Contradiction and (in)Consistency

Carnielli, Walter; Coniglio, Marcelo E.

doi:10.1007/978-3-319-33205-5_1

Cited by 1 publication

(6 citation statements)

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“…Condition (iii) is required in order to satisfy condition (ii) in a non-trivial way. The hierarchy of LFI s studied in [15, 20] starts from a logic called mbC , which extends positive classical logic CPL by adding a negation and a primitive consistency operator satisfying minimal requirements in order to define an LFI .…”

Section: The Logic Rmbcmentioning

confidence: 99%

“…The logic mbC ( ciw ) was considered in [15] under the name mbCciw . This logic was introduced in [2] under the name , presented by means of a standard Gentzen calculus such that B is a Gentzen calculus for mbC .…”

Section: Adding Replacement To Extensions Of Mbc: a Solution To An Op...mentioning

confidence: 99%

“…Now, let RCila be the logic ; then, this logic derives . Indeed, as shown in [15, Proposition 3.1.10], axiom is derivable from mbC plus axiom . This being so, by Example 3.25 and the fact that is a bottom formula in Cila (hence in RCila ) for any (i.e., implies any other formula), the connective could be eliminated from the signature of RCila , and so the logic RCila could be defined over the signature (recall Definition 2.2).…”

Section: Limits For Replacement While Preserving Paraconsistencymentioning

confidence: 99%

“…In [24, Theorem 3.51(iv)] it was proved that cannot hold in any paraconsistent extension of Ci in which the schema : is valid. On the other hand, in [15, Theorem 3.6.4] it was proved that the logic obtained from mbCciw by adding is equivalent to the logic obtained from mbCciw by adding the schema axiom . Since RCila derives and , it also derives the schema .…”

Section: Limits For Replacement While Preserving Paraconsistencymentioning

confidence: 99%

“…This result was generalized in [24, Theorem 3.83] to Cila , the presentation of in the language of the LFI s (where the consistency connective is primitive instead of being defined in terms of the other connectives). We obtain as a consequence that no such algebraization is possible for any other of the LFI s weaker than Cila to be discussed in Section 3 (i.e., the systems mbC , mbCciw , bC and Ci studied in [15, 20, 24]). The same reasoning applies to every calculus in the infinite da Costa’s hierarchy, since they are weaker than .…”

Section: Introduction: the Quest For The Algebraic Counterpart Of Par...mentioning

confidence: 99%

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Logics of Formal Inconsistency Enriched With Replacement: An Algebraic and Modal Account

Carnielli

Coniglio

Fuenmayor³

2021

The Review of Symbolic Logic

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One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C 1 , each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E ⊕ E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics.

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