Recovery operators, paraconsistency and duality

Abstract

recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices.

On Barrio, Lo Guercio, and Szmuc on Logics of Evidence and Truth

“…From items (1) and (2), LET s are Logics of Formal Inconsistency and Undeterminedness [cf. Carnielli et al, 2020;Carnielli and Rodrigues, 2017a]. Item (3) expresses the fact that a formula •A implies that A and •-free formulas formed with A behave classically.…”

Semantical Investigations on Non-classical Logics with Recovery Operators: Negation

“…It is worth remarking that we have leveraged automated provers (via Sledgehammer) together with model finder Nitpick to uncover minimal semantic conditions (modulo existence of 'not-too-big' finite countermodels) under which the above relationships hold. 10 In fact, surprisingly few of an operators' conditions (axioms) are actually required in most cases (we refer the reader to [15] for details). As a side remark, we are not aware of any works in the literature aiming to systematic uncover this sort of minimal conditions (questions like, e.g., "which of the four Kuratowski conditions are actually necessary to prove .…”

“…Let us now introduce a corresponding algebraic operation • ∶=  fp ( ), which we shall call a consistency operator inspired by the paraconsistent Logics of Formal Inconsistency (LFIs; introduced in [11], cf. also [9,10]) from which we took inspiration. Moreover, we can do the same 'recovering' exercise for the (dual) TND(¬ ) property, thus obtaining an operator ☆ ∶=  fp ( ), such that ☆ → ∨ ¬ ≈ ⊤, i.e., ⊢ ★ ∨ ∨ ¬ (for , arbitrary).…”

Measuring evidence: a probabilistic approach to an extension of Belnap–Dunn logic

“…Fact 11) is analogous to the notion of undeterminedness in LFU s. Actually, in our view, except for the same acronym of LFI s, LFU s could well be called Logics of Formal Incompleteness. The name LFU was established in Marcos (2005) and adopted in Carnielli and Rodrigues (2017) and Carnielli, Coniglio, and Rodrigues (2019).…”