True(X) and ~Provable(X) is Impossible

A Fact(English) is Only known to be True(Math) when it is verified(English).
Analytic Facts(English) are Only Verified(English) by their Proof(Math).

An analytic Fact is an expression of language that is completely verified as True entirely based on the meaning of its words (or symbols).

Some expressions of language are defined to be necessarily True.  This forms the foundation of all Truth. To keep things simple we will call these expressions VERIFIED_FACTS.

To provide mathematical rigor to the definition of VERIFIED_FACTS we apply the David Hilbert formalist approach and specify a set of finite strings that are defined to have the semantic property of: True.

True(L, Y, X) means that there is an inference chain from a set Y of one or more VERIFIED_FACTS of L that derive X.

False(L, Y, X) means that there is an inference chain from a set Y of one or more VERIFIED_FACTS of L that derive ~X.

Provable(L, Y, X means that there is an inference chain from a set Y of one or more expressions of L that derive X.

Refutable(L, Y, X) means that there is an inference chain from a set Y of one or more expressions of L that derive ~X.

Statement(L, Y, X) ↔ ( True(L, Y, X) ∨ False(L, Y, X) ) ∴
~◇( Statement(L, Y, X) ∧ ~Provable(L, Y, X) ∧ ~Refutable(L, Y, X) )

This is a refinement to the 1997 Mathematical Mapping Theory of Truth

Copyright 1997 2004, 2015, 2016, 2017, 2018 Pete Olcott