Vagueness and uncertainty

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

Unrestricted Composition: the argument from the semantic theory of vagueness?

I’ve seen the following claim made quite a lot in and out of print, so I’m wondering if I’m missing something. The claim is that Lewis’s argument for unrestricted composition relies on a semantic conception of vagueness. In particular, people seem to think epistemicists can avoid the argument.

Maybe I’m reading Lewis’s argument incorrectly, but I can’t see how this is possible. The argument seems to have three premisses

1. If a complex expression is vague, then one of it’s constituents is vague.
2. Neither the logical constants, nor the parthood relation are vague.
3. Any answer to the special composition question that accords with intuitions must admit vague instances of composition.

By 3. one has that there (could be) a vague case of fusion: suppose it’s vague whether the xx fuse to make y. Thus it must be vague whether or not . By 1. this means either parthood, or one of the logical constants is vague, which contradicts 2.

I can’t see any part of the argument that requires me to read `vague’ as `semantically indeterminate’. These seem to be all plausible principles about vagueness, and if, say, epistemicism doesn’t account for one of these principles, so much the worse for epistemicism.

That said, I think epistemicists should be committed to these principles. Since it would be a pretty far off world where we used English non-compositionally, the metalinguistic safety analysis of vagueness ensures that 1. holds. Epistemicists, like anyone else, think that the logical constants are precise. Parthood always was the weak link in the argument, but one might think you could vary usage quite a bit without changing the meaning of parthood since it refers to a natural relation, and is a reference magnet. Obviously the conclusion that the conditions for composition to occur are sharp isn’t puzzling for an epistemicist. But epistemicists think that vagueness is a much stronger property than sharpness (the latter being commonplace), and the conclusion that circumstances under which fusion occurs do not admit vague instances should be just as bad for an epistemicist as for anyone else who takes a medium position on the special composition question.

The most I can get from arguments that epistemicism offers a way out is roughly: “Epistemicists are used to biting bullets. Lewis’s argument requires you to bite bullets. Therefore we should be epistemicists.” Is this unfair?

Truth Functionality

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, , is truth functional just in case the truth value of depends only on the truth values of .

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

• iff, necessarily, .
• iff or .

Why, in the second example but not the first, does the truth value of 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 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 is truth functional and 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 then they’ll agree on . 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 T is not an acceptable assignment. There are assignments which give the same truth values to p and q, but different truth values to , 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 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 .

• : reads T just in case the second column reads T, if the the first column does.
• : reads T just in case the first or the second column reads T.
• : reads T if the first column doesn’t read T.

The classical logician claims that the truth table for should be the same as the truth table for . 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 will determinately read T. But the statement that 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 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 ? 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 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.

Field on Restall’s Paradox

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

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

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

1. 
2. If then

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

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

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

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

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

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

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

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

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

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

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

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

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

Size and Modality

There’s this thing that’s been puzzling me for a while now. It’s kind of related to the literature on indefinite extensibility, but the thing that puzzles me has nothing to do with sets, quantification or Russell’s paradox (or at least, not obviously.) I think it is basically a puzzle about infinities, or sizes.

First I should get clear on what I mean by size. Size, as I am thinking about it, is closely related to what set theorists call cardinality. But there are some important differences.

(i) Cardinality is heavily bound up with set theory, whereas I take it that size talk does not commit us to sets. For example, I believe I can truly say there are more regions than open regions of spacetime, even if I’m a staunch nominalist. Think of size talk as analogous to plural quantification: I am not introducing new objects into the domain (sizes/pluralities), I am just quantifying over the existing individuals in a new way.

(ii) Only sets have cardinalities. I believe you can talk about the sizes of proper class sized pluralities.

(iii) Points (i) and (ii) are compatible with a Fregean theory of size. But Fregean sizes, as well as cardinalities, are thought to be had by pluralities (concepts, sets) of individuals in the domain. In particular: every size, is the size of some plurality/set. I reject this. I think there are sizes which no plurality has – I think there could have been more things than there in fact are, and thus, that there are sizes which no plurality in fact has. So sizes are inherently bound up with modality on this view – sizes are had by possible pluralities.

(iv) Frege and the set theorists both believe sizes are individuals. I’m not yet decided on this one, but Frege’s version of Hume’s principle forces the domain to be infinite, which contradicts (i) – that size talk isn’t ontologically committing. Interestingly, the plural logic version of HP is satisfiable on domains of any size – thus size’s can be always be construed as objects, if needs be. But I’m inclined to think that size talk is fundamentally grounded in certain kinds of quantified statements (e.g., “there are countably many F’s”.)

