Van Deemter, WORD, May 2010 Not Exactly Vagueness as Original Sin? Kees van Deemter University of...
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Transcript of 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
van Deemter, WORD, May 2010
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?
van Deemter, WORD, May 2010
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?
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
Ensatina (Stebbins 1949, Dawkins 2004)
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
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}
van Deemter, WORD, May 2010
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?
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
Are you and a the same species?
Formal Response: “No; the interbreeding criterion should be used”
Many overlapping species
Standard Response: “Yes; species should be defined via the transitive closure of i”
van Deemter, WORD, May 2010
Are you and a the same species?
Formal Response: “No; the interbreeding criterion should be used”
Many overlapping species
Standard Response: “Yes; species should be defined via the transitive closure of i”
All living beings are one species
van Deemter, WORD, May 2010
Interim conclusion
Key concepts of science resist precise definition
van Deemter, WORD, May 2010
Vagueness as original sin? (with thanks to Tintoretto)
van Deemter, WORD, May 2010
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:
van Deemter, WORD, May 2010
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”
van Deemter, WORD, May 2010
van Deemter, WORD, May 2010
• 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?
van Deemter, WORD, May 2010
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)
van Deemter, WORD, May 2010
Second answer
11m 12m
van Deemter, WORD, May 2010
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”
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
3. Vagueness is a problem
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
van Deemter, WORD, May 2010
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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
van Deemter, WORD, May 2010
Eubulides in the audio lab
-30dB is inaudible
-30dB is indistinguishable from -29.5dB, so
-29.5dB is inaudible
van Deemter, WORD, May 2010
Eubulides in the audio lab
-29.5dB is inaudible
-29.5dB is indistinguishable from -29dB, so
-29dB is inaudible
van Deemter, WORD, May 2010
Eubulides in the audio lab
...........
99.5dB is inaudible
99.5dB is indistinguishable from 100dB, so
100dB is inaudible !!
van Deemter, WORD, May 2010
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 !!
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
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?
van Deemter, WORD, May 2010
So, …
Vagueness is not just a matter of ignorance
Models of logic and language ought to embrace vagueness
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
Window in Lincoln Cathedral
van Deemter, WORD, May 2010
“audible” in classical logic
audible
inaudible
x dB
van Deemter, WORD, May 2010
“audible” in classical logic
x dB
audible
inaudible
Indistinguishable
x+
x-
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
We have seen:
1. Vagueness is everywhere
2. We are vague for a reason
3. Vagueness is a problem
van Deemter, WORD, May 2010
4. How to model vagueness?
van Deemter, WORD, May 2010
Some like it crisp
Blastland & Dilnot (2008): false clarity Substances that are “poisonous” Genes that “cause” a condition
Another example: Vagueness as ignorance
van Deemter, WORD, May 2010
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!
van Deemter, WORD, May 2010
“Continuous” logics
Date back to J.Łukasiewicz 1920 and M.Black 1937
Map statements to numbers between 0 and 1
van Deemter, WORD, May 2010
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],[])
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
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:
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
Boole’s 2-valued paradise was such an attractive place
van Deemter, WORD, May 2010
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
van Deemter, WORD, May 2010
Questions for linguists, logicians, philosophers, computer scientists
We’d better rise to the challenge!
van Deemter, WORD, May 2010
“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
van Deemter, WORD, May 2010
The End
www.csd.abdn.ac.uk/~kvdeemte/NotExactly
With thanks to
Judith Masthoff (for Homer Simpson’s coiffure)
van Deemter, WORD, May 2010
van Deemter, WORD, May 2010
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”)
van Deemter, WORD, May 2010
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!