Formal Systems, Common Logic and LBase
Folks might enjoy the Soloman Feferman lecture Goedel, Nagel, Minds and Machines. After recounting an exchange between Godel and Nagel circa 1956, Feferman describes the implications of the minds vs. machines dichotomy ensuing from the exchange. To avoid the impass resulting from the dichotomy, Feferman proposes the redefinition of a formal system to an “open ended schematic axiomatic system.” He claims this redefinition is a constructive step towards an “informative, systematic account at a theoretical level of how the mathematical mind works that squares with experience.” He stresses the importance of a subject neutral theory of operations with basic schemata for language, arithmetic, set theory that would amount to a version of an untyped lambda calculus. Feferman concludes by rejecting any effective machine representation of mind as contemplated by Nagel, Penrose & others.
So, what does this mean to Common Logic and LBase ? Seems to me that efforts like Common Logic and LBase would either have to a) be defined within this type of an open ended system, let’s say as the natural language description of the constraints to which the axioms that make up the theory of such a system would adhere; or b) evolve into an open ended system that exhibits characteristics of transformation across languages, logics, models and theories.