I’m going to mostly ignore (iv) from hereon and talk about sizes like they were objects, because as noted, you can consistently do this if needs be (given global choice.) That said, I can’t adopt HP because of point (iii). It’s built into the notation of HP that every size is the size of some plurality. Furthermore, Hume’s principle entails there is a largest size. (Cardinality theory say there is no largest cardinality, but this is because of an expressive failure on it’s part – proper classes don’t have cardinalities.) However, if we accept the following principle:

• Necessarily, there could have been more things.

it follows from (iii) that there is no largest size.

I think this is right. It just seems weird and arbitrary to think that there could be this largest size, . Why and not ? Clearly, it seems, there are worlds, that have this many things (think of, e.g. Forrest-Armstrong type constructions.) If not, what metaphysical fact could possibly ground this cutoff point?

What I don’t object to is there being a largest size of an actual plurality. I’m fine with arbitrariness, so long as it’s contingent. But to think that there is some size that limits the size of all possible worlds seems really strange. Just to state the existence of a limit seems to commit us to larger sizes – it’s like saying there are sizes which no possible world matches.

Here is a second principle about sizes I really like. Any collection of sizes has an upperbound. This is something that Fregean, and in a certain sense, cardinality theories of size share with me, so I’m not going to spend as long defending it. But intuitively, if you can have possible worlds with domains of sizes for each , then there should be a world containing the union of all these domains – a world with at least things.

So this is what I mean by size. Here is the puzzle: this conception of size seems to be inconsistent. To see this we need to formalise a bit further. Take as our primitive a binary relation over sizes, < (informally “smaller than”.) For simplicity, assume we are only quantifying over sizes. Here are some principles. You can ignore 3. and 4. if you want, 1. and 2. are obvious, and 5. and 6. we have just argued for.

1. 
2. 
3. 
4. 
5. 
6. 

The first three principles say that < than is a total order, which is pretty much self evident. The fourth says it’s a well order. (The inconsistency to follow doesn’t require (3) or (4).) The fifth encodes the principle that there is no largest size, and the sixth says that every collection of sizes has an upper bound.

These principles are jointly inconsistent: let xx be the plurality of self-identical things. By (6) xx has an upper bound, k. By (5) there is a size larger than k, k<k+. Since k+ is in xx, and k is an upperbound for xx, k+ k. Thus k<k by (2) and logic, which is impossible by (1).

There are roughly three ways out of this usually considered. Fregean theories reject (5), cardinality theory (with unrestricted plural quantifiers) deny (6) and indefinite extensibilists do something funky with the quantifiers (I’ve never really worked out how that helps, but it’s there for completeness.) Also note, the version of (6) restricted to “small” (roughly, “set-sized”) pluralities is consistent.

My own diagnosis is that the above formulation of size theory simply fails to take account of the modal nature of sizes. If we are pretending that sizes are objects at all (which, I think, is also not an innocent assumption), we should remember that just because there could be such a size, doesn’t mean in fact there is such a size. This is the same kind of fallacious reasoning encoded in the Barcan formula and its converse  (this is partly why it is very unhelpful to think of sizes as objects; we are naturally inclined to think of them as abstract, necessarily existing objects.)

Anyway – a natural way to formulate (1)-(6) in modal terms would be in a second order modal logic, perhaps with a primitive second level size comparison relation. For example (1) would be ‘necessarily, if the xx are everything, then there aren’t more xx than xx‘, (2) would be ‘necessarly for all xx, necessarily for all yy, necessarily for all zz, if there are more zz’s than yy’s and more yy’s than zz’s there are more zz’s than xx’s’ and (5) would be ‘necessarily, there could have been more things’. The only problem is, how would we state (6)?

I’ve been toying around with propositional quantification. Let me change the primitives slightly: instead of using to talk about possibility and necessity, I’ll interpret them as saying p is true in some/every accessible world with a larger domain than the current world. Also, since I don’t care about anything about a world except the size of it’s domain, let us think of the worlds not as representing maximally specific ways for things to be, but as sizes themselves. Thus the intended models of the theory will be Kripke frames of the following form: where (i) the transitive closure of R is a well order on W, and (ii) for each w in W, R is a well order on R(w). (We’re going to have to give up S4, so we mustnt assume R is transitive on W, although it’s locally transitive on R(w) for each w in W.) Propositions are sets of worlds, so the range of the propositional quantifiers differ from world to world, since R is non-trivial.

