Logic, Metalogic and Neutrality

Erkenntnis 79 (Suppl 2):211-231 (2013)
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Abstract

The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own. A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds, with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral.

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References found in this work

Modal Logic as Metaphysics.Timothy Williamson - 2013 - Oxford, England: Oxford University Press.
Vagueness.Timothy Williamson - 1994 - London and New York: Routledge.
Philosophy of Logic.W. V. Quine - 1970 - Harvard University Press.
Saving Truth From Paradox.Hartry Field - 2008 - Oxford, England: Oxford University Press.

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