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The Metaphysics of Logic
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    Cole, Julian 2017. Humanizing Mathematics and its Philosophy. p. 151.

    Stotts, Megan Henricks 2017. Walking the tightrope: unrecognized conventions and arbitrariness. Inquiry, Vol. 60, Issue. 8, p. 867.

    Atiq, Emad H. 2018. On Ground as a Guide to Realism. Ratio, Vol. 31, Issue. 2, p. 165.

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    The Metaphysics of Logic
    • Online ISBN: 9781139626279
    • Book DOI: https://doi.org/10.1017/CBO9781139626279
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Book description

Featuring fourteen new essays from an international team of renowned contributors, this volume explores the key issues, debates and questions in the metaphysics of logic. The book is structured in three parts, looking first at the main positions in the nature of logic, such as realism, pluralism, relativism, objectivity, nihilism, conceptualism, and conventionalism, then focusing on historical topics such as the medieval Aristotelian view of logic, the problem of universals, and Bolzano's logical realism. The final section tackles specific issues such as glutty theories, contradiction, the metaphysical conception of logical truth, and the possible revision of logic. The volume will provide readers with a rich and wide-ranging survey, a valuable digest of the many views in this area, and a long overdue investigation of logic's relationship to us and the world. It will be of interest to a wide range of scholars and students of philosophy, logic, and mathematics.

Reviews

'Rush's volume will be recommended as important reading for anyone interested in the philosophy of logic. Each article is interesting and well edited, doing a service to the study of logic.'

James Cargile - University of Virginia

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Contents

  • Chapter 5 - A Second Philosophy of logic
    pp 93-108
  • View abstract

    Summary

    Logic might chart the rules of the world itself; the rules of rational human thought; or both. Husserl had a very broad concept of logic that embraces our usual modern idea of logic as well as something he called pure logic, which we can loosely characterise as something like the fundamental forms of experience. For Husserl, the fundamental forms of pure logic are an in-eliminable part of experience: i.e. experience encompasses direct apprehension of these inferential relationships. The apprehended structures are abstract and platonic; discovered, rather than constructed. Theory, empirical observation, and experience are in this sense fallible: they may or may not get it right and reveal the actual independent structure of logic. Both logic and mathematics as they are characterised by Husserl, should encounter the realist problem of independence, neither are the sort of thing we can simply take as part of human cognition.
  • Chapter 6 - Logical nihilism
    pp 109-127
  • View abstract

    Summary

    Our logical practices, it seems, already exhibit "truth by convention". A visible part of contemporary research in logic is the exploration of non classical logical systems. It's sad that almost no one notices that Quine's refutation of the conventionality of logic is a dilemma. The famous Lewis Carroll infinite regress assails only one horn of this dilemma, the horn that presupposes that the infinitely many needed conventions are all explicit. One of the oldest ways of begging the question against proponents of alternative logics (as well as a popular way of begging the question against logical conventionalism) is to implicitly adopt a lofty metalanguage stance, and then use the very words that are under contention against the opponent. That doing this is so intuitive evidently contributes to the continued popularity of the fallacy.
  • Chapter 7 - Wittgenstein and the covert Platonism of mathematical logic
    pp 128-144
  • View abstract

    Summary

    This chapter discusses a kind of relativism or pluralism concerning logic. It explores a core metaphysical issue concerning logic, the extent to which logic is objective. The chapter adopts a Hilbertian perspective, either the original version where consistency is the only formal, mathematical requirement on legitimate theories, or the liberal orientation where there are no formal requirements on legitimacy at all. It explores the ramifications for what the author takes to be a longstanding intuition that logic is objective. This chapter explains the matter of objectivity with the present folk-relativism concerning logic in focus. Sometimes it concentrates on general logical matters, such as validity and consistency, as such, and sometimes it deals with particular instances of the folk-relativism, such as classical validity, intuitionistic consistency, and the like. The chapter limits the discussion to Wright's axes of epistemic constraint and cognitive command.
  • Chapter 8 - Logic and its objects: a medieval Aristotelian view
    pp 147-159
  • View abstract

    Summary

    In general, the Second Philosopher's epistemological investigations take the form of asking how human beings, as described in biology, physiology, psychology, linguistics, and so on come to have reliable beliefs about the world as described in physics, chemistry, botany, astronomy, and so on. Second Philosopher's focus is somewhat broader; not only does she study how people come to form beliefs about the world, she also takes it upon herself to match these beliefs up with what her other inquiries have told her about how the world actually is, and to assess which types of belief-forming processes, in which circumstances, are reliable. After all, even Second Philosophy and Second Philosopher are used to describe her and her behavior. In any case, philosophy or not, the Second Philosopher's investigations do tell us something about the nature of the inference about the foreign coin.
  • Chapter 9 - The problem of universals and the subject matter of logic
    pp 160-177
  • View abstract

    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
    pp 178-188
  • View abstract

    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.
  • Chapter 12 - Revising logic
    pp 211-223
  • View abstract

    Summary

    The semantic notions of truth and logical validity in predicate logic, being dependent on what the correlates of our universal terms are, demand at least a certain semantic clarification of the issue of universals. Apparently, the primary issue concerning universals is ontological. It should be clear that these objective concepts are non-conventionally objective. It should also be clear that the laws of logic in the framework are supposed to be fundamentally different from the laws of psychology. For while the former are the laws of the logical relations among objective concepts, the latter are the laws of the causal relations among formal concepts. Thus, whereas logic can be normative, prescribing the laws of valid inference, cognitive psychology can only be descriptive, describing and perhaps explaining the psychological mechanisms that can make us prone to certain types of logical errors.
  • Chapter 13 - Glutty theories and the logic of antinomies
    pp 224-232
  • View abstract

    Summary

    A transcendental philosophy as described and practiced by Kant is itself a logic. It is not intended to decide such factual questions as whether there is a God or humans are free, but to address semantical issues like what the meaning of God or freedom is. Within the semantical space where the (transcendental) logical enterprise is located, one can take different words as primitives and establish a network of semantical relations and dependencies based on those primitives. A logic is a self-organizing structure, self-enclosed and self-referential, that provides the bare scaffolding of a world and, if given enough data, even a large part of its actual construction. Logic is a highly ambitious theory: one that attempts to construct a universal language. In and by itself, this theory will be found persuasive only by those who are already committed to the particular view it expresses and articulates.
  • Chapter 14 - The metaphysical interpretation of logical truth
    pp 233-248
  • View abstract

    Summary

    Bolzano's Theory of Science presents the first explicit and methodical espousal of internal logical realism. It also contains a formidable number of theoretical innovations. They include: the first account of the distinction between sense and reference; definitions of analyticity and consequence, i.e. deducibility based on a new substitutional procedure that anticipates Quine's and Tarski's, respectively; and an account of mathematical knowledge that excludes, contra Kant. In Bolzano's case, one of the main purposes in introducing propositions in themselves is to achieve precise and satisfactory definitions. By way of consequence, on Bolzano's own account the success of the endeavour depends on whether his commitment to propositions allows him to deliver a good theory of logic, or at least one that is preferable to its rivals. Bolzano did have views on epistemic modality, though unfortunately, there is no place for a discussion of the latter here.

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