46 research outputs found
On formal aspects of the epistemic approach to paraconsistency
This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelsonâs logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed
Recovery operators, paraconsistency and duality
There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common coreâ in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with 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
An epistemic approach to paraconsistency: a logic of evidence and truth
The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true.
The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE), and extends it to the Logic of Evidence and Truth (LETj). The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation
An epistemic approach to paraconsistency: a logic of evidence and truth
The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true.
The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE), and extends it to the Logic of Evidence and Truth (LETj). The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation
On Barrio, Lo Guercio, and Szmuc on Logics of Evidence and Truth
The aim of this text is to reply to criticisms of the logics of evidence and truth and the epistemic approach to paraconsistency advanced by Barrio [2018], and Lo Guercio and Szmuc [2018]. We also clarify the notion of evidence that underlies the intended interpretation of these logics and is a central point of Barrioâs and Lo Guercio & Szmucâs criticisms
On philosophical motivations for paraconsistency: an ontology-free interpretation of the logics of formal inconsistency
In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non- contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formal inconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ÂŹA does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A
On six-valued logics of evidence and truth expanding Belnap-Dunn four-valued logic
The main aim of this paper is to introduce the logics of evidence and truth
LETK+ and LETF+ together with a sound, complete, and decidable six-valued
deterministic semantics for them. These logics extend the logics LETK and LETF-
with rules of propagation of classicality, which are inferences that express
how the classicality operator o is transmitted from less complex to more
complex sentences, and vice-versa. The six-valued semantics here proposed
extends the 4 values of Belnap-Dunn logic with 2 more values that intend to
represent (positive and negative) reliable information. A six-valued
non-deterministic semantics for LETK is obtained by means of Nmatrices based on
swap structures, and the six-valued semantics for LETK+ is then obtained by
imposing restrictions on the semantics of LETK. These restrictions correspond
exactly to the rules of propagation of classicality that extend LETK. The logic
LETF+ is obtained as the implication-free fragment of LETK+. We also show that
the 6 values of LETK+ and LETF+ define a lattice structure that extends the
lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional
values mentioned above, intuitively interpreted as positive and negative
reliable information. Finally, we also show that LETK+ is Blok-Pigozzi
algebraizable and that its implication-free fragment LETF+ coincides with the
degree-preserving logic of the involutive Stone algebras.Comment: 38 page
On the philosophical motivations for the logics of formal consistency and inconsistency
We present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account of paraconsistency
Measuring evidence: a probabilistic approach to an extension of Belnap-Dunn Logic
This paper introduces the logic of evidence and truth LETF as an extension of the Belnap-Dunn four-valued logic F DE. LETF is a slightly modified version of the logic LETJ, presented in Carnielli and Rodrigues (2017). While LETJ is equipped only with a classicality operator â, LETF is equipped with a non-classicality operator â as well, dual to â. Both LETF and LETJ are logics of formal inconsistency and undeterminedness in which the operator â recovers classical logic for propositions in its scope.
Evidence is a notion weaker than truth in the sense that there may be
evidence for a proposition α even if α is not true. As well as LETJ,
LETF is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for LETF where statements P(α) and P(âα) express, respectively, the
amount of evidence available for α and the degree to which the evidence for α is expected to behave classically - or non-classically for P(âα). A probabilistic scenario is paracomplete when P(α) + P(α) 1, and in both cases, P(âα) < 1. If P(âα) = 1, or P (âα) = 0, classical probability is recovered for α. The proposition âα âš âα, a theorem of LETF , partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theor