By Pucci P., Rigoli M.
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Extra info for Entire solutions of singular elliptic inequalities on complete manifolds
Suppose that, following my decision to adopt a certain set-theoretic semantics for arithmetical language, I interpret others—Euler, Gauss and so on—in the very same way. On this semantics, Gauss’s statement of the quadratic reciprocity theorem comes out true. What are the objections to this? One could accuse me of putting words in Gauss’s mouth. His proof was about prime numbers, not sets.
In a nutshell, the Bad Company Objection is merely the observation that some abstraction principles, such as BLV, are not candidates for acceptability since they are incompatible with presumably acceptable principles such as (HP). Stated this way, a partial solution to the problem is already evident: Acceptable implicit deﬁnitions must be consistent, and consistent with (HP). Unfortunately, this is not enough since there are pairs of abstraction principles that are (individually) consistent with (HP), but not consistent with each other.
The problem, of course, is that irenicity does not seem to have the same sort of epistemological motivation as conservativeness. On the contrary, the inference underlying the restriction to irenic abstraction principles seems to be little more than a fallacy: faced with pairwise incompatible, conservative abstraction principles, logic dictates that at least one must be unacceptable, but as of yet we have no data determining which of the two is unacceptable. 25 The problem, I think, is that when faced with this extremely difﬁcult problem – drawing a sharp distinction between good and bad abstraction principles, attention has strayed from the philosophical question regarding what epistemological characteristics might make an abstraction principle a good one, to what formal characteristics might supply us with a powerful yet consistent collection of abstraction principles.