Here is the final post (for the time being) on structured propositions. As promised, this is to be an account of the truth-conditions of structured propositions, presupposing a certain reasonably contentious take on the metaphysics of linguistic representation (metasemantics). It's going to be compatible with the view that structured propositions are nothing but certain n-tuples: lists of their components. (See earlier posts if you're getting concerned about other factors, e.g. the potential arbitriness in the choice of which n-tuples are to be identified with the structured proposition that Dummett is a philosopher.)
Here's a very natural way of thinking of what the relation between *sentences* and truth-conditions are, on a structured propositions picture. It's that metaphysically, the relation of "S having truth-conditions C" breaks down into two more fundamental relations: "S denoting struc prop p" and "struc prop p having truth-conditions C". The thought is something like: primarily, sentences express thoughts (=struc propositions), and thoughts themselves are the sorts of things that have intrinsic/essential representational properties. Derivatively, sentences are true or false of situations, by expressing thoughts that are true or false of those situations. As I say, it's a natural picture.
In the previous posting, I've been talking as though this direction-of-explanation was ok, and that the truth-conditions of structured propositions should have explanatory priority over the truth-conditions of sentences, so we get the neat separation into the contingent feature of linguistic representation (which struc prop a sentence latches onto) and the necessary feature (what the TCs are, given the struc prop expressed).
The way I want to think of things, something like the reverse holds. Here’s the way I think of the metaphysics of linguistic representation. In the beginning, there were patterns of assent and dissent. Assent to certain sentences is systematically associated with certain states of the world (coarse-grained propositions, if you like) perhaps by conventions of truthfulness and trust (cf. Lewis's "Language and Languages"). What it is for expressions E in a communal language to have semantic value V is for E to be paired with V under the optimally eligible semantic theory fitting with that association of sentences with coarse-grained propositions.
That's a lot to take in all at one go, but it's basically the picture of linguistic representation as fixed by considerations of charity/usage and eligibility/naturalness that lots of people at the moment seem to find appealing. The most striking feature---which it shares with other members of the "radical interpretation" approach to metasemantics---is that rather than starting from the referential properties of lexical items like names and predicates, it depicts linguistic content as fixed holistically by how well it meshes with patterns of usage. (There's lots to say here to unpack these metaphors, and work out what sort of metaphysical story of representation is being appealed to: that's something I went into quite a bit in my thesis---my take on it is that it's something close to a fictionalist proposal).
This metasemantics, I think, should be neutral between various semantic frameworks for generating the truth conditions. With a bit of tweaking, you can fit in a Davidsonian T-theoretic semantic theory into this picture (as suggested by, um... Davidson). Someone who likes interpretational semantics but isn't a fan of structured propositions might take the semantic values of names to be objects, and the semantic values of sentences to be coarse-grained propositions, and say that it's these properties that get fixed via best semantic theory of the patterns of assent and dissent (that's Lewis's take).
However, if you think that to adequately account for the complexities of natural language you need a more sophisticated, structured proposition, theory, this story also allows for it. The meaning-fixing semantic theory assign objects to names, and structured propositions to sentences, together with a clause specifying how the structured propositions are to be paired up with coarse-grained propositions. Without the second part of the story, we'd end up with an association between sentences and structured propositions, but we wouldn't make connection with the patterns of assent and dissent if these take the form of associations of sentences with *coarse grained* propositions (as on Lewis's convention-based story). So on this radical interpretation story where the targetted semantic theories take a struc prop form, we get a simultaneous fix on *both* the denotation relation between sentences and struc props, and the relation between struc props and coarse-grained truth-conditions.
Let's indulge in a bit of "big-picture" metaphor-ing. It’d be misleading to think of this overall story as the analysis of sentential truth-conditions into a prior, and independently understood, notion of the truth-conditions of structured propositions, just as it's wrong on the radical interpretation picture to think of sentential content as "analyzed in terms of" a prior, and independently understood, notion of subsentential reference. Relative to the position sketched, it’s more illuminating to think of the pairing of structured and coarse-grained propositions as playing a purely instrumental role in smoothing the theory of the representational features of language. It's language which is the “genuine” representational phenomenon in the vicinity: the truth-conditional features attributed to struc propositions are a mere byproduct.
Again speaking metaphorically, it's not that sentences get to have truth-conditions in a merely derivative sense. Rather, structured propositions have truth-conditions in a merely derivative sense: the structured proposition has truth-conditions C if it is paired with C under the optimal overall theory of linguistic representation.
For all we've said, it may turn out that the same assignment of truth-conditions to set-theoretic expressions will always be optimal, no matter which language is in play. If so, then it might be that there's a sense in which structured propositions have "absolute" truth-conditions, not relative to this or that language. But, realistically, one'd expect some indeterminacy in what struc props play the role (recall the Benacerraf point King makes, and the equally fitness of [a,F] and [F,a] to play that "that a is F" role). And it's not immediately clear why the choice to go one way for one natural language should constrain way this element is deployed in another language. So it's at least prima facie open that it's not definitely the case that the same structured propositions, with the same TCs, are used in the semantics of both French and English.
It's entirely in the spirit of the current proposal that we think of we identify [a,F] with the structured proposition that a is F only relative to a given natural language, and that this creature only has the truth-conditions it does relative to that language. This is all of a piece with the thought that the structured proposition's role is instrumental to the theory of linguistic representation, and not self-standing.
Ok. So with all this on the table, I'm going to return to read the book that prompted all this, and try to figure out whether there's a theoretical need for structured propositions with representational properties richer than those attributed by the view just sketched.
[update: interestingly, it turns out that King's book doesn't give the representational properties of propositions explanatory priority over the representational properties of sentences. His view is that the proposition that Dummett thinks is (very crudely, and suppressing details) the fact that in some actual language there is a sentence of (thus-and-such a structure) of which the first element is a word referring to Dummett and the second element is a predicate expressing thinking. So clearly semantic properties of words are going to be prior to the representational properties of propositions, since those semantic properties are components of the proposition. But more than this, from what I can make out, King's thought is that if there was a time where humans spoke a language without attitude-ascriptions and the like, then sentences would have truth-conditions, and the proposition-like facts would be "hanging around" them, but the proposition-like facts wouldn't have any representational role. Once we start making attitude ascriptions, we implicitly treat the proposition-like structure as if it had the same TCs as sentences, and (by something like a charity/eligibility story) the "propositional relation" element acquires semantic significance and the proposition-like structure gets to have truth-conditions for the first time.
That's very close to the overall package I'm sketching above. What's significant dialectically, perhaps, is that this story can explain TCs for all sorts of apparently non-semantic entities, like sets. So I'm thinking it really might be the Benacerraf point that's bearing the weight in ruling out set-theoretic entities as struc propns---as explained previously, I don't go along with *that*.]
Tuesday, December 18, 2007
Structured propositions and truth conditions.
In the previous post, I talked about the view of structured propositions as lists, or n-tuples, and the Benacerraf objections against it. So now I'm moving on to a different sort of worry. Here's King expressing it:
“A final difficulty for the view that propositions are ordered n-tuples concerns the mystery of how or why on that view they have truth conditions. On any definition of ordered n-tuples we are considering, they are just sets. Presumably, many sets have no truth conditions (eg. The set of natural numbers). But then why do certain sets, certain ordered n-tuples, have truth-conditions? Since not all sets have them, there should be some explanation of why certain sets do have them. It is very hard to see what this explanation could be.”
I feel the force of something in this vicinity, but I'm not sure how to capture the worry. In particular, I'm not sure whether the it's right to think of structured propositions' having truth-conditions as a particularly "deep" fact over which there is mystery in the way King suggests. To get what I'm after here, it's probably best simply to lay out a putative account of the truth-conditions of structured propositions, and just to think about how we'd formulate the explanatory challenge.
Suppose, for example, one put forward the following sort of theory:
(i) The structured proposition that Dummett is a philosopher = [Dummett, being a philosopher].
(ii) [Dummett, being a philosopher] stands in the T relation to w, iff Dummett is a philosopher according to w.
(iii) bearing the T-relation to w=being true at w
Generalizing,
(i) For all a, F, the structured proposition that a is F = [a, F]
(ii) For all individuals a, and properties F, [a, F] stands in the T relation to w iff a instantiates F according to w.
