Many-valued logic

Remarks

With $V = \set{0, 1}$, $D = \set{1}$ and the usual tables, the tautologies are those of classical propositional logic; clause 1 excludes this two-valued case.

Example: Łukasiewicz's three-valued logic $\mathrm{L}_3$ has $V = \set{0, \tfrac12, 1}$ and $D = \set{1}$. Its tables are $\neg x = 1 - x$, $x \to y = \min(1, 1 - x + y)$, $x \lor y = \max(x, y)$ and $x \land y = \min(x, y)$. The formula $p \to p$ is a tautology, since $\min(1, 1 - x + x) = 1$ for every $x$.

Non-example: $p \lor \neg p$ is a classical tautology but not a tautology of $\mathrm{L}_3$. The valuation $p \mapsto \tfrac12$ gives $\max(\tfrac12, \tfrac12) = \tfrac12 \notin D$.

Infinite-valued logics use, for example, $V = [0, 1]$ with the same tables.