van deemter, word, may 2010 not exactly vagueness as original sin? kees van deemter university of...

van Deemter, WORD, May 2010

Not ExactlyVagueness as Original Sin?

Kees van Deemter

University of Aberdeen


Plan of the talk

1. Vagueness is hard to avoid

2. We are often vague for good reasons

3. Vagueness is a problem

4. How to model vagueness formally?


1. Vagueness is hard to avoid

Vague words have borderline cases

An Aberdeen afternoon in May at 3PM

22 C warm

8 C not warm

15 C ¿warm?


Vague adjectives: warm, cold, large, ... Vague nouns: girl, giant, island, ...

and so on …

Most words in ordinary English are vague

Vagueness is prevalent in science too

Example: species terms


What makes a species?

Long thought unproblematic (e.g. Linnaeus 1750)

The interbreeding criterion(Mayr, Dobzhansky, 1940)

x is same species as y x interbreeds with y


Ensatina (Stebbins 1949, Dawkins 2004)


Ensatina’s habitat and interbreeding

Called a ring species. Logically:

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

For example, not i(eschscholtzii,klauberi)

Interbreeding predicts overlapping species:

{esch,x} {x,p} {p,o} {o,c} {c,klau}


Our own ancestry

you stand in relation i with your parents, grandparents, ...

Let a = the first ancestor such that not i(a,you)

Do you and a belong to same species?


Are you and a the same species?

Formal Response: “No; the interbreeding criterion should be used”

Many overlapping species

s..s6

s5s4

s3s2

s1

time





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





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

All living beings are one species


Interim conclusion

Key concepts of science resist precise definition


Vagueness as original sin? (with thanks to Tintoretto)


2. We are often vague for good reasons

“Strategic” vagueness

Why are we often more vague than we need to be? (Game theorists, e.g., B. Lipman 2000, 2006)

Some tentative answers:


First answer

Suppose I say about a day in May: “The temperature is 15 C. The rain probability is 20%. Wind speed is 10mph, humidity 55%”

Easier to digest: “A nice-enough Spring day, with light winds and a chance of rain”


• The numbers use an old-fashioned scale (inches of Mercury)

• Words like Very Dry and Much Rain help us to understand the scale

• These words are vague:

Does 22.8 count as Rain or Much Rain?


Aberdeen Computing dept. build programsInput: numbers or formulas (15 C, …)

Output: words (“Mild, … A nice Spring day’’)

Medical applications too (e.g. BABYTALK)

Open questions: What’s best understood?Remembered? Acted on?

(Peters et al. 2009, Zikmund-Fisher et al 2007)


Second answer

11m 12m


Height of house 1 =11m Height of house 2 =12m

- “the 12m house needs to be demolished”- “the tall house needs to be demolished”

Comparison is easier than measurement

Therefore, we might prefer “the tall house”


Third answer

A politician promising “drastic budget cuts”, or “stable government”

Game-theory models predict benefits from vague promises (Aragones & Neeman 2000) Unforeseen contingencies could make

concrete promises difficult to honour Disappointed voters could hold politician

to account



Sorites puzzle (Eubulides, 450 BC)

One of the top ten unsolved problems of science (“The list universe”, 2007 AD)

0 hairs is bold (x hairs is bold) (x+1 hairs is bold) therefore, 106 hairs is bold Yet 106 hairs is not bold


Sorites enhanced by science

Eubulides in the audio lab

Decibel (dB): measures the loudness of sounds

-30dB is inaudible 100dB is very loud differences of 0.5dB cannot be discerned



-30dB is inaudible

-30dB is indistinguishable from -29.5dB, so

-29.5dB is inaudible



-29.5dB is inaudible

-29.5dB is indistinguishable from -29dB, so

-29dB is inaudible



...........

99.5dB is inaudible

99.5dB is indistinguishable from 100dB, so

100dB is inaudible !!


The new sorites argument as a whole

-30dB is inaudible-30dB is indistinguishable from -29.5dB, so

-29.5dB is inaudible-29.5dB is indistinguishable from -29dB, so

-29dB is inaudible

...........

99.5dB is inaudible99.5dB is indistinguishable from 100dB, so

100dB is inaudible !!


Vagueness as ignorance

The concept “bald” does have sharp boundaries, but speakers do not know them

A surprisingly popular view (Williamson 1994, Sorensen 2001, Tuck 2009)

Contradicted by empirical evidence


We’re all different

Colour terms like “red” (Hilbert 1987, R.Parikh 2000) People cannot distinguish the same colours

pigment on lens and retina; sensitivity of photo receptors

Time words like “evening” (Reiter et al. 2005) Is dinner time relevant? The time of year?


