Presuppositions and Modal Operators

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?

The 69th Philosophers’ Carnival

Welcome to the 69th Philosophers’ carnival! I had a lot of good submissions this time, but I had to cut them down, so sorry if your submission doesn’t appear here. If you would like to submit a post to a future carnival use our online submission form.

epistemology

Over at Movement of Existence Bryan Norwood draws attention to a similarity between Heidegger and Kuhn in Normal and Revolutionary Science in Heidegger.

logic and language

Matt Ward suggests a counterexample to Abduction as Logical Inference with a simple quantum mechanical experiment. Posted at A Mind for Madness.

metaphysics

Oisin Deery questions whether compatibilism is committed to prepunishment in Compatibilism and Prepunishment: A Response to Smilansky? posted at The Fog Blog.

Terrance Tomkow discusses Blackburn, Truth and other Hot Topics at Tomkow.com.

mind

Gualtiero Piccinini explains the difference between Computation vs. Information Processing at Brains.

moral philosophy

“Q” the Enchanter discusses ethical universalism in Ethical Universalism and the Problem of Natural Moral Variation posted at Meta and Meta.

Over at Philosophy, et cetera Richard Chappell defends philosophy from the accusation of irrelevance in In Defence of Impractical Philosophy.

Well - that’s the end of this edition. Thanks everyone for your submissions. For the next edition, submit your blog article to the philosophers’ carnival submission form.

Is ‘determinately’ a logical constant?

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 (’determinately’) operator is added to the language. For example, if validity is thought of as preservation of supertruth, then the deduction theorem fails.

• If then

Any model in which p is supertrue is one in which 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, 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, , 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 fixed across models. If we’re not, then the deduction theorem holds again - interpret 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 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 fails (but I still need to check up on what the MacFarlane tests are here.)

Another reason to think that 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, 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 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 is vague. But nonetheless the existence of higher order vagueness (in p, say - so ) strikes me as compelling evidence that there is vagueness located in rather than in p. I was hoping to run a more precise version of this argument by constructing a sorites for 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 and 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 with precise operations yields a vague term, then is vague. Thus, if

• 

is vague, then 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, must be vague.

Could There Be Exactly Two Things?

Lewis’s theory of worlds is sometimes criticized because it invalidates certain true modal claims. Could there be no objects what so ever? According to Lewis no, if worlds are maximally self-connected regions of spacetime the smallest world consists of one spacetime point. But intuitively, yes: we can keep subtracting objects until we don’t have anything left. Or to put it otherwise, once you’ve gotten to the last two objects you can either delete the first, or delete the second. Since both operation are possible, why aren’t both together?

Similarly, one might have thought it possible that there be two disconnected regions of spacetime - this idea is common in science fiction. But again according to Lewis there is no such world, if they are truly spatiotemporally disconnected, they are two worlds.

What I find slightly weird is that not only is there no world with no objects, but there is no world with exactly two objects¹ (or indeed any finite or countable number.) First note it is not possible for two closed sets to be touching without overlapping. Secondly note that our mereological atoms (spacetime points in this case) are closed. A world w is a maximally self connected region, which means if x and y fuse to w, then x and y must be connected. So if there was a world containing exactly two points and nothing else, they would have to be touching. Since they are closed they must be overlapping, and since they are atomic they must be identical (if x and y are atomic and overlap, then x = y.) This contradicts the assumption that there were two objects.

Now of course Lewis later refines his view in the Plurality of Worlds. There he says that possible worlds aren’t all the possibilities - possible individuals are possibilities as are worlds supplemented with information about the de re representation of various objects (see here.) Since the fusion of two worlds is a possible object we can get our spatially disconnected epochs. What about our two disconnected atoms? Well, a similar story can be told - but its not quite that simple. We need to decide a principle that modal realism is silent about - namely that one can have two qualitatively indistinguishable worlds. If you can, there should be no problem with helping yourself to two point worlds and taking their fusion. Much more controversial is the possibility of no things whatsoever. If the null fusion (the fusion of just those things which are not self-identical) is legitimate, there is a possible individual corresponding to this possibility. My point is that Lewis can account for these problems, but only at the expense of some controversial metaphysics.

¹I’m ignoring their fusion for now, technically I mean there is no world with exactly three objects but this obscures my point. That there is no world with exactly two objects for the mereological reason is also worrisome, but not relevant.

