We introduce Arianean Arithmetic, a semantic extension of classical Peano Arithmetic based on Belnap's four-valued logic FOUR = T, F, N, B. The syntactic structure and proof theory of arithmetic remain entirely classical. The extension operates at the semantic level: arithmetic formulas are evaluated over collections of classical and partial models, yielding four epistemic values that represent confirmed truth, confirmed falsehood, absence of information, and contradictory information. The central contribution is the Ibrahim Induction Principle, a parametric family of four induction schemes indexed by the values of FOUR. Classical mathematical induction is recovered as the special case at threshold T. The principles at thresholds N and B are genuinely new and have no counterparts in classical arithmetic. We establish three main results. First, a conservativity theorem: every theorem of classical Peano Arithmetic receives value T under every admissible valuation. Second, a parity duality result: epistemic ignorance about whether a number is even is always symmetric with ignorance about whether it is odd. Third, an epistemic asymmetry between addition and multiplication: while addition preserves epistemic uncertainty, multiplication by zero annihilates it completely, converting any epistemic value into T. This asymmetry, which we call the Molina Asymmetry, is invisible in the classical two-valued setting and becomes visible only in the four-valued framework. This is the first paper in the Arianean Arithmetic series.
Ariana Karina Molina Campoverde (Sat,) studied this question.