Many Valued Logic (MVL)
By: Shay Erov - 01/11/2007
Many Valued Logic
• Today’s lecture:
– What is MVL?
– Motivation
– MVL structure
– Some 3-valued logics
– Consequence relations
– n-valued logics
– Dunn/Belnap 4-valued logic
– Extension to the first-order level
Many Valued Logic
• Today’s lecture (continued) :
– Alternative
– Fuzzy Logic (∞-valued logics)
– Applications
– History review
– Summary
What is MVL?
• A truth-functional logic containing more
than 2 truth values
• We will mainly focus on 3-valued logics,
and Fuzzy logics (∞-valued )
Motivation
• Motivation for Many Valued Logic came
mainly from paradoxes raised by
philosophers
• MVL usually comes to give an answer to
questions where it is hard / impossible to
determine the truth values
• In the next couple of slides we will cover
some of those motivations
Motivation – Inconsistent law
• Let assume that in a certain country, the
following rules apply:
– No aborigine shall have the right to vote
– All property holders shall have the right to
vote
• Now, let’s say John is an aborigine that did
manage to buy a property.
Is John allowed to vote?
Motivation – self reference
• Consider the following two paradoxes:
– The Liar’s paradox : This sentence is false
– Russell's paradox: Define a group r in the
following way: {x| xx}. is rr?
• Any truth value assigned to one of the
above, will result in it having both true and
false values.
• Extended paradox : this sentence is either
false or neither true nor false
Motivation – denotation failure
• There may be a sentence where neither A
nor A can be verified
• “The present king of Canada was born in
Italy”
• “Sherlock Holmes had 3 aunts”
• Sentences containing noun references
that do not refer to anything
• Frege suggested that all such sentences
are neither true nor false. Is that right?
Motivation – Future Contingents
• The sea battle paradox
– There will be a sea battle tomorrow
• If there’s a truth value, then what will
happen is of necessity, and it’s not
possible since it’s still contingent matter.
Motivation – Future Contingents
• Łukasiewicz :
If statements about the future events are
already true or false, then the future is as
much determined as the past and differs
from the past only in so far as it has not
yet come to pass
MVL – Definitions
• Language (atomic propositions,
Connectives)
• Assignment / Interpretation
• Valid formula
• Designated values
MVL – Structure
•
•
•
•
{V , D,{ fc | c  C}
V - Set of truth degrees
D - Set of designated truth values
f c - for every n-place connective c, an
n-place function on V to denote it
• Truth degrees meaning
• How do we define the classical logic this
way?
3-valued logics
• Consist of 3 truth values, 2 main types for
the third value, i,:
– Neither true nor false
– Both true and false
• and now for some examples…
3-valued logics – L3
• Invented by Łukasiewicz
• i stands for ‘possible’
• {1} as the only designated value
f
f 1 i 0
f 1 i 0
f 1 i 0
1 0
1 1 i 0
1 1 1 1
1 1 i 0
i i
i i i 0
i 1 i i
i 1 1 i
0 1
0 0 0 0
0 1 i 0
0 1 1 1
• The tables for {, , } are intuitive
• Why is i  i also 1 ?
3-valued logics – K3
• Invented by Kleene
• i stands for ‘possible’
• {1} as the only designated value
f
f 1 i 0
f 1 i 0
f 1 i 0
1 0
1 1 i 0
1 1 1 1
1 1 i 0
i i
i i i 0
i 1 i i
i 1 i i
0 1
0 0 0 0
0 1 i 0
0 1 1 1
• Same as L3 besides i  i
• What is the problem here?
3-valued logics – LP
i stands for ‘both true and false’
{1, i} as the designated values
Tables the same as K3
Follows the rule of excluded middle

p p
• Does not follow Modus Ponens
p, p  q q
•
•
•
•
3-valued logics – RM 3
• Based on LP, change in table of f 
f 1 i 0
1 1 0 0
i 1 i 0
0 1 1 1
• Follows Modus Ponens
3-valued logics – Bochvar
• Trying to avoid the paradoxes
• i stands for ‘meaningless’
• {1} as the only designated value
f
f 1 i 0
1 0
1 1 i 0
i i
i i i i
0 1
0 0 i 0
• Any formula involving i is meaningless
• No tautologies, introducing ‘assertion’
3-valued logics – Bochvar
• Łukasiewicz
• Kleene
• Bochvar
f 1 i 0
f 1 i 0
f 1 i 0
1 1 i 0
1 1 i 0
1 1 i 0
i 1 1 i
i 1 i i
i i i i
0 1 1 1
0 1 1 1
0 1 i 1
f 1 i 0
f 1 i 0
f 1 i 0
f 1 i 0
f 1 i 0 f 1 i 0
1 1 i 0
1 1 1 1
1 1 i 0
1 1 1 1
1 1 i 0
1 1 i 0
i i i 0
i 1 i i
i i i 0
i 1 i i
i i i i
i i i i
0 0 0 0
0 1 i 0
0 0 0 0
0 1 i 0
0 0 i 0
0 0 i 0
• What are the main differences?
Consequence Relations
• To simplify notation we work with
– 1 two-place connective ( * )
– 1 one-place connective 
• A matrix for language L consist of
– A non empty set, V of truth values
– A non empty subset of V, D of the designated
values.
– A set of interpretations for the connectives of
L (for every n-place connective c, an n-place
function f c )
Consequence Relations
• g is a substitution for a language L if it is a
function from the set of variables into L
g
We write  for the result of applying the
substitution g to the variables in 
•  is consistent with respect to if there’s a
formula  so that  
Consequence Relations
• A is a consequence of Σ with respect to
the matrix M (Σ MA) if for every
assignment g, if g(B)  D for all BΣ then
g(A)  D
• A is a tautology with respect to the matrix
M if  MA
Consequence Relations
• A consequence relation for a matrix M
satisfies the following conditions:
–  M if  
– if ,  M and 
M
 , then , 

