Kees van Deemter. For Institut Nicod, Jan 2009 Vagueness as Original Sin from measurement to semantic theory Kees van Deemter University of Aberdeen Scotland, United Kingdom Paris, Institut Nicod, 14 Jan 2009

Kees van Deemter. For Institut Nicod, Jan 2009 Introduction Not Exactly : in Praise of Vagueness Oxford University Press, Winter 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

Kees van Deemter. For Institut Nicod, Jan 2009 Plan of the talk 1.Original sin 2.The trouble with variety 3.Expulsion from Booles paradise

Kees van Deemter. For Institut Nicod, Jan Vagueness as original sin

Kees van Deemter. For Institut Nicod, Jan 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]

Kees van Deemter. For Institut Nicod, Jan 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

Kees van Deemter. For Institut Nicod, Jan 2009 History of the metric system To minimise variation, the bar was –made of platinum for low oxidation –kept at a fixed temperature (0 0 Celcius) –supported at fixed distances to minimise contact with other substances –given a sturdy profile to minimise bending

Kees van Deemter. For Institut Nicod, Jan 2009

Parisian method was extremely successful: used for about 100 years This method had an error of approximately 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

Kees van Deemter. For Institut Nicod, Jan 2009 just like an ordinary tool...

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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 wont notice the difference (Sorites argument is easily constructed)

Kees van Deemter. For Institut Nicod, Jan 2009 What to conclude? Situation pre 1900: metre is vague. This is true regardless of which bedrock you choose: –metre = 1/40,000,000 of Earths circumference vague because the Earth varies in size –metre = size of bar vague because bar varies in size

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 Whatever your view on the theoretical perspective... The predicate x.size(x) 1 metre can 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

Kees van Deemter. For Institut Nicod, Jan 2009 Further consequences If metre is not crisp then what is crisp? –many metrics depend on distance: volume, temperature, pressure,...

Kees van Deemter. For Institut Nicod, Jan 2009

Further consequences 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

Kees van Deemter. For Institut Nicod, Jan 2009 Further consequences Many of the things we usually call crisp are (a little bit) vague: Greater or equal to 1 metre x-axis: bars of increasing sizes (not metres !) y-axis: truth values

Kees van Deemter. For Institut Nicod, Jan 2009 Further consequences 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.

Kees van Deemter. For Institut Nicod, Jan 2009 Stepping back Vagueness in measurement is like original sin

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 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?

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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. And so on. But there is a trickier problem, to do with vagueness

Kees van Deemter. For Institut Nicod, Jan 2009 Ensatina salamanders A kind of salamanders living in the hills around Californias Central Valley Studied by biologists such as Stebbins (1949), popularised by Dawkins (2004) Ensatina salamanders look rather different, depending on where they live

Kees van Deemter. For Institut Nicod, Jan 2009

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 p x eschscholtzii klauberi CENTRAL VALLEY

Kees van Deemter. For Institut Nicod, Jan 2009 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} }

Kees van Deemter. For Institut Nicod, Jan 2009 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:

Kees van Deemter. For Institut Nicod, Jan 2009 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!

Kees van Deemter. For Institut Nicod, Jan 2009 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 Ancestors Tale) A reasonable compromise, but it leaves the concept of species undefined

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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,...)

Kees van Deemter. For Institut Nicod, Jan 2009 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)

Kees van Deemter. For Institut Nicod, Jan The trouble with variety Book: Part I: Vagueness in the world Part II: Theories of vagueness Part III: Vagueness in language generation/production From Part II: Knowledge as Ignorance Degree models

Kees van Deemter. For Institut Nicod, Jan 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?

Kees van Deemter. For Institut Nicod, Jan 2009 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.

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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 cm en 180cm

Kees van Deemter. For Institut Nicod, Jan 2009 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 x 1,...,x n such that x i+1 >x i yet x i+1 is indistinguishable from x i –This means the threshold cannot be within that range

Kees van Deemter. For Institut Nicod, Jan 2009 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?)

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 Sorensen: Epstemicism may be counter-intuitive, but it allows us to use classical logic –The history of deviant logics is without a single success Sorensens 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)

Kees van Deemter. For Institut Nicod, Jan 2009 Discussion Difficult to see the virtue of Epistemicism What is its appeal?

