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- TL;DR Summary
- 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:
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.
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
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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