Call R a local well order on W iff it satisfies (i) and (ii). I’m going to assert without defence (for the time being) that the formulae valid over the class of local well orders, will be the modal equivalent of (1)-(4) holding (I expect it would be fairly easy to come up with an axiomatisation of this class directly and that this axiomatisation would correspond to (1)-(4). For example, the complicated one, (4), would correspond to .)

The important thing is that it is possible to state (5) and (6) directly, and, it seems, consistently (although we’ll have to give up on unrestricted S4.) [Note: I may well have made some mistakes here, so apologies in advance.]

1. 
2. 
3. …
4. 
5. 
6. 

(I decided halfway through writing this post it was simpler to axiomatise a reflexive well order, so the modal (1)-(4) above don’t correspond as naturally to the original (1)-(4) – I’ll try and neaten this up at some point).

What is slightly striking is the failure of S4. Informally, if I were to have S4 I would be able to quantify over the universal proposition of all worlds, take its supremum by (6), and find a world not in the proposition by (5). This would just be a version of the inconsistency given for the extensional size theory above.

Instead, we have a picture on which worlds can only see a limited number of world sizes – to see the larger sizes you have to move to larger worlds. At no point can you “quantify” over all collections of worlds – so, at least in this sense, the view is quite close to the indefinite extensibility literature. But of course, the non-modal talk is misleading: worlds are really maximally specific propositions, and the only propositions that exist are those in the range of our propositional quantifiers at the actual world – the worlds inaccessible to the actual world in the model should just be thought of as a useful picture for characterising which sentences in the box and diamond language are true at the actual world.

Greatest Philosopher of the 20th-Century?

You can find out here.

But seriously: Lewis came second to Wittgenstein? (I could understand how LW might rank top in a poll involving the general public, but the first ranking was supposedly based mostly on the Leiter readership!)

Update: some interesting thoughts on Russell’s ranking here and here.

Fitch’s paradox and self locating belief

It’s been a while since I last posted here – which is bad seeing as I’ve had much less going on recently. I hope to return to regular blogging soon!

For now just a little note on something I’ve been thinking about to do with a version of the knowabality principle for rational belief. Back in this post I considered a version of Fitch’s paradox for rational belief, which shows the following believability principle cannot hold in full generality (C stands for rational certainty)

• 

Here’s another route to that conclusion if you accept something like Adam Elga’s indifference principle. Suppose p is the proposition that you are in a Dr. Evil like scenario: that (a) you are Dr. Evil and (b) you have just received a message from entirely reliable people on Earth saying they have created an exact duplicate of Dr. Evil, whose situation is epistemically indistinguishable from Dr. Evils (including having him receive a duplicate message like this one) who will be tortured unless Dr. Evil deactivates his super laser. Notice that p includes self locating information.

If you accept Elga’s version of the indifference principle, once you’ve become certain of (b) you’re rationally required to lower your credence that you’re Dr. Evil to 1/2 and give credence 1/2 to the hypothesis that you’re the clone. So suppose for reductio that you could be certain that p. Since p is the conjunction of (a) and (b) you must be certain in both (a) and (b). But this is impossible, since indifference requires anyone who is certain in (b) to give credence 1/2 (or less) to (a).

It is impossible to be certain in p (p is probably unknowable too.) And since p is clearly possibly true, the principle given above is at best contingently true.

Links

• JC Beall has started what he describes as a “logic-leaning philosophy blog” which looks like it should be of interest to readers here when it gets going.
• Not as recently, Jeff Russell started a new blog which is looking very interesting so far.
• Lastly, Wolfgang Schwarz has an interesting post on decision theory and probability in EQM over at Wo’s Weblog.

Cardinality and the intuitive notion of size

According to mathematicians two sets have the same size iff they can be put in one-one correspondence with one another. Call this Cantor’s principle:

• CP: X and Y have the same size iff there is a bijection

Replace ’size’ by ‘cardinality’ in the above and it looks like we have a definition: an analytic truth. As it stands, however, CP seems to be a conceptual analysis – or at the very least an extensionally equivalent charaterisation. In what follows I shall call the pretheoretic notion ’size’ and the technical notion ‘cardinality. CP thus states that two sets have the same size iff they have the same cardinality.

Taken as a conceptual analysis of sizes of sets, as we ordinarily understand it, people often object. For example, according to this definition the natural numbers are the same size as the even numbers, and the same size as the square numbers, and many more sets even sparser than these. This is an objection to the right to left direction of CP.

