### Monday, December 12, 2005

## Boundaryless Concepts and Set Theory

It seems appropriate to open this blog with a post on boundarylessness and language. In 'Concepts Without Boundaries', Mark Sainsbury argues against the common (if not ubiquitous) assumption that vague concepts classify by making set-theoretic divisions. The epistemicist, for example, holds that there is a particular number of grains of sand that marks the boundary between the heaps and the non-heaps. Rival pictures of the semantics of vague concepts try to avoid this committment by positing more divisions; so supervaluationists hold that vague concepts divide objects into an extension, an anti-extension, and those that have borderline status, while supporters of many-valued logics think that they parition the domain into even more sets.

Sainsbury thinks that this whole approach is radically misguided. There are borderline cases of 'red', i.e. things that are neither red nor not red. But sets are sharp, so there is no set of red things. Positing a set of objects that are borderline only helps if we ignore the phenomenon of higher-order vagueness; once we appreciate that there are borderline cases of 'borderline red', we can see that the problem just recurs. Positing yet more boundaries, and so partioning the domain even more finely, is just to repeat the original mistake a bunch more times. Vague concepts are boundaryless.

It's worth noting that the paper was delivered and published several years before people really started taking epistemicism seriously, and the argument that there is no set of red things looks undermined to the extent that characterising vagueness as an epistemic phenomenon is a viable option. More generally, Roy Cook has argued at length that Sainsbury's attack on many-valued approaches rests on bad philosophy of logic ('Vagueness and Mathematical Precision', Mind 2002). To formalise a region of natural langauge is to build a mathematical model, and if our model introduces sharpness that isn't present in the concepts we were trying to capture,we can see that as just useful idealisation - idealisation of the sort that has facilitated the success of formal semantics. We don't have to, and shouldn't, see our mathematical treatment of natural language as mis-describing its workings in the way Sainsbury alleges.

I don't want to evaluate either of these responses to Sainsbury, but rather to draw attention to a point in his paper that I think hasn't been sufficiently discussed in this debate. He observes that:

'Boundaryless concepts tend to come in systems of contraries: opposed pairs like child/adult, hot/cold, weak/strong, true/false, and the more complex systems exemplified by our colour terms.' (p258 in the Keefe and Smith reader).

Further down the same page he writes:

'Not just any clear case of the non-applicability of a concept will serve to help a learner see what the concept excludes. Television sets, mountains and French horns are all absolutely definite cases of non-children; but only the contrast with adult will help the learner grasp what child excludes. So it is no accident that boundaryless concepts come in groups of contraries.'

This observation goes awol on the picture Sainsbury is attacking. If we think of 'red' as classifying by partisioning the domain into an extension and an anti-extension, we don't seem to be able to explain the relationship between contraries. The anti-extension of a vague concept would contain all the junk that Sainsbury observes plays no role in grasping what the concept excludes.

I think Sainsbury has pointed out an important feature of vague concepts, and one that may be difficult to capture on the set-theoretic approach. Of course, Cook's response can still be given; just as our formalisation can introduce useful artefacts, it can abstract away from certain features of the discourse we want to study. But it would be interesting to see how far Sainsbury's observation can be pushed; if the set theoretic framework really does just lack the resources to model the feature he draws attention to, that provides some support for Sainsbury's conclusion that we've a bad picture of how vague concepts classify.

Sainsbury thinks that this whole approach is radically misguided. There are borderline cases of 'red', i.e. things that are neither red nor not red. But sets are sharp, so there is no set of red things. Positing a set of objects that are borderline only helps if we ignore the phenomenon of higher-order vagueness; once we appreciate that there are borderline cases of 'borderline red', we can see that the problem just recurs. Positing yet more boundaries, and so partioning the domain even more finely, is just to repeat the original mistake a bunch more times. Vague concepts are boundaryless.

It's worth noting that the paper was delivered and published several years before people really started taking epistemicism seriously, and the argument that there is no set of red things looks undermined to the extent that characterising vagueness as an epistemic phenomenon is a viable option. More generally, Roy Cook has argued at length that Sainsbury's attack on many-valued approaches rests on bad philosophy of logic ('Vagueness and Mathematical Precision', Mind 2002). To formalise a region of natural langauge is to build a mathematical model, and if our model introduces sharpness that isn't present in the concepts we were trying to capture,we can see that as just useful idealisation - idealisation of the sort that has facilitated the success of formal semantics. We don't have to, and shouldn't, see our mathematical treatment of natural language as mis-describing its workings in the way Sainsbury alleges.

I don't want to evaluate either of these responses to Sainsbury, but rather to draw attention to a point in his paper that I think hasn't been sufficiently discussed in this debate. He observes that:

'Boundaryless concepts tend to come in systems of contraries: opposed pairs like child/adult, hot/cold, weak/strong, true/false, and the more complex systems exemplified by our colour terms.' (p258 in the Keefe and Smith reader).

Further down the same page he writes:

'Not just any clear case of the non-applicability of a concept will serve to help a learner see what the concept excludes. Television sets, mountains and French horns are all absolutely definite cases of non-children; but only the contrast with adult will help the learner grasp what child excludes. So it is no accident that boundaryless concepts come in groups of contraries.'

This observation goes awol on the picture Sainsbury is attacking. If we think of 'red' as classifying by partisioning the domain into an extension and an anti-extension, we don't seem to be able to explain the relationship between contraries. The anti-extension of a vague concept would contain all the junk that Sainsbury observes plays no role in grasping what the concept excludes.

I think Sainsbury has pointed out an important feature of vague concepts, and one that may be difficult to capture on the set-theoretic approach. Of course, Cook's response can still be given; just as our formalisation can introduce useful artefacts, it can abstract away from certain features of the discourse we want to study. But it would be interesting to see how far Sainsbury's observation can be pushed; if the set theoretic framework really does just lack the resources to model the feature he draws attention to, that provides some support for Sainsbury's conclusion that we've a bad picture of how vague concepts classify.

Labels: Mark Sainsbury, Philosophy of Logic, Vagueness

Comments:

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Hello--

Let me add my welcome to Matt's. One quick question: maybe I've misunderstood you, but it seems to me that the point you're making is not particular to vague words/ concepts.

Any non-vague predicate will have an extension and anti-extension, but (unlike vague predicates) the union of those two sets will exhaust the domain. And the anti-extension of those non-vague predicates will also fail to separate the 'proper contraries' from the other members of the anti-extension. For example: "... is a natural number" is (I think)a non-vague predicate. This predicate has you and me and Saturn in its anti-extension, but we only really grasp(as you say) what is meant by 'natural number' when learn that fractions etc. are not natural numbers. But the anti-extension of 'is a natural number' contains both Saturn and 3/4... and there is no formal correlate in the model for that distinction.

Anyway, maybe I've missed something. I'm looking forward to future posts!

-Greg

Let me add my welcome to Matt's. One quick question: maybe I've misunderstood you, but it seems to me that the point you're making is not particular to vague words/ concepts.

Any non-vague predicate will have an extension and anti-extension, but (unlike vague predicates) the union of those two sets will exhaust the domain. And the anti-extension of those non-vague predicates will also fail to separate the 'proper contraries' from the other members of the anti-extension. For example: "... is a natural number" is (I think)a non-vague predicate. This predicate has you and me and Saturn in its anti-extension, but we only really grasp(as you say) what is meant by 'natural number' when learn that fractions etc. are not natural numbers. But the anti-extension of 'is a natural number' contains both Saturn and 3/4... and there is no formal correlate in the model for that distinction.

Anyway, maybe I've missed something. I'm looking forward to future posts!

-Greg

Thanks Greg, interesting comment. I'm not actually at all signed up to Sainsbury's conclusion, I'm just wondering how far the little observation I highlight can be pushed. So if it turned up to generalize in the way you suggest, that would seem to be a clear signal that we've demanded too much somewhere - it seems pretty obvious the right conclusion to draw isn't that the natural numbers don't form a set.

That said, I don't think it's crazy to hold, as Sainsbury seems to be suggesting, that there's something distinctive about the way grasp of his boundaryless concepts tends to come in these pairs that requires some explanation. I personally find it difficult to see that natural number/rational number is a pair of this sort (not to mention irrational, complex, etc.). I'm not convinced that mastery of 'is a natural number' requires appreciation of the contrast with fractions, reals, etc.

Firstly, it just looks way too demanding. Sainsbury suggests that we acquire our colour concepts as a system, and perhaps that itself isn't plausible. But it seems clear that grasp of the various kinds of numbers, unlike the colour case, require varying degrees of mathematical sophistication. I don't think that we really want to withhold grasp of 'is a natural number' from a subject who otherwise seems to competently classify with the concept just because they don't have a handle on the contrast with 'is a real number'.

There's more I should say in response, but real life is interfering just now.

That said, I don't think it's crazy to hold, as Sainsbury seems to be suggesting, that there's something distinctive about the way grasp of his boundaryless concepts tends to come in these pairs that requires some explanation. I personally find it difficult to see that natural number/rational number is a pair of this sort (not to mention irrational, complex, etc.). I'm not convinced that mastery of 'is a natural number' requires appreciation of the contrast with fractions, reals, etc.

Firstly, it just looks way too demanding. Sainsbury suggests that we acquire our colour concepts as a system, and perhaps that itself isn't plausible. But it seems clear that grasp of the various kinds of numbers, unlike the colour case, require varying degrees of mathematical sophistication. I don't think that we really want to withhold grasp of 'is a natural number' from a subject who otherwise seems to competently classify with the concept just because they don't have a handle on the contrast with 'is a real number'.

There's more I should say in response, but real life is interfering just now.

I'll try finish the reply I started earlier. Firstly, as I managed to write earlier, I find it somewhat implausible that we only really master the concept 'natural number' once we've understood the contrast with certain other kinds of objects the concept excludes, namely the other kinds of numbers. This just seems too demanding, since understanding the relevant contrasts between the natural numbers and the reals seems to require way more mathematical sophistication than intuitively it takes to possess the concept 'natural number'. Whatever one thinks of Sainsbury's suggestion that we tend to acquire our colour concepts as a system, there's no analogous problem there I think.

Secondly, I have some ill-formed worries due to the possibility that our different number concepts are not exclusive. Now, there are certain philosophies of mathematics on which that's not a worry; certain versions of structuralism, for example, will distinguish the natural number one and the real number one since they feature in different structures. But if we set such views aside for the moment, it seems that there's an awkwardness here, not present in Sainsbury's cases. (One might think that something akin to this awkwardness arise with, say, 'blue' and 'purple'. I don't think there's a real problem here for the point I've just made, though saying why would take longer than I can stomach right now.)

That said, perhaps ordinal/cardinal is a pair of concepts which one grasps partly in virtue of the contrast with the other in the way Sainsbury suggests of his examples, and which plausibly don't display any relevant vagueness. I'll give that some thought. If so, it would seem Greg is right to suggest that Sainsbury's point seriously overgenerates.

(Ps. The assumption that the union of the extension and anti-extension of a vague predicate won't exhaust the domain is contentious so long as treating borderline-hood as some kind of epistemic status is an option. One doesn't have to be an epistemicist to hold that; witness Crispin Wright in 'On Being in a Quandary'.)

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Secondly, I have some ill-formed worries due to the possibility that our different number concepts are not exclusive. Now, there are certain philosophies of mathematics on which that's not a worry; certain versions of structuralism, for example, will distinguish the natural number one and the real number one since they feature in different structures. But if we set such views aside for the moment, it seems that there's an awkwardness here, not present in Sainsbury's cases. (One might think that something akin to this awkwardness arise with, say, 'blue' and 'purple'. I don't think there's a real problem here for the point I've just made, though saying why would take longer than I can stomach right now.)

That said, perhaps ordinal/cardinal is a pair of concepts which one grasps partly in virtue of the contrast with the other in the way Sainsbury suggests of his examples, and which plausibly don't display any relevant vagueness. I'll give that some thought. If so, it would seem Greg is right to suggest that Sainsbury's point seriously overgenerates.

(Ps. The assumption that the union of the extension and anti-extension of a vague predicate won't exhaust the domain is contentious so long as treating borderline-hood as some kind of epistemic status is an option. One doesn't have to be an epistemicist to hold that; witness Crispin Wright in 'On Being in a Quandary'.)

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