The logic of ordered questions.

The laws of logic are a priori and scientific theories are a posteriori. This is the received view. Here we generalize the classical laws of logic. The logic follows inevitably from a tautology involving order-dependent questions of both orders. It reduces to Boolean algebra for order invariant questions. We derive the inevitable logical conjunction that generalizes the classical conjunction. It can take negative values. Commutative conjunctions can be represented by a special Jordan algebra. The algebra is formally real, which means that it can be represented by a direct sum of simple Jordan algebras in terms of real, complex or quaternionic Hilbert spaces. Thus, formally real Jordan algebras and the Hilbert space foundations of quantum theory are manifestations of a priori logic. It shows that there are no physical, informational or other fundamental principles of quantum theory. A logical joint probability follows directly from the conjunction. It is a generalization of the real part of the Kirkwood-Dirac distribution in quantum mechanics, which is known to take negative values. The formalism applies both to microscopic and macroscopic systems. We demonstrate order-invariant, non-negative logical joint probabilities in order dependent opinion polls.