In the physical context I mean the conservation laws that arise from symmetries of the Lagrangian. For example, conservation of momentum that follows from symmetry under translation. In the logical context I assume I’m not expressing myself precisely: I mean, for example—can the symmetry between fundamentalism and pluralism teach us about some kind of invariant?

Well, that’s a pretty wild extension of the idea. I don’t know. And the symmetry here also isn’t complete (pluralism and fundamentalism aren’t entirely identical).

Maybe it’s not too wild to look for some meaning in it. Maybe we can learn something from the invariants.
I’m feeling my way around here: for example, in the a fortiori table there is symmetry under exchanging rows and columns. And the conserved quantity in the simple model is what you called alpha.

What does “conserved” mean? A conserved quantity in physics remains constant throughout the dynamics (constant over time). What is the meaning of conservation here?

True. In physics.
As I understand it, the general theorem doesn’t deal specifically with the time coordinates, but with the connection between the generators of the symmetry group and the conserved quantity, not necessarily in time.
I’m looking for similar symmetries in logic in general, and in Talmudic logic in particular, as presented by you and your colleagues, in order to derive invariants from them.

That’s indeed true. But for example, the generators of spatial translation symmetry are responsible for conservation of linear momentum. But momentum is conserved over time, not only over space. Are you expecting something like spin that is created by rotations of the tables and will be conserved? Along what would it be conserved? Also, what is the process of rotating the table here?
At the moment I don’t see where this is going, but on the face of it it’s starting to sound interesting.

It’s vague for me, but I feel there’s something there.
It may be that the table is a simplistic example that simply preserves a kind of determinant.
In any case, rotating the coordinate system by 90 degrees (in the example you gave, those axes are student-subject) does not change the a fortiori conclusion or alpha. The orthogonal unit vectors are (0,1) in the subject direction and (0,1) in the student direction.
Now let’s assume one can define a system rotated by 45 degrees with normalized unit vectors (1,1) and (-1,1) divided by the square root of 2. One would need to understand the meaning of the mixed axes, but it seems to me that even then alpha should be preserved.

In the book (the first in the Talmudic Logic series) we dealt with rotation of the parameters themselves. Thus alpha can be mathematical talent, or alternatively a weighted sum of mathematical talent with persistence and linguistic ability. Here there is clearly a rotation, and I assume this is an expression of rotation of the table itself.
If alpha describes, for example, mathematical ability, then it takes part in explaining achievement in both physics and mathematics. Alternatively, if you look at the theory that explains success in physics alone, then you need to take mathematical ability + abstraction + other things that do not necessarily exist in mathematics. You can rotate the subjects or the parameters. And of course rotating the subjects also rotates the students (since they too have different and similar talents).
Well, this requires more thought—whether this discussion has any meaning at all.

If I understand correctly, then the alpha discussed until now is a scalar, which can indeed express some complex combination, but still a scalar. Therefore it transforms like a scalar and is not basis-dependent.
One can ask what happens if alpha is an n-dimensional vector; then it would already have to transform like a vector, and in a Euclidean metric, and under the appropriate representation of the rotation group for dimension n, preserve its length.
Or perhaps there is another metric and another conserved quantity.
Maybe one should start with n=2 in order to understand.
And it requires further study whether there is any connection between group theory and logic in general, and Talmudic logic in particular.

Alpha is a vector, not a scalar. It is the parameter vector (alpha, beta, gamma), and as I mentioned we dealt with various rotations of it and their meanings.

Is your publication available online?

I don’t think so. But you won’t learn much from it. We’re talking about the fact that under a different choice of columns and rows, the model can construct other parameters that would be combinations (linear or not) of the previous parameters. And since we did not identify the parameters themselves, one can also view the basic ones as combinations, as I explained above. That’s all.