(iii) bearing the T-relation to w=being true at w
In a full generality, I guess we’d semantically ascend for an analogue of (i), and give a systematic account of what structured propositions are associated with which English sentences (presumably a contingent matter). For (ii), we’d give a specification (which there’s no reason to make relative to any contingent facts) about which ordered n-tuples stand in the T-relation to which worlds. (iii) can stay as it is.
The naïve theorist may then claim that (ii) and (iii) amount to a reductive account of what it is for a structured proposition to have truth-conditions. Why does [1,2] not have any truth-conditions, but [Dummett, being a philosopher] does? Because the story about what it is for an ordered pair to stand in the T-relation to a given world, just doesn’t return an answer where the second component isn’t a property. This seems like a totally cheap and nasty response, I’ll admit. But what’s wrong with it? If that’s what truth-conditions for structured propositions are, then what’s left to explain? It doesn't seem that there is any mystery over (ii): this can be treated as a reductive definition of the new term "bearing the T-relation". Are there somehow explanatory challenges facing someone who endorses the property-identity (iii)? Quite generally, I don't see how identities could be the sort of thing that need explaining.
(Of course, you might semantically ascend and get a decent explanatory challenge: why should "having truth conditions" refer to the T-relation. But I don't really see any in principle problem with addressing this sort of challenge in the usual ways: just by pointing to the fact that the T-relation is a reasonably natural candidate satisfying platitudes associated with truth-condition talk.)
I'm not being willfully obstructive here: I'm genuinely interested in what the dialectic should be at this point. I've got a few ideas about things one might say to bring out what's wrong with the flat-footed response to King's challenge. But none of them persuades me.
Some options:
(a)Earlier, we ended up claiming that it was indefinite what sets structured propositions were identical with. But now, we’ve given a definition of truth-conditions that is committal on this front. For example, [F,a] was supposed to be a candidate precisification of the proposition that a is F. But (ii) won’t assign it truth conditions, since the second component isn’t a property but an individual.
Reply: just as it was indefinite what the structured propositions were, it is indefinite what sets have truth-conditions, and what specification of those truth-conditions is. The two kinds of indefiniteness are “penumbrally connected”. On a precisification on which the prop that a is F=[a,F], then the clause holds as above; but on a precisification on which that a is F=[F,a], a slightly twisted version of the clause will hold. But no matter how we precisify structured proposition-talk, there will be a clause defining the truth-conditions for the entities that we end up identifying with structured propositions.
(b) You can’t just offer definitional clauses or “what it is” claims and think you’ve evaded all explanatory duties! What would we think of a philosopher of mind who put forward a reductive account whereby pain-qualia were by definition just some characteristics of C-fibre firing, and then smugly claimed to have no explanatory obligations left.
Reply: one presupposition of the above is that clauses like (ii) “do the job” of truth-conditions for structured propositions, i.e. there won’t be a structured proposition (by the lights of (i)) whose assigned “truth-conditions” (by the lights of (ii)) go wrong. So whatever else happens, the T-relation (defined via (ii)) and the truth-at relation we’re interested in have a sort of constant covariation (and, unlike the attempt to use a clause like (ii) to define truth-conditions for sentences, we won’t get into trouble when we vary the language use and the like across worlds, so the constant covariation is modally robust). The equivalent assumption in the mind case is that pain qualia and the candidate aspect of C-fibre firing are necessarily constantly correlated. Under those circumstances, many would think we would be entitled to identify pain qualia and the physicalistic underpinning. Another way of putting this: worries about the putative “explanatory gap” between pain-qualia and physical states are often argued to manifest themselves in a merely contingent correlation between the former and the latter. And that’d mean that any attempt to claim that pain qualia just are thus-and-such physical state would be objectionable on the grounds that pain qualia and the physical state come apart in other possible worlds.
In the case of the truth-conditions of structured propositions, nothing like this seems in the offing. So I don’t see a parody of the methodology recommended here. Maybe there is some residual objection lurking: but if so, I want to hear it spelled out.
(c)Truth-conditions aren’t the sort of thing that you can just define up as you please for the special case of structured propositions. Representational properties are the sort of things possessed by structural propositions, token sentences (spoken or written) of natural language, tokens of mentalese, pictures and the rest. If truth-conditions were just the T-relation defined by clause (ii), then sentences of mentalese and English, pictures etc couldn’t have truth-conditions. Reductio.
Reply: it’s not clear at all that sentences and pictures “have truth-conditions” in the same sense as do structured propositions. It fits very naturally with the structured-proposition picture to think of sentences standing in some “denotation” relation to a structured proposition, through which may be said to derivatively have truth-conditions. What we mean when we say that ‘S has truth conditions C’ is that S denotes some structured proposition p and p has truth-conditions C, in the sense defined above. For linguistic representation, at least, it’s fairly plausible that structured propositions can act as a one-stop-shop for truth-conditions.
Pictures are a trickier case. Presumably they can represent situations accurately or non-accurately, and so it might be worth theorizing about them by associating them with a coarse-grained proposition (the set of worlds in which they represent accurately). But presumably, in a painting that represents Napolean’s defeat at waterloo, there doesn’t need to be separable elements corresponding to Napolean,Waterloo
None of this does anything to suggest that we can’t give an ok story about pairing pictures with (e.g.) coarse-grained propositions. It’s just that the relation between structured propositions and coarse-grained propositions (=truth conditions) and the relation between pictures and coarse-grained propositions can’t be the same one, on this account, and nor is even obvious how the two are related (unlike e.g. the sentence/structured proposition case).
So one thing that may cause trouble for the view I’m sketching is if we have both the following: (A) there is a unified representation relation, such that pictures/sentences/structured propositions stand in same (or at least, intimately related) representation relations to C. (B) there’s no story about pictorial (and other) representations that routes via structured propositions, and so no hope of a unified account of representation given (ii)+(iii).
The problem here is that I don’t feel terribly uncomfortable denying (A) and (B). But I can imagine debate on this point, so at least here I see some hope of making progress.
Having said all this in defence of (ii), I think there are other ways for the naïve, simple set-theoretic account of structured propositions to defend itself that don't look quite so flat-footed. But the ways I’m thinking of depend on some rather more controversial metasemantic theses, so I’ll split that off into a separate post. It’d be nice to find out what’s wrong with this, the most basic and flat-footed response I can think of.
“A final difficulty for the view that propositions are ordered n-tuples concerns the mystery of how or why on that view they have truth conditions. On any definition of ordered n-tuples we are considering, they are just sets. Presumably, many sets have no truth conditions (eg. The set of natural numbers). But then why do certain sets, certain ordered n-tuples, have truth-conditions? Since not all sets have them, there should be some explanation of why certain sets do have them. It is very hard to see what this explanation could be.”
I feel the force of something in this vicinity, but I'm not sure how to capture the worry. In particular, I'm not sure whether the it's right to think of structured propositions' having truth-conditions as a particularly "deep" fact over which there is mystery in the way King suggests. To get what I'm after here, it's probably best simply to lay out a putative account of the truth-conditions of structured propositions, and just to think about how we'd formulate the explanatory challenge.
Suppose, for example, one put forward the following sort of theory:
(i) The structured proposition that Dummett is a philosopher = [Dummett, being a philosopher].
(ii) [Dummett, being a philosopher] stands in the T relation to w, iff Dummett is a philosopher according to w.
(iii) bearing the T-relation to w=being true at w
Generalizing,
(i) For all a, F, the structured proposition that a is F = [a, F]
(ii) For all individuals a, and properties F, [a, F] stands in the T relation to w iff a instantiates F according to w.
(iii) bearing the T-relation to w=being true at w
In a full generality, I guess we’d semantically ascend for an analogue of (i), and give a systematic account of what structured propositions are associated with which English sentences (presumably a contingent matter). For (ii), we’d give a specification (which there’s no reason to make relative to any contingent facts) about which ordered n-tuples stand in the T-relation to which worlds. (iii) can stay as it is.
