Maximality, uniqueness, and reliability

What is a definition of logical inevitability that is better than

(1)  T® á$T

?

We want T to be reliable in the sense of Kaplan []. T is reliable if

(2) from kT infer káT

where k means “knowing”. The paradigmatic example is a speaker speaking “I am here now.” I don’t know if (2) is the best interpretation of reliable.   

What can it mean for a logical system to “know” something? This is the structure the system is “aware” of. And this is the image of implication within the system, given all of T as axioms. Let k_T be what T knows. Then (1) gives

(3) T k_T á$T

There are arguments elsewhere that this relational existence is sufficient for actual existence. A big question is the one of the uniqueness of T. There had better be a maximal T, where

(4) T is maximal if for all theories S such that S® á$S, S Ì T.

If there is no maximal T, then there would be many things that have an equal right to exist. In this case a reinterpretation of (1) would be called for.

Maximality results usually proceed…