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

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. This version updates the paper that originally appeared as Chapter 13 in "Beyond Faith and Rationality: Essays on Logic, Religion and Philosophy" published by Springer to respond to criticisms by Oppenheimer and Zalta.