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

There are about seven reasons this work is revolutionary.

1) First, it is a full-featured extension of classical logic into the realm of the uncertain. The arithmetic of the integers is an extension of the ordinary arithmetic of whole numbers into the negative, and the arithmetic of common factions is an extension of ordinary arithmetic into the realm of parts of objects. Both are fully compatible and include the whole numbers as a special case. In a similar fashion, all the laws of classical two-valued logic remain true in the two-valued case, but some of them must be modified in the three valued case.

2) It is truth functional and the same methods of truth truth tables and algebraic manipulation apply in this logic as in two valued logic. It follows the associative, commutative, distributive, identity, annihilator, idempotent, double negation, and De Morgan laws of Boolean algebra but does not follow the complementation laws x & ~x= F and x ∨ ~x = T. This is called a De Morgan algebra.

3) It is a truth functional system of modal logic. It differs from the Lewis systems S1-S5, because it uses a different version of the strict conditional, and because it does not include the law of the excluded middle. However, theorems of L3M that are parallel to the axioms of S5 can be formulated, therefore anything that can be proven in S5 can be proven with this logic. It has been established that S5 itself cannot be reduced to 3 values, but this is a narrow result and can probably be traced to the difference in conditionals.
It can also be established that that if the strict material conditional []( P =) Q) holds, then so does the strict Lukasewicz conditional [](P -> Q), but not conversely.

4) It is a truth functional version of intuitionism. In similar fashion to modal logic, if all the axioms of intuitionism are replaced using the strong negation ~<>P instead of the ordinary negation, and using the strict Lukasewicze conditional instead of the ordinary material conditional, the resulting axioms are theorems of this system. Therefore, anything that can be proven using these axioms is also true in this systems.

5) It clarifies long-standing controversies in logic, notably the meaning of the conditional and its relation with concepts of implication and entailment, and doubts about the universal validity of modus ponens. Along with fuzzy logic, it offers a resolution to the Sorites paradox. It addresses some of the same concerns as relevance logic. The Lukasiewicz conditional can be defined as (~Q \/ P \/ P==Q), if P== Q has been defined as P and Q having the same truth value.

6) It is connected to Fuzzy logic. It has some of the same features, although this logic includes all the values between T and F in the one value U, where Fuzzy logic gives each truth value a distinct number.

7) It shares features with paraconsistent logic. (P & ~P) => Q is not a tautology, so it is not explosive, and is not sufficient to prove any other proposition at all. However, this logic does include methods of indirect proof, which work by establishing contradictions. It is simply necessary to establish a contradiction and not merely a contrary. Expressions such as (p & ~<>P) “true and impossible”, ([]p & ~P) “certain and false”, ([]p & ~[]P) “certain and not certain”, (<>P & ~<>P) “possible and impossible”, and ([]P & ~<> P) “certain and impossible” all work.

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.