| > Sure, and abductive reasoning can be formalized in certain modal logics with Kripke semantics. No, it can't. Abductive reasoning is probabilistic modelling, and notably, there's a line of research by Cristian Calude showing that you can soundly, non-paradoxically place probabilities on Halting questions. (Computational tractability is still an obstacle with his current approach, but it has been shown not to generate paradoxes, which is already a major step forward.) >you might want to take a look at Gödel, Escher, Bach, where Hofstadter discusses how the second incompleteness theorem applies to Turing machines. This is backwards: halting problems and Kolmogorov complexity for Turing machines give us the two Incompleteness Theorems for proof systems, via Chaitin's Incompleteness Theorem. Which also neatly gives a way around the Second Incompleteness Theorem: a hierarchical-probabilistic reasoner can create an abstract, compressed model of themselves which consists of small-enough amounts of information that they can reason about its behavior without becoming subject to Chaitin Incompleteness. |