CM3112 Artificial Intelligence
Fuzzy logic Introduction
Steven Schockaert
SchockaertS1@cardiff.ac.uk
School of Computer Science & Informatics Cardiff University
Natural categories are fuzzy
McCLOSKEY AND GLUCKSBERG
Typi- Exemplar cality
ustpan 6.25 oap 5.71 pices 5.38
aste Basket 5.13
ashing Machine 3.13 encil 2.96 ammer 2.71
modal response
percentage of non-
Candidate
MR* NR* Wl*
Y .35 .30 N .28 .17
N ,13 .13 N .22 .30 N .15 .27 N .10 .13 N .08 .10 N .00 .00
Candidate Exemplar
Typi- cality
modal responses
elevision 1.71
Natural Earth Formation Category
ountain olcano
lateau ault
liff
land esert alley ontinent
trait eninsula
lacier aterfall
each talagmite
9.75 Y 9.13 Y 9.04 Y 8.87 Y 8.50 Y 8.50 Y 8.43 Y 8.42 Y 8.08 Y 8.04 Y 8.00 Y 7.42 Y 7.42 Y 7.38 Y 7.36 Y 7.33 Y 7.25 Y 7.17 Y
.00 .00 .03 .07 .00 .00 .09 .03 .00 .00 .00 .00 .13 .07 .02 .03 .03 .07 .07 .07 .02 .03 .15 .17 .05 .10 .00 .00 .02 .04 .07 .07 .12 .10 .03 .07
Science
Biology Physics
Astronomy Oceanography
Medicine Anatomy
Meteorology Mineralogy Psychology Engineering Metallurgy Dentistry Nursing Pharmacy Nutrition
Archaeology Anthropology Geometry Agriculture Criminology Economics Geography Architecture Sociology Linguistics Politics Philosophy History Advertising
Ocean Liner
Y .00 .00 .00 .00 9.33 Y .02 .03
8,87 Y .02 .03 8.62 Y .06 .10 8.50 Y .07 .06 8,37 Y .00 .00 8.12 Y .00 .00 7.79 Y .06 .10
7.70 Y .09 .03 7,62 Y .08 .14 7.12 Y .20 .13 6,70 Y .49 .10 6.66 Y .32 .27 6.62 Y .16 .24 6,50 Y .15 .16
6,41 Y .22 .03 6.29 Y .45 .34
6,28 Y .09 .10 6,16 Y .19 .10 6.04 Y .29 .10 5,66 Y .30 .26 5,16 Y .50 .24 5.00 Y .35 .30 4,83 Y .22 .16 4,00 N .48 .13 3,79 N .44 .06 3.54 N .32 .17 3A2 N .30 .20
Ship Category
9.92 Y .00 .00
9.87
9.83 Y
MR* NR* Wl*
iver
eef
eyser oulde(rMcCloskey&7G.0l8ucksYberg1.19278).17
cean tone
eberg
7.08 Y .08 .10 7.04 Y .17 .27 6.58 Y .33 .24
0
D S S
W W P H
T
M V P F C
s D V C
S P G W B S R R G B O
S c
Y
.00
e
e
s
n
1.33 N
.00 .00
Contact Lens
3.58
N .07 .00
N
.45 .30
Trout
9 91
Y .00
.00
1.00 N .00 .00
Btrd Category
10.00 Y .00 .00
9.96 Y .00 .00 9.58 Y .00 .00
Briefcase Dentures Suitcase
3.21
3.08 2.92 2.71 2.67
2.17 1.79
9.75 9.67 9.25 8.42 8.13 8.04
7 75 7.67 7.08 6.88 6.71 6.58 6.50 6.29 6.29 6.25
6.04 5.88 5.75 5.50
5.46 4.71 4.71 4.71 4.04
4.04
N .00 .00
N .03 .00 N .02 .03 N .00 .00 N .05 .03
N .02 .03 N .03 .00
Disease Category
Y .00 .00
Y .00 .00 Y .05 .10 Y .00 .00 Y .02 .03 Y .07 .07 Y .12 .17 Y .07 .13 N .48 .23 Y .27 .20 Y .08 .10 Y .45 .30 Y .22 .17 Y .42 .23 Y .48 .37 N .35 .23 Y .05 .03 Y .33 .20
Y .22 .17 Y .30 .13 Y .47 .27 N .43 .40
