What difference does it make: Three truth-values or two plus gaps? (Q700692)

A 3-valued logic is called strongly 3-valued iff the equivalence connective (primitive or defined in the classical way) satisfies the property \(v(\alpha \equiv \beta) = 1\) iff \(v(\alpha) = v(\beta)\). A table is called regular if a given column (row) contains a 1 at the \(\square\)-row (column) only if the column (row) consists entirely of 1's, and likewise for 0 (where \(\square\) denotes the third value or gap). A logic is called regular if the tables of all its primitive connectives are regular. The author concludes that Łukasiewicz's logics are strongly 3-valued but lack the feature of monotonicity, which disqualifies them from epistemological applications. However, since the sets of tautologies are never identical to the set of classical tautologies, this provides a promising basis for ontological applications. Kleene's logics are weakly 3-valued logics, but are regular and thus monotonic and as such well suited for epistemological applications.