What are Quantum Logics and What Ought They to be

We, and our students and colleagues at the University of Massachusetts, have erected the foundations of a general scientific language capable of elucidating, in the spirit of Leibniz, the physical theories that are of concern to us here. We feel that this has now been accomplished with a clarity and precision that has been wanting up to this point. As a consequence of this work, we feel qualified to submit our answer to the question in the title of this paper. It is both important and illuminating to understand the roots of any work of such a fundamental nature. Therefore, to begin with, we sketch the mundane pragmatic background of our own work and the history of the subject -- as we see it.