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This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead (...) |
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This article discusses the proof theory, semantics and meta-theory of a class of adaptive logics, called hierarchic adaptive logics. Their specific characteristics are illustrated throughout the article with the use of one exemplary logic HKx, an explicans for reasoning with prioritized belief bases. A generic proof theory for these systems is defined, together with a less complex proof theory for a subclass of them. Soundness and a restricted form of completeness are established with respect to a non-redundant semantics. It is (...) |
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In this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is $\Pi _1^1 - complete$ . So, the complexity results in (Horsten and Welch, Synthese 158:41–60,2007) are mistaken for adaptive logics using the minimal abnormality strategy. |
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A logic is said to be contraction free if the rule from A→(A→B) to A→B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there is another contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to be robustly contraction free if (...) |
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This paper answers the philosophical contentions defended in Horsten and Welch . It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts. |
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This paper answers the philosophical contentions defended in Horsten and Welch (2007, Synthese, 158, 41–60). It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts. |
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. In this paper, adaptive logics are studied from the viewpoint of universal logic (in the sense of the study of common structures of logics). The common structure of a large set of adaptive logics is described. It is shown that this structure determines the proof theory as well as the semantics of the adaptive logics, and moreover that most properties of the logics can be proved by relying solely on the structure, viz. without invoking any specific properties of the (...) |
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The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads (...) |
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On the one hand, the absence of contraction is a safeguard against the logical (property theoretic) paradoxes; but on the other hand, it also disables inductive and recursive definitions, in its most basic form the definition of the series of natural numbers, for instance. The reason for this is simply that the effectiveness of a recursion clause depends on its being available after application, something that is usually assured by contraction. This paper presents a way of overcoming this problem within (...) |
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