N .32 °30 N .45 .34 N .45 .34 N .27 .20
N .00 .00 N .00 .00
Fish Category
Natural categories are fuzzy
8.42 8.38 8,29 8.25
8.23 8.17 8.08 7.96 7.92 7.88 7.75 7.52 7.43
7.29
7.25 Y 7.13 Y 6.96 Y
tyl 4.96 Y
3.63 N Egg 2.96 N
quirrel 2.63 N 2.38 N 2.29 N 2.04 N 1.83 N
Bandaid
Cancer Tuberculosis Measles Leprosy Epilepsy Asthma
Gangrene Arthritis
Heart Attack
High Blood Pressure Alcoholism Paralysis
Drug Addiction Allergy
Stroke Fever
Schizophrenia Ulcer
Neurosis Tooth Decay
Food Poisoning Blindness
Dandruff Deafness Depression Nearsightedness
8.71
Y .00 .00 Y .00 .00
¥ .00 .00 Y .03 .07 Y .03 .07 Y .00 .00
Y .00 .00 Y .02 .03 Y .03 .07
Y .00 .00
Y .00 .00
Y .05 .03
Y .05 .10
Y .00 .00
Hair brush
Beard Hearing Aid
er ch
8,54 Y .03 .07
Disease
Y
.03 .00 .03 .07
.02 .03 .08 .13 .44 .13 .17 .13 .27 .27
.05 .03 .05 .03
.13 .13 .03 .00 .09 .10
Carpenter’s Tool Category
9.83 Y .00 .00
(McCloskey & Glucksberg 1978)
Friendliness 1.17 I tappiness 1.17
8.46 Y 8.13 Y 7.88 Y
.20 .27 .32 .23 .00 .00
Oyster
4.83 N .45 .23
Chair
9.95 Y
.00 .00
Refrigerat
p
e
m
d e
e r
Natural categories are fuzzy
Fruit
.00 .00 .07 .07 .02 .03
.00 .00 .03 .07 .00 .00 .02 .03
.00 .00 .02 .03 .03 .07 .05 .03 .00 .00 .15 .17 .28 .17
.25 .17 .47 .27 .20 .20 .40 .33 .40 .27 .28 .23 .15 o10 .20 .20 .00 o00 .12 .10 .23 .27 .20 .27
.05 .03 .02 .03 .08 .10
Sea Cow 4.57 N .28 .05
Fly
Mosquito Ant Wasp
Beetle Bee Flea
Moth Locust Firefly Grasshop
Termite Butterfly Caterpilla Centiped
Millipede Spider Louse
Tarantula Silkworm Scorpion Leech
Worm
Bacteriu Amoeba
Salaman Sea Hors
Bat Mouse Dog
Fork
Spatula Pan
Can Open
Pot Holde Toaster Blender Plate Stove
Salamander 4.13 Sea Anemone 3.96 Sponge 3.45 Plankton 3.12
N .23 .07 N .45 .07 N .38 .32 N .32 .21
Alligator
2.71 N 2.70 N
.10 .20 .13 .07
Otter
Apple
Banana Pineapple Strawberry Canteloupe
Mango Watermelon Papaya
Fig Cranberry Raisin Pomegranate Coconut
Avocado Orange Juice Pumpkin Tomato Olive
Acorn Cucumber Eggplant
Squash
Sweet Potato Beet
Sunflower Seed Peanut
Carrot Onion
Corn Chicken
Fruit Category
9.92 Y
9.58 Y 9.00 Y
8.96 Y 8.25 Y 8.13 Y 7.88 Y 7.64 Y 7.42 Y
7.38 Y 7.25 Y 7.05 Y 6.83 Y 5.58 Y
5.58 N 5.42 Y
5.17 Y 4.04 Y
3.79 N 3.42 N
3.38 N 3.25 N
3.25 N 3.17 N
3.00 N 2.92 N
2.88 N 2.88 N 2.75 N
1.04 N
Furniture Category
(McCloskey & Glucksberg 1978)
.00 .00
ark
8.25 Y
.20 .13 .18 .23 .40 .27 .38 .30 .13 .13 .40 .20
.47 .27 .17 .07 .45 .17 .33 .27
.43 .20 .13 .14 .20 .22 .45 .10
.38 .23 .42 .30 .23 .20 .45 .23 .28 .05
.23 .07 .45 .07 .38 .32
cality MR*
5.07 N
4.78 N 4.70 N 4.53 N 4.32 N
4.20 N 4.12 N 3.91 N 3.78 N
3.62 N 3.45 N 3.25 N 2.87 N 2.53 N