I’m not inclined to give these intuitions too much weight. In fact, I think the intuitive principles behind these judgements are inconsistent. Here are two principles that seem to be at work: (i) if X is a proper subset of Y then X is smaller than Y, (ii) if by uniformly shifting X you get Y, then X and Y have the same size. For example (i) is appealed to when it’s argued that the set of evens is smaller than the set of naturals. (ii) is appealed to when people argue that the evens and the odds have the same size. Furthermore, both principles are solid when we are dealing with finite sets. However (i) and (ii) are clearly inconsistent. If the evens and the odds have the same size, so do the odds and the evens\{2}. This is just an application of (ii), but intuitively, the evens\{2} stand in exactly the same relation to the odds, as the odds to the evens. By transitivity, the evens and the evens\{2} are the same size – but this contradicts (i) since one is a proper subset of the other.

In fact Gödel gave a very convincing argument for the right to left direction: (a) changing the properties of the elements of a set does not change its size, (b) two sets which are completely indistinguishable have the same size and (c) if , each can morph its properties so that x and are indistinguishable.  Thus, if is a bijection, X can be transformed in such a way that it is indiscernable from Y, and must have the same size. (Kenny has a good discussion of this at Antimeta.)

The direction of CP I think there is a genuine challenge to is the left to right. And without it, we cannot prove there is more than one infinite size! (That is, if we said every infinite set had the same size, that would be consistent with the right to left direction of CP alone.)

What I want to do here is justify the left to right direction of CP. The basic idea is to do with logical indiscernability. If two sets have the same size, I claim, they should be logically indiscernable in the following sense: any logical property had by one, is had by the other. Characterising the logical properties as the permutation invariant ones, we can see that if two sets have the same cardinality, then they are logically indiscernable. Since we accept the inference from having the same cardinality to having the same size, this partially confirms our claim.

But what about the full claim? If two sets have the same size, how can they be distinguished logically? There must be some logically relevant feature of the set which is distinguishing them, but has nothing to do with the size. But what could that possibly be? Surely size tells us everything we can know about a set without looking at the particular characteristics of  its elements (i.e. its non-logical properties.) If there is any natural notion of size at all, it must surely involve logical indiscernability.

The interesting thing is that if we have the principle that sameness in size entails logical indiscernability we get CP in full. The logical properties over the first layer of sets of the urelemente are just those sets invariant under all permutations of the urelemente. Logical properties of these sets are just unions of collections sets of the same size. Thus logically indiscernable sets are just sets with the same cardinality!

Ignore sets for a moment. The usual setting for permutation invariance tests is on the quantifiers. A variant of the above argument can be given. This time we assume that size quantifiers are maximally specific logical quantifiers. There are two ways of spelling this out, both of which will do:

• For every logical quantifier, Q, or
• For every logical quantifier, Q, if then

The justification is exactly the same as before: the size of the ’s tells us everything we can possibly know about the ’s without looking at the particular characteristics of the individuals ’s – without looking at their non-logical properties. Since the cardinality quantifiers have this property too, we can show that every size quantifier is logically equivalent to some cardinality quantifier and vice versa.

I take this to be a strong reason to think that cardinality is the only natural notion of size on sets. That said, there’s still the possibility that the ordinary notion of size is simply underdetermined when it comes to infinite sets. Perhaps our linguistic practices do not determine a unique extension for expressions like ‘X is the same size as Y’ for certain X and Y. One thing to note is that the indeterminacy view seems to be motivated by our wavering intuitions about sizes. But as we saw earlier, a lot of these intuitions turn out to be inconsistent, so there won’t even exist precisifications of ’size’ corresponding to these intuitions. On the other hand, if we are to think of the size of a set as the most specific thing we can say about that set, without appealing to the particular properties of its members, then there is a reason to think this uniquely picks out the cardinality precisification.

The Sorites paradox and non-standard models of arithmetic

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 -inconsistent (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.

-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 -inconsistency. a numeral 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 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.

Composition as identity, part II

Aside from Leibniz’s law, there are various other constraints identity must obey. For example, every object is identical with at most one thing, so every object presumably is identical* to at most one plurality. But here we have a disanalogy with the relation “x is the fusion of the yy’s”, for x is the fusion of many pluralities. If there may be more than one plurality *identical to x, then our notation for pluralizing x, x*, isn’t justified: * isn’t a function.

A way of sharpening this problem, pointed out by Jeff in the comments, is that you’d want (*)identity(*) to be transitive. For example, my upper and lower body are *identical to me, and I’m identical* to my arms, legs head and torso. Does that mean my lower and upper body are *identical* to my arms, legs, head and torso? How do you state that they’re different pluralities?

