Beall, Jc; Forster, Thomas; Seligman, Jeremy A note on freedom from detachment in the logic of paradox. (English) Zbl 1272.03115 Notre Dame J. Formal Logic 54, No. 1, 15-20 (2013). Summary: We shed light on an old problem by showing that the logic LP cannot define a binary connective \(\odot\) obeying detachment in the sense that every valuation satisfying \(\varphi\) and \((\varphi \odot \psi)\) also satisfies \(\psi\), except trivially. We derive this as a corollary of a more general result concerning variable sharing. Cited in 8 Documents MSC: 03B53 Paraconsistent logics 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03B80 Other applications of logic Keywords:detachment-free logics; detachable connective; relevance logics; variable-sharing; paraconsistent logic; logic of paradox PDFBibTeX XMLCite \textit{J. Beall} et al., Notre Dame J. Formal Logic 54, No. 1, 15--20 (2013; Zbl 1272.03115) Full Text: DOI Euclid References: [1] Arieli, O., A. Avron, and A. Zamansky, “Maximally paraconsistent three-valued logics,” pp. 310-18 in Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning (KR ’10) , AAAI Press, Palo Alto, Calif., 2010. [2] Asenjo, F. G., “A calculus of antinomies,” Notre Dame Journal of Formal Logic , vol. 7 (1966), pp. 103-5. · Zbl 0145.00508 · doi:10.1305/ndjfl/1093958482 [3] Beall, J., Spandrels of Truth , Oxford University Press, Oxford, 2009. · doi:10.1093/acprof:oso/9780199285495.003.0009 [4] Beall, J., “Multiple-conclusion LP and default classicality,” Review of Symbolic Logic , vol. 4 (2011), pp. 326-36. · Zbl 1252.03008 · doi:10.1017/S1755020311000074 [5] Belnap, N. D., “A useful four-valued logic,” pp. 5-37 in Modern Uses of Multiple-Valued Logics (Bloomington, Ind., 1975) , edited by J. Dunn and G. Epstein, vol. 2 of Episteme , Reidel, Dordrecht, 1977. [6] Dowden, B., “Accepting inconsistencies from the paradoxes,” Journal of Philosophical Logic , vol. 13 (1984), pp. 125-30. · Zbl 0543.03005 · doi:10.1007/BF00453017 [7] Dunn, J. M., The Algebra of Intensional Logics , Ph.D. dissertation, University of Pittsburgh, 1966. · Zbl 0145.45104 [8] Goodship, L., “On dialetheism,” Australasian Journal of Philosophy , vol. 74 (1996), pp. 153-61. [9] Kleene, S., Introduction to Metamathematics , D. Van Nostrand, New York, 1952. · Zbl 0047.00703 [10] Lewis, D., “Logic for equivocators,” Noûs , vol. 16 (1982), pp. 431-41. · Zbl 1366.03090 · doi:10.2307/2216219 [11] Middelburg, C. A., “A survey of paraconsistent logics,” preprint, [cs.LO]. 1103.4324 [12] Priest, G., “The logic of paradox,” Journal of Philosophical Logic , vol. 8 (1979), pp. 219-41. · Zbl 0402.03012 · doi:10.1007/BF00258428 [13] Priest, G., In Contradiction: A Study of the Transconsistent , Martinus Nijhoff, Dordrecht, 1987; 2nd edition, Oxford University Press, Oxford, 2006. · Zbl 0682.03002 [14] Takano, M., “Interpolation theorem in many-valued logics with designated values,” Kodai Mathematical Journal , vol. 12 (1989), pp. 125-31. · Zbl 0697.03009 · doi:10.2996/kmj/1138039033 [15] Woodruff, P., “Paradox, truth and logic,” Journal of Philosophical Logic , vol. 13 (1984), pp. 213-32. · Zbl 0546.03006 · doi:10.1007/BF00453022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.