##
**Noncontractive classical logic.**
*(English)*
Zbl 1472.03020

Summary: One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference.

### MSC:

03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |

03A05 | Philosophical and critical aspects of logic and foundations |

### Software:

Pesca
PDF
BibTeX
XML
Cite

\textit{L. Rosenblatt}, Notre Dame J. Formal Logic 60, No. 4, 559--585 (2019; Zbl 1472.03020)

### References:

[1] | Barrio, E., L. Rosenblatt, and D. Tajer, “Capturing naive validity in the Cut-free approach,” Synthese, published electronically September 1, 2016. |

[2] | Beall, J., and J. Murzi, “Two flavors of Curry’s paradox,” Journal of Philosophy, vol. 110 (2013), pp. 143-65. |

[3] | Caret, C. R., and Z. Weber, “A note on contraction-free logic for validity,” Topoi, vol. 34 (2015), pp. 63-74. · Zbl 1382.03050 |

[4] | Cintula, P., and F. Paoli, “Is multiset consequence trivial?,” Synthese, published electronically September 8, 2016. |

[5] | Cobreros, P., P. Égré, D. Ripley, and R. van Rooij, “Reaching transparent truth,” Mind, vol. 122 (2013), pp. 841-866. |

[6] | Da Ré, B., and L. Rosenblatt, “Contraction, infinitary quantifiers, and omega paradoxes,” Journal of Philosophical Logic, vol. 47 (2018), 611-29. · Zbl 1436.03143 |

[7] | Field, H., Saving Truth from Paradox, Oxford University Press, New York, 2008. · Zbl 1225.03006 |

[8] | Fjellstad, A., “How a semantics for tonk should be,” Review of Symbolic Logic, vol. 8 (2015), pp. 488-505. · Zbl 1382.03016 |

[9] | Fjellstad, A., “\( \omega \)-inconsistency without cuts and nonstandard models,” Australasian Journal of Logic, vol. 13 (2016), pp. 96-122. · Zbl 1422.03122 |

[10] | Fjellstad, A., “Non-classical elegance for sequent calculus enthusiasts,” Studia Logica, vol. 105 (2017), pp. 93-119. · Zbl 1417.03200 |

[11] | Girard, J. Y., “Linear logic: Its syntax and semantics,” pp. 1-42 in Advances in Linear Logic (Ithaca, NY, 1993), edited by J. Y. Girard, Y. Lafont, and L. Regnier, vol. 222 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1995. · Zbl 0828.03003 |

[12] | Hinnion, R., and T. Libert, “Positive abstraction and extensionality,” Journal of Symbolic Logic, vol. 68 (2003), pp. 828-36. · Zbl 1056.03027 |

[13] | Hjortland, O. T., “Theories of truth and the maxim of minimal mutilation,” Synthese, published electronically November 10, 2017. |

[14] | Humberstone, L., “Heterogeneous logics,” Erkenntnis, vol. 29 (1989), pp. 395-435. |

[15] | Kripke, S., “Outline of a theory of truth,” Journal of Philosophy, vol. 72 (1975), pp. 690-716. · Zbl 0952.03513 |

[16] | Mares, E., and F. Paoli, “Logical consequence and the paradoxes,” Journal of Philosophical Logic, vol. 43 (2014), pp. 439-69. · Zbl 1302.03021 |

[17] | Negri, S., and J. von Plato, Structural Proof Theory, Cambridge University Press, Cambridge, 2001. · Zbl 1113.03051 |

[18] | Priest, G., “The structure of the paradoxes of self-reference,” Mind, vol. 103 (1994), pp. 25-34. |

[19] | Priest, G., An Introduction to Non-Classical Logic: From If to Is, 2nd edition, Cambridge University Press, Cambridge, 2008. · Zbl 1148.03002 |

[20] | Ripley, D., “Conservatively extending classical logic with transparent truth,” Review of Symbolic Logic, vol. 5 (2012), pp. 354-78. · Zbl 1248.03012 |

[21] | Ripley, D., “Paradoxes and failures of cut,” Australasian Journal of Philosophy, vol. 91 (2013), pp. 139-64. |

[22] | Ripley, D., “Anything goes,” Topoi, vol. 34 (2015), pp. 25-36. · Zbl 1382.03078 |

[23] | Ripley, D., “Comparing substructural theories of truth,” Ergo, vol. 2 (2015), pp. 299-328. |

[24] | Ripley, D., “Contraction and closure,” Thought, vol. 4 (2015), pp. 131-38. |

[25] | Rosenblatt, L., “Naive validity, internalization, and substructural approaches to paradox,” Ergo, vol. 4 (2017), pp. 93-120. |

[26] | Rosenblatt, L., “On structural contraction and why it fails,” Synthese, published electronically May 14, 2019. |

[27] | Shapiro, L., “Naive structure, contraction and paradox,” Topoi, vol. 34 (2015), pp. 75-87. · Zbl 1382.03043 |

[28] | Steinberger, F., Harmony and Logical Inferentialism, Ph.D. dissertation, Cambridge University, Cambridge, 2009. |

[29] | Teijeiro, P., “What is tonk?,” preprint, 2018. |

[30] | Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, 2nd edition, vol. 43 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2000. · Zbl 0957.03053 |

[31] | Yaqub, A., The Liar Speaks the Truth, Oxford University Press, New York, 1993. |

[32] | Zardini, E., “Truth without contra(di)ction,” Review of Symbolic Logic, vol. 4 (2011), pp. 498-535. · Zbl 1252.03018 |

[33] | Zardini, E., “Breaking the chains: Following-from and transitivity,” pp. 221-75 in Foundations of Logical Consequence, edited by C. R. Caret and O. T. Hjortland, Oxford University Press, Oxford, 2015. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.