So there appear to be two ways you can go. One is to say that every object is really identical* to only one plurality, and the other is to say that every plurality is *identical to exactly one object. I’ll call the two approaches (a) unique decomposition, and (b) unique composition.

The first seems to be the most natural. By way of analogy, note that the pluralities, over a domain of objects, have many of the formal properties of mereology. (i) there’s no null fusion/no empty plurality, (ii) pluralities are closed under ‘unioning’ and ‘non-empty intersecting’ (fusion and products) (iii) they’re closed under complements (supplementation.)

In fact, they form a complete Boolean algebra under ’subplurality’, and thus model the standard mereological axioms. However, there are some drawbacks. Firstly, you can form pluralities of mereological objects, in standard mereological theories (that’s how unrestricted composition is usually stated.) However, on this picture, you can’t, for it would amount to forming pluralities of pluralities – which is nonsense.

You might think this is not too much of a cost; after all, you can always talk about superpluralities when the standard mereologist talks of pluralities.

So this seems to answer our original problem, which was to ensure that many-one identity really associated each object with one plurality. What we have is unique decomposition: there is a unique plurality associated with each object, and that plurality fuses to that object (is *identical to it.) The way we have achieved unique decomposition in this case is by identifying xx with x’s atoms.

There may be other ways to achieve unique decomposition, but it seems they’ll all fall to the following problem. There are some situations where unique decomposition can’t be achieved, at least according to the standard mereologist. One of these is the possibility of a gunky world: a world where everything has a proper part. Formally, we have an atomless Boolean algebra. But if the points in our algebra are pluralities, what are they pluralities of? There cannot be any singleton pluralities, and if there can’t be singleton pluralities, there can’t be objects for there to be pluralities of.

[Side note: by Stone's representation theorem, any gunky world a standard mereologist can conjure up may be represented by a 'Henkin' model of plural logic. Thus, you may feel like you're in a gunky world - but only because your plural quantifiers are restricted. You're failing to quantify over all pluralities (in particular, the singletons.)]

The other approach was what I labelled unique composition. Every plurality is *identical to exactly one object. In particular, the tables legs and surface are *identical to exactly one object, x, and the tables atoms are *identical to exactly one object, y. Since they two pluralities aren’t *identical*, neither is x and y. But now we should be worried: this seems to mean we must be able to uniquely assign one object to every plurality in the domain. Since we already have a condition for plurality identity, namely we get the following:

• 

This is essentially Frege’s infamous Basic Law V, which entails that there is exactly one object. (Actually, in Frege’s logic it entailed a contradiction, but he allowed there to be empty pluralities.)

In Frege’s system you could derive this, via Russell’s paradox, and I (probably) haven’t written out enough axioms for you to be able to derive the paradox formally. But the problem is still there in the form of Cantor’s theorem, which says you cannot uniquely assign an object to each plurality.

(Note: I never said this, but can bind and vice versa.)

Composition as identity, part I

I’ve been thinking a bit about the (somewhat radical) thesis that an object is literally identical with its parts. So, for example, these things, my parts, are identical to me. One nice thing about this is that you seem to get unrestricted composition for free: you get it from the plural comprehension schema.

However, its main drawback is it requires you to be able to make sense of many one identity. Lewis notes one problem with this, namely: my parts are many, whereas I am not. There are a couple of responses out there: Baxter takes this to be a failure of Leibniz’s law, and Sider has a language where plurals and and singular terms are intersubstituteable. Predicates are polymorphic and you can say truly that I’m both one, and many.

Both these views have crazy consequences (see Sider’s paper “Parthood” to see why.) So I’ve been trying to come up with a more natural way for the composition as identity theorist to go.

Note firstly that Alice, Bob and Fred are human iff Alice is human, Bob is human, and Fred is human. ‘Human’ is a distributive property. Consequently, the atoms that compose me are human iff each atom individually is human. They’re not, so the atoms that compose me aren’t human. However, there is a non-distributive property my atoms have, being human*, which some things have, roughly, if they compose a human. Thus I am human iff my atoms are human*.

So that’s the first step: every monadic predicate of the language, F, has a pluralised homonym, F*. For example, ‘one*’ is short for ‘many’: I am one, the atoms that compose me are one* (they’re many.) The second step: for every singular variable (or name), x,  there is a pluralised version, x*. I shall follow the tradition in plural logic, and use xx for x*. So, for example, ‘Andrew*’ is short for ‘Andrew’s parts’. Finally identity. We have one-one identity, =, many-one identity, *=, one-many identity, =*, and many-many identity, *=*. For n-place relations, well, you can work out your own notation, but it’s the same idea as identity.