2.03 N 1.87 N
NR* WI*
.18 .23
.30 .33 .31 .30 .43 .20
.16 .13
.28 .30 .31 .30 .10 .06
.13 .06 .08 .10 .21 .10 .15 .16 .10 .13
.00 .00 .00 .00 .00 .00
Natural categories are fuzzy
Candidate Typi-
Exemplar cality MR* NR* Wl*
Candidate Exemplar
Refrigerator
Curtains Waste Basket Bookends Ironing Board Candlestick Pillow
Potted Plant Electric Fan Telephone Ashtray
Blackboard Door Window Ceiling
Fence
Fly
Mosquito Ant Wasp
Beetle Bee Flea
Moth
FUZZY CATEGORIES 469 Typi-
6.83 Y uid 6.30 Y rimp 6.17 Y
l
ngray 6.09 Y lyfish 5.75 Y
bster 5.71 N hale 5.71 N topus 5.66 N rpoise 5.63 N
arfish 5.58 Y al 5.48 N mprey 5.47 Y am 5.25 N
a Horse 5.09 Y ab 4.95 N dpole 4.87 N ster 4.83 N a Cow 4.57 N
lamander 4.13 N a Anemone 3.96 N onge 3.45 N
ankton 3.12 N
Insect Category
Furniture
(McCloskey & Glucksberg 1978)
ligator 2.71 N .10 .20 tter 2.70 N .13 .07
Fruit Category
.32 .21
9.79 Y
9.79 Y 9.42 Y 9.21 Y 8.96 Y
8.88 Y 8.83 Y 8,71 Y
.00 ,00 .00 .00 .00 .00
.00 .00 .00 .00 .03 .07
.02 .03 ,02 .03
h
e q
h i l
o
c o
e a l
e r a
y e
a e p
l
Locust
8.63 Y
.02 ’ .03
Natural categories are fuzzy
Candidate Exemplar
Shark
Eel Squid
Shrimp Stingray Jellyfish
Lobster Whale Octopus
Typi-
cality MR*
8.25 Y 6.83 Y 6.30 Y 6.17 Y
NR* Wl*
.20 .13 .18 .23 .40 .27 .38 .30 .13 .13 .40 .20
Refri
Curta Waste Book Ironi Candl
Fish Porpoise
5.71 N 5.66 N 5.63 N
.17 .07 .45 .17 .33 .27
Potte Elect Telep
Starfish Seal Lamprey Clam
Sea Horse Crab Tadpole Oyster
Sea Cow
Salamander Sea Anemone Sponge
Plankton Alligator
Otter Apple
5.58 Y .43
.20 Ashtr
6.09 Y
5.75 Y
5.71 N .47
.27 Pillo
(McCloskey & Glucksberg 1978)
5.48 N 5.47 Y 5.25 N
5.09 Y 4.95 N 4.87 N 4.83 N 4.57 N
4.13 N 3.96 N 3.45 N 3.12 N
2.71 N 2.70 N
.13 .14 .20 .22 .45 .10
.38 .23 .42 .30 .23 .20 .45 .23 .28 .05
.23 .07 .45 .07 .38 .32
.32 .21
.10 .20 .13 .07
Black Door Wind Ceili
Fence
Fly
Mosq Ant Wasp
Beetl Bee Flea
Moth Locus Firefl
Fruit Category
9.92 Y .00 .00
Banana
9.58 Y
.07 .07
C
g i
e n
w
d r
h a
o n
u e
y Grassh
Vagueness: sorites paradox
1 grain of sand is not a heap
If n grains of sand is not a heap, then n+1 grains of sand is not a heap
There are no heaps of sand
Classical sets
Sets are entirely characterised by their membership function
Fuzzy sets
Fuzzy set in universe X = mapping from X to [0,1]
Vagueness: sorites paradox
1 grain of sand is a heap to degree 0
Adding one grain of sand to a collection increases the degree to which that collection is a heap by 10-12 (until the degree reaches 1)
Fuzzy logic: truth degrees vs. epistemic states
truth(“The glass is filled with wine”) = 0.5
