kees van deemter. for institut nicod, jan 2009 vagueness as original sin from measurement to...

Kees van Deemter. For Institut Nicod, Jan 2009

Vagueness as Original Sinfrom measurement to semantic theory

Kees van Deemter

University of Aberdeen

Scotland, United Kingdom

Paris, Institut Nicod, 14 Jan 2009


Introduction

• Not Exactly : in Praise of VaguenessOxford University Press, Winter 2009-2010

• Vagueness for wide audience, from different perspectives:– Daily life, politics, and science– Philosophical logic, linguistic theory– Applications in AI and NLG; Game Theory

Red: Paris, Institut Nicod, Jan 2009 Blue: ENLG-2009, Athens, March 2009


Plan of the talk

1. Original sin

2. The trouble with variety

3. Expulsion from Boole’s paradise


1. Vagueness as original sin


1. Vagueness as original sin

1. History of the metric system

2. The fiction of species

3. [If time allowed it: the notion of an object]


1.1 History of the metric system

• (1790) French government aims to define measurements “pour tous les temps, pour tous les peuples”

• 1 metre = 1/40,000,000 of the length of the meridian running through the Panthéon– originally a church, now a “temple of reason”

• Definition made practical by producing a metal bar of approximately this length


History of the metric system

• To minimise variation, the bar was– made of platinum for low oxidation– kept at a fixed temperature (00 Celcius)– supported at fixed distances to minimise

contact with other substances– given a sturdy profile to minimise bending


• Parisian method was extremely successful: used for about 100 years

• This method had an error of approximately 0.00005 mm (0.05 -m)

• This is no longer precise enough (astronomy, medical tools, etc.)– nuts and bolts need to fit each other

• Main cause of lack of precision: where does the bar end?

• Standard solution: make bar longer, mark start and end point by small incisions


just like an ordinary tool ...


Later solutions

• Hunt for a better “bedrock” than Earth (or a bar!)• After 1900: Wavelength of light

– Cadmium (around 1927) – Krypton 86 (around 1960)

• (1983) Distance travelled by light in 1/299,792,458 of a second

• These measures are better– higher precision– same across the Earth (?)– available everywhere, so

no need to make copies of (...) of copies


But ...

• Suppose you find a treasure in your attic: a platinum bar of 1 metre length

• Every morning you check its size, using the best measurement method available

• At night a thief shaves off 1/10000 M

• You won’t notice the difference

• (Sorites argument is easily constructed)


What to conclude?

• Situation pre 1900: “metre” is vague.

• This is true regardless of which bedrock you choose:– “metre = 1/40,000,000 of Earth’s

circumference” vague because the Earth varies in size

– “metre = size of bar” vague because bar varies in size


What to conclude?

• Situation post 1900: “metre” is vague. Two perspectives:– Practical: to verify whether a distance

exceeds “x times the wave length of Krypton” involves low-level measurements like those discussed. vague

– Theoretical: do we have any guarantee that this wave length is always equal? (How do we measure this?) vague


Whatever your view on the theoretical perspective ...

• The predicate x.size(x) 1 metrecan only be used vaguely

• There exist cases where hearer cannot know whether speaker intended it to hold

• “I offer you 1 billion if you give me a platinum bar of at least 1 metre” has borderline cases where a judge could not say confidently whether 1 billion should be awarded.

• All of this holds inside the physics lab, but even more so in the garage, the kitchen and the kindergarten


Further consequences

• If “metre” is not crisp then what is crisp? – many metrics depend on distance: volume,

temperature, pressure, ...



• Some concepts may be entirely crisp (`grandmother`, `subset`, ...)– pace property theory

• But many of the things we usually call crisp are (just a little bit) vague



Many of the things we usually call crisp are

(a little bit) vague: 1

0

0.5

“Greater or equal to 1 metre”

x-axis: bars of increasing sizes (not metres !)

y-axis: truth values



• Some concepts are clearly vague (e.g. `tall`)

• Some concepts are clearly crisp (e.g. `grandmother`)

• Some concepts are none of the two(e.g. `metre`)

• So vagueness itself is a degree concept. We tend to use it vaguely.


Stepping back

• Vagueness in measurement is like original sin


Stepping back

• Some authors have asked why human language is vague – Gary Marcus, Kluge: vagueness is a leftover

from primitive days

• Vagueness in measurement is like original sin: a stain that can be diminished but never removed


1.2 The fiction of species

• We know measurement is tricky

• Surely, classification is often easier?

• How about the notion of a common Chimpanzee, for example, or a person (Homo sapiens)?

• Surely, species-denoting terms are crisp?


