Equivalence

This is a continuation of a series of posts on three-valued logic which began with “The Learned Professors”

The strict Lukasiewicz conditional, unlike the ordinary material condition, the strict Lewis conditional, or the ordinary Lukasiewicz conditional, meets the condition for an ordering relation: It is symmetric (A => A), antisymmetric (A => B & B=> A if and only if A = B), and transitive (if A => B and B => C, then A => C). It also satisfies the requirements for a desirable logical entailment relation: if A => B is true, B is not less true than A.

In classical two-valued logic, the notion of logical equivalence is expressed by the biconditional; “if A then B, and if B then A”, or “A if and only if B”. Because of the simplicity of 2-valued logic, this also expresses the happy property of being truth functional, that is, it is possible to evaluate a logical expression algebraically, by plugging in truth values for the variables and evaluating the expressions to determine whether they are true or false. This also expresses the notion that if two expressions are logically equivalent, they have the same truth value. The laws of logic are tautologies, that is, they are expressions that are always true, whatever truth values are substituted for the logical variables. This also a mathematical equivalence, because it is reflexive, (A = A), Symmetric (if A = B then B = A, and transitive (if A = B and B = C, then A = C).

This becomes more difficult in three valued logic. Most of the “logical equivalences” used in three valued logic are biconditionals, but not mathematical equivalences, and do not reflect the notion that two expressions have the same truth value. This is particularly obnoxious with the Lukasiewicz biconditional.
<-> P
* T U F
* T T U F
Q U U T U
* F F U T

However, it is simple enough to apply the “Definite” operator, [](P <-> Q) to define a new strict Lukasiewicz biconditional
<=> P
* T U F
* T T F F
Q U F T F
* F F F T

This does have the same desirable properties of being both a logical equivalence and a mathematical equivalence.
It’s simple enough, but it hasn’t been done and does not appear in the standard references or surveys of multi-valued logic. Kleene’s 3-valued logic does have this equivalence, but it uses the standard definition of the conditional P -> Q = (~P \/ Q), which doesn’t work in three values.

Next, I will discuss some of the more revolutionary consequences and implications of this logic.

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