The naïve theorist may then claim that (ii) and (iii) amount to a reductive account of what it is for a structured proposition to have truth-conditions. Why does [1,2] not have any truth-conditions, but [Dummett, being a philosopher] does? Because the story about what it is for an ordered pair to stand in the T-relation to a given world, just doesn’t return an answer where the second component isn’t a property. This seems like a totally cheap and nasty response, I’ll admit. But what’s wrong with it? If that’s what truth-conditions for structured propositions are, then what’s left to explain? It doesn't seem that there is any mystery over (ii): this can be treated as a reductive definition of the new term "bearing the T-relation". Are there somehow explanatory challenges facing someone who endorses the property-identity (iii)? Quite generally, I don't see how identities could be the sort of thing that need explaining.
(Of course, you might semantically ascend and get a decent explanatory challenge: why should "having truth conditions" refer to the T-relation. But I don't really see any in principle problem with addressing this sort of challenge in the usual ways: just by pointing to the fact that the T-relation is a reasonably natural candidate satisfying platitudes associated with truth-condition talk.)
I'm not being willfully obstructive here: I'm genuinely interested in what the dialectic should be at this point. I've got a few ideas about things one might say to bring out what's wrong with the flat-footed response to King's challenge. But none of them persuades me.
Some options:
(a)Earlier, we ended up claiming that it was indefinite what sets structured propositions were identical with. But now, we’ve given a definition of truth-conditions that is committal on this front. For example, [F,a] was supposed to be a candidate precisification of the proposition that a is F. But (ii) won’t assign it truth conditions, since the second component isn’t a property but an individual.
Reply: just as it was indefinite what the structured propositions were, it is indefinite what sets have truth-conditions, and what specification of those truth-conditions is. The two kinds of indefiniteness are “penumbrally connected”. On a precisification on which the prop that a is F=[a,F], then the clause holds as above; but on a precisification on which that a is F=[F,a], a slightly twisted version of the clause will hold. But no matter how we precisify structured proposition-talk, there will be a clause defining the truth-conditions for the entities that we end up identifying with structured propositions.
(b) You can’t just offer definitional clauses or “what it is” claims and think you’ve evaded all explanatory duties! What would we think of a philosopher of mind who put forward a reductive account whereby pain-qualia were by definition just some characteristics of C-fibre firing, and then smugly claimed to have no explanatory obligations left.
Reply: one presupposition of the above is that clauses like (ii) “do the job” of truth-conditions for structured propositions, i.e. there won’t be a structured proposition (by the lights of (i)) whose assigned “truth-conditions” (by the lights of (ii)) go wrong. So whatever else happens, the T-relation (defined via (ii)) and the truth-at relation we’re interested in have a sort of constant covariation (and, unlike the attempt to use a clause like (ii) to define truth-conditions for sentences, we won’t get into trouble when we vary the language use and the like across worlds, so the constant covariation is modally robust). The equivalent assumption in the mind case is that pain qualia and the candidate aspect of C-fibre firing are necessarily constantly correlated. Under those circumstances, many would think we would be entitled to identify pain qualia and the physicalistic underpinning. Another way of putting this: worries about the putative “explanatory gap” between pain-qualia and physical states are often argued to manifest themselves in a merely contingent correlation between the former and the latter. And that’d mean that any attempt to claim that pain qualia just are thus-and-such physical state would be objectionable on the grounds that pain qualia and the physical state come apart in other possible worlds.
In the case of the truth-conditions of structured propositions, nothing like this seems in the offing. So I don’t see a parody of the methodology recommended here. Maybe there is some residual objection lurking: but if so, I want to hear it spelled out.
(c)Truth-conditions aren’t the sort of thing that you can just define up as you please for the special case of structured propositions. Representational properties are the sort of things possessed by structural propositions, token sentences (spoken or written) of natural language, tokens of mentalese, pictures and the rest. If truth-conditions were just the T-relation defined by clause (ii), then sentences of mentalese and English, pictures etc couldn’t have truth-conditions. Reductio.
Reply: it’s not clear at all that sentences and pictures “have truth-conditions” in the same sense as do structured propositions. It fits very naturally with the structured-proposition picture to think of sentences standing in some “denotation” relation to a structured proposition, through which may be said to derivatively have truth-conditions. What we mean when we say that ‘S has truth conditions C’ is that S denotes some structured proposition p and p has truth-conditions C, in the sense defined above. For linguistic representation, at least, it’s fairly plausible that structured propositions can act as a one-stop-shop for truth-conditions.
Pictures are a trickier case. Presumably they can represent situations accurately or non-accurately, and so it might be worth theorizing about them by associating them with a coarse-grained proposition (the set of worlds in which they represent accurately). But presumably, in a painting that represents Napolean’s defeat at waterloo, there doesn’t need to be separable elements corresponding to Napolean,
None of this does anything to suggest that we can’t give an ok story about pairing pictures with (e.g.) coarse-grained propositions. It’s just that the relation between structured propositions and coarse-grained propositions (=truth conditions) and the relation between pictures and coarse-grained propositions can’t be the same one, on this account, and nor is even obvious how the two are related (unlike e.g. the sentence/structured proposition case).
So one thing that may cause trouble for the view I’m sketching is if we have both the following: (A) there is a unified representation relation, such that pictures/sentences/structured propositions stand in same (or at least, intimately related) representation relations to C. (B) there’s no story about pictorial (and other) representations that routes via structured propositions, and so no hope of a unified account of representation given (ii)+(iii).
The problem here is that I don’t feel terribly uncomfortable denying (A) and (B). But I can imagine debate on this point, so at least here I see some hope of making progress.
Having said all this in defence of (ii), I think there are other ways for the naïve, simple set-theoretic account of structured propositions to defend itself that don't look quite so flat-footed. But the ways I’m thinking of depend on some rather more controversial metasemantic theses, so I’ll split that off into a separate post. It’d be nice to find out what’s wrong with this, the most basic and flat-footed response I can think of.
Structured propositions and Benacerraf
I’ve recently been reading Jeff King’s book on structured propositions. It’s really good, as you would expect. There’s one thing that’s bothering me though: I can’t quite get my head around what’s wrong with the simplest, most naïve account of the nature of propositions. (Disclaimer: this might all turn out to be very simple-minded to those in the know. I'd be happy to get pointers to the literature (hey, maybe it'll be to bits of Jeff's book I haven't got to yet...)
The first thing you encounter when people start talking about structured propositions is notation like [Dummett, being a philosopher]. This is supposed to stand for the proposition that Dummett is a philosopher, and highlights the fact that (on the Russellian view) Dummett and the property of being a philosopher are components of the proposition. The big question is supposed to be: what do the brackets and comma represent? What sort of compound object is the proposition? In what sense does it have Dummett and being a philosopher as components? (If you prefer a structured intension view, so be it: then you’ll have a similar beast with the individual concept of Dummett and the worlds-intension associated with “is a philosopher” as ‘constituents’. I’ll stick with the Russellian view for illustrative purposes.)
For purposes of modelling propositions, people often interpret the commas as brackets as the ordered n-tuples of standard set theory. The simplest, most naïve interpretation of what structured propositions are, is simply to identify them as n-tuples. What’s the structured proposition itself? It’s a certain kind of set. What sense are Dummett and the property of being a philosopher constituents of the structured proposition that Dummett is a philosopher? They’re elements of the transitive closure of the relevant set.
So all that is nice and familiar. So what’s the problem? In his ch 1. (and, in passing, in the SEP article here) King mentions two concerns. In this post, I’ll just set the scene by talking about the first. It's a version of a famous Benacerraf worry, which anyone with some familiarity with the philosophy of maths will have come across (King explicitly makes the comparison). The original Benacerraf puzzle is something like this: suppose that the only abstract things are set like, and whatever else they may be, the referents of arithmetical terms should be abstract. Then numerals will stand for some set or other. But there are all sorts of things that behave like the natural numbers within set theory: the constructions known as the (finite) Zermelo ordinals (null, {null}, {{null}}, {{{null}}}...) and the (finite) von Neumann ordinals (null, {null}, {null,{null}}…) are just two. So there’s no non-arbitrary theory of which sets the natural numbers are.
