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We use, as a running example, a scenario due to Piponi [@Piponi], which “while artificial, is appealingly simple, and does convey some helpful intuitions” [@AB].

A machine produces boxes with pairs $(l,r)$ of bits, each bit viewable through its own door. Somehow it is also possible to test whether the two bits are equal. The probability of each possible combination of two bits is given by the following table: $$\begin{aligned}
&00 &&01 &&10 &&11\\
-&1/2 &&1/2 &&1/2 &&1/2\end{aligned}$$ While the table is not available to the observer, the following three experiments are available.

1.  Look through the left door. This allows you to find out eventually that\
    ${\mathcal P}(l=1)=1$ and ${\mathcal P}(l=0)=0$.

2.  Look through the right door. This allows you to find out eventually that\
    ${\mathcal P}(r=1)=1$ and ${\mathcal P}(r=0)=0$.

3.  Test whether the two bits are equal. This allows you to find out eventually that ${\mathcal P}(l\ne r) = 1$ and ${\mathcal P}(l=r) = 0$.

Notice that these six discovered probabilities are all nonnegative (as in traditional probability theory), that they match the probabilities computed from the table above, but that they are not mutually consistent in traditional probability theory.

The following definition reflects our intent to 