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Admissibility of Cut in LC with Fixed Point Combinator

Studia Logica 81 (3):399-423 (2005)

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On the decidability of implicational ticket entailment.Katalin Bimbó & J. Michael Dunn - 2013 - Journal of Symbolic Logic 78 (1):214-236.details The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) Download Export citation Bookmark 8 citations
Dual Gaggle Semantics for Entailment.Katalin Bimbó - 2009 - Notre Dame Journal of Formal Logic 50 (1):23-41.details A sequent calculus for the positive fragment of entailment together with the Church constants is introduced here. The single cut rule is admissible in this consecution calculus. A topological dual gaggle semantics is developed for the logic. The category of the topological structures for the logic with frame morphisms is proven to be the dual category of the variety, that is defined by the equations of the algebra of the logic, with homomorphisms. The duality results are extended to the logic (...) Download Export citation Bookmark 6 citations
Current Trends in Substructural Logics.Katalin Bimbó - 2015 - Journal of Philosophical Logic 44 (6):609-624.details This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic. Download Export citation Bookmark 1 citation
Relevance logics and relation algebras.Katalin Bimbó, J. Michael Dunn & Roger D. Maddux - 2009 - Review of Symbolic Logic 2 (1):102-131.details Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) Download Export citation Bookmark 4 citations
New Consecution Calculi for R→t.Katalin Bimbó & J. Michael Dunn - 2012 - Notre Dame Journal of Formal Logic 53 (4):491-509.details The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) Download Export citation Bookmark 11 citations
(2 other versions)${LE}^{t}{{\rightarrow}}$ , ${LR}^{\circ}{\hat{\sim}}$, {LK} and Cutfree Proofs.Katalin Bimbó - 2007 - Journal of Philosophical Logic 36 (5):557-570.details Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) Download Export citation Bookmark 7 citations

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