Define the truth degree as the percentage of the volume that is filled
Vagueness results from the finite nature of natural languages
Fuzzy logic: truth degrees vs. epistemic states
probability(“The glass is filled with wine”) = 0.5
or
Probability degree depends on
Randomness: if the outcome of a coin flip is tails, somebody will fill the glass with wine.
Epistemic state: I am prepared to take a bet, as soon as the betting odds are such that
expected gain = 0.5 X moneyWin – (1-0.5) x moneyLose > 0
Uncertainty depends on randomness or on the epistemic state of an agent
Historical context
“Every proposition is either true or false”
Aristotle
The principle of bivalence
Historical context
Aristotle
“Every proposition is either true or false”
‣There will be a sea battle tomorrow ‣There will not be a sea battle tomorrow
Historical context
∧
0
0.5
1
0
0
0
0
0.5
0
0.5
0.5
1
0
0.5
1
∨
0
0.5
1
0
0
0.5
1
0.5
0.5
0.5
1
1
1
1
1
➝
0
0.5
1
0
1
1
1
0.5
0.5
1
1
1
0
0.5
1
¬
0
1
0.5
0.5
1
0
Jan Łukasiewicz
V(p) = 0.5 “it is possible that p is true”
Historical context
“Every proposition is either true or false”
There is no set whose cardinality is strictly between that of the integers and that of the real numbers
Stephen Cole Kleene
Historical context
∧
0
0.5
1
0
0
0
0
0.5
0
0.5
0.5
1
0
0.5
1
∨
0
0.5
1
0
0
0.5
1
0.5
0.5
0.5
1
1
1
1
1
➝
0
0.5
1
0
1
1
1
0.5
0.5
0.5
1
1
0
0.5
1
¬
0
1
0.5
0.5
1
0
Stephen Cole Kleene
V(p) = 0.5 “the truth of p is undefined/underspecified”
Historical context
Bertrand Russell
“Every proposition is either true or false” The king of France is bald
Russell: proposition is false, because it implicitly means “there is a king of France, and he is bald”
Historical context
“Every proposition is either true or false” The king of France is bald
Peter Strawson
Strawson: the sentence has no truth value (it is not a proposition)
The meaning of truth degrees
Bruno De Finetti
“Propositions are assigned only two values, true and false, and no other, not because there exists an a priori truth called excluded middle law, but because we call ‘propositions’ logical entities built in such a way that only a yes/no answer is possible … A logic, similar to the usual one, but leaving room for three or more values, cannot aim but at compressing several ordinary propositions into a single many-valued logical entity, which may very well turn out to be very useful”
What fuzzy logic offers is the ability to distinguish between a continuum of different states. As such, it is a vehicle to model continuous phenomena in a logic, rather than a technique to model vagueness per se.