The phenomenon crops up all over the place. Think of ordered n-tuples themselves. Famously, within an ontology of unordered sets, you can define up things that behave like ordered pairs: either [a,b]={{a},{a,b}} or {{{a},null},{{b}}}. (For details see http://en.wikipedia.org/wiki/Ordered_pair). It appears there’s no non-arbitrary reason to prefer a theory that ‘reduces’ ordered to unordered pairs one way or the other.
Likewise, says King, there looks to be no non-arbitrary choice of set-theoretic representation of structured propositions (not even if we spot ourselves ordered sets as primitive to avoid the familiar ordered-pair worries). Sure, we *could* associate the words “the proposition that Dummett is a philosopher” with the ordered pair [Dummett, being a philosopher]. But we could also associate it with the set [being a philosopher, Dummett] (and choices multiply when we get to more complex structured propositions).
One reaction to the Benacerrafian challenge is to take it to be a decisive objection to an ontological story about numbers, ordered pairs or whatever that allows only unordered sets as a basic mathematical ontology. My own feeling is (and this is not uncommon, I think) that this would be an overreaction. More strongly: no argument that I've seen from the Benacerraf phenomenon to this ontological conclusion seems to me to be terribly persuasive.
What we should admit, rather, is that if natural numbers or ordered pairs are sets, it’ll be indefinite which sets they are. So, for example, [a,b]={{a},{a,b}} will be neither definitely true nor definitely false (unless we simply stipulatively define the [,] notation one way or another rather than treating it as pre-theoretically understood). Indefiniteness is pervasive in natural language---everyone needs a story about how it works. And the idea is that whatever that story should be, it should be applied here. Maybe some theories of indefiniteness will make these sort of identifications problematic. But prominent theories like Supervaluationism and Epistemicism have neat and apparently smooth theories of what it we’re saying when we call that identity indefinite: for the supervaluationist, it (may) mean that “[a,b]” refers to {{a},{a,b}} on one but not all precisifications of our set-theoretic language. For the epistemicist, it means that (for certain specific principled reasons) we can’t know that the identity claim is false. The epistemicist will also maintains there’s a fact of the matter about which identity statement connecting ordered and unordered sets is true. And there’ll be some residual arbitrariness here (though we’ll probably have to semantically ascend to find it)---but if there is arbitriness, it’s the sort of thing we’re independently committed to to deal with the indefiniteness rife throughout our language. If you’re a supervaluationist, then you won’t admit there’s any arbitriness: (standardly) the identity statement is neither true nor false, so our theory won’t be committed to “making the choice”.
If that’s the right way to respond to the general Benacerraf challenge, it’s the obvious thing to say in response to the version of that puzzle that arises for the Benacerraf case. And this sort of generalization of the indefiniteness maneuver to philosophical analysis is pretty familiar, it’s part of the standard machinery of the Lewisian hoardes. Very roughly, the programme goes: figure out what you want the Fs to do, Ramsify away terms for Fs and you get a way to fix where the Fs are amidst the things you believe in: they are whatever satisfy the open sentence that you’re left with. Where there are multiple, equally good satisfiers, then deploy the indefiniteness maneuver.
I’m not so worried on this front, for what I take to be pretty routine reasons. But there’s a second challenge King raises for the simple, naïve theory of structured propositions, which I think is trickier. More on this anon.
The first thing you encounter when people start talking about structured propositions is notation like [Dummett, being a philosopher]. This is supposed to stand for the proposition that Dummett is a philosopher, and highlights the fact that (on the Russellian view) Dummett and the property of being a philosopher are components of the proposition. The big question is supposed to be: what do the brackets and comma represent? What sort of compound object is the proposition? In what sense does it have Dummett and being a philosopher as components? (If you prefer a structured intension view, so be it: then you’ll have a similar beast with the individual concept of Dummett and the worlds-intension associated with “is a philosopher” as ‘constituents’. I’ll stick with the Russellian view for illustrative purposes.)
For purposes of modelling propositions, people often interpret the commas as brackets as the ordered n-tuples of standard set theory. The simplest, most naïve interpretation of what structured propositions are, is simply to identify them as n-tuples. What’s the structured proposition itself? It’s a certain kind of set. What sense are Dummett and the property of being a philosopher constituents of the structured proposition that Dummett is a philosopher? They’re elements of the transitive closure of the relevant set.
So all that is nice and familiar. So what’s the problem? In his ch 1. (and, in passing, in the SEP article here) King mentions two concerns. In this post, I’ll just set the scene by talking about the first. It's a version of a famous Benacerraf worry, which anyone with some familiarity with the philosophy of maths will have come across (King explicitly makes the comparison). The original Benacerraf puzzle is something like this: suppose that the only abstract things are set like, and whatever else they may be, the referents of arithmetical terms should be abstract. Then numerals will stand for some set or other. But there are all sorts of things that behave like the natural numbers within set theory: the constructions known as the (finite) Zermelo ordinals (null, {null}, {{null}}, {{{null}}}...) and the (finite) von Neumann ordinals (null, {null}, {null,{null}}…) are just two. So there’s no non-arbitrary theory of which sets the natural numbers are.
The phenomenon crops up all over the place. Think of ordered n-tuples themselves. Famously, within an ontology of unordered sets, you can define up things that behave like ordered pairs: either [a,b]
Likewise, says King, there looks to be no non-arbitrary choice of set-theoretic representation of structured propositions (not even if we spot ourselves ordered sets as primitive to avoid the familiar ordered-pair worries). Sure, we *could* associate the words “the proposition that Dummett is a philosopher” with the ordered pair [Dummett, being a philosopher]. But we could also associate it with the set [being a philosopher, Dummett] (and choices multiply when we get to more complex structured propositions).
One reaction to the Benacerrafian challenge is to take it to be a decisive objection to an ontological story about numbers, ordered pairs or whatever that allows only unordered sets as a basic mathematical ontology. My own feeling is (and this is not uncommon, I think) that this would be an overreaction. More strongly: no argument that I've seen from the Benacerraf phenomenon to this ontological conclusion seems to me to be terribly persuasive.
What we should admit, rather, is that if natural numbers or ordered pairs are sets, it’ll be indefinite which sets they are. So, for example, [a,b]={{a},{a,b}} will be neither definitely true nor definitely false (unless we simply stipulatively define the [,] notation one way or another rather than treating it as pre-theoretically understood). Indefiniteness is pervasive in natural language---everyone needs a story about how it works. And the idea is that whatever that story should be, it should be applied here. Maybe some theories of indefiniteness will make these sort of identifications problematic. But prominent theories like Supervaluationism and Epistemicism have neat and apparently smooth theories of what it we’re saying when we call that identity indefinite: for the supervaluationist, it (may) mean that “[a,b]” refers to {{a},{a,b}} on one but not all precisifications of our set-theoretic language. For the epistemicist, it means that (for certain specific principled reasons) we can’t know that the identity claim is false. The epistemicist will also maintains there’s a fact of the matter about which identity statement connecting ordered and unordered sets is true. And there’ll be some residual arbitrariness here (though we’ll probably have to semantically ascend to find it)---but if there is arbitriness, it’s the sort of thing we’re independently committed to to deal with the indefiniteness rife throughout our language. If you’re a supervaluationist, then you won’t admit there’s any arbitriness: (standardly) the identity statement is neither true nor false, so our theory won’t be committed to “making the choice”.
If that’s the right way to respond to the general Benacerraf challenge, it’s the obvious thing to say in response to the version of that puzzle that arises for the Benacerraf case. And this sort of generalization of the indefiniteness maneuver to philosophical analysis is pretty familiar, it’s part of the standard machinery of the Lewisian hoardes. Very roughly, the programme goes: figure out what you want the Fs to do, Ramsify away terms for Fs and you get a way to fix where the Fs are amidst the things you believe in: they are whatever satisfy the open sentence that you’re left with. Where there are multiple, equally good satisfiers, then deploy the indefiniteness maneuver.
I’m not so worried on this front, for what I take to be pretty routine reasons. But there’s a second challenge King raises for the simple, naïve theory of structured propositions, which I think is trickier. More on this anon.
Wednesday, December 12, 2007
Public service announcements (updated)
There's some interesting conferences being announced these days. A couple have caught my eye/been brought to my attention.
