The Axiom of Non-Identity
You cannot avoid paradox at the general level when paradox is the foundation for all phenomenon. All measurement begins with a localization of some facet of "being". This in itself is an act of seperation resulting in a base dualistic state. This is observe in the "New Laws of Identity II" thread. The problem occurs in the respect that contradiction is unavoidable in these respects, a foundation, but as such negates itself into a "meaningful" solution is this is the foundation of truth; hence all non-contradictory statements are true and all contradictory statements are approximations of the truth with this itself being a contradiction.
1) "Contradiction Axiom 1" as "◬ → ⊡"
where "→" observes the direction of one axiom towards another axiom resulting in a state of opposition by progressive variation in which one axiom is directed into another, and "◬" equivocates to a variable and "⊡" to a variation of the another variable. This is contradictorly reflected in Point 2, by the progression of Point 1 to point 2 and point 3.
2) "Contradiction Axiom 2" as "◬ ← ⊡"
where "←" observes that the variable, as directed to another variable necessitates the "variation" directed back to its root variable due to this "variation" existing through the prior variable. This is contradictorly reflected in Point 2, by the progression of Point 2 to Points 1 and 3.
3) "Contradiction Axiom 3" as "⊕"
The polarity between a truth and its variant is inherently oppository in nature, where one is directed towards the other resulting in a perpetual cycle thus necessitating all variables and there variations to exist as a perpetual Circularity which in itself goes to Identity Axioms as a Variable in itself. One variable is directed towards another, observing there continual cycling to exist as a third variable in itself. This is contradictorly cycles between Points 1 and 2 and Cycles between Points 1 and 2.