Remember when I asked a question about ternary logic? It was my first question here.
Let $F$, $U$, $T$ be the truth values, where $F$ is designated for false and $T$ is designated for true. Though the truth tables in above question were absurd, it turns out that it was because I assumed $\neg U \doteq U$.
I dropped $\neg U \doteq U$, and assumed that the logic reduces to classical logic if only $F$ and $T$ are picked from the truth tables. Also, there is no modality anymore.
That leaves 4 possible systems:
System #0: $$ \begin{array}{l|l} p & ¬p \\ \hline F & T \\ U & F \\ T & F \end{array} \begin{array}{l|l l} p \to q & F & U & T \\ \hline F & T & U & T \\ U & F & T & T \\ T & F & T & T \end{array} $$
System #1: $$ \begin{array}{l|l} p & ¬p \\ \hline F & T \\ U & F \\ T & F \end{array} \begin{array}{l|l l} p \to q & F & U & T \\ \hline F & T & T & T \\ U & F & T & T \\ T & F & T & T \end{array} $$
System #2: $$ \begin{array}{l|l} p & ¬p \\ \hline F & T \\ U & T \\ T & F \end{array} \begin{array}{l|l l} p \to q & F & U & T \\ \hline F & T & T & T \\ U & T & T & T \\ T & F & F & T \end{array} $$
System #3: $$ \begin{array}{l|l} p & ¬p \\ \hline F & T \\ U & T \\ T & F \end{array} \begin{array}{l|l l} p \to q & F & U & T \\ \hline F & T & T & T \\ U & T & T & T \\ T & F & U & T \end{array} $$
And finally, I could give $U$ a meaningful name. For System #0 and #1, $U$ would be called Semitrue. That is, "assumable as an axiom, but not actually assumed." For System #2 and #3, $U$ would be in dual notion, Semifalse.
Yet some questions arise:
Is there any way to formally define $\land$ and $\lor$ in these systems? They turn out to be non-functional. For example, let $p$ be the continuum hypothesis in ZFC, and let $q$ be "continuum hypothesis is false in ZFC." Then $p$ and $q$ are $U$. $p \land p$ is $U$ due to the idempotency of $\land$, yet $p \land q$ is $F$ due to the law of contradiction.
Will any of these systems resolve Gödel's incompleteness? That is, is every proposition designatable into a truth value?
My conclusion from reviewing MVL, the Lewis systems, and intuitionist logic has been that is not possible to consistently retain:
At least one of these has to go.
Łukasiewicz logic rejects a and b and keeps c and d. In my view, the conditional it does use is inferior for logical purposes. An alternate and more satisfactory conditional can be defined within the system. Lewis logics keep a and b, at the cost of c and d. Intuitionism rejects a, retains b, and loses c and d.
The most difficult to sacrifice is the law of the excluded middle. This should be no great sacrifice if a third value is being considered at all, but the idea that (not possible for $P ∧ \neg P$) is a conditional statement that is true for some propositions, but not a universal one that applies to all is very hard to accept. To deny it in any case yields an irreducible "maybe, and maybe not" as a valid logical category, which is an intellectual irritant and significantly complicates the problem of how to prove or disprove propositions when it must be accounted for.
Next is the definition of the material conditional. $P → Q ≡ \neg P ∨ Q$. While this works in classical two-valued logic, this is not the essential property of the conditional. It is instead an artifact of having only two values. It does not work and confuses the essential issue in working with three or more values. The essential issue is that $P → Q$ should assert a relationship between the truth values of $P$ and $Q$, such that $Q$ is not less true than $P$.