First is the Semantics and Philosophy in Europe CFP. This looks really like a really excellent event... one of those events where I think: If I'm not there, I'll be regretting not being there...
The second event is the 2008 Wittgenstein Symposium. It's remit seems far wider than the name might suggest... looks like a funky set of topics. I reproduce the CFP below...
[Update: a third is a one-day conference on the philosophy of mathematics in Manchester. Announcement at the bottom of the post.]
CALL FOR PAPERS:
31st International Wittgenstein Symposium 2008 on
Reduction and Elimination in Philosophy and the Sciences
Kirchberg am Wechsel, Austria, 10-16 August 2008
<http://www.alws.at/>
INVITED SPEAKERS:
William Bechtel, Ansgar Beckermann, Johan van Benthem, Alexander Bird, Elke
Brendel, Otavio Bueno, John P. Burgess, David Chalmers, Igor Douven, Hartry
Field, Jerry Fodor, Kenneth Gemes, Volker Halbach, Stephan Hartmann, Alison
Hills, Leon Horsten, Jaegwon Kim, James Ladyman, Oystein Linnebo, Bernard
Linsky, Thomas Mormann, Carlos Moulines, Thomas Mueller, Karl-Georg
Niebergall, Joelle Proust, Stathis Psillos, Sahotra Sarkar, Gerhard Schurz,
Patrick Suppes, Crispin Wright, Edward N. Zalta, Albert Anglberger, Elena
Castellani, Philip Ebert, Paul Egre, Ludwig Fahrbach, Simon Huttegger,
Christian Kanzian, Jeff Ketland, Marcus Rossberg, Holger Sturm, Charlotte
Werndl.
ORGANISERS:
Alexander Hieke (Salzburg) & Hannes Leitgeb (Bristol),
on behalf of the Austrian Ludwig Wittgenstein Society.
SECTIONS OF THE SYMPOSIUM:
Sections:
1. Wittgenstein
2. Logical Analysis
3. Theory Reduction
4. Nominalism
5. Naturalism &Physicalism
6. Supervenience
Workshops:
- Ontological Reduction & Dependence
- Neologicism
More detailed information on the contents of the sections and workshops can
be found in the "BACKGROUND" part further down.
DEADLINE FOR SUBMITTING PAPERS: 30th April 2008
Instructions for authors will soon be available at <http://www.alws.at/>.
All contributions will be peer-reviewed. All submitted papers accepted for
presentation at the symposium will appear in the Contributions of the ALWS
Series. Since 1993, successive volumes in this series have appeared each
August immediately prior to the symposium.
FINAL DATE FOR REGISTRATION: 20th July 2008
Further information on registration forms and information on travel and
accommodation will be posted at <http://www.alws.at/>.
SCHEDULE OF THE SYMPOSIUM:
The symposium will take place in Kirchberg am Wechsel (Austria) from 10-16
August 2008. Sunday, 10th of August 2008 is supposed to be the day on which
speakers and conference participants are going to arrive and when they
register in the conference office. In the evening, we plan on having an
informal get together. On the next day (11 August, 10:00am) the first
official session of presentations will start with Professor Jerry Fodor's
opening lecture of the symposium. The symposium will end officially in the
afternoon of 16 August 2008.
BACKGROUND:
Philosophers often have tried to either reduce "disagreeable" entities or
concepts to (more) acceptable entities or concepts, or to eliminate the
former altogether. Reduction and elimination, of course, very often have to
do with the question of "What is really there?", and thus these notions
belong to the most fundamental ones in philosophy. But the topic is not
merely restricted to metaphysics or ontology. Indeed, there are a variety
of attempts at reduction and elimination to be found in all areas (and
periods) of philosophy and science.
The symposium is intended to deal with the following topics (among others):
- Logical Analysis: The logical analysis of language has long been regarded
as the dominating paradigm for philosophy in the modern analytic tradition.
Although the importance of projects such as Frege's logicist construction
of mathematics, Russell's paraphrasis of definite descriptions, and
Carnap's logical reconstruction and explicatory definition of empirical
concepts is still acknowledged, many philosophers now doubt the viability
of the programme of logical analysis as it was originally conceived.
Notorious problems such as those affecting the definitions of knowledge or
truth have led to the revival of "non-analysing" approaches to
philosophical concepts and problems (see e.g. Williamson's account of
knowledge as a primitive notion and the deflationary criticism of Tarski's
definition of truth). What role will -- and should -- logical analysis play
in philosophy in the future?
- Theory Reduction: Paradigm cases of theory reduction, such as the
reduction of Kepler's laws of planetary motion to Newtonian mechanics or
the reduction of thermodynamics to the kinetic theory of gases, prompted
philosophers of science to study the notions of reduction and reducibility
in science. Nagel's analysis of reduction in terms of bridge laws is the
classical example of such an attempt. However, those early accounts of
theory reduction were soon found to be too naive and their underlying
treatment of scientific theories unrealistic. What are the state-of-the-art
proposals on how to understand the reduction of a scientific theory to
another? What is the purpose of such a reduction? In which cases should we
NOT attempt to reduce a theory to another one?
- Nominalism: Traditionally, nominalism is concerned with denying the
existence of universals. Modern versions of nominalism object to abstract
entities altogether; in particular they attack the assumption that the
success of scientific theories, especially their mathematical components,
commit us to the existence of abstract objects. As a consequence,
nominalists have to show how the alleged reference to abstract entities can
be eliminated or is merely apparent (Field's Science without Numbers is
prototypical in this respect). What types of "Constructive Nominalism" (a
la Goodman & Quine) are there? Are there any principal obstacles for
nominalistic programmes in general? What could nominalistic accounts of
quantum theory or of set theory look like?
- Naturalism & Physicalism: Naturalism and physicalism both want to
eliminate the part of language that does not refer to the "natural facts"
that science -- or indeed physics -- describes. Metaphysical Naturalism
often goes hand in hand with (or even entails) an epistemological
naturalism (Quine) as well as an ethical naturalism (mainly defined by its
critics), so that also these two main disciplines of philosophy should
restrict their investigations to the world of natural facts. Physicalist
theses, of course, play a particularly important role in the philosophy of
mind, since neuroscientific findings seem to support the view that,
ultimately, the realm of the mental is but a part of the physical world.
Which forms of naturalism and physicalism can be maintained within
metaphysics, philosophy of science, epistemology and ethics? What are the
consequences for philosophy when such views are accepted? Is philosophy a
scientific discipline? If naturalism or physicalism is right, can we still
see ourselves as autonomous beings with morality and a free will?
- Supervenience: Mental, moral, aesthetical, and even "epistemological"
properties have been said to supervene on properties of particular kind,
e.g., physical properties. Supervenience is claimed to be neither reduction
nor elimination but rather something different, but all these notions still
belong to the same family, and sometimes it is even assumed that reduction
is a borderline case of supervenience. What are the most abstract laws that
govern supervenience relations? Which contemporary applications of the
notion of supervenience are philosophically successful in the sense that
they have more explanatory power than "reductive theories" without leading
to unwanted semantical or ontological commitments? What are the logical
relations between the concepts of supervenience, reduction, elimination,
and ontological dependence?
The symposium will also include two workshops on:
- Ontological Reduction & Dependence: Reducing a class of entities to
another one has always been regarded attractive by those who subscribe to
an ideal of ontological parsimony. On the other hand, what it is that gets
reduced ontologically (objects or linguistic items?), what it means to be
reduced ontologically, and which methods of reduction there are, is
controversial (to say the least). Apart from reducing entities to further
entities, metaphysicians sometimes aim to show that entities depend
ontologically on other entities; e.g., a colour sensation instance would
not exist if the person having the sensation did not exist. In other
philosophical contexts, entities are rather said to depend ontologically on
other entities if the individuation of the former involves the latter; in
this sense, sets might be regarded to depend on their members, and
mathematical objects would depend on the mathematical structures they are
part of. Is there a general formal framework in which such notions of
ontological reduction and dependency can be studied more systematically? Is
ontological reduction really theory reduction in disguise? How shall we
understand ontological dependency of objects which exist necessarily? How
do reduction and dependence relate to Quine's notion of ontological
commitment?
