K3, Ł3, LP, RM3, A3, FDE, M: How to Make Many-Valued Logics Work for You

Abstract

We investigate some well-known (and a few not-so-well-known) many-valued logics that have a small number (3 or 4) of truth values. For some of them we complain that they do not have any logical use (despite their perhaps having some intuitive semantic interest) and we look at ways to add features so as to make them useful, while retaining their intuitive appeal. At the end, we show some surprising results in the system FDE, and its relationships with features of other logics. We close with some new examples of "synonymous logics." An Appendix contains a natural deduction system for our augmented FDE, and proofs of soundness and completeness.

4-Valued Semantics Under the OWA: A Deductive Database Approach

“…For instance, assuming that the truth value of Φ is n, then Φ ⇒ Φ is not valid since its truth value is n. Notice in this respect that Figure 1 shows that Φ → Φ, Φ → Φ, Φ * → Φ and Φ * → Φ are valid for any truth assignment regarding Φ. As argued in [2,12,16], considering ⇒ as an implication does not satisfy the deduction theorem, because the formula Φ defined by (…”

“…3. The database semantics is defined based on the four valued semantics as defined in [12,16]. These semantics reflect the Open World Assumption (OWA), contrary to most database models that assume the Closed World Assumption (CWA) ( [14]).…”

“…We define semantics for databases containing contradictions in the context of OWA. To do so, we review the formalism introduced in [3] and further discussed in the literature [2,7,12,16], in particular regarding implication. 2.…”

“…For example, Figure 1 shows that formulas of the form Φ∨¬Φ are not always true, independently from the truth value of Φ. More importantly, it has been argued in [2,12,16] that defining the implication Φ 1 ⇒ Φ 2 as usual in two-valued logics, that is by ¬Φ 1 ∨ Φ 2 , is problematic.…”

“…To see this, we consider as in [2,3,12,16], that t and b are the two designated truth values, and a formula Φ is said to be valid if its truth value is designated. For instance, assuming that the truth value of Φ is n, then Φ ⇒ Φ is not valid since its truth value is n. Notice in this respect that Figure 1 shows that Φ → Φ, Φ → Φ, Φ * → Φ and Φ * → Φ are valid for any truth assignment regarding Φ.…”