Fitch’s Paradox and True Believability

Fitch’s paradox shows us that if all truths are knowable, then all truths must be known. (Proof sketch. Show the contrapositive: if there is an unknown truth, there is an unknowable truth. Let p be an unknown truth. Claim: that p is an unknown truth is unknowable. Suppose that it was known - then it must both be known and known to be unknown, for to know than both p and that p is unknown you must know each seperately. But of course by factivity, if it’s known to be unknown it must be unknown. Therefore its known and unknown. The supposition that that p is an unknown truth is known leads to contradiction, so knowing it is impossible. I.e. it’s unknowable.) The above can be made precise by formalising the knowability principle as the following schema for p:

• WVER:

Of course, this argument does not rely on knowledge in particular, it is true of any operator that is both factive, and distributes over conjunctions

• 
• 

For example, if all truths are possibly necessary, then all truths are necessary. Indeed, it holds for any operator whose accessibility relation is reflexive.

Clearly not all substitutions for K are interesting. E.g. it yields results for truth, actuality, determinateness (in the context of vagueness), necessity, provability and the operator true only of eternal truths, but not all these results are particularly philosophically interesting.

The strongest version of Fitch’s paradox I know of results from substituting O for true belief (I shall write for .) And by strongest, I mean has the weakest premise a verificationist would accept, and has an unsavoury Fitch like consequence: if it is possible to truly believe any truth, then all truths are in fact believed by someone or other. Whatever notion of verifiability the anti-realist uses, if the possibility of X-ing p entails the possibility of truly believing p we are in trouble. For example, the possibility of knowing p entails the possibility of truly believing p, since knowledge entails true belief.¹

One tactic an anti-realist might take is to reject the analysis of knowability in terms of possibility and knowledge (or: some modal notion plus some epistemic notion) and take knowabality as a primitive epistemic notion, , over a set of epistemic states or worlds. The knowability principle would then be

• 

Of course, we must then be careful that doesn’t entail . For example we might take to be where is intuitionistic negation (note: negation is an epistemic operator in intuitionist logic.) This gives us some insight into what the accessibility relation for might look like, taken over a set of situations (states of potential knowledge):

• 

This is only one way to go, others have suggested , and no doubt there are many more. I just feel that most of these approaches are wrong headed. It seems to me that an anti-realist would already have a notion of “verifiability” that is conceptually prior to metaphysical possibility and knowledge. And I see no reason why that notion of verifiability should be definable in terms of more familiar operators. It seems to me the best approach is to start with a notion of verifiability, see what principles hold, and work backwards from there. E.g. we’d plausibly want

• 
• If , infer

and so on. We’d then need to think about how this would interact with knowledge and true belief. If truth just is knowability, then the claim that knowability entails the possibility of true belief should be harmless. Anyway, this requires some careful thought, and its all a little unclear in my head right now - but I hope to post on this stuff again (so stay tuned!)

¹ I guess we could make the paradox even stronger if we interpreted as consistency (or, if we’re not treating K as a logical constant, consistency with a minimal logic of knowledge/true belief.) Then the knowability principle is: if p is true, then it is consistent that it be truly believed. Note also that we don’t need full necessitation for the proof, only: if and does not contain box or diamond, then . So it doesn’t matter whether we treat consistency as logical.

A Theory of Duplication

Normally when people define a duplicate they’ll say something like x is a duplicate of y iff x and y have exactly the same intrinsic properties. This might be a fruitful way to think about intrinsic properties, but it seems to me that a theory of duplication can be theoretically separated from our theory of intrinsic properties. For example, as I understand it, duplicates are always spatial entities, yet you might still want to make room for talk about abstract objects having intrinsic properties (unless you’re a structuralist, I guess.)
Also, some people have argued that shape is not an intrinsic property, yet clearly no two duplicates can have different shapes.

What I want to do here is outline a theory of duplication from a spatial perspective. There are quite a lot of potential principles we might adopt, but I’m only going to discuss a small subset of them. Let’s start off by adding to our language of mereology, another primitive, (to be read: x is a duplicate of y.) Here are some axioms/interesting principles worth thinking about:

D1.
D2.
D3.
D4.
D5.
D6.
D7.
D8.
D9.
D10.

