CP=PC Revisited: Structural Obstructions in Implicational Logics

This paper addresses the CP=PC problem: whether the probability of a conditional formula can be identified with the corresponding conditional probability.We show that this identification is obstructed already at the structural level. In the lattice [C0, S] of implicational logics of order, every implicational connective validates a strong inferential congruence principle (p4). Probabilistic conditioning fails this principle and therefore cannot serve as an implicational connective. This yields a structural no-go result: CP=PC is not merely false for particular connectives, but incompatible with the inferential role of implication as such. We further show that even if this incompatibility is ignored and CP=PC is forcibly imposed on an implicational connective, then under natural semantic assumptions for order models the addition of global symmetry leads to a collapse of the positive-probability fragment. This collapse is presented as a reductio illustrating the pathology of identifying implication with conditional probability.The positive lesson is a separation of roles: the connective − governs purely inferential behavior, while a distinct internal conditional (B | A) is interpreted semantically within a classical probability space so that its probability coincides with Pr(B | A). This reframes Stalnaker’s program by locating CP=PC in the semantics of an internal conditional rather than in the implicational fragment .