Kees van Deemter. For Institut Nicod, Jan 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

Kees van Deemter. For Institut Nicod, Jan 2009 Indistinguishability as a crisp relation Examples: Kamp 1981, Veltman & Muskens, Van Deemter An often-used mechanism stems from Goodman and Dummett:

Kees van Deemter. For Institut Nicod, Jan 2009 Suppose x~y, but h << x h~y The observer can deduce that y<x, so y and x become distinguishable. Short(y) no longer implies Short(x) A B x y h

Kees van Deemter. For Institut Nicod, Jan 2009 h is called a help element Sorites introduces more and more things into the argument that can serve as help elements A B x y h

Kees van Deemter. For Institut Nicod, Jan 2009 Problem: Help elements assume that `~` is crisp A B x y h

Kees van Deemter. For Institut Nicod, Jan 2009 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 (etc)...

Kees van Deemter. For Institut Nicod, Jan 2009 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:

Kees van Deemter. For Institut Nicod, Jan 2009 The diamond in the harem The emperors 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.Parikhs story of Bob and Ann, who use different crisp notions of blue; the story highlights the concept of utility)

Kees van Deemter. For Institut Nicod, Jan 2009 The diamond in the harem The emperors 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

Kees van Deemter. For Institut Nicod, Jan 2009 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 and p(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...

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 For example Suppose, as before, we assume that half of the staff can be ruled out. Then p(referent = Mr190)=3/6 1 search p(referent = Mr185)=2/6 2 searches p(referent = Mr180)=1/6 no more search Expected number of searches = (3/6)*1+(2/6)*2=1 1/6 This is better than the other strategies. This solution hinges on degrees!

Kees van Deemter. For Institut Nicod, Jan Expulsion from Booles paradise What happens if bivalence is abandoned? Whats outside the Boolean Gates?

Kees van Deemter. For Institut Nicod, Jan 2009 Life does become harder...

Kees van Deemter. For Institut Nicod, Jan 2009 Drawbacks of Degrees Truthfulness and lying become problematic –We didnt 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)

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 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

Kees van Deemter. For Institut Nicod, Jan 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...

Kees van Deemter. For Institut Nicod, Jan 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

Kees van Deemter. For Institut Nicod, Jan 2009

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

Kees van Deemter. For Institut Nicod, Jan 2009 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!

Kees van Deemter. For Institut Nicod, Jan 2009 RESERVE SLIDES ON DEGREE THEORIES

Kees van Deemter. For Institut Nicod, Jan 2009 Degree Models Fuzzy Logic A drastic deviation from Classical Logic.

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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]-[ ])

Kees van Deemter. For Institut Nicod, Jan 2009 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)]

Kees van Deemter. For Institut Nicod, Jan 2009 Conclusions of sorites arguments get increasingly low values small(150) [Some high value, lets 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

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 Other forms of same problem (Sorensen) Suppose we dont 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

Kees van Deemter. For Institut Nicod, Jan 2009 Other forms of the same problem Lets 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

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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 Its unclear what the choice depends on This is a problem for engineers as well as theoreticians

Kees van Deemter. For Institut Nicod, Jan 2009 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?

Kees van Deemter. For Institut Nicod, Jan 2009 verity defined v( )=1- v( ) v( )=v( )*v( | ) v( )=v( | ) v( )=(v( )+v( ))-v( ) u=uncertainty=1-v

Kees van Deemter. For Institut Nicod, Jan 2009 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 |= A B is valid because v(A)=0.98 and v(B)=0.98 imply that v(A B) 0.96

Kees van Deemter. For Institut Nicod, Jan 2009 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...

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 atomic verities: the darts mataphor Where the dart can land: 175cm v=0 150cm v=1

Kees van Deemter. For Institut Nicod, Jan 2009 Atomic verities: the darts metaphor Where the dart can land: 175cm v=0 150cm v=1 Mr160 Mr162.5 v(Short(Mr160)) > v(Short(Mr162.5)) = 0.5

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 Atomic verities: the darts metaphor thresholds that make Mr160 short 175cm v=0 150cm v=1 Mr160 Mr162.5 v(Short(Mr160)) > v(Short(Mr162.5)) = 0.5

Kees van Deemter. For Institut Nicod, Jan 2009 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 Isnt this just as crude as Classical Logic? Mark Sainsbury: You dont improve a bad idea by iterating it infinitely many times

Kees van Deemter. For Institut Nicod, Jan 2009 Put differently The abstract probabilistic account cannot be an accurate model of any persons language use If this objection worries you, then here is a more empirically-oriented one

Kees van Deemter. For Institut Nicod, Jan 2009 Basic idea: group together all of a subjects judgments P(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:

Kees van Deemter. For Institut Nicod, Jan 2009 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))

Kees van Deemter. For Institut Nicod, Jan 2009 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

Kees van Deemter. For Institut Nicod, Jan 2009 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