Theory is inconsistent

Martin Raza December 01, 2020 at 19:43 1625 views 2 comments Logic & Philosophy of Mathematics

Suppose T is a formal AI theory for Knowledge Representation. T contains a distinguished 1-place predicate symbol K (included in the Gödel numbering scheme), where K(???) is intended to formalize the idea that sentence ? is known. Suppose further that the diagonal function is representable in the system, and that the logic of the knowledge predicate includes the general principles:
(*) ?T K(???) ? ? , since knowledge is considered to be factive, and
(**) if ?T ?, then ?T K(???) , which is intended to capture the idea that if a statement is proven then it is known.
How do we show that the resulting formal theory of Knowledge Representation is inconsistent.

Comments (2)

fdrake December 01, 2020 at 20:06 ¶ #476038 0 likes

Looks like a Tarski's undefinibility theorem problem. The strategy there will probably be using the diagonal lemma to set up a derivable contradiction using K as the truth predicate - since we have that K(phi) and phi are interderivable.

fishfry December 03, 2020 at 22:01 ¶ #476680 0 likes

Quoting Martin Raza

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Attack of the killer robots?

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