- Neologicism: Classical Logicism aimed at deriving every true mathematical
statement from purely logical truths by reducing all mathematical concepts
to logical ones. As Frege's formal system proved to be inconsistent, and
modern set theory seemed to require strong principles of a genuinely
mathematical character, the programme of Logicism was long regarded as
dead. However, in the last twenty years neologicist and neo-Fregean
approaches in the philosophy of mathematics have experienced an amazing
revival (Wright, Boolos, Hale). Abstraction principles, such as Hume's
principle, have been suggested to support a logicist reconstruction of
mathematics in view of their quasi-analytical status. Do we have to
reconceive the notion of reducibility in order to understand in what sense
Neologicism is able to reduce mathematics to logic (as Linsky & Zalta have
suggested recently)? What are the abstraction principles that govern
mathematical theories apart from arithmetic (in particular: calculus and
set theory)? How can Neo-Fregeanism avoid the logical and philosophical
problems that affected Frege's original system -- cf. the problems of
impredicativity and Bad Company?
If you know philosophers or scientists, especially excellent graduate
students, who might be interested in the topic of Reduction and Elimination
in Philosophy and the Sciences, we would be very grateful if you could
point them to the symposium.
With best wishes,
Alexander Hieke and Hannes Leitgeb
********************************************************************************************
Announcing a one-day conference....
Metaphysics and Epistemology: Issues in the Philosophy of Mathematics
Saturday 15 March 2008
Chancellors Hotel and Conference Centre, University of Manchester
Speakers to include:
Joseph Melia (University of Leeds)
Alexander Paseau (University of Oxford)
Philip Ebert (University of Stirling)
For registration details, see
http://www.socialsciences.manchester.ac.uk/disciplines/philosophy/events/conference/index.html
This conference is organised with financial support from the Royal Institute of
Philosophy.
First is the Semantics and Philosophy in Europe CFP. This looks really like a really excellent event... one of those events where I think: If I'm not there, I'll be regretting not being there...
The second event is the 2008 Wittgenstein Symposium. It's remit seems far wider than the name might suggest... looks like a funky set of topics. I reproduce the CFP below...
[Update: a third is a one-day conference on the philosophy of mathematics in Manchester. Announcement at the bottom of the post.]
CALL FOR PAPERS:
31st International Wittgenstein Symposium 2008 on
Reduction and Elimination in Philosophy and the Sciences
Kirchberg am Wechsel, Austria, 10-16 August 2008
<http://www.alws.at/>
INVITED SPEAKERS:
William Bechtel, Ansgar Beckermann, Johan van Benthem, Alexander Bird, Elke
Brendel, Otavio Bueno, John P. Burgess, David Chalmers, Igor Douven, Hartry
Field, Jerry Fodor, Kenneth Gemes, Volker Halbach, Stephan Hartmann, Alison
Hills, Leon Horsten, Jaegwon Kim, James Ladyman, Oystein Linnebo, Bernard
Linsky, Thomas Mormann, Carlos Moulines, Thomas Mueller, Karl-Georg
Niebergall, Joelle Proust, Stathis Psillos, Sahotra Sarkar, Gerhard Schurz,
Patrick Suppes, Crispin Wright, Edward N. Zalta, Albert Anglberger, Elena
Castellani, Philip Ebert, Paul Egre, Ludwig Fahrbach, Simon Huttegger,
Christian Kanzian, Jeff Ketland, Marcus Rossberg, Holger Sturm, Charlotte
Werndl.
ORGANISERS:
Alexander Hieke (Salzburg) & Hannes Leitgeb (Bristol),
on behalf of the Austrian Ludwig Wittgenstein Society.
SECTIONS OF THE SYMPOSIUM:
Sections:
1. Wittgenstein
2. Logical Analysis
3. Theory Reduction
4. Nominalism
5. Naturalism &Physicalism
6. Supervenience
Workshops:
- Ontological Reduction & Dependence
- Neologicism
More detailed information on the contents of the sections and workshops can
be found in the "BACKGROUND" part further down.
DEADLINE FOR SUBMITTING PAPERS: 30th April 2008
Instructions for authors will soon be available at <http://www.alws.at/>.
All contributions will be peer-reviewed. All submitted papers accepted for
presentation at the symposium will appear in the Contributions of the ALWS
Series. Since 1993, successive volumes in this series have appeared each
August immediately prior to the symposium.
FINAL DATE FOR REGISTRATION: 20th July 2008
Further information on registration forms and information on travel and
accommodation will be posted at <http://www.alws.at/>.
SCHEDULE OF THE SYMPOSIUM:
The symposium will take place in Kirchberg am Wechsel (Austria) from 10-16
August 2008. Sunday, 10th of August 2008 is supposed to be the day on which
speakers and conference participants are going to arrive and when they
register in the conference office. In the evening, we plan on having an
informal get together. On the next day (11 August, 10:00am) the first
official session of presentations will start with Professor Jerry Fodor's
opening lecture of the symposium. The symposium will end officially in the
afternoon of 16 August 2008.
BACKGROUND:
Philosophers often have tried to either reduce "disagreeable" entities or
concepts to (more) acceptable entities or concepts, or to eliminate the
former altogether. Reduction and elimination, of course, very often have to
do with the question of "What is really there?", and thus these notions
belong to the most fundamental ones in philosophy. But the topic is not
merely restricted to metaphysics or ontology. Indeed, there are a variety
of attempts at reduction and elimination to be found in all areas (and
periods) of philosophy and science.
The symposium is intended to deal with the following topics (among others):
- Logical Analysis: The logical analysis of language has long been regarded
as the dominating paradigm for philosophy in the modern analytic tradition.
Although the importance of projects such as Frege's logicist construction
of mathematics, Russell's paraphrasis of definite descriptions, and
Carnap's logical reconstruction and explicatory definition of empirical
concepts is still acknowledged, many philosophers now doubt the viability
of the programme of logical analysis as it was originally conceived.
Notorious problems such as those affecting the definitions of knowledge or
truth have led to the revival of "non-analysing" approaches to
philosophical concepts and problems (see e.g. Williamson's account of
knowledge as a primitive notion and the deflationary criticism of Tarski's
definition of truth). What role will -- and should -- logical analysis play
in philosophy in the future?
- Theory Reduction: Paradigm cases of theory reduction, such as the
reduction of Kepler's laws of planetary motion to Newtonian mechanics or
the reduction of thermodynamics to the kinetic theory of gases, prompted
philosophers of science to study the notions of reduction and reducibility
in science. Nagel's analysis of reduction in terms of bridge laws is the
classical example of such an attempt. However, those early accounts of
theory reduction were soon found to be too naive and their underlying
treatment of scientific theories unrealistic. What are the state-of-the-art
proposals on how to understand the reduction of a scientific theory to
another? What is the purpose of such a reduction? In which cases should we
NOT attempt to reduce a theory to another one?
- Nominalism: Traditionally, nominalism is concerned with denying the
existence of universals. Modern versions of nominalism object to abstract
entities altogether; in particular they attack the assumption that the
success of scientific theories, especially their mathematical components,
commit us to the existence of abstract objects. As a consequence,
nominalists have to show how the alleged reference to abstract entities can
be eliminated or is merely apparent (Field's Science without Numbers is
prototypical in this respect). What types of "Constructive Nominalism" (a
la Goodman & Quine) are there? Are there any principal obstacles for
nominalistic programmes in general? What could nominalistic accounts of
quantum theory or of set theory look like?
- Naturalism & Physicalism: Naturalism and physicalism both want to
eliminate the part of language that does not refer to the "natural facts"
that science -- or indeed physics -- describes. Metaphysical Naturalism
often goes hand in hand with (or even entails) an epistemological
naturalism (Quine) as well as an ethical naturalism (mainly defined by its
critics), so that also these two main disciplines of philosophy should
restrict their investigations to the world of natural facts. Physicalist
theses, of course, play a particularly important role in the philosophy of
mind, since neuroscientific findings seem to support the view that,
ultimately, the realm of the mental is but a part of the physical world.
Which forms of naturalism and physicalism can be maintained within
metaphysics, philosophy of science, epistemology and ethics? What are the
consequences for philosophy when such views are accepted? Is philosophy a
scientific discipline? If naturalism or physicalism is right, can we still
see ourselves as autonomous beings with morality and a free will?
