I Axioms of Fuzzy Logic

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Boolean algebra operates on two truth values, true and false, and is governed by various axioms such as involution, commutation, and absorption. Fuzzy logic extends this concept by allowing truth values to range between 0 and 1, introducing new axioms like monotonicity while preserving many properties from Boolean logic. Key operations such as conjunction, disjunction, and negation are adapted to maintain commutativity and associativity, with specific functions like Gödel-Zadeh norms illustrating these relationships. The discussion highlights the implications of distributiveness, absorption, and idempotence in fuzzy logic, revealing that these properties cannot coexist simultaneously. Overall, the exploration of negation and exclusive or functions in fuzzy logic showcases the complexity and adaptability of logical frameworks.
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Extensions of axioms of Boolean algebra, but what extensions are possible?
Boolean algebra, or crisp logic, involves functions of two truth values: true (T) and false (F). These functions satisfy various interrelationships: Boolean algebra (structure) - Wikipedia Some of these functions:

Negation: not ¬ -- conjunction: and ∧ -- disjunction: or ∨ -- exclusive or: xor ⊕ (exclusive disjunction, symmetric difference) -- (symbols from List of logic symbols - Wikipedia)

They satisfy several properties, which can be treated as axioms:
  • Involution (self-inversion): ¬ (¬ x) = x
  • Commutation, association: ∧, ∨, ⊕ -- ∧, ∨ distributive over each other
  • Absorption: x ∨ (x ∧ y) = x -- x ∧ (x ∨ y) = x
  • Idempotence: x ∧ x = x ∨ x = x
  • Identity: ∧ T, ∨ F, ⊕ F -- Zero: ∧ F, ∨ T
  • DeMorgan inversion: ¬ (x ∧ y) = (¬ x) ∨ (¬ y) -- ¬ (x ∨ y) = (¬ x) ∧ (¬ y)
  • XOR Inversion: ¬ (x ⊕ y) = (¬ x) ⊕ y -- x ⊕ y = (¬ x) ⊕ (¬ y)
  • Complementation: non-contradiction: x ∧ (¬ x) = F -- excluded middle: x ∨ (¬ x) = T
In fuzzy logic, we generalize the variables' domain from {T,F} to real numbers [0,1] where 0 ~ F and 1 ~ T. We attempt to carry over as much of these axioms as we can, and we introduce a new one: monotonicity: for operator "op", if x <= u and y <= v, then (x op y) <= (u op v). We make and ∧, or ∨, xor ⊕ commutative, associative, and monotonic, with the same identities and zeros as in crisp logic. For more, T-norm - Wikipedia. These functions can be expressed as changes of variables applied to other commutative, associative, and monotonic functions. For op -> op' with function f and its inverse fi:

x op' y = fi( f(x) op f(y) )

Common choices for op are addition and multiplication, themselves related with the exp and log functions.

Negation we can carry over by making it satisfy involution, making it a bijection, and if it is continuous, then it is monotonic: x < y implies (¬ x) > (¬ y). A non-monotonic discontinuous negation function that is an involution is, for argument x, (x rational: 1 - x), (x irrational: x). One with piecewise continuity is (x < 1/3 and x > 2/3: 1 - x), (1/3 <= x <= 2/3: x). It may also be carried over by imposing monotonic decrease with increasing argument, however.

With negation, we can relate conjunction (and ∧) and disjunction (or ∧) with DeMorgan inversion.

Let us consider idempotence and complementation with a continuous negation function. If negation is continuous, then there is some self-negating "middle" value m between 0 and 1 such that m = ¬ m. If ∧ and ∨ satisfy idempotence, then x ∧ (¬ x) = x ∨ (¬ x) = m, equaling neither 0 nor 1. Thus, self-negation, idempotence and complementation cannot coexist.
 
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Continuing further, let us consider the consequences of fuzzy logic having the distributive property.

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and likewise for ∧ and ∨ interchanged.

Set z = 1 (interchanged: z = 0). This reduces distributiveness to absorption:

x = (x ∧ y) ∨ x = (x ∨ y) ∧ x

Set y = 0 (interchanged: y = 1). This reduces absorption to idempotence:

x = x ∨ x = x ∧ x

So distributiveness -> absorption -> idempotence. This is equvalent to: not idempotence -> not absorption -> not distributiveness.