The fiction of species

• We know measurement is tricky

• Surely, classification is often easier?

• How about the notion of a common Chimpanzee, for example, or a person (Homo sapiens)?

• Surely, species-denoting terms are crisp?

• We shall see that these terms are not only vague but also incoherently defined


What makes a species?

• Thought to be unproblematic until 1900 (?)– Platonic world view: there “just are” different

species ... (e.g. Linnaeus 1750)

• Evolution theory: species evolve gradually

• (Mayr, Dobzhansky, 1940) Modern theory of species, based on interbreeding:

same-species(x,y) x interbreeds with y


Problems with this definition

• Problem 1: animals of same sex can belong to same species. Solution: i(x,y) i(x,z) i(y,z)

• Problem 2: distance in time or space.Solution: disregard these; specieshood is about being able to interbreed in principle– For example: you “interbreed” with your great

grandparents


Problems with this definition

• Problem 1: animals of same sex can belong to same species. Solution: i(x,y) i(x,z) i(y,z)

• Problem 2: distance in time or spaceSolution: disregard these; specieshood is about being able to interbreed in principle.

• And so on. But there is a trickier problem, to do with vagueness


Ensatina salamanders

• A kind of salamanders living in the hills around California’s Central Valley

• Studied by biologists such as Stebbins (1949), popularised by Dawkins (2004)

• Ensatina salamanders look rather different, depending on where they live


Habitat and interbreeding

Because of this shape, Ensatina is called a ring species. Logically, the ordering is not ring-like:

eschscholtzii i x i p i o i c i klauberi

c

o

px

eschscholtzii

klauberi

CENTRAL VALLEY


escholtzii i x i p i o i c i klauberi

• The point: i(eschscholtzii,klauberi)

• Observe: i is non-transitive

• Definition of `species` predicts proliferation of overlapping species: { {esch,x}, {x,p}, {p,o}, {o, c}, {c,klau} }


Dawkins also asks:

• How about our own ancestry?

• Each person p stands in relation i with his/her parents, grandparents, great grandparents, etc. ...

• But at some time there was an ancestor a such that i(a,p) (perhaps an ape?)

• Do p and a belong to same species?

• Possible responses:


Are p and a the same species?

• Response 1: “No; interbreeding should be used strictly, as in the original definition of `species`”

• Implication: many overlapping species

• Response 2: “Yes; species should be defined via the transitive closure of i”

• NB This appears to be the standard response• Implication: All animals are one big species. The

concept becomes meaningless!


Dawkins: thinking in crisp terms amounts to a “tyranny of the discontinuous mind”

“Let us use names as if they really reflected a discontinuous reality, but let's privately remember that, at least in the world of evolution, it is no more than a convenient fiction, a pandering to our own limitations”. (Dawkins 2004, “The Ancestor’s Tale”)

A reasonable compromise, but it leaves the concept of species undefined


Why does this compromise work so often?

• Often, many of the links between different species have gone extinct

• Ensatina in the year 2000:

• Imagina Ensatina in the year 3000: esch i xan i pi i oreg i cro i klau• Three separate species!

esch i xan i pi i oreg i cro i klau


Lessons

• Coherence can depend on the facts on the ground– Philosophers know this. E.g., sorites only “bites” when

some objects cannot be distinguished from each other (with respect to a given quality)

• Species are fictions, which are only useful in some situations

• Similar things can be argued about the notion of a car, a person, and so on (Parsons, Forbes, ...)


Analogous arguments

• Problems with the notion of a language are analogous to those with species

• The relevant definition: “idiolects x and y belong to the same language if their speakers can understand each other (without previous exposure)”


2. The trouble with variety

Book:Part I: Vagueness “in the world”Part II: Theories of vaguenessPart III: Vagueness in language

generation/productionFrom Part II:• Knowledge as Ignorance• Degree models


2.1 Knowledge as ignorance

• “Concepts like tall do have sharp boundaries, but speakers do not know exactly what they are”

• If this is true then, strictly speaking, tall is not vague (because it has no boundary cases)

• Pure epistemicism: Same for all other “vague” expressions. Vagueness is only apparent. Advantage:– Epistemicism allows us to use Classical Logic

• Epistemicism is popular (Williamson 1994, Bonini et al 1999, Sorensen 2001)

• How plausible is it?


Possible objections against epictemicism

• A. Inconsistent usage: “Different people use words like tall, blue, evening in different ways”.

• Some evidence of differences in usage:• Reiter et al: weather forecasters use the word

evening in bafflingly different ways. Interviews suggest lots of explanations– Is dinner time relevant– Does the season matter?– Etc.


