In English: “This sentence is not true”.

English Translated to Minimal Type Theory:
LP @ ∀L ∈ Formal_Systems ~∃Γ ⊂ L (Γ ⊢ LP)

Minimal Type Theory (MTT) Knowledge Tree of the Liar Paradox:
[00] LP                              (01)(05)(09)
[01] FOR_ALL                   (02)
[02] ELEMENT_OF          (03)(04)
[03] L
[04] Formal_Systems
[05] NOT                           (06)
[06] THERE_EXISTS        (07)
[07] SUBSET_OF              (08)(03)
[08] Γ
[09] PROVES                     (08)(00) // 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