Let us go further with monotonicity. For y < z, x ∧ y <= x ∧ z.

Consider x <= y <= 1. Then,

x ∧ x <= x ∧ y <= x ∧ 1 (= x)

If ∧ is idempotent, then

x <= x ∧ y <= x

and we find the Gödel-Zadeh norm: x ∧ y = min(x,y) with conorm x ∨ y = max(x,y). This can be shown to satisfy distributiveness and absorption, though not complementation.

Negation is only constrained to be monotonically decreasing, however.
 
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Having found exactly one conjunction-disjunction set that is distributive, let us see what one can find of complementation.

Here, however, there is more than one conjunction-disjunction set that satisfies this property. For example,
  • Łukasiewicz: x ∧ y = max(x+y-1, 0) and x ∨ y = min(x+y, 1) with negation ¬ x = 1 - x.
  • Drastic: x ∧ y = y if x = 1, x if y = 1, 0 otherwise. x ∨ y = y if x = 0, x if y = 0, 1 otherwise. Negation can be any that is strictly monotonic.
What else might satisfy this property?
 
Let's look at negation more closely. Most fuzzy-logic work assumes simple reflection:

¬ x = 1 - x

It is easy to show involution, self-inversion, that ¬ (¬ x) = x. But Wikipedia's pages contain bilinear reflection, with parameter p:

¬ x = (1 - x) / (1 + p*x)

It is also an involution.

In analogy with the binary operations, one can create a function f and its inverse fi that do this mapping:

{0, m, 1} - f -> {-1, 0, 1} - fi -> {0, m, 1}

where ¬ x = fi(- f(x))

For reflection, f(x) = 2x-1 and fi(x) = (1+x)/2.
 
These functions can be used to set up an exclusive-or function that satisfies the xor-inversion axioms:

fxor(x) = - fneg(x), fixor(x) = fineg(-x)

x ⊕ y = fi( f(x) * f(y) )

For plain reflection, x ⊕ y = x + y - 2*x*y

For bilinear reflection, x ⊕ y = (x + y - 2*x*y) / (1 + p*x*y)
 
As I'd mentioned earlier, (distributiveness) > (absorption) > (idempotence) > (Gödel-Zadeh minmax and-or) > (Idempotence), (absorption), (distributiveness) -- one can prove idempotence, absorption, and distributiveness from minmax. Since these implications go in both directions, that means that these properties have equivalent truth value:

(distributiveness) == (absorption) == (idempotence) == (and-or is minmax)

I was unable to track down the reference, but I earlier found an extended or super kind of absorption:

x ∧ ((¬ x) ∨ y) = x ∧ y
x ∨ ((¬ x) ∧ y) = x ∨ y

This can be verified for crisp logic, so let us see what it implies in fuzzy logic.

First one, y = 0 ... x ∧ (¬ x) = 0
Second one, y = 1 ... x ∨ (¬ x) = 1
One finds complementation.

y = x ... x = x ∧ x = x ∨ x
One also finds idempotence.

But these two properties cannot coexist in fuzzy logic, therefore, this extended absorption is not satisfied in fuzzy logic.
 
Exclusive or again. The previous versions:
  • Quadratic (simple negation): x ⊕ y = x + y - 2*x*y
  • Biquadratic (bilinear negation): x ⊕ y = (x + y - 2*x*y) / (1 + p*x*y)
I have discovered a piecewise linear version that satisfies the axioms with simple negation. For (condition 1) and (condition 2), it has (value).
Cond 1Cond 2Value
y >= xy >= 1-x1-x
y >= xy <= 1-xy
y <= xy >= 1-x1-y
y <= xy <= 1-xx

If some xor satisfies the axioms with some negation, then

0 ⊕ x = x ⊕ 0 = x
1 ⊕ x = x ⊕ 1 = ¬ x
For m = ¬ m,
m ⊕ x = x ⊕ m = m
 
This piecewise linear version is equivalent to

x ⊕ y = min( max(x,y), max(1-x,1-y) ) = max( min(x,1-y), min(y,1-x) )

and all three definitions can be extended to alternative negations, by replacing 1-x with some other ¬ x.
 

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