We are now in a position to state Leibniz’s law. There are actually lots of versions, I’ll just state a couple

• 
• 

(you must also add suitable identity axioms such as , , etc…). So, for example, Fred is one, Fred is Fred’s parts (that is, Fred =* Fred*), therefore Fred’s parts are one* (Fred* are one*.) So, Fred’s parts are many. I’m human, I’m my parts, so my parts are human*. That’s the idea.

So much for identity. How do we get mereology out of this? Define x is a part of y, iff the xx’s are among the yy’s. Supposing is parthood, we have the following definition

• 

where is the ‘is one of’ relation from plural logic. Thus is defineable in purely logical vocabulary, so if is truly a homonym parthood is logical. What’s more, unrestricted composition falls out from plural comprehension as desired.

But the other good thing about this formulation is that it avoids some of the crazy consequences Sider claims they get. For example, allegedly the principle: x is one of iff or … or , fails. But his argument required moving between (in my language) ‘x is part of y’ and ‘x is part* of y’, rather than ‘x is a part* of yy’. Similarly he had to move between ‘x is-one-of xx’ and ‘x *is-one-of yy’ rather than ‘xx *is-one-of yy’ (his argument is just ungrammatical in this framework.)

Similarly, because he doesn’t pay attention to the difference between parthood, parthood*, *parthood and *parthood*, he gets all kinds of weird things coming out, e.g. ‘Tom, Dick and Harry carried the basket’ iff ‘Dom, Hick and Tarry carried the basket’, where Dom is the fusion of Dicks head and Toms body, Hick the fusion of Harry’s head and Dick’s bady, and Tarry the fusion of Toms head and Harry’s body. Following in the spirit of my rules, you can get from the LHS to ‘Tom*, Dick* and Harry* (carried the bucket)*’, where ‘(carried the bucket)*’ is a superplural predicate. But you can’t then swap bits from the plural terms ‘Tom*’, ‘Dick*’ and ‘Harry*’ and expect it to still satisfy (carried the basket)*.

Lastly, a predicate, P, is distributive iff . Sider claims there are no distributive predicates if you’re a composition as identity theorist. But again, the argument seems to rely on being able to freely move between plural and singular terms, without moving between the corresponding plural and singular predicates.

Ok, so it seems to be a natural way to formulate the position. That said, I think the position is ultimately incoherent, so I’ll talk a bit about that in the next post…

Supertask decision making

I have a little paper writing up the supertask puzzle I posted recently. I’ve added a second puzzle that demonstrates the same problem, but doesn’t use the axiom of choice (it’s basically just a version of Yablo’s paradox), and I’ve framed the puzzles in terms of failures of the deontic Barcan formulae.

Anyway – if anyone has any comments, I’d be very grateful to hear them!

New rules for overseas students

Ross Cameron from metaphysical values:

New rules on overseas students are being introduced by the Government, which will involve academics having to report to the Border Agency when such students have missed a certain number of contact hours. If you are a British citizen or resident and think it is not our role to act as immigration officers, that such rules threaten the autonomy of Universities, that they will make it harder for us to attract overseas students, or that this is generally a Bad Idea, please sign the petition below:

http://petitions.number10.gov.uk/Overseasstudent/

Or just click here.

Indeterminacy and knowledge

What do people think of this principle: determinate implication preserves indeterminacy? Formally¹:

• 

If this principle is ok, and we accept that factivity of knowledge is determinate, it seems we can make trouble for the epistemicist, ignorance view of vagueness. That is, given:

• 

we can infer that : whenever p is indeterminate, it is indeterminate whether you know p. This, I take it, is incompatible with (determinate) ignorance concerning p.

[¹ Note that, although this looks similar, it's not quite the same as , which is a theorem of the weakest normal modal logic, K. and don't stand in the same relation as and .]

A Paradox of Choice

I’ve been thinking about variations on the coin tossing puzzle I posted about a month or so back. This is one I find particularly weird, and seems to violate principles of free choice. You can have a two player game where both players have a winning strategy, but only one player can win. In particular, this implies that if one player follows her winning strategy, the other player can’t. So, although at every point in the game the second player is free to follow the strategy, she is not free to follow the strategy at every point in the game. (I intend there to be some kind of scope difference there.)

The games I am interested in are defined as follows. First I shall define a round: player one chooses 1 or 0, then player two chooses 1 or 0 (having heard player one’s choice.) Player one wins if player two chooses the same number as he did, player two wins if her number is different. Next, a game is a sequence of rounds. Player 2 wins if she wins every round, player 1 wins otherwise.

