Notre Dame Journal of Formal Logic

Peter van Inwagen, Substitutional Quantification, and Ontological Commitment

William Craig

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Peter van Inwagen has long claimed that he doesn’t understand substitutional quantification and that the notion is, in fact, meaningless. Van Inwagen identifies the source of his bewilderment as an inability to understand the proposition expressed by a simple sentence like “(Σx) (x is a dog),” where “Σ” is the existential quantifier understood substitutionally. I should think that the proposition expressed by this sentence is the same as that expressed by “(x) (x is a dog).” So what’s the problem? The problem, I suggest, is that van Inwagen takes traditional existential quantification to be ontologically committing and substitutional quantification to be ontologically noncommitting, which requires that the two quantifiers have different meanings—but no different meaning for the substitutional quantifier is forthcoming. What van Inwagen fails to appreciate is that substitutional quantification is directed at a criterion of ontological commitment, namely, W. V. O. Quine’s, which is quite different from van Inwagen’s criterion. Substitutional quantification successfully avoids the commitments Quine’s criterion would engender but has the same commitments as existential quantification given van Inwagen’s criterion. The question, then, is whether the existential quantifier is ontologically committing, as van Inwagen believes. The answer to that question will depend on whether the ordinary language “there is/are,” which is codified by the existential quantifier, is ontologically committing. There are good reasons to doubt that it is.

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Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 553-561.

First available in Project Euclid: 7 November 2014

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Peter van Inwagen philosophy of logic ontological commitment substitutional quantification existential quantifiers


Craig, William. Peter van Inwagen, Substitutional Quantification, and Ontological Commitment. Notre Dame J. Formal Logic 55 (2014), no. 4, 553--561. doi:10.1215/00294527-2798736.

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