The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzmüller and Scott's Notes on Gödel's and Scott's variants of the ontological argument, reaches beyond the modal object language: its property quantifiers range over domains containing the nominals and satisfaction operators of hybrid logic. A proof using one proves a theorem of the embedding, not necessarily of the modal logic. That the framework affords this is not new. Whether a result is one of the modal logic can be settled by replaying it in an explicit calculus, one result at a time. More cheaply, and for all results at once, it can be approached by analysing what the embedding's proofs substitute. Every statement the Notes prove has a proof in the embedding substituting nothing outside the object language: 294 of them explicit proofs, machine-checked, none using a nominal. The Notes' own proofs substitute none either; what the check flags are terms a prover introduced. The three questions the Notes leave open are settled too, without nominals, but by the empty instance of the conjunction axiom, too cheaply to convince. As the Notes generalise it, to cover Gödel's "any number of summands", the axiom covers the empty conjunction and the singleton. This article restricts it to at least two different conjuncts, a non-degenerate reading of Gödel's footnote, and settles the questions again, without nominals. In effect the restriction excludes only the empty conjunction; the one-element case comes back with the other axioms. In one of the three variants the restriction refutes the possible existence of a God-like being; the answers return, with no side condition, once the Notes' non-emptiness clause is confined to Gödel's essence. Every theorem stated here is verified in Isabelle/HOL and independently in Lean 4; the countermodels are Nitpick's, checked by the build.

翻译：暂无翻译