# Boolos's Curious Inference in Isabelle/HOL

June 20, 2022

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### Abstract

In 1987, George Boolos gave an interesting and vivid concrete example of the considerable speed-up afforded by higher-order logic over first-order logic. (A phenomenon first noted by Kurt Gödel in 1936.) Boolos's example concerned an inference $I$ with five premises, and a conclusion, such that the shortest derivation of the conclusion from the premises in a standard system for first-order logic is astronomically huge; while there exists a second-order derivation whose length is of the order of a page or two. Boolos gave a short sketch of that second-order derivation, which relies on the comprehension principle of second-order logic. Here, Boolos's inference is formalized into fourteen lemmas, each quickly verified by the automated-theorem-proving assistant Isabelle/HOL.