M
• Matrix M is a characteristic matrix of an
abstract sequence relation if it
coincides with M
M
Consequence Relations
•
is a structural consequence relation if it
satisfies the rule of uniform substitution:
M
– If 
•
M
g

and
g
is
a
substitution
then

M
g

M
is uniform if it satisfies :
– If ,   , Var (  {})  Var ()   and  is
consistent, then  
• If is a uniform structural consequence
relation then has a characteristic matrix.
n-valued logics – Ln
• Generalization of L3 for n truth values
•
 k

V 
| 0  k  n  1
 n 1

• truth functions :
x  1  x
x  y  min{1,1  ( x  y )}
x  y  min( x, y )
x  y  max( x, y )
n-valued logics - Post
• Truth values set = {0 … m-1}
n
• Tm
• Designated values – {0 … n}
x  y  min( x, y )

x  x  1 (mod m)
• Have a proof for completeness
Dunn/Belnap 4-valued logic
• V  {,{F },{T },{F , T }}
• The two additional values represent
uncertainty / inconsistency
• The truth degrees ordering is non-linear,
two natural ways to order it are :
Extension to first-order level
• We will use Lnas an example
• We extend the language to contain:
– Individual variables
– Predicates
– The quantifiers , 
• Given non empty set of individuals I
– Constants a,b,c… to correlate with them
Extension to first-order level
• An m-valued structure is an assignment of
a truth-value to the atomic sentences in
the Language.
• For a1...an in I and an n-place predicate P
we give the value Pa1..an
• And for the quantifiers:
x  min   a / x  | a  I 

x  max   a / x  | a  I

Alternative - Supervaluations
• Given a K3 interpretation v, v’ is a
supervaluation of v if it is the same as v
except where v(p)=i, in which case v’(p) is
either 1 or 0.
• v’ represents one of the ways things could
turn
Alternative - Supervaluations
• We define
s
in the following way:
– Σ sA iff for every interpretation v, every
supervaluation v’ that makes every B 
true, makes A true.
• This definition reduces to truth
preservation under all classical
interpretations
• s is just classical logic, although allowing
truth-value gaps.
Fuzzy Logic (∞-valued logics)
• Fuzzy Logic is a type of MVL where the
set of truth values is infinite
• Usually the set of all real values in [0,1]
are used, where 1 is true.
Fuzzy Logic – Motivation
• Sorites paradox (the bald man / the heap)
– (i) One grain of wheat does not make a heap
– (ii) Adding one grain of wheat to something
that is not a heap, does not make it a heap
– Apply (ii) 9,999 times and you’ll get that
10,000 grains of wheat do not make a heap.
Fuzzy Logic – Motivation
• Can be displayed as a sequence of
Modus Ponens:
M0
M 0  M1
M1
M1  M 2
M2
M k 1
M k 1  M k
Mk
Fuzzy Logic – Motivation
• M 2 is definitely true. M k is definitely false
• If there exist a unique i for which M i is true
and M i 1 is false, then M i  M i 1 is false.
• It is counterintuitive that one small hair will
make the person bald.
The continuum-valued logic - Ł
• Truth values – all real numbers in [0,1]
• Extension of Ln for infinite truth-values
• Same truth functions
x  1  x
x  y  min{1,1  ( x  y )}
x  y  min( x, y )
x  y  max( x, y )
The continuum-valued logic - Ł
• Designated values
– Are context dependent
– We choose ε and Dwill be {x | x   }
• Consequence relations
– Σ  A iff for all interpretation v, if v(B)  
for all B Σ, then v(A)  
– Context independent : Σ A iff for all  Σ  A

The continuum-valued logic - Ł
• Modus Ponens is invalid when 
• v(p)=½ → v(^p)=½ →
1

p

p)=½
v(
• Let’s assume there are 3 balls (a,b)
– Red(a)=1, Small(a)=½
– Red(b)=½, Small(b)=½
– Which ball is better for Red & Small?
Applications of MVL
• Linguistics – treating sentences with
presuppositions (assumptions created by
the sentence)
• Logic – understanding other systems of
logic, merging modalities, and modeling
truth-values gaps
• Philosophical – understanding the
meaning of “truth” in paradoxes
Applications of MVL
• Hardware design – designing many
valued switches (more than two voltage
levels)
• Artificial Intelligence – using fuzzy logic
for vague notions and commonsense
reasoning.
• Mathematics – using the logical matrices
as a technical tool for consistency proofs
in the set theory
Some history
•
•
•
•
Aristotle
[~1920] Łukasiewicz L3 & Ln
[1952] Kleene K3
[1975] Kripke suggested i should be lack
of truth value and not a third one
• [1979] Priest created LP
• [1969] van Fraassen invented
Supervaluations
Summary
• Last words and room for questions