'Strenge' Arithmetics

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We consider the virtues of relevant arithmetic couched in the logic E of entailment as opposed to R of relevant implication. The move to a stronger logic allows us to construct a complete system of true arithmetic, in which whenever an entailment A → B is not true (an example: 0=2 → 0=1 is not provable) then its negation ~(A → B) is true.