Explanation below the fold.

A Game Theoretic Semantics for Vagueness

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 as follows.

• : the verifier chooses between and then the game continues with where is the chosen formula.
• : the falsifier chooses between and then the game continues with where is the chosen formula.
• : the verifier and falsifier swap roles and the game continues with
• : the verifier chooses an assignment that differs from v at most in its assignment to x, and the game continues with .
• : the falsifier chooses an assignment that differs from v at most in its assignment to x, and the game continues with .
• : if then the player playing the role of verifier wins. Otherwise the falsifier wins.

We can then say that a formula is true (in M, on v) if the player who starts off playing the verifier has a winning strategy for , 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 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 . 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 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 .

In this case we get different results, for example 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 , 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

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 . 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 be the regular open set lattice from the standard ball topology over . Elements of 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 , 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 , the composition of the interior and the closure operation. Let be the domain of discourse.

• 
• 
• 
• 
• 
• 

The logic will be classical since 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

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.

Philosophers’ Carnival #65

Is here.

Plural Reference

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

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 .) 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 , 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 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. . 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 is a total function. We set one particular pair to be the actual possibility. The crucial truth clauses are as follows

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 where . We can give a translation schema of for QML+@ as follows:

(Note in the clause, are the free terms in .)

64th Philosophers Carnival

Is here. Also, the blog has passed the 1,000 hits mark! Thank you readers…

Youtube links

I came across this cool video on moebius transformations by Douglas Arnold and Jonathan Rogness today which I thought was worth sharing. Makes it all look very intuitive. Also found a nice video for Autechre’s “Dropp” - pretty good for a fan video I thought.

More non-well-founded gunk!

Aaron has an nice response to my last post on non-well-founded gunk. He notes that my definition of proper parthood (iff:) is too restrictive for the kinds things normally associated with non-well-foundedness. In particular objects can’t be proper parts of themselves, diverging from the set theoretic analogue of non-well-foundedness where is a paradigmatic kind of non-well-founded set.

I think maybe I should put my reasons for discussing this particular kind of non-well-foundedness in a broader context. I was thinking of cases a bit like Goliath and the lump of clay. In this case they are not identical, nonetheless they are both parts of each other. So on my definition they are proper parts of each other, and so form an instance of NWF gunk. This case poses weird problems for mereology, for example neither Goliath nor Lump have a part disjoint from the other, even though they are distinct, they have several fusions, and no sum (see my last post for reasons.)

Nonetheless I think there should be another way of setting up NWF gunk which is more closely related to the non-well-founded set theory. One that’s transitive unlike my earlier definition, and e.g. captures examples like “Borges Aleph” - here the intuition is that Borges Aleph is a proper part of itself, while, of course, remaining identical with itself.

At the end of his post Aaron poses the problem of what the definition of proper parthood should be to allow this. Here are some candidates, all of which are equivalent to my definition in standard mereology

• 
• 
• 
• 
• 

(Here + means mereological fusion, and means mereological sum. See last post for definitions.) However, it’s quick to check that non of these allow an object to be a proper part of itself, and I’m not convinced that there will be such a definition in terms of parthood.

I think the best way forward would be to take proper parthood as a primitive, a good starting point would be

• A1
• A2

Just take the two element digraph where every node is connected to every other node in each direction to see this is at least consistent. (It’s a bit of a mouthful, so call it “Borges Aleph” for short.)

Of course, fusions will never be unique, nor will supplementation hold, but I think you are pretty much committed to this however you set it up. We can obviously add, for each axiom of standard mereology, an axiom to restricted to well-founded-objects (Define WF(x) as . My question now is, what substantial principles can we consistently add?

Non-well-founded mereology

I was wondering recently if there has been any work on non-well-founded mereology. A preliminary google search didn’t bring up much except the SEP article on mereology, which informed me there has so far been no systematic study of well founded mereology in the literature so far.

I thought a little about what it might look like, but the prospects don’t look particularly bright to me. By a non-well-founded mereology, I mean one in which a particularly weird kind of gunk is allowed - gunk with finitely many parts. Although this is sufficient for the mereology to be non-well-founded, it is not necessary: let us simply say that a mereology is non-well-founded iff . (The SEP article quoted “Borges Aleph” as an example of this: “I saw the earth in the Aleph and in the earth the Aleph once more and the earth in the Aleph… (Borges 1949: 151)”)

The first thing to note is that anti-symmetry fails (this follows straight from the definition.) So already there is a worry that we are not talking about a parthood relation here - if there are any analytic constraints on parthood, it must surely include being a partial order.

