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Self-implications in BCI

Tomasz Kowalski

Notre Dame Journal of Formal Logic 49 (3):295-305 (2008)

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Variations on a Theme of Curry.Lloyd Humberstone - 2006 - Notre Dame Journal of Formal Logic 47 (1):101-131.details After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which closes with (...) Download Export citation Bookmark 8 citations
Fragments of R-Mingle.W. J. Blok & J. G. Raftery - 2004 - Studia Logica 78 (1-2):59-106.details The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Download Export citation Bookmark 8 citations
A Note on Monothetic BCI.Tomasz Kowalski & Sam Butchart - 2006 - Notre Dame Journal of Formal Logic 47 (4):541-544.details In "Variations on a theme of Curry," Humberstone conjectured that a certain logic, intermediate between BCI and BCK, is none other than monothetic BCI—the smallest extension of BCI in which all theorems are provably equivalent. In this note, we present a proof of this conjecture. Download Export citation Bookmark 5 citations
On a property of BCK-identities.Misao Nagayama - 1994 - Studia Logica 53 (2):227 - 234.details A BCK-algebra is an algebra in which the terms are generated by a set of variables, 1, and an arrow. We mean by aBCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two termss andt, if neithers=1 nort=1 is a BCK-identity, ands=t is a BCK-identity, then the rightmost variables of the two terms are identical.This theorem was conjectured firstly in [5], and then in [3]. As a corollary of this theorem, we (...) Download Export citation Bookmark 2 citations

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