A strategy for one of these games is a function taking sequences of 1’s and 0’s (provided the order type of the sequences are initial segments of the game order type) to {0, 1}. A winning strategy for a player is a strategy , such that, if at each point in the game, s, you played then you would win.

Now clearly player one does not have a winning strategy for any game that is a finite sequence of rounds – and indeed, this holds for any game that is a well founded sequences of rounds. Obviously, player two has a winning strategy, since she may always say the opposite to what player one says. Since on well founded games, only one player can have a winning strategy, player one never has a winning strategy.

Bizarrely, however, player on does have winning strategies on non-well founded games. Suppose they play on a backwards omega sequence, e.g. a move takes place at each 1/n hours past 12pm, and the game ends at 1pm. Then you divide the possible sequences that player two might play into equivalence classes according to whether they differ by at most finitely many moves. If player one picks a representative from each class, then at each point in the game he can work out what class he’s in, and he can play the same move that the representative sequence predicts player two will play. At the end he must have won all but finitely many moves (I discussed the strategy a bit more here.

So both player one and player two have a winning strategy. But clearly, they can’t both win – so it follows that at least one of them can’t follow their strategy in a given game. This is particularly weird, since at each point in the game they are free to follow their strategy – there’s nothing physically preventing them from them from doing so – but they are not free to to follow it at all of the moves.

This contradicts what I shall call the ‘free choice principle’, that if a rational agent is free and able to do something, and wants to do it, she will do it. For the game above we can formulate this as follows. Let be read roughly as ‘player i (i = 1 or 0) is free to make it the case that’, and let say ‘at round n, player i (i=1 or 0) follows his/her strategy’. Round n is the n’th round from the end of the game. The free choice principle reads:

• 

If at a given round each player is free to follow their strategy, then each player does follow their strategy. We assume tacitly that the players we are concerned with want to follow their strategy, and are physically able to carry it out, etc… We may formulate the principle that at each point in the game, both players are free to follow their strategy as follows

• 

But this entails the impossible conclusion: . At least one player has to lose.

As far as I can see, the premise that at each point in the game each player is free to play according to her strategy is fine. It’s been stipulated that nothing is preventing them from following the strategy, and there are no other relevant limitations.

So it has to be the principle of free choice that goes. There will be a round such that one of the two perfectly rational players wants to follow her strategy, intends to follow it, can follow it in the sense that nothing is preventing her, yet doesn’t follow it. Strange.

Philosophers’ Carnival #82

is here.

Time Travel in “Primer”

So instead of working on exams for the last few days, I’ve been watching a lot of films. I’ve just watched “Primer” again, after having watched it a few years back. It’s an excellent, almost Heinleinesque, time travel movie, albeit quite low budget (and you can watch it here for free!) I’m not entirely sure its consistent, but its fun all the same.

Time travel in this film is fairly limited. You switch on the machine and wait for the number of hours you want to go back to, get in the machine, wait that number of hours again, and you’ll come out the time you started. For example, suppose you switch the machine on at 12pm, and you want to go back four hours, you climb into the machine at 4pm, wait -4 hours (4 hours personal time) and get out at 12pm again. If you wait for longer you just bounce back and forth between 12pm and 4pm (e.g. if you wait 5 hours personal time you get out at 1pm, etc…)

Just a few points of clarification. Time travel in Primer, unlike, say, in Heinlein’s stories, is continuous. Informally, this means that when God drew the time travellers trajectory, her pencil never left the four-dimensional page.

[Digression. Note, that this definition is *not* the same as saying that for each time traveller the function from personal time to location in fourspace is continuous: we must distinguish three kinds of temporal distance. (i) temporal displacement: if I meet my time travelling self I have a displacement of zero from him (ii) temporal distance: if I meet my time travelling self who, this time tomorrow, gets in a time machine and zips continuously to 3008AD and back to today in a space of an hour, his distance from me is 6000 years and (iii) personal time: in the last case my time travelling self has a personal time difference of one day and an hour.

I'm imagining the map from your personal timeline to your location can be discontinuous while your worm is perfectly continuous. E.g. suppose over the next hour I undergo a strange experience: for each x < 60, x minutes into the hour I can't remember the first x minutes, but I can remember the future last x minutes of the hour in reverse order. After the hour is over I experience everything normally again. It seems plausible that you could fill enough details into this story so that it constituted a discontinuity in my personal time, even though my worm is continuous throughout.

