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4138 results in Logic

Page 14 of 207

  • 9 - Topology

  • from Part III - Where Are the Paradoxes?
  • Zach Weber, University of Otago, New Zealand
  • Book:
    Paradoxes and Inconsistent Mathematics
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 256-282

  • Summary

    This chapter develops some topology, the abstract geometry ofcloseness, that manages to capture properties of nearness withoutany appeal to distance. The previous chapter studied the shape ofthe linear continuum, focusing on what happens when one tries to cutor tear it in two. This chapter offers a qualitative generalizationof these ideas, about continuity and connectedness, but now withoutany metrics. The focus is on closed sets, boundaries, andeventually, continuous transformations and their fixed points,bringing ideas from analysis back to set theory. Concluding theoremsare proven about retractions and some lemmas about absolutelydisconnected space, leading to a parameterized version ofBrouwer’s fixed point theorem.

  • 4 - Prolegomena to Any Future Inconsistent Mathematics

  • from Part II - How to Face the Paradoxes?
  • Zach Weber, University of Otago, New Zealand
  • Book:
    Paradoxes and Inconsistent Mathematics
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 110-148

  • Summary

    This chapter looks at the constraints on a logic for inconsistentmathematics. Curry’s paradox is presented in several forms,leading to problems for conditionals and validity. Grisin’sparadox and the Hinnion–Libert paradox lead to problems foridentity (equality) and the axiom of set extensionality. Inresponse, a broadly substructural response is recommended, where allforms of “contraction” are dropped, and a“relevant” logic is adopted to preserveextensionality. This leads to presentation of the official logicused in the second half of the book, and its main properties areoutlined, including some further methodological considerations forworking without contraction.

Paradoxes and Inconsistent Mathematics
  • Zach Weber
  • Published online:
    08 October 2021
    Print publication:
    21 October 2021
  • Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.

Page 14 of 207