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9.3 MV-Algebras

167

9.3 MV-Algebras

In Section 9.1 we gave Lukasiewicz’s bold disjunction the numeric truth-condition V(P Q) = min(1, V(P) + V(Q)). Bold conjunction can be defined as P & Q =def ¬(¬P ¬Q), from which it follows that V(P & Q) = max (0, V(P) + V(Q) – 1). The algebraic structure L3MV induced by L3 bold disjunction, bold conjunction, and negation, that is, the structure <{1, 1/2, 0}, L, L, 1, 1, 0> where L, L, and 1are L3 bold disjunction, bold conjunction, and negation, is an MV-algebra. An MV-algebra is a an algebra <M, , , , unit, zero> that meets the following conditions for all x, y, and z in M:4

i.

x y = y x, and x y = y x

(commutation)

ii.

x (y z) = (x y) z, and x (y z) = (x y) z

(association)

iii.

x zero = x, and x unit = x

(identity for join and meet)

iv.

x unit = unit, and x zero = zero

(unit and zero consumption)

v.

x x = unit, and x x = zero

(complementation)

vi.

(x y) = x y , and (x y) = x y

(DeMorgan’s Laws)

vii.

x = x

(Double Negation)

viii. zero = unit

(duality of zero and unit)

ix.

(x y) y = (y x) x

(lattice meet commutation)

(The reason for calling condition ix lattice meet commutation will be explained later.) We’ll show that bold disjunction and bold conjunction are both commutative and associative operations on {1, 1/2, 0}, thus meeting the first two conditions on MV-algebras:

Commutation: x L y = x L y, and x L y = x L y

Proof of first equation: For all x, y {1, 1/2, 0}, min (1, x + y) = min (1, y + x) since x + y = y + x.

Proof of second equation: Left as an exercise.

Association: x L (y L z) = (x L y) L z, and x L (y L z) = (x L y) L z

Proof of first equation: For all x, y, z {1, 1/2, 0}, min (1, x + min (1, y + z)) = min (1, min (x + 1, x + y + z)) = min (1, x + 1, x + y + z), and since x 0 it follows that x + 1 1, so min (1, x + 1, x + y + z) = min (1, x + y + z). Similarly, min (1, min (1, x + y) + z) = min (1, min (1 + z, x + y + z)) = min (1, 1 + z, x + y + z) = min (1, x + y + z).

Proof of second equation: Left as an exercise.

4

MV-algebras were first developed in Chang (1958b, 1959). MV is short for many-valued.

 

 

These are not Chang’s original conditions but are a variation of an equivalent formulation

 

given in Mangani (1973). An excellent monograph that explores various formulations of MV-

 

algebras is Cignoli, D’Ottaviano, and Mundici (2000).

168

Alternative Semantics for Three-Valued Logic

It is also left as an exercise to show that the remaining conditions on MV-algebras hold for L3MV.

We noted in Section 9.2 that Lukasiewicz/Kleene negation cannot serve as a complementation operation forLKL because it doesn’t satisfy the complementation conditions with respect to LKL’s lattice join and meet. However, as will be verified in the exercises, Lukasiewicz/Kleene negation does serve as complementation with respect to the MV-algebraic join and meet operations L and L. This isn’t surprising since we already knew that the bold disjunction Law of Excluded Middle A ¬A is an L3-tautology, while the bold conjunction A & ¬A is an L3-contradiction.

The first complementation condition for MV-algebras isn’t required as a separate condition since it can be derived using the other conditions:

x x = x x

(commutation)

= (x zero) x

(identity for join)

= (zero x) x

(commutation)

= (zero x) x

(Double Negation)

= (x zero ) zero

(lattice meet commutation)

= (x zero ) unit

(duality of zero and unit)

= unit

(unit consumption)

But this equality is so important that we have included it as a separate condition. We know that we can also derive the second complementation condition from

the other conditions because there is a general principle of duality for MV-algebras similar to that for Boolean algebras. Each of the conditions i–vi consists of a pair of equations such that join and meet have been interchanged and so have unit and zero. We can derive unit = zero, the dual to condition viii, from conditions vii and viii (see Exercise 7 for Section 9.2), and the dual to lattice meet commutation (which is lattice join commutation) is also derivable from other conditions—this is left as an exercise.

Every Boolean algebra is an MV-algebra, where the Boolean join serves as the MV-algebra join and the Boolean meet serves as the MV-algebra meet . We have already shown (in Chapter 4) that most of the conditions for MV-algebras hold for Boolean algebras; we leave it as an exercise to show that the duality of zero and unit and lattice meet commutation both hold for Boolean algebras as well. On the other hand, not all MV-algebras are Boolean algebras. For example, L3MV is an MV-algebra but it isn’t a Boolean algebra. In particular, idempotence fails since 1/21/2 = 1, not 1/2 as would be required by idempotence, and 1/2 1/2 = 0. Distribution also fails—for example, 1/2 (1 1) = 1/2, but (1/2 1) (1/2 1) = 1. (If distribution held then idempotence would hold as well—we showed in Chapter 4 how to derive idempotence from distribution, complementation, and identity for meet and join, and the latter conditions both hold in MV-algebras). In fact, an MV-algebra is a Boolean algebra if and only if idempotence and/or distribution holds for the MValgebra’s meet and join.

