Assume we have a generator of statements (black box type). We generated a long list of $N$ statements ($N \gg 1$).
For theory $T_1$, approximately $x_1N$ of the statements are true and $y_1N$ are false, $x_1+y_1=1$. Then there is another theory, $T_2$, and it is somehow correlated with theory $T_1$ such that, for $T_2$, the fraction of true and false statements are \begin{equation} x_2=\psi_{12}x_1+(1-\mu_{12})y_1,~~~ y_2=(1-\psi_{12})x_1+\mu_{12}y_1 \end{equation}
Graphically the relation between theories $T_1$ and $T_2$ can be represented as a point on the unit square. Note that the inverse relation, between $T_2$ and $T_1$ can be different, i.e., described with $\psi_{21}\neq \psi_{12}$ and $\mu_{21}\neq \mu_{12}$.
There are ultimate cases, e.g., $\psi_{12} = \mu_{12} = 1$ (all statements that are true/false for $T_1$ are true/false for $T_2$), or $\psi_{12} = \mu_{12} = 0$ (converse). There are cases when the theories are not correlated at all, $\psi_{12} = 1 - \mu_{12}$ (antidiagonal of the square).
Now I need to add a third theory, $T_3$ and define coefficients, $\psi_{13}, \psi_{31}, \mu_{13}, \mu_{31},..., \mu_{32}$, relating it to the already existing two theories. But not all values of the coefficients are legitimate because some of them can violate some common sense logics (these conditions I do not know how to formulate explicitly). For example, if $\psi_{12} = \mu_{12} = 1$ and $\psi_{31} = \mu_{32} = 1$, there is only one (probably) choice $\psi_{32} = \mu_{32}=1$.
The question is how to sample for three theories a random set of coefficients $\psi_{ij}, \mu_{ij}$, $i,j = 1,2,3,~i \neq j$ without violating some basic principles (of common-sense logics?). What conditions to fulfill? What is possible sampling algorithm(s)? And, finally, how to generalize this to the case of $n$ theories?
This set-up is not so specific that the problem should be considered in the framework of mathematical logics or model theory. One could think of $N$ pathogen cells and different drugs, $\{T_i\}$, $j=1,..,n$, so for drug $T_j$ on average $x_jN$ cells are sensitive and $y_jN$ cells are resistant.
This all becomes easier to handle if you consider the fractions $n_{t_1t_2t_3}$ of statements that have truth value $t_i$ in theory $i$. If you want, you can express all your coefficients in terms of these values. For instance,
$$\psi_{12}=\frac{n_{11}}{n_{11}+n_{10}}=\frac{n_{110}+n_{111}}{n_{110}+n_{111}+n_{100}+n_{101}}$$
(where the two-index values $n_{t_1t_2}=n_{t_1t_20}+n_{t_1t_21}$ sum over the third index). The only constraints imposed on these $8$ fractions are that they not be negative and sum to $1$, and your coefficients are admissible iff they can be represented by a valuation of these fractions that fulfils those constraints.
To sample, generate $7$ independent uniform random samples in the unit interval $[0,1]$ and use the $8$ interval lengths that these $7$ points cut the interval into as the $8$ fractions.
In the case of $n$ theories, there are $2^n$ fractions, and you can sample them by cutting the unit interval with $2^n-1$ random numbers.