But even granting this, many other natural principles are inconsistent with the existence of non-well-founded gunk (as usual we help ourselves to reflexivity and transitivity.) For example, take weak supplementation.

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Suppose . By weak supplementation there should be a part of b, z say, disjoint from a. But since , by transitivity, which contradicts the assumption that was disjoint from . (Ironically, I think the so called “strong supplementation” axiom is consistent.)

Similarly uniqueness of fusions fail radically in the presence of non-well-founded gunk. A reminder: u is a fusion of the F’s iff

For example is a fusion of iff . Clearly, if then both and are fusions of . Even worse, whenever , then is a fusion of . Similarly, whenever is a fusion of and , is a fusion of .

Perhaps we could make do with “sums” instead of fusions. Say that is a sum of the ’s iff

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In standard mereology, sums play exactly the same role as fusions do, but they come apart in many non-standard mereologies. However, even here we have problems. For NWF gunk, sums fail to exist: take the and from above and take . Both and are upperbounds for but neither are least, so there is no sum.

It seems, then, that there isn’t anything much mereological about the resulting system. After all, where is mereology without fusions or remainders? Maybe there are other ways of setting up the axioms (e.g. I think that strong supplementation is still consistent.) Can anyone think of an alternative to play the rule of fusions? Maybe something that coincides with fusions in a standard mereology?

Philosophers Carnival

Is here.

There are just as many F’s as G’s

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 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?

Philosoper’s Carnival

Is here.

Fregean concepts

I’m currently reading up on Frege’s theory of concepts for one of my courses. There are several things that really puzzle me about the theory though.

The first confusing thing is to do with relations. Frege says that relations are doubly unsaturated. You can fill a diadic relation with an object, and get a monadic relation. For example if I have the relation I may fill it out with four to get , the monadic concept of being less than four, or alternatively I may put four in the other slot and get the monadic concept of being greater than four. And it clearly made a difference, for the former but not the later is true of 3. But what if we consider ? What about and - are they the same or different? If we think analogously to the last case, we might think they were different. Relations are doubly unsaturated, they have two “object shaped holes”*, in the first case the first hole was filled, in the second the second hole was filled - they denote different concepts because in each case a different hole is covered. But this cannot be the case, since for Frege, concepts are extensional. If they’re true of the same things, they are the same concept.

There is the puzzling section 70 in the Grundlagen where he suggests that relations might actually be monadic concepts taking composite objects as arguments. I assume the composite objects would have to have as much structure as ordered pairs if they were to do the job. But I don’t think that really helps us, for if was really a monadic concept of pairs, the notion of filling in only one of the argument places doesn’t make too much sense. The most natural way to do that would be to compose with , the function that pairs an object with 4. But this means treating < as a second level concept.

That brings me onto the second thing I find confusing. In “Function and Concept” Frege tries to compositionally derive the truth values of various formulae from the Begriffsschrift from the concepts/functions each expression denotes. This works fine for the simple formulae he tries like . But what about: ? It should be derived compositionally from the following pieces:

1. The second level concept true of a first level concept F(x) iff F gives true for each object.
2. The first level function which for each object gives the number four.
3. the number four.
4. , the first level diadic relation-concept that gives the true when supplemented with x and y, where x and y are identical.

However, I cannot see any plausible way to piece these concepts together - they just don’t fit. Crucially is not type correct, since = is a first level relation, it cannot take a first level function for its left argument, only an object. But even if you try combining (1) - (4) in other ways (i.e. not in the obviously compositional fashion), you don’t seem to get the right results.

Quite generally, there are all kinds of typing problems with the way Frege uses concepts. For example, in , is supposed to take a monadic concept as argument like does, but instead it gets a diadic relation-concept.

I guess I don’t really know the literature, or Frege’s writings well enough to know whether he addressed this, or if there is a straightforward way to get around the type mismatches. Does anyone have any ideas? Any pointers to the literature would be very welcome :D.

*This is my metaphor, not Frege’s. You should probably that argument with a pinch of salt.