So, …

Vagueness is not just a matter of ignorance

Models of logic and language ought to embrace vagueness


For analysing the meaning of language, mathematical logic is the tool of choice

Classical logic is built on crisp dichotomies George Boole (1815-1864) gave

the first mathematical account A statement is either true or false (1 or 0) Nice and simple: Boole’s paradise


Window in Lincoln Cathedral


“audible” in classical logic

audible

inaudible

x dB


“audible” in classical logic

x dB

audible

inaudible

Indistinguishable

x+

x-


Semi-classical logics use dichotomies too

Context-aware logics (Kamp 1981) use Just-Noticeable Difference

E.g., loudness: JND 1dB

JNDs mistakenly modelled as crisp

Crispness contradicted by empirical evidence

Subtler models are needed


We have seen:

1. Vagueness is everywhere

2. We are vague for a reason



4. How to model vagueness?


Some like it crisp

Blastland & Dilnot (2008): false clarity Substances that are “poisonous” Genes that “cause” a condition

Another example: Vagueness as ignorance


Two cultures (compare C.P. Snow)

Engineers & psychophysicists: approximations, real numbers,Gaussian distributions,

Philosophers, linguists, and most logicians: crisp dichotomies (true/false, 1/0).

They inhabit Boole’s Paradise!


“Continuous” logics

Date back to J.Łukasiewicz 1920 and M.Black 1937

Map statements to numbers between 0 and 1


Fuzzy logic (L.Zadeh 1975)

[φ] = degree of truth of φ

[1000 hairs is bald] < [100 hairs bald]

Negation: [not φ] = 1- [φ]

Disjunction: [φ or ] = max([φ],[])

Conjunction: [φ & ] = min([φ],[])


sorites paradox in Fuzzy Logic

As x increases, Bald(x) becomes less true:

[Bald(0)] = 1

[Bald(103)] 0.5

[Bald(106)] 0

Each premiss

Bald(x) Bald(x+1)is almost true


Problems for Fuzzy Logic

Is 1000 hairs bald or somewhat bald?

[Bald(1000)] = 0.5

[SwBald(1000)] = 0.5

Consider Bald(1000) or SwBald(1000)

Fuzzy Logic assigns a strangely low value:


Problems for Fuzzy Logic

Is 1000 hairs bald or somewhat bald?

[Bald(1000)] = 0.5

[SwBald(1000)] = 0.5

Consider Bald(1000) or SwBald(1000)

Fuzzy Logic assigns a strangely low value:

[Bald(1000) or SwBald(1000)] = max(0.5, 0.5) = 0.5


A better way (e.g., Edgington 1992,1996)

[] = probability of someone agreeing with

[ or ] = [] + [] - [&]

[Bald(1000) or SwBald(1000)] = 0.5 + 0.5 - 0 = 1


Boole’s 2-valued paradise was such an attractive place


When vagueness is taken seriously ...

Truthfulness becomes problematic

“We didn’t know that smoking causes cancer” Not exactly true

Falsification & Belief Revision

“Are all ravens black? What about this grey-black one?” Not exactly black


Questions for linguists, logicians, philosophers, computer scientists

We’d better rise to the challenge!


“Not Exactly: in Praise of Vagueness” Oxford University Press, Jan. 2010

Part 1: Vagueness in science and daily life

Part 2: Theories of vagueness

Part 3: Vagueness in Artificial Intelligence


The End

www.csd.abdn.ac.uk/~kvdeemte/NotExactly

With thanks to

Judith Masthoff (for Homer Simpson’s coiffure)


Dawkins on species terms

“Let us use names as if they really reflected a discontinuous reality, but let's privately remember that (...) it is no more than a convenient fiction, a pandering to our own limitations”.

“Tyranny of the discontinuous mind”.

(Dawkins 2004, “The Ancestor’s Tale”)


Why is the fiction of species convenient?

Links between species have gone extinct

When xan and oreg are extinct:

esch i xan i pi i oreg i cro i klau

Three separate species!

van deemter, word, may 2010 not exactly vagueness as original sin? kees van deemter university of...

Documents

inaudible slide

bold slide

discerned slide

tintoretto slide

tall house slide

chance of rain slide

vague vagueness

relation i