Reality Is Impossible

Things that are true are not possibly true. They are just true!

Likewise, things that are false are not possibly false. They are just false!

If something is necessarily true, then it is true, which means that it is not possibly true. And if something is necessarily false, then it is false, which means that it is not possibly false.

Yet Alethic Modal Logic tells us that something being necessarily true just means that it is not possibly not true.

Does this definition, or pair of definitions for Necessity N and Possibility P, [(Na = -P-a) & (Pa = -N-a)], make sense, given our two Reality Principles, (1) [(a > -Pa) & (-a > -P-a)] and (2) [(Na > a) & (N-a > -a)]?

Let's see:  

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.

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