Possible objections against epistemicism

• Differences between people are also enforced by the differences in their senses

• Example: Colour. Hilbert 1987: – “Standard Observer Model” (Commission

Internationale de l’ Éclairage) gives JNDs • JND: Just Noticeable Difference

– Based on averages between normally sighted people– But even normally sighted people do not distinguish

between exactly the same pairs of colours • Differences densities of pigment layers on lens and retina;

different sensitivities of photo receptors


• Possible response against objection A:– Language community as a whole defines

concepts like tall, blue, and evening.– Speakers try to converge

• Evidence of alligment in language use

– Without this, no communication is possible

• Not sure how forceful this response is.– Essentially, it says that sameness of meaning

is a useful illusion


• Another response to objection A: “Differences between people could coexist with crisp boundaries”

• But if everyone speaks differently, how could a child learn a crisp concept?

• Moreover, a concept like tall, blue, or evening could never be used crisply because of indistinguishables– Suppose tall means 180cm– Speaker and hearer cannot discriminate between

179.999cm en 180cm


So, a second objection:

• Objection B: “In psychologically continuous domains, sharp concepts cannot be used sharply”

• Mathematical continuity is not required• Even mathematical density is not required• What is required?

– Across the range of interest, there exists a chain x1,...,xn such that xi+1>xi

yet xi+1 is indistinguishable from xi

– This means the threshold cannot be within that range


But there is more

• Objection C against Epistemicism: “New usage could never be sharp”

• Example: the word flibbery:– Consider the feeling of rhubarb in your

mouth?– I now decide to call this fibberiness:

“My mouth feels fibbery now”.– Have I defined the threshold? How? (And how

do you know what it is?)


Summarising objections against epistemicism

• A: usage data is inconsistent, so how can we learn a sharp boundary

• B: indistinguishables cannot be allowed to cross a sharp boundary

• C: usage data may be too sparse to sustain a sharp boundary


• Sorensen: Epstemicism may be counter-intuitive, but it allows us to use classical logic– “The history of deviant logics is without a single success”

• Sorensen’s claim may have been defensible in 1960 (except for three-valued logics (eg Łucasiewics 1920), but certainly not after … – Nonmonotonic logics (circumscription, PROLOG)– Linear logic (e.g. Troelstra 1991)– Logic of argumentation (e.g. Dung 1995)– Dynamic Logic (e.g. Harel, Kozen and Tiuryn 2000)

• Overview of non-classical logic: Gabbay and Guenthner (1984)


Discussion

• Difficult to see the virtue of Epistemicism

• What is its appeal?


2.2 Degree Models

What model of vagueness is best?

A (far-too-brief) roundup:• Classical models do not do justice to

indistinguishables• Partial Logic + Supervaluations

replaces one crisp threshold with two crisp thresholds same problem

• Context-based models treat indistinguishability as if this notion were crisp


Indistinguishabilityas a crisp relation

• Examples: Kamp 1981, Veltman & Muskens, Van Deemter 1996.

• An often-used mechanism stems from Goodman and Dummett:


Suppose x~y, but h << x h~y

The observer can deduce that y<x, so y and x become distinguishable. Short(y) nolonger implies Short(x)

184.999

A

B

185.001xy

184.000h


h is called a help element

Sorites introduces more and more things into the argument that can serve as help elements.

184.999

A

B

185.001xy

184.000h


Problem:

Help elements assume that `~` is crisp

184.999

A

B

185.001xy

184.000h


If you ask a subject to compare x,y and h 1000 times, you get different responses, e.g.

1. x~y, h<<x, h~y. 2. x~y, h~x, h~y.3. y<<x, h<<x, h<<y.4. y<<x, h<<x, h~y.5. x<<y, h<<x, h~y.6. x~y, h<<x, h~y. 7. ... (etc) ...


If you ask a subject to compare x,y and h many times, you get different responses

A sharp JND is a simplification– the right account would show an S-curve

A good model of vagueness should do justice to degrees/probabilities

Another argument with the same conclusion:


The diamond in the harem

• The emperor’s diamond was stolen. Someone catches the perpetrator, but is shaken off and badly wounded. With his last gasp he says “The thief was tall”. Then he dies.

• (A bit like R.Parikh’s story of Bob and Ann, who use different crisp notions of blue; the story highlights the concept of utility)


The diamond in the harem

• The emperor’s diamond was stolen. Someone catches the perpetrator, but is shaken off and badly wounded. With his last gasp he says “The thief was tall”. Then he dies.

