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Self-Referential Probabilistic Logic Admits the Payor's Lemma — LessWrong

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In summary: A probabilistic version of the Payor's Lemma holds under the logic proposed in the Definability of Truth in Probabilistic Logic. This gives us modal fixed-point-esque group cooperation even under probabilistic guarantees. Payor's Lemma: If ⊢ □ ( □ x → x ) → x , then ⊢ x . We assume two rules of inference: Proof: The Payor's Lemma is provable in all normal modal logics (as it can be proved in K , the weakest, because it only uses necessitation and distributivity). Its proof sidesteps the assertion of an arbitrary modal fixedpoint, does not require internal necessitation ( ⊢ □ x ⟹ ⊢ □ □ x ), and provides the groundwork for Lobian handshake-based cooperation without Lob's theorem. It is known that Lob's theorem fails to hold in reflective theories of logical uncertainty. However, a proof of a probabilistic Payor's lemma has been proposed, which modifies the rules of inference necessary to be: The question is then: does there exist a consistent formalism under which these

x Self-Referential Probabilistic Logic Admits the Payor's Lemma — LessWrong Payor's Lemma Logic & Mathematics Decision theory Löb's theorem Rationality World Modeling Frontpage 85 Self-Referential Probabilistic Logic Admits the Payor's Lemma by yudhister 28th Nov 2023 AI Alignment Forum 7 min read 14 85 Ω 36 In summary: A probabilistic version of the Payor's Lemma holds under the logic proposed in the Definability of Truth in Probabilistic Logic . This gives us modal fixed-point-esque group cooperation even under probabilistic guarantees. EDIT 10/24/24: I think the way the way this post is fra

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