A Mechanically Assisted Examination of Vacuity and Question Begging in Anselm's Ontological Argument

John Rushby

Appears as Chapter 13 in the book "Beyond Faith and Rationality: Essays on Logic, Religion and Philosophy" published by Springer; Ricardo Silvestre, Paul Gocke, Jean-Yves Beziau and Purushottama Bilimoria, eds.

This updates a paper published in IfCoLog 5(7), 2018 but see below for further update

DOI: https://doi.org/10.1007/978-3-030-43535-6_13. See below for updated version

Abstract

I use mechanized verification to examine several first- and higher-order formalizations of Anselm's Ontological Argument against the charge of begging the question. I propose three different but related criteria for a premise to beg the question in fully formal proofs and find that one or another applies to all the formalizations examined. I also show that all these formalizations entail variants that are vacuous, in the sense that they apply no interpretation to "than which there is no greater" and are therefore vulnerable to Gaunilo's refutation.

My purpose is to demonstrate that mechanized verification provides an effective and reliable technique to perform these analyses; readers may decide whether the forms of question begging and vacuity so identified affect their interest in the Argument or its various formalizations.

Paper

Available at Springer Link (likely paywalled) but see below for updated version.

Update (September 2020):

A recent paper by Oppenheimer and Zalta criticizes my formulation of "begging the question" and its application to the Ontological Argument, so in this update I give more intuitive explanations for my choices and conclusions and hope that readers will find my case persuasive.

Updated Paper

Available here: Updated PDF, also available as arXiv:2205.14071

Publication History

This version of the paper extends that presented at the 2nd World Congress on Logic and Religion, Warsaw in June 2017 and published in IfCoLog 5(7) 2018 by showing that all versions of the argument considered in the paper entail variants that are vacuous, in the sense that they apply no interpretation to "than which there is no greater," and are therefore vulnerable to Gaunilo's refutation. The further update available here (above) adds intuitive explanations for my question begging claims.

Another (unpublished) paper extends this analysis to modal versions of the Proslogion II argument. Please email me if you would like to receive a draft copy.

An earlier paper examined the first-order treatment of Oppenheimer and Zalta, which uses definite descriptions, in tutorial detail.

Another recently published paper analyzes the Proslogion III (modal) argument.

For those not familiar with these topics, St. Anselm was Archbishop of Canterbury and a contemporary of William the Conqueror. His Ontological Argument is a 3-line proof of the existence of God that has fascinated logicians for nearly 1,000 years.

BibTeX Entry


@inbook{Rushby:ontargbegsvac20,
  AUTHOR = {John Rushby},
  TITLE = {A Mechanically Assisted Examination of Vacuity and
      Question Begging in {Anselm's Ontological Argument}},
  BOOKTITLE = {Beyond Faith and Rationality: Essays on Logic, Religion
      and Philosophy},
  PUBLISHER = {Springer},
  YEAR = 2020,
  EDITOR = {Silvestre, R.S. and Göcke, B.P. and Beziau, J.-Y.
      and Bilimoria, P.},
  CHAPTER = 13,
  PAGES = {229--253},
  MONTH = sep,
  SERIES = {Sophia Studies in Cross-cultural Philosophy of Traditions
      and Cultures},
  VOLUME = 34
}
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