Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation

Vakarelov, Dimiter

doi:10.1007/s11225-006-9004-y

Cited by 9 publications

(2 citation statements)

References 92 publications

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“…A few words in connection with Vakarelov (2005Vakarelov ( , 2006 are in order at this point. He considers a weak negation in combination with a strong negation in the sense of Nelson (1949), which makes it somewhat difficult to compare to our work.…”

Section: P1: To Show: δ ∈ N (U ) If and Only Ifmentioning

confidence: 99%

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Subminimal negation

2016

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Minimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the most important ones are contraposition logic and negative ex falso, a weak form of the ex falso principle. Completeness is proved, and the created semantics is further studied. The negative translation of classical logic into intuitionistic logic is made part of a chain of translations by introducing translations from minimal logic into contraposition logic and intuitionistic logic into minimal logic, the latter having been discovered in the correspondence between Johansson and Heyting. Finally, as a bridge to the work of Franco Montagna a start is made of a study of linear models of these logics.Dedicated to the memory of Franco Montagna.

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Section: P1: To Show: δ ∈ N (U ) If and Only Ifmentioning

confidence: 99%

“…So, we use the term subminimal negation in a non-technical sense. This term has been used before by Vakarelov (2005Vakarelov ( , 2006) with a more restricted meaning. We will return to this point later.…”

Section: Introductionmentioning

confidence: 99%