False by Definition

With the Liar Statement—i.e. “This statement is false,” which we can translate as “p = this statement (p) is false = (p = ~p)”— the definition of p is false, not p itself. What does it mean for a definition to be false? It means that it is formally invalid.  This in turn means that it is not permitted within our logic to use a definition of this form. Note that the statement of a definition implies the statement that it defines, but it is not identical to the statement it defines. I.e. Def. p > (p = -p).

Conjecture: ∀p [(Def. p > (p = -p)) > -(Def. p)]          (1)

Def. q > ( q = -q)                                  S

Def. q                                   S

q = -q                                MP

q                                  S

-q                                ES

q & -q                        CONJ

-q                                      IP

q                                      ES

q & -q                             CONJ

-(Def. q)                                            IP

(Def. q > (q = -q)) > -(Def. q)                    CP

∀p [(Def. p > (p = -p)) > -(Def. p)]                       UG

(1), QED                                                     

We might say that all statements of self-reference involve such a definition statement implicitly because every self-reference statement necessarily requires a self-stipulation, which is of course a type of stipulation, i.e. a definition. However, some self-referential statements, e.g. “This statement is a written statement,” i.e. “p = this statement (p) is a written statement (w) = (p = w),” involve a valid form of definition, Def. p > (p = w), such that the definition is not false. 

In any case, the above proves that the Liar Statement is falsidical and not a real antinomic paradox.