Ramana Kumar comments on Controllables and Observables, Revisited

[Ramana Kumar]

Counterexample to your claim that $Ctrl(C\oplus D)=Ctrl\left(C\right)\cup Ctrl\left(D\right)$ (with both $C$ and $D$ not $null$):

Take $C$ and $D$ to be the following Cartesian frames over world $\{x,y\}$ with agent and environment both $\{a,b\}$ as the following matrices:

$$\begin{array}{ccc}& a& b\\ a& y& x\\ b& x& x\end{array}$$$$\begin{array}{ccc}& a& b\\ a& x& y\\ b& y& y\end{array}$$

Then $Ctrl\left(C\right)=Ctrl\left(D\right)=\varnothing$ but $Ctrl(C\oplus D)\ni \left\{x\right\}$.

(Sorry for my formatting—I haven’t done LaTeX in these comments before.)

[Scott Garrabrant]

Thank You! You are correct. Oops!

I deleted that claim from the post. That was quite a mistake. Luckily it seems self-contained. This is the danger of “Trivial” proofs.

Initial versions of the theory only talked about ensurables, and the addition of (and shift of attention to) controllables came much later.