A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory
01 Jan 2015-pp 165-183
TL;DR: Soundness and completeness theorems are established in a three-valued paraconsistent logic obtained from some algebra-valued model of set theory.
Abstract: This paper presents a three-valued paraconsistent logic obtained from some algebra-valued model of set theory. Soundness and completeness theorems are established. The logic has been compared with other three-valued paraconsistent logics.
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08 Jan 2015
TL;DR: It is proved that the collection of all ordinals is not a set in this model which is dissimilar to the other existing paraconsistent set theories.
Abstract: This paper deals with ordinal numbers in an algebra-valued model of a paraconsistent set theory. It is proved that the collection of all ordinals is not a set in this model which is dissimilar to the other existing paraconsistent set theories. For each ordinal α of classical set theory α-like elements are defined in the mentioned algebra-valued model whose collection is not singleton. It is shown that two α-like elements (for same α) may perform conversely to validate a given formula of the corresponding paraconsistent set theory.
7 citations
TL;DR: In this paper, the authors show that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.
Abstract: An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.
6 citations
TL;DR: In this paper, the authors extend the standard strategy of proving independence using Boolean-valued models to non-classical set theories by combining algebras (by taking their product), which is able to provide product-algebravalued models of set theories.
Abstract:
In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of
$\\mathsf {CH}$
); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse.
4 citations
TL;DR: Two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) andParaconsistent pre-rough logic (PPRL) are presented as examples of logics with some rules of inference that have variable sharing restrictions imposed on them.
Abstract: In this paper, we study two companions to a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems - Intuitionistic Paraconsistent Weak Kleene logic (IPWK) and Paraconsistent Pre-Rough logic (PPRL) - are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of Intuitionistic Propositional logic (IPC) and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of Pre-Rough logic (PRL) but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Plonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.
3 citations
Cites background from "A Paraconsistent Logic Obtained fro..."
...It has been proved in [41] that the Deduction theorem holds in LPS3....
[...]
...of a formula algebra of type L, the logic LPS3 can be described syntactically as the Hilbert-style logic 〈Fm,⊢LPS3〉 with the following axioms and rules [41]....
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...However, as noted in [41], LPS3 is paraconsistent....
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...Comparisons between some well-known paraconsistent systems can be found in [2, 18, 19, 41]....
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...2 LPS3 Our second example is the logic LPS3 introduced in [41]....
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TL;DR: In this article, the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values.
Abstract: We present a generalization of the algebra-valued models of $$\mathrm {ZF}$$
where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate $$\mathrm {ZF}$$
.
2 citations
References
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507 citations
01 Jan 2007
TL;DR: The Logicas da Inconsistencia Formal (LIFs) as mentioned in this paper form a classe of logicas paraconsistentes particularmente expressivas, in which a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to.
Abstract: Segundo a pressuposicao de consistencia classica, as contradicoes tem um cara[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocinio sensato pode entao ter lugar. Uma logica e paraconsistente se ela rejeita uma tal pressuposicao, e aceita ao inves que algumas teorias inconsistentes conquanto nao-triviais facam perfeito sentido. A? Logicas da Inconsistencia Formal, LIFs, formam uma classe de logicas paraconsistentes particularmente expressivas nas quais a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to. Como consequencia, as LIFs sao capazes de recapturar o raciocinio consistente pelo acrescimo de assuncoes de consistencia apropriadas. Assim, por exemplo, enquanto regras classicas tais como o silogismo disjuntivo (de A e {nao-,4)-ou-13, infira B) estao fadadas a falhar numa logica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presuncao de que estamos raciocinando em um ambiente consistente (neste caso, pelo acrescimo de (consistente-.A) como uma hipotese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustracoes destas logicas e de suas propriedades, mostrando que tais logicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocinio consistente dentro de tais sistemas inconsistentes sao tambem ilustradas Em cada caso, interpretacoes em termos de semânticas polivalentes, de traducoes possiveis ou modais sao fornecidas, e os problemas relacionados a provisao de contrapartidas algebricas para tais logicas sao examinados. Uma abordagem formal abstra[cjta e proposta para todas as definicoes relacionadas e uma extensa investigacao e feita sobre os principios logicos e as propriedades positivas e negativas da negacao
Abstract
348 citations
164 citations
TL;DR: New logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic are introduced and form part of a much larger family of similar logics.
Abstract: This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.
116 citations
01 Jan 1979
86 citations