- Supervenience: Mental, moral, aesthetical, and even "epistemological"
properties have been said to supervene on properties of particular kind,
e.g., physical properties. Supervenience is claimed to be neither reduction
nor elimination but rather something different, but all these notions still
belong to the same family, and sometimes it is even assumed that reduction
is a borderline case of supervenience. What are the most abstract laws that
govern supervenience relations? Which contemporary applications of the
notion of supervenience are philosophically successful in the sense that
they have more explanatory power than "reductive theories" without leading
to unwanted semantical or ontological commitments? What are the logical
relations between the concepts of supervenience, reduction, elimination,
and ontological dependence?
The symposium will also include two workshops on:
- Ontological Reduction & Dependence: Reducing a class of entities to
another one has always been regarded attractive by those who subscribe to
an ideal of ontological parsimony. On the other hand, what it is that gets
reduced ontologically (objects or linguistic items?), what it means to be
reduced ontologically, and which methods of reduction there are, is
controversial (to say the least). Apart from reducing entities to further
entities, metaphysicians sometimes aim to show that entities depend
ontologically on other entities; e.g., a colour sensation instance would
not exist if the person having the sensation did not exist. In other
philosophical contexts, entities are rather said to depend ontologically on
other entities if the individuation of the former involves the latter; in
this sense, sets might be regarded to depend on their members, and
mathematical objects would depend on the mathematical structures they are
part of. Is there a general formal framework in which such notions of
ontological reduction and dependency can be studied more systematically? Is
ontological reduction really theory reduction in disguise? How shall we
understand ontological dependency of objects which exist necessarily? How
do reduction and dependence relate to Quine's notion of ontological
commitment?
- Neologicism: Classical Logicism aimed at deriving every true mathematical
statement from purely logical truths by reducing all mathematical concepts
to logical ones. As Frege's formal system proved to be inconsistent, and
modern set theory seemed to require strong principles of a genuinely
mathematical character, the programme of Logicism was long regarded as
dead. However, in the last twenty years neologicist and neo-Fregean
approaches in the philosophy of mathematics have experienced an amazing
revival (Wright, Boolos, Hale). Abstraction principles, such as Hume's
principle, have been suggested to support a logicist reconstruction of
mathematics in view of their quasi-analytical status. Do we have to
reconceive the notion of reducibility in order to understand in what sense
Neologicism is able to reduce mathematics to logic (as Linsky & Zalta have
suggested recently)? What are the abstraction principles that govern
mathematical theories apart from arithmetic (in particular: calculus and
set theory)? How can Neo-Fregeanism avoid the logical and philosophical
problems that affected Frege's original system -- cf. the problems of
impredicativity and Bad Company?
If you know philosophers or scientists, especially excellent graduate
students, who might be interested in the topic of Reduction and Elimination
in Philosophy and the Sciences, we would be very grateful if you could
point them to the symposium.
With best wishes,
Alexander Hieke and Hannes Leitgeb
********************************************************************************************
Announcing a one-day conference....
Metaphysics and Epistemology: Issues in the Philosophy of Mathematics
Saturday 15 March 2008
Chancellors Hotel and Conference Centre, University of Manchester
Speakers to include:
Joseph Melia (University of Leeds)
Alexander Paseau (University of Oxford)
Philip Ebert (University of Stirling)
For registration details, see
http://www.socialsciences.manchester.ac.uk/disciplines/philosophy/events/conference/index.html
This conference is organised with financial support from the Royal Institute of
Philosophy.
Tuesday, December 04, 2007
Two problems of the many.
Here's a paradigmatic problem of the many (Geach and Unger are the usual sources cited, but I'm not claiming this to be exactly the version they use.) Let's take a moulting cat. There are many hairs that are neither clearly attached, nor clearly unattached to the main body of the cat. Let's enumerate them 1---1000. Then we might consider the material objects which are the masses of cat-arranged matter that include half of the thousand hairs, and exclude to the other half. There are many ways to choose the half that's included. So by this recipe we get many many distinct masses of cat-arranged matter, differing only over hairs. The various pieces of cat-arranged matter change their properties over time in very much the way that cats do: they are now in a sitting-shape, now in a standing-shape, now in a lapping-milk shape, now in an emitting-meows configuration. They each seem to have everything intrinsically required for being a cat.
If you're inclined to think (and I am) that a cat is a material object identical to some piece of cat-arranged matter, then the problem of the many arises: which of the various distinct pieces of cat-arranged matters is the cat? Various answers have been suggested. Some of the most obvious (though not necessarily the most sensible) are: (i) nihilism: none of the cat-candidates are cats. (ii) brutalism: exactly one is a cat, and there is a brute fact of the matter which it is; (iii) vague cat: exactly one is a cat, and it's a vague matter which it is; (iii) manyism: lots of the cat-candidates are cats.
(By the way, (ii) and (iii) may not be incompatible, if you're an epistemicist about vagueness. And those who are fans of many-valued logics for vagueness should have a think about whether they can really support (iii). Consider the best candidates to be a cat, c1....c1000. Suppose these are each cats to an equal degree. Then "one of c1...c1000 is a cat" will standardly have a degree of truth equal to the disjunction=the maximum of the disjuncts=the degree of truth of "c1 is a cat". And the degree of truth of the conjunction: "all of c1...c1000 is a cat" will standardly have a degree of truth equal to the conjunction=the minimum of the conjuncts=the degree of truth of "c1 is a cat". So to the extent that the (determinately distinct) best candidates aren't all cats, to exactly that extent there's no cat among them (and since we chose the best candidates, we won't get a higher degree of truth for "the cat is present" by including extra disjuncts. Conclusion: if you're tempted by response (iii) to the problem of the many, you've got strong reason not to go for many-valued logic. [Edit (see comments): this needs qualification. I think you've reason not to go for many-valued logics that endorse the (fairly standard, but not undeniable) max/min treatment of disjunction/conjunction; and in which the many values are linearly arranged].)
What I'd really like to emphasize is the above leaves open the following question: Is there a super-cat-candidate, i.e. a piece of cat-arranged matter of which every other cat-candidate is a proper part? Take the Tibbles case above, and suppose that the candidates only differ over hairs. Then a potential super-cat-candidate would be the piece of matter that's maximally generous: that includes all the 1000 not-clearly-unattached hairs. If this particular fusion isn't genuinely a cat-candidate, then it's open that if you arrange the cat-candidates by which is a part of which, you'll end up with multiple maximal cat-candidates none of which is a part of the other. Perhaps they each contain 999 hairs, but differ amongst themselves which hair they don't include.
If there is a super-cat-candidate, let's say the problem of the many is of type-1, and if there's no super-cat-candidate, let's say that the problem of the many is of type-2.
My guess is that our description of cases like Tibbles leaves is simply underspecified as to whether it's of type-1 or type-2. But I certainly don't see any principled reason to think that the actual cases of the POM we find around us are always of type-1. There's certainly no a priori guarantee that the sort of criterion that rules in some things as parts of a cat won't also dismiss other things as non-parts. So for example, perhaps we can rank candidates for degrees of integration: some unintegrated parts are ok, but there's some cut-off where an object is just too unintegrated to count as a candidate. One cat-candidate includes some borderline-attached skin cells, and is to that extent unintegrated. Another cat-candidate includes some borderline-attached teeth, and is to that extent unintegrated. But plausibly the fusion that includes both skin cells and teeth is less integrated: enough to disqualify it from being a cat-candidate. It's hard to know how to argue the case further without going deeply into feline biology, but I hope you get the sense of why type-2 POM need to be dealt with.
Now, one response to the standard POM is to appeal to the "maximality" allegedly built into various predicates (like "rock", "cat", "conscious" etc): things that are duplicates of rocks, but which are surrounded by extra rocky stuff, become merely parts of rocks (and so forth). There are presumably intrinisic duplicates of rocks embedded as tiny parts at the centre of large boulders: but there's no intuitive pressure to count them as rocks. Likewise a cat might survive after it's limbs are destroyed by a vengeful deity, but it's unintuitive to think of the duplicate head-and-torso part of Tibbles as itself a cat-candidate. So there's some reasons independently of paradigmatic problem of the many scenarios to think of "cat" and "rock" etc as maximal. (For more discussion of maximality, see Ted Sider's various papers on the topic).
