  We propose a conjugate logic that can capture the behavior of quantum and
quantum-like systems. The proposal is similar to the more generic concept of
epistemic logic: it encodes knowledge or perhaps more correctly, predictions
about outcomes of future observations on some systems. For a quantum system,
these predictions are statements about future outcomes of measurements
performed on specific degrees of freedom of the system. The proposed logic will
include propositions and their relations including connectives, but importantly
also transformations between propositions on conjugate degrees of freedom of
the systems. A key point is the addition of a transformation that allows to
convert propositions about single systems into propositions about correlations
between systems. We will see that subtle choices of the properties of the
transformations lead to drastically different underlying mathematical models;
one choice gives stabilizer quantum mechanics, while another choice gives
Spekkens' toy theory. This points to a crucial basic property of quantum and
quantum-like systems that can be handled within the present conjugate logic by
adjusting the mentioned choice. It also enables a discussion on what behaviors
are properly quantum or only quantum-like, relating to that choice and how it
manifests in the system under scrutiny.