9.3 MV-Algebras

169

There is a natural ordering on MV-algebras defined as

x y if and only if x y = unit

Like its lattice counterpart, this ordering is reflexive, antisymmetric, and transitive. Alternatively, we can define an MV-algebra operation that satisfies the conditions defining lattice meet operations:

x y =def x (x y)

and then define the ordering as we did for lattices:

x y if and only if x y = x

It is left as an exercise to prove that when the lattice meet operation is defined as x (x y), x y = x if and only if x y = unit, so that the two definitions for the MV-relation coincide. An operation satisfying the conditions defining lattice join operations can also be defined within an MV-algebra, for example, as

x y =def (x y )

MV-algebras are said to contain lattices as substructures because these meet and join operations and together with the MV-algebraic operation form a lattice over the MV-algebra’s domain. And because of these substructures, the MV-algebra meet and join operations and are sometimes called bold meet and join to distinguish them from the lattice operations.

To prove that the operations and as defined in the previous paragraph do indeed form a lattice, we need to show that lattice conditions i–iv hold, that is, that these meet and join operations are commutative, associative, and idempotent, and that the absorption conditions hold. The name for condition ix in the definition of MV-algebras—lattice meet commutation—refers to the fact that this condition guarantees that the meet operation as we have just defined it is a commutative operation:

x y = x (x y)

=(x (x y))

=(x (x y) )

=((y x ) x )

=((y x ) x )

=((x y ) y )

=((x y ) y )

=(y (y x) )

=(y (y x))

=y (y x)

=y x

(definition) (Double Negation) (DeMorgan’s Law)

(commutation, twice) (Double Negation)

(lattice meet commutation) (Double Negation) (commutation, twice) (DeMorgan’s Law)

(Double Negation) (definition)

170

Alternative Semantics for Three-Valued Logic

(Note the second half of this derivation reverses the steps of the first half, but on different operands.) An equivalent derivation without lattice meet commutation cannot be constructed.

The associativity of can be derived as follows (we use lattice join commutation, which is the dual of lattice meet commutation, the derivation of which is left as an exercise). This derivation is tricky to follow, and it may help the reader to note that once again the second half of the derivation reverses the steps of the first half:

x (y z) = (y z) x

(lattice commutation, just established)

= (y (y z)) ((y (y z)) x)

(definition)

= y ((y z) ((y (y z)) x))

(association)

= y ((y z) ((y (y z) ) x))

(DeMorgan’s Law)

= y ((y z) (((y z) y ) x))

(commutation)

= y ((y z) ((y z) (y x)))

(association)

= y ((y z) ((y z) (y x)) )

(Double Negation)

= y ((y z) ((y z) (y x) ) )

(DeMorgan’s Law)

= y ((y z) ((y z) (y x) ) )

(Double Negation)

= y (((y x) (y z)) (y z))

(commutation, twice)

= y (((y z) (y x)) (y x))

(lattice join commutation)

= y ((y x) ((y x) (y z) ) )

(commutation, twice)

= y ((y x) ((y x) (y z) ) )

(Double Negation)

= y ((y x) ((y x) (y z)) )

(DeMorgan’s Law)

= y ((y x) ((y x) (y z)))

(Double Negation)

= y ((y x) (((y x) y ) z))

(association)

= y ((y x) ((y (y x) ) z))

(commutation)

= y ((y x) ((y (y x)) z))

(DeMorgan’s Law)

= (y (y x)) ((y (y x)) z)

(association)

= (y x) z

(definition)

= (x y) z

(commutation)

We leave it as an exercise to establish the remaining lattice properties for MV-algebra lattice meet and join operations.

Given the lattice substructure definable in an MV-algebra, it should come as no surprise that L3’s weak conjunction and disjunction connectives, corresponding to the meet and join operations of this lattice substructure, are definable using negation and the L3 bold connectives. Of course, we already knew this because we showed how to use theL3 bold connectives and negation to define the L3 conditional, and how to use this conditional and negation to define L3 weak disjunction and conjunction, but this shows how algebraic structures can also be used to support claims about logical systems. Using the definition of lattice meet in MV-algebras, for example, we conclude that P Q may be defined as P & (¬P Q).

9.3 MV-Algebras

171

We can also use bold join to define L, the algebraic counterpart of the Lukasiewicz conditional, completely for any MV-algebra as

x L y = x y

for all x and y, for recall that in L3 we have V(P Q) = V(¬P Q). In L3MV this gives us exactly the truth-conditions for a Lukasiewicz conditional.

We can now relate L3 tautologies and MV-algebra tautologies. We will say that a formula of propositional logic is a tautology of an MV-algebra if the formula evaluates to unit under every algebraic interpretation based on that algebra. First, we have

Result 9.2: Every three-valued MV-algebra MV = <{unit, zero, other}, , , , unit, zero> generates the following truth-tables for assignments of unit, zero, or other to each atomic formula of propositional logic when , , and respectively define the bold disjunction, bold conjunction, and negation operations (we use u, z, and o to stand, respectively, for unit, zero, and other):

P ¬P

u z o o z u

P Q

P \ Q

 

u o

z

 

u

 

u u u

o

 

u u o

z

 

u o z

P & Q

P \ Q

 

u

o z

 

u

 

u

o z

o

 

o

z z

z

 

z z z

Proof: Left as an exercise.

Given definitions of the L3 conditional and biconditional in terms of negation, bold disjunction, and bold conjunction we therefore have

Result 9.3: For any three-valued MV-algebra BA, the set of formulas of propositional logic that are MV-tautologies is exactly the set of L3 tautologies under the standard semantics {1, 1/2, 0}.

In Chapter 4 we proved a stronger result relating Boolean algebras and classical propositional logic: that for any Boolean algebra BA, the set of formulas that are BA-tautologies is exactly the set of classical tautologies. There is not an analogous result for relating L3 and MV-algebras, for some L3 tautologies are not tautologies in every MV-algebra. This will become clear when we study fuzzy propositional logics and their algebras in Chapters 11 and 12.