Paraconsistent and Multivalued logics vs Gödel's Theorem: A Question
As people reading this may know, Gödel's Theorem, which places limits on the consistency/strength of axiomatic systems like mathematics, is based upon demonstrating the existence of a necessary contradiction in any sufficiently powerful such system.
(That is, for a strong axiomatic system, there must be at least one possible statement P of the form "P is not provable".)
While this seems to be a problem for axiomatic systems within traditional bivalent / consistent logics, I'm not sure if it is a problem for (dialetheistic) paraconsistent logics (in which we allow things to be both "true and not true", in some cases) or multivalent logics (such as fuzzy logic, in which we can have statements with varying degrees of truth, not simply true/false). Does anyone know if Gödel's Theorem can be reformulated to apply to these extensions of formal logic, or, conversely, if such extensions, being free of the restrictions of Gödel, are capable of being both powerful and consistent?
(As an extension, does this mean that, if logical systems exist in which Gödel's Theorem is not true, then Roger Penrose's objection to the potential computability of human thought is also invalid, assuming that the human mind can only be represented in such a formal system?)
(That is, for a strong axiomatic system, there must be at least one possible statement P of the form "P is not provable".)
While this seems to be a problem for axiomatic systems within traditional bivalent / consistent logics, I'm not sure if it is a problem for (dialetheistic) paraconsistent logics (in which we allow things to be both "true and not true", in some cases) or multivalent logics (such as fuzzy logic, in which we can have statements with varying degrees of truth, not simply true/false). Does anyone know if Gödel's Theorem can be reformulated to apply to these extensions of formal logic, or, conversely, if such extensions, being free of the restrictions of Gödel, are capable of being both powerful and consistent?
(As an extension, does this mean that, if logical systems exist in which Gödel's Theorem is not true, then Roger Penrose's objection to the potential computability of human thought is also invalid, assuming that the human mind can only be represented in such a formal system?)