On the Completion of the Calculus Ratiocinator: An Intensional Arithmetic of Categorical Logic

This paper formalizes an intensional arithmetic for categorical logic over the square-free positive integers (Z+ sf ), extending G.W. Leibniz’s Calculus Ratiocinator and clarifying the boundary at which it can and cannot be completed. A concept must be defined by a set of atomic predicates (primes); a predicate is contained in a subject when dividing the subject by the predicate yields a square-free integer. The result is passed through a valuation block V (n) that outputs 1 if the quotient is square- free, otherwise 0. Within this system, I prove that negation and contradiction can be handled via the Greatest Common Divisor (gcd), and I formalize Leibniz’s Law of Identity with its falsification, the Barbara syllogism, and conjunction. Like any bounded arithmetic, this system has explicit theoretical boundaries chiefly non-coprime premise overlaps and compositional negation parity marking the precise points at which this positive fragment halts.