Valuation Semantics for First-Order Logics of Evidence and Truth

A conventional expansion of first-order Belnap-Dunn logic

“…The connectives that have been added to BD in its studied expansions include the unary is-designated connective ∆ and the unary classicality connective ⊛ (also knows as the normality connective). The connective ∆ is for example found in the expansion of BD known as BD∆ [12] and the connective ⊛ is for example found in the expansion of BD known as QLET F [2]. 4 In the setting of BD, the unary consistency connective ○ that is for example found in the expansion of the first-order version of Priest's logic of paradox known as LP ○ [10] and its counterpart in the setting of Kleene's strong three-valued logic, the unary determinedness connective ⭐, are closely related to the classicality connective.…”

“…The logics BD ⊃,F , BD∆ [12], and QLET F [2] are the same as the logics BD[{⊃, F}], BD[{∆}], and BD[{⊛}], respectively.…”

“…Moreover, it has been shown in Section 5.1 that several interesting nonclassical connectives added to Belnap-Dunn logic in its studied expansions are definable in BD ⊃,F , viz. the is-designated connective ∆ from BD∆ [12], the classicality connective ⊛ from QLET F [2], and the consistency connective ○ and determinedness connective ⭐ studied in the setting of BD in [4].…”

“…However, these studies primarily concern expansions with connectives that are not familiar from classical logic. Some examples of such expansions are BD∆ [12], E fde ⊛ [4], F4CC [5], and QLET F [2].…”