AN ANALYSIS TO DETERMINE THE VALIDITY OF THE DICHOTOMY OF
CONTRADITIONS AND TAUTOLOGIES
N. Key, Jr
M. Alaclypse III
Let c and t represent the set of contradictions and tautologies, respectively. To determine whether a contradiction and a tautology are equivalent, one calculates:
( c ∪ t ) ∪ ( c ∩ t ) →
~( ( c ∩ t ) ∩ ( c ∪ t ) ) →
~( (c ∩ t ) ∪ ( c ∩ c ) ∪ ( t ∩ t) ∪ ( t ∩ t) ) →
~( c ∩ t ) ∩ ~( c ∩ c ) ∩ ~( t ∩ t)
This brings up interesting points:
As expected, no contradiction is also a tautology. However, if one does not assume trivial identity, one recognizes that the outcome implies that no contradiction is a contradiction, and no tautology is a tautology.
This proves, mathematically, that no equals are the same. It also disproves all of symbolic logic through symbolic logic.
Furthermore, we can simplify using identity:
~( c ∩ t) ∩ ~c ∩ ~t →
( c ∪ t) ∩ ~c ∩ ~t →
( c ∩ ~c ∩ ~t ) ∪ (t ∩ ~c ∩ ~t) →
( ( c ∩ ~c ) ∩ ~t) ∪ ( ( t ∩ ~t ) ∩ ~c ) →
( F ∩ ~t ) ∪ ( F ∩ ~c ) →
F ∪ F →
In other words, if we use the identity property, our initial statement (that contradictions and tautologies are either mutually dependent or mutually exclusive) reveals itself as a contradiction, whereas without the identity property, it reveals the identity property as a contradiction.
To prove an obvious truism as a contradiction is the result in either case, and the unique factoring method does not break DeMorgan's Law, nor any other rule of formal symbolic logic. The fact that we pursued operations in a non-canonical order upon a statement that seems initially to be simplified maximally is irrelevant in determining the validity of our results, since no rules were broken.
In the end, it seems that the basic tenets of symbolic logic are inherently flawed. We have, at this point, no suggestions on how to remedy this, aside from those suggestions that invalidate the rules of symbolic and formal logic - suggestions which are irrational and do not correspond to the traditional concepts of reason - however, as our proof invalidated reason as it now stands, we must state our conclusion: nothing is true, and everything is permissible. There are no logical relationships that cannot be proven to be contradictions, even those that are proven to be correct elsewhere.