∀L ∈ Formal_Systems   True(L, X)(    ∃Γ ⊂ L (Γ ⊢   X) )
∀L ∈ Formal_Systems ~True(L, X)( ~∃Γ ⊂ L (Γ ⊢   X) )
∀L ∈ Formal_Systems   False(L, X)(   ∃Γ ⊂ L (Γ ⊢ ~X) )

“This sentence is not true.”
Formalized as: L ∈ Formal_Systems LP( ~∃Γ ⊂ L (Γ ⊢ LP) ) 

Minimal Type Theory (MTT) Knowledge Tree of Liar Paradox
01 ∀      (2)(5)     
02 ∈      (3)(4)     
03 L      
04 Formal_Systems
05 ~      (6)     // LP is an alias for this node
06 ∃      (7)(9)
07 ⊂      (8)(3)
08 Γ       
09 ⊢      (8)(5) // cycle indicates error: evaluation infinite loop

Minimal Type Theory Directed Acyclic Graph

The above formalism shows that the Liar Paradox gets stuck in an infinite evaluation loop and is thus not a Truth Bearer. Any expression that is impossibly true and impossibly false is not a truth bearer and therefore not a member of any Formal System even though it may otherwise be a syntactically WFF.

Copyright 2016 and 2017 Pete Olcott