• If the emperor is a classical logician, he separates his staff (e.g. 6) in two groups: the tall ones (e.g. 3) and the others (3). Suppose the latter group can be ruled out. expect to search half of 3 = 1.5


A short story

• If the emperor is a partial logician, he separates his staff into three groups: the tall ones, the not-tall ones and the doubtful ones.

• Assume each group has 2 members andp(Thief Tall)=70%

p(Thief Doubtful)=30%p(Thief not-Tall)=0%

• Expect to search (0.7*1)+(0.3*2)=1.3• Better than 1.5. But ...


The diamond and the harem

• If the emperor is smart, he assumes that the taller a person is, the more likely he is to be called tall. (Degrees!)

• He has his staff arranged in order of height, starting with the tallest

• The tallest one is searched first, etc.• If his assumption is correct then this

strategy leads to the smallest expected number of searches


For example

Suppose, as before, we assume that half of the staff can be ruled out. Then

p(referent = Mr190)=3/6 1 searchp(referent = Mr185)=2/6 2 searchesp(referent = Mr180)=1/6 no more searchExpected number of searches =

(3/6)*1+(2/6)*2=11/6

This is better than the other strategies.This solution hinges on degrees!


3. Expulsion from Boole’s paradise

• What happens if bivalence is abandoned?

• What’s outside the Boolean Gates?


Life does become harder ...


Drawbacks of Degrees

• Truthfulness and lying become problematic– “We didn’t know there was a link between smoking

and cancer”• Verification and falsification

– “All ravens black? What about this grey-black one”• Belief revision

– No longer just the removal of possible worlds• Truth degrees should interact with

– “real” probability (What number to associate with “the patient may have had a brachycardia”?)

– context (If context affects truth then context should affect truth degrees)


Conclusions

1. Many everyday and scientific concepts are vague, even where we normally speak in crisp terms– white/black (skin)– poisonous substances– plagiarism– Ensatina, Homo sapiens – more than one metre long

Vagueness itself comes in degrees


2. Epistemicism seems hard to defend• Difficult to reconcile with variety within and

between subjects• Difficult to reconcile with the existence of

indistinguishables• Makes no sense for novel usage• Classicality should not be a goal in itself in a

logic– but admittedly, many applications can afford to

simplify. Example in book: weather forecasting


3. Degree Theories have plusses and minusses

• Fuzzy Logic has many shortcomings, and these may have given Degree Theories a bad name– For example, truthfunctionality looses the Law of

Excluded Middle

• [These shortcomings can be resolved by a probabilistic approach: Black, Edgington, ...]

• But indistinguishables remain a slight problem ...


3. Degree Theories have plusses and minusses• Fuzzy Logic has many shortcomings, and these

may have given Degree Theories a bad name• [These shortcomings can be resolved by a

probabilistic approach (Black, Edgington)]• But indistinguishables remain a slight problem ...

• And life does become harder


Concluding suggestion

• Perhaps we have to learn that a theory/model is not true or false but– a more or a less accurate approximation of

the world– which is more or less useful for a particular

application


Concluding suggestion

• Perhaps we have to learn that a theory/model is not true or false but– a more or a less accurate approximation of

the world– which is more or less useful for a particular

application

• Degrees of truth at a meta level!


RESERVE SLIDES

ON DEGREE THEORIES


Degree Models

Fuzzy Logic

A drastic deviation from Classical Logic.


• Given that ‘crisp boundaries’ are a disadvantage of 3-valued logic, how about using real numbers as values?

• Best known version is fuzzy logic(Zadeh 1975): [φ] ε [0,1], where

[φ] ≤ [] is at least as true as φ

• Fuzzy logic does not say how truth values may be obtained


Truth definition (fuzzy logic)One example:

1. Negation: [¬φ] = 1- [φ]

2. Disjunction [φv] = max([φ] , [])

3. Conjunction [φ] = min([φ] , [])

4. Implicaton [φ]If [φ] ≤ [] then [φ] = 1,

otherwise [φ] = 1- ([φ]-[])


If [φ] ≤ [] then [φ] = 1 otherwise, [φ] = 1- ([φ]-[])

Assume that [small(i)]- [small(i+1)] = ε,

for some small constant ε.

Then [small(i)small(i+1)] =

1- [small(i)]-[small(i+1)]


Conclusions of sorites arguments get increasingly low values

small(150) [Some high value, let’s say 1] small(151) [1-ε] small(152) [1-2ε] small(153) [1-3ε], etc.

This is great !

