I have the following definition:
Def: (Conditionality Principle (CP)). If two experiments on the parameter $\theta$, $\mathcal{E}_1$ and $\mathcal{E}_2$, are available and if one of these two experiments is selected with probability $p$, the resulting inference on $\theta$ should only depend on the selected experiment.
My book provides the following example:
Suppose you work in a laboratory that contains three thermometers, $T_1$, $T_2$, and $T_3$. All three thermometers produce measurements that are normally distributed about the true temperature being measured. The variance of $T_1$'s measurements is equal to that of $T_2$'s but much smaller than that of $T_3$'s. $T_1$ belongs to your colleague John, so he always gets to use it. $T_2$ and $T_3$ are common lab property, so there are frequent disputes over the use of $T_2$. One day, you and another colleague both want to use $T_2$, so you toss a fair coin to decide who gets it. You win the toss and take $T_2$. That day, you and John happen to be performing identical experiments that involve testing whether the temperature of your