Also, I'm imagining that a continuous time traveller could live an infinite amount of personal time while having only a finite distance and displacement from her birth (after 1/2 an hour she's aged 1 year, after 3/4 of an hour she's aged 2 years, etc...). Similarly, a continuous time traveller can have a finite amount of displacement and have had an infinite amount of distance from her birth (imagine the sine wave that diminishes in amplitude according to the sequence 1/2, 1/3, 1/4, 1/5 etc... the line cannot be assigned a finite length, but will still be continuous.) Finally, you can have an infinite (or at least unbounded) amount of distance and displacement from your birth while only a finite amount of personal time has passed. Imagine you're in an accelerating time machine: in the first half hour of personal time you travel to 4000AD, in the next quarter of an hour you're at 8000AD, in the next sixteenth 16000AD etc... Actually, I think this is another example where the personal time to location function will have to be discontinuous while your worm is continuous.

Ok that was longer than expected! End digression.]

Anyway. One cool thing about the film, which I think was intended, was that when they came out of the machine for the first time they couldn’t write properly. The only way I could make sense of the bouncing back and forth process involved rotation in time, which would mean they would come out left handed, if they went in right handed. (The arrows represent the direction of personal time.)

[Analogy: in twospace you can rotate a 'p' onto a 'q' going into the third dimension.]

Another weird thing: in the 12pm to 4pm example I gave above, if you decided to wait 4n hours personal time before getting out, you would expect to have n colocated copies of yourself in the time machine (since the machine doesn’t move.) This seems to be quite different from continuous time travel as I had always previously thought about it (where you would go in large arcs to avoid hitting yourself.)

Is second order logic really first order?

Nowadays, I guess, a lot more people are sympathetic to the idea that second order logic is real logic than in Quine’s day due to the popularity of plural logic. However, this falls short of full second order logic by quite a long way due to the fact that it can’t quantify over relations. For example, you can’t state various facts about sizes or the axiom of choice.

In the first order case, the question seems to be more tractable. If we identify the logical vocabulary as those terms that are not sensitive to the particular identities of the individuals (i.e. whose extensions remain unchanged if you permute the domain) then we get the cardinality quantifiers and arbitrary unions of the cardinality quantifiers as logical terms. McGee confirms the intuition that these truly are logical by showing that the permutation invariant (first order) vocabulary are precisely those defineable from intuitively logical operations: negation, identity, arbitrary conjunctions, universal quantification with respect to an arbitrary block of variables. Admittedly, this language () is not a language that anyone can speak, but that is a deficiency on our part, and should not place constraints on logic. Thus, first order quantification seems to be ontologically innocent, even for quantifiers like ‘there are uncountably many F’s’.

Indeed, similar results hold if we allow second order quantifiers. They are also permutation invariant, and conversely, the permutation invariant second order quantifiers is precisely those that can be defined in the equivalent of with arbitrary blocks of second order quantifiers too. But the difference here, it seems, is that it is not clear that second order quantification over relations is ontologically innocent. Sure, plural quantifiers are, but as soon as we leave the realm of monadic quantification there is less reason to think so (although some have suggested that you can get around it: e.g. Burgess, and Rayo and Linnebo.)

Anyway, I was wondering if it would be possible to reduce second order quantification to first order quantification in our infinitary language. If this were possible then we could happily use the second order quantifiers and safely know that the are not ontologically committing, because they are definable using first order vocabulary.

I think you can do it, but I’m not entirely sure so this might be wrong. Let be antizero – the size of everything. For each second order variable X of the language keep aside many variables: for . Then define a translation schema as follows: [UPDATE: I reformulated it slightly so that it wasn't quite so confusing.] For a subset of the domain, I, we define the translate of with repsect to I as follows:

For the other connectives and quantifiers translation just commutes in the natural way. A couple of notes: this isn’t like in that it must allow truly arbitrary disjunctions and quantifications (including proper class length conjunctions.) Secondly, it’s not really as simple a translation as it looks because in the first clause I left I “free”, to be later “bound” by an earlier application of the second translation clause. What this really means is that the length of the disjunction in the first clause is really determined by when it is called in the second clause. Lastly – that’s just monadic quantification, which we already had – but it seems it will extend nicely to polyadic second order quantifiers (this time we disjoin instead.)

Exams

I’m afraid it’s going to be quite quiet around here for a bit. I have started the “BPhil exams” – three and a half months to write six 5,000 word essays! And they have this ridiculous rule that you’re not allowed talk about philosophy (if it can be construed as related to your questions) which means, I guess, I’m not allowed to blog about it either.

So if I have any philosophy thoughts that are unrelated to metaphysics and epistemology, logic and language or Frege’s philosophy, then I’ll be sure to put them up here. Don’t cross your fingers.