If we've got a type-1 problem of the many, then one might think that the maximality of "cat" or "rock" or whatever gives a principled answer to our original question: the super-cat-candidate (/super-rock-candidate) is the one uniquely qualified to be the cat (/rock). For we've then got an explanation for why all the others, though intrinsically qualified just like cats, aren't cats: being a cat is a maximal property, and all the rival cat-candidates are parts of the one true cat in the vicinity.
But the type-2 problem of the many really isn't addressed by maximality as such. There's no unique super-cat-candidate in this setup, rather a range of co-maximal ones. So maximality won't save our bacon here.
The difference between the two cases is important when we consider other things. For example, in the light of the (fairly widely accepted) maximality of "house" and "cat" and "rock" and the like, few would say that any duplicate of a house must be a house (even setting aside extrinsicality due to social setting). But there's an obvious fall back position, which is floating around the literature: that any duplicate of a house must be a (proper or improper) part of a house (holding fixed social setting etc). That is, any house possesses the property of being part of a house intrinsically (so long as we hold fixed social setting etc). And the same goes for cat: at least holding fixed biological origin, it's plausible that any cat is intrinsically at least part of a cat, and any rock is intrinsically at least part of a rock.
These claims aren't threatened by maximality. But appealing to them in a type-2 problem of the many gets us an argument directly for response (iv): manyism. For plausibly if you took a duplicate of one of the co-maximal cat candidates, T, while eliminating from the scene those bits of matter that are not part of T but are part of one of the other co-maximal cat candidates, then you get something T* that's (determinately) a cat. And so, any duplicate of T* must be at least part of a cat. And since T is a duplicate of T*, T must be at least part of a cat. But T isn't proper part of anything that's even a cat-candidate. So T must itself be a cat.
So the type-2 POM is harder to resolve than the type-1 kind. Maybe some extra weakening of the properties a cat-candidate has intrinsicality are called for. Or maybe (very surprisingly) type-2 POMs never arise. But either way, more work is needed.
If you're inclined to think (and I am) that a cat is a material object identical to some piece of cat-arranged matter, then the problem of the many arises: which of the various distinct pieces of cat-arranged matters is the cat? Various answers have been suggested. Some of the most obvious (though not necessarily the most sensible) are: (i) nihilism: none of the cat-candidates are cats. (ii) brutalism: exactly one is a cat, and there is a brute fact of the matter which it is; (iii) vague cat: exactly one is a cat, and it's a vague matter which it is; (iii) manyism: lots of the cat-candidates are cats.
(By the way, (ii) and (iii) may not be incompatible, if you're an epistemicist about vagueness. And those who are fans of many-valued logics for vagueness should have a think about whether they can really support (iii). Consider the best candidates to be a cat, c1....c1000. Suppose these are each cats to an equal degree. Then "one of c1...c1000 is a cat" will standardly have a degree of truth equal to the disjunction=the maximum of the disjuncts=the degree of truth of "c1 is a cat". And the degree of truth of the conjunction: "all of c1...c1000 is a cat" will standardly have a degree of truth equal to the conjunction=the minimum of the conjuncts=the degree of truth of "c1 is a cat". So to the extent that the (determinately distinct) best candidates aren't all cats, to exactly that extent there's no cat among them (and since we chose the best candidates, we won't get a higher degree of truth for "the cat is present" by including extra disjuncts. Conclusion: if you're tempted by response (iii) to the problem of the many, you've got strong reason not to go for many-valued logic. [Edit (see comments): this needs qualification. I think you've reason not to go for many-valued logics that endorse the (fairly standard, but not undeniable) max/min treatment of disjunction/conjunction; and in which the many values are linearly arranged].)
What I'd really like to emphasize is the above leaves open the following question: Is there a super-cat-candidate, i.e. a piece of cat-arranged matter of which every other cat-candidate is a proper part? Take the Tibbles case above, and suppose that the candidates only differ over hairs. Then a potential super-cat-candidate would be the piece of matter that's maximally generous: that includes all the 1000 not-clearly-unattached hairs. If this particular fusion isn't genuinely a cat-candidate, then it's open that if you arrange the cat-candidates by which is a part of which, you'll end up with multiple maximal cat-candidates none of which is a part of the other. Perhaps they each contain 999 hairs, but differ amongst themselves which hair they don't include.
If there is a super-cat-candidate, let's say the problem of the many is of type-1, and if there's no super-cat-candidate, let's say that the problem of the many is of type-2.
My guess is that our description of cases like Tibbles leaves is simply underspecified as to whether it's of type-1 or type-2. But I certainly don't see any principled reason to think that the actual cases of the POM we find around us are always of type-1. There's certainly no a priori guarantee that the sort of criterion that rules in some things as parts of a cat won't also dismiss other things as non-parts. So for example, perhaps we can rank candidates for degrees of integration: some unintegrated parts are ok, but there's some cut-off where an object is just too unintegrated to count as a candidate. One cat-candidate includes some borderline-attached skin cells, and is to that extent unintegrated. Another cat-candidate includes some borderline-attached teeth, and is to that extent unintegrated. But plausibly the fusion that includes both skin cells and teeth is less integrated: enough to disqualify it from being a cat-candidate. It's hard to know how to argue the case further without going deeply into feline biology, but I hope you get the sense of why type-2 POM need to be dealt with.
Now, one response to the standard POM is to appeal to the "maximality" allegedly built into various predicates (like "rock", "cat", "conscious" etc): things that are duplicates of rocks, but which are surrounded by extra rocky stuff, become merely parts of rocks (and so forth). There are presumably intrinisic duplicates of rocks embedded as tiny parts at the centre of large boulders: but there's no intuitive pressure to count them as rocks. Likewise a cat might survive after it's limbs are destroyed by a vengeful deity, but it's unintuitive to think of the duplicate head-and-torso part of Tibbles as itself a cat-candidate. So there's some reasons independently of paradigmatic problem of the many scenarios to think of "cat" and "rock" etc as maximal. (For more discussion of maximality, see Ted Sider's various papers on the topic).
If we've got a type-1 problem of the many, then one might think that the maximality of "cat" or "rock" or whatever gives a principled answer to our original question: the super-cat-candidate (/super-rock-candidate) is the one uniquely qualified to be the cat (/rock). For we've then got an explanation for why all the others, though intrinsically qualified just like cats, aren't cats: being a cat is a maximal property, and all the rival cat-candidates are parts of the one true cat in the vicinity.
But the type-2 problem of the many really isn't addressed by maximality as such. There's no unique super-cat-candidate in this setup, rather a range of co-maximal ones. So maximality won't save our bacon here.
The difference between the two cases is important when we consider other things. For example, in the light of the (fairly widely accepted) maximality of "house" and "cat" and "rock" and the like, few would say that any duplicate of a house must be a house (even setting aside extrinsicality due to social setting). But there's an obvious fall back position, which is floating around the literature: that any duplicate of a house must be a (proper or improper) part of a house (holding fixed social setting etc). That is, any house possesses the property of being part of a house intrinsically (so long as we hold fixed social setting etc). And the same goes for cat: at least holding fixed biological origin, it's plausible that any cat is intrinsically at least part of a cat, and any rock is intrinsically at least part of a rock.
These claims aren't threatened by maximality. But appealing to them in a type-2 problem of the many gets us an argument directly for response (iv): manyism. For plausibly if you took a duplicate of one of the co-maximal cat candidates, T, while eliminating from the scene those bits of matter that are not part of T but are part of one of the other co-maximal cat candidates, then you get something T* that's (determinately) a cat. And so, any duplicate of T* must be at least part of a cat. And since T is a duplicate of T*, T must be at least part of a cat. But T isn't proper part of anything that's even a cat-candidate. So T must itself be a cat.
So the type-2 POM is harder to resolve than the type-1 kind. Maybe some extra weakening of the properties a cat-candidate has intrinsicality are called for. Or maybe (very surprisingly) type-2 POMs never arise. But either way, more work is needed.
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