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2 results

  • Chapter 9 - The problem of universals and the subject matter of logic

  • from Part II - History and Authors
  • Edited by Penelope Rush, University of Tasmania
  • Book:
    The Metaphysics of Logic
    Published online:
    05 October 2014
    Print publication:
    16 October 2014, pp 160-177

  • Summary

    The idea that there may be more than one correct logic has recently attracted considerable interest. The most notorious bone of contention in the discussion of logical correctness is the law of excluded middle. The connections between sequent calculus, constructive proof transformation, structural completeness, and lem are fixtures from logical knowledge store, but they cannot seriously be thought of as a network of consequences in some allegedly correct logic. The author indicates logic's modern contours, highlighting the fact that the deepest observations logic has to offer come with no ties to preconceptions about its essence. The richness of logic comes into view only when we stop looking for such an essence and focus instead on the accumulation of applications and conceptual changes that have made current logical investigations possible. The study of logic might be the best practical antidote to the view of it that we have inherited.

  • Chapter 10 - Logics and worlds

  • from Part II - History and Authors
  • Edited by Penelope Rush, University of Tasmania
  • Book:
    The Metaphysics of Logic
    Published online:
    05 October 2014
    Print publication:
    16 October 2014, pp 178-188

  • Summary

    Wittgenstein's thought on mathematics had undergone a major, if often undetected, change. The idea that adopting an algorithm like "plus" determines in some physical, mental, or metaphysical way one's response to infinitely many exercises is nothing but covert Platonism, in many ways worse than the Platonism of objects. Wittgenstein agrees entirely with the Intuitionist critique of the law of excluded middle. For the Goldbach conjecture to be true in the sense of classical mathematics, we have to say that the operations of arithmetic determine in advance that every even number, no matter how large, can be partitioned into two primes. The law of excluded middle cannot be regarded as a hardened regularity in cases in which it is applied it to a putative infinite totality. But precisely because of this, there is no direct comparison possible between empirical observations and mathematical theorems in this type of proof.