilities and it should then be clear that it will validate exactly the same inferences as classical @ would.

We let the set of states be the set of possibilities, i.e. S := \{ \langle w, \sigma \rangle \mid w \in W \wedge \sigma : Ind \rightarrow Ind(w), \sigma \subseteq C \}. Let Ind be the set of individuals from any world, Ind(w) the individuals from w, and C the counterpart relation (I’ve relaxed the constraint that the function must go from and to individuals in the world.) I’ve idealised and assumed that \sigma is a total function. We set one particular pair s^* := \langle w^*, \sigma^* \rangle to be the actual possibility. The crucial truth clauses are as follows

\langle w, \sigma, v \rangle \models Px_1, \ldots, x_n \Leftrightarrow \langle \sigma(v(x_1)), \ldots, \sigma(v(x_1)) \rangle \in [[P]]
\langle w, \sigma, v \rangle \models @\phi \Leftrightarrow \langle w^*, \sigma^*, v \rangle \models \phi
\langle w, \sigma, v \rangle \models \Diamond\phi \Leftrightarrow \langle w^\prime, \sigma^\prime, v \rangle \models \phi \mbox{ for some } \langle w^\prime, \sigma^\prime \rangle \in S
\langle w, \sigma, v \rangle \models \exists x\phi \Leftrightarrow \langle w, \sigma, \sigma \circ v^\prime \rangle \models \phi
\mbox{ for some } v^\prime \mbox{ which differs only from v in its assignment to x.}

Since the clause for @, is exactly the same as in the standard semantics where we intepret S as the set of possible worlds, and the other truth clauses are sufficiently similar – we should get exactly the same inferences for @ as in the ordinary case.

Of course, counterpart theorists don’t like to use a language with primitive modal operators like QML+@, and will, if they can, phrase it all in first order logic. Standardly counterpart theorists will need the two primitive symbols: Iwx and Cxy. I is the relation of being a part of a world, C is the counterpart relation. We shall use one primitive, Rsxy, interpreted as x = \sigma(y) where s = \langle w, \sigma \rangle. We can give a translation schema of for QML+@ as follows:

(Px_1,\ldots,x_n)^s \mapsto
\exists y_1,\ldots,y_n(Rsy_1x_1 \wedge \ldots \wedge Rsy_nx_n \wedge Py_1, \ldots, y_n)
(\neg \phi)^s \mapsto \neg(\phi^s)
(\phi \wedge \psi)^s \mapsto (\phi^s \wedge \psi^s)
(\exists x \phi)^s \mapsto \exists x \exists y(Rsxy \wedge \phi^s)
(\Diamond \phi)^s \mapsto \exists s^\prime \exists y_1\ldots y_n \exists z_1\ldots z_n(Rsz_1x_1 \wedge \ldots \wedge Rsz_nx_n
\wedge Rs^\prime y_1z_1 \wedge \ldots \wedge Rs^\prime y_nz_n \wedge \phi^{s^\prime})
(@\phi)^s \mapsto \phi^{s^*}

(Note in the \Diamond clause, x_1, \ldots, x_n are the free terms in \phi.)


There are just as many F’s as G’s

February 4, 2008

Since Boolos, many people have come to believe that second order plural quantifiers are logical quantifiers. According to Boolos, they are topic neutral, ontologically innocent, and well understood.

One way to cash out topic neutrality would be to say that plural quantifiers are invariant under arbitrary permutations of the domain, but this seems right even if we go for a pretheoretic notion of topic neutrality. One might argue that plural quantification is simply quantification over sets. This would undermine ontological innocence if it where correct. Take for example:

(1) Some cats were making a racket outside

The proposal here being that there is a set of cats such that each cat in the set is making a racket. So according to this analysis, (1) entails the existence of sets. However, (1) doesn’t appear to entail any such thing, indeed the truth of (1) is presumably independent of the ontological status of sets.

Things get even worse the theory when we consider the apparent truism:

(2) There are some sets which are all and only the non self membered sets.

On the current proposal (2) would entail the existence of a set of all non self-membered sets. But as we know, no such set exists. The quantification at hand is irreducibly plural – we must take pains to note that we are quantifying plurally over individuals, not quantifying singularly over pluralities (or sets or classes.)

That plural quantification is well understood is probably the most controversial claim. The logicality of plural quantifiers is partly motivated by the legitimacy of plural quantification in English. This is why, for example, super plural quantification (the analogue of third order quantification, if you like) is not widely considered logical, despite being topic neutral and ontologically innocent (if it even makes sense.)

What does this entail for logic? Well, if we add plural quantifiers to regular first order logic you get a logic which is considerably stronger. For example, the Geach-Kaplan sentence can be used to give a categorical axiomatisation of PA, something which no first order sentence could do.

Ok, nothing new so far. What I wanted to think about, was whether there are any other English expressions which have this property. I.e. are topic neutral, ontologically innocent, well understood and when added to plural logic increase its strength.

The expression I had in mind was:

There are just as many F’s as G’s

First off: it seems to be topic neutral. It is definable in second order logic, so it is certainly invariant under arbitrary permutations of the domain.

One could argue that it’s not ontologically innocent. Someone taking this line might say that it involves full second order quantification over relations, plurally quantifying over pairs, or singularly quantifying over bijections (i.e. sets.) But this doesn’t sound right to me at all, for basically the same reasons as with the plural quantifiers. Consider

(3) There are just as many knifes as forks

(3) doesn’t seem to entail the existence of relations, ordered pairs or functions. Intuitively, (3) will remain true, even if there are no abstract objects. Similarly, (3) is independent of the consistency of ZFC – if set theorists discover an inconsistency somewhere, there will still be as many knifes as forks in my drawer. All (3) depends on is the knifes and the forks, and whether there are as many knifes as forks.

What happens when you add “there are just as many F’s as G’s” to plural logic? That is, when we add the relation xx \approx yy to plural logic, to be given the same truth conditions as the obvious second order formula. The resulting system is somewhere in between plural logic and full second order logic in strength: you cannot define equinumerosity in plural logic, and you cannot emulate full second order quantification over sets using only plural quantifiers and equinumerosity.

The real question is whether this English expression is as well understood as plural quantification. The problem is when F and G are infinite. For example:

(4) There are just as many rationals as naturals

This is a classic example where things go wrong: mathematicians take this sentence to be true, while an ordinary man on the street would take it to be false. Is the mathematician right and the man on the street wrong? Or is the mathematician using (4) in a very stipulative way. I’m inclined to think that the mathematicians use, even if it is different from the way ordinary people talk, is still ontologically innocent (thus sidestepping the issue.) However, this is a difficult claim to back up, since the mathematicians use is grounded heavily on an understanding of the set theoretic definition of equinumerosity in terms of bijections.

Perhaps there are many ways to extend the meaning of “there are just as many F’s as G’s” to infinite cases – this indeterminacy shouldn’t change the point at hand. For even if “there are just as many F’s as G’s” is false whenever F or G is infinite, we can still get a more powerful logic than plural logic, since we can define equinumerosity for finite pluralities.

Either way, it would be interesting to see what other English expressions have these properties?