But some truth values seem unnatural For example, for most i, we have

[small(i) v ¬small(i)] ≠ 1


The problem with truthfunctionality

Truthfunctionality implies:

If [q] = [¬p] then

[p V ¬p] = [p V q]

Truth definition of `V` cannot take into account whether the disjuncts are related


Other forms of same problem (Sorensen)

• Suppose we don’t know whether to call Mr165 “short” or “somewhat short”[Short(Mr165)]=0.5 and

[Somewhatshort(Mr165)]=0.5

• Do we really want [Short(Mr165) Somewhatshort(Mr165)]=0.5?

• This value should be much higher


Other forms of the same problem

Let’s take a vote:Which one is the big expensive car?

x. [car(x)]=1[big(x)]=0.5[expensive(x)]=0.5

y. [car(y)=1[big(y)]=0.49[expensive(y)]=1


Other forms of the same problem

x. [car(x)]=1[big(x)]=0.5[expensive(x)]=0.5

y. [car(y)=1[big(y)]=0.49[expensive(y)]=1

Fuzzy Logic says, counter-intuitively, that [car(x) big(x) exp(x)]=0.5[car(y) big(y) exp(y)]=0.49


• To address these problems, Fuzzy Logic allows different kinds of conjunction, e.g.

Conjunction 1: [φ] = min([φ] , [])Conjunction 2: [φ] = [φ] * []

• Every new application may require its own definition

• It’s unclear what the choice depends on• This is a problem for engineers as well

as theoreticians


Is there a better way?

• Non-truthfunctional accounts have been proposed, e.g. Edgington 1992, 1996

• Basic intuition: [] = probability of someone agreeing with

• Consequences:[small(i) v ¬small(i)] =1

• Why?


“verity” defined

v()=1- v()

v()=v()*v(|)

v()=v(|)

v()=(v()+v())-v()

u=uncertainty=1-v


validity defined

“An argument is valid iff it is impossible for the uncertainty of the conclusion to exceed the sum of uncertainties of the premisses”

For example, A,B |= AB

is valid because v(A)=0.98 and v(B)=0.98 imply that v(AB)0.96


applied to sorites

Consider a sorites conditional, e.g.

Short(Mr160)Short(Mr161)

• v(Short(Mr161)|Short(Mr160)) is close to 1• Uncertainty is very small, say • So if v(Short(Mr160)) = 0.8

then v(Short(Mr161) =0.8 - And so on ...


One problem

What does v() mean?

• Edgington suggests an abstract account, resembling supervaluations

• v(Short(x))=a priori chance Short being defined in such a way that Short(x) is true

• Think of a darts game, where the position of the dart is the threshold for Short


atomic verities: the darts mataphor

Where the dart can land:

175cmv=0

150cmv=1


Atomic verities: the darts metaphor

Where the dart can land:

175cmv=0

150cmv=1

Mr160

Mr162.5

v(Short(Mr160)) > v(Short(Mr162.5)) = 0.5


Drawbacks of this abstract account

Conditional probability:

v(Short(Mr162.5)|Short(Mr160)) =

section of thresholds that make Mr162.5 short /section of thresholds that make Mr160 short


Atomic verities: the darts metaphor

thresholds that make Mr160 short

175cmv=0

150cmv=1

Mr160

Mr162.5

v(Short(Mr160)) > v(Short(Mr162.5)) = 0.5


Drawbacks of this abstract account

• Conditional probability is not very well motivated

• Any two heights are separated by a possible dart throw, even if the heights are indistinguishable

• Isn’t this just as crude as Classical Logic?

• Mark Sainsbury: “You don’t improve a bad idea by iterating it infinitely many times”


Put differently

• The abstract probabilistic account cannot be an accurate model of any person’s language use

• If this objection worries you, then here is a more empirically-oriented one


• Basic idea: group together all of a subject’s judgmentsP(Short(x)) = probability that x is called Short

P(Short(y)|Short(x)) = given that s calls x Short, what is the probability that s calls y Short?

• For example:


Rosanna's judgements:S(149), ..., S(153), S(154), S(155), S(156), ..., S(175))Roy's judgements:S(149), ..., S(154), S(155), S(156,) S(157), ..., S(175))Hans' judgements:S(149), ..., S(155), S(156,) S(157), S(158), ..., S(175))Joe's judgements:S(149), ..., S(154), S(155), ..., S(175))Tim's judgements:S(149), ..., S(157), S(158), ..., S(175))


How sorites conditionals work out

p(Short(150)Short(151))=1

p(Short(153)Short(154))=4/5

p(Short(154)Short(155))=3/4


Some properties

• The community defines the meaning of tall• This account can sometimes separate

indistinguishables• But without claiming that any person

knows which of two indistinguishables is taller– Different subjects may judge them differently– On a more refined account, the same is even

true for each given indivisual

kees van deemter. for institut nicod, jan 2009 vagueness as original sin from measurement to...

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