tag:blogger.com,1999:blog-2631035637795172582.post5153737354625138775..comments2023-01-24T10:06:57.212-08:00Comments on (Blog&~Blog): A Simplified Version of My Revenge Paradox for Paracomplete Solutions to the Liar ParadoxBenhttp://www.blogger.com/profile/06702722560438833244noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-2631035637795172582.post-4843074257850927442010-10-13T13:29:46.854-07:002010-10-13T13:29:46.854-07:00These posts are very interesting. It gives me the...These posts are very interesting. It gives me the idea that people in philosophy departments (the various -ists you mentioned, not you yourself) are very confused.<br /><br />I think 99% of the confusion around liar's paradox seems to come from "true" having two different senses, which are confused endlessly.<br /><br />Given a language with an unary predicate T (and optionally an unary predicate F), and a model of that language, one says that a formula phi is "true" if the Godel number of phi is an element of the model's interpretation of T. One says that phi "holds in the model", or that phi is "satisfied", or whatever, if (insert inductive definition of a model satisfying a formula, here). All the confusion and "paradoxes" seem to arise from confusing these notions, as if they were equivalent.<br /><br />(In "mainstream" logic there is no danger of using "true" to mean "holds in the model", it's only when the language has that added truth predicate that trouble arises)<br /><br />Now as for the interesting paradox you introduced in this post... there are three ways I can think of to formalize it:<br /><br />1. Introduce predicates about which (Godel numbers of) formulas the omniscient being accepts. Say, A for accept, R for reject. Omniscience might be interpreted axiomatically as "phi iff A(phi)" and "~phi iff R(phi)" [I abuse notation and write phi for the Godel number of phi inside predicates]. Sadly, these axioms are inconsistent (e.g., by the plain vanilla liar's paradox), no model exists, and everything follows vacuously.<br /><br />2. Instead, introduce modal operators A and R. Then liar's paradoxes don't arise because of the lack of free variables needed to invoke Godel. But then, neither can you formalize the revenge paradox.<br /><br />Or 3. Use 2nd order logic, and formalize it as something like "for every set X, if X contains precisely the Godel numbers of formulas which hold, then X doesn't contain the Godel number of this sentence". Unfortunately I don't know enough about 2nd order logic to know whether you can actually use fixed point lemmas to pull this off...Glowing Face Manhttps://www.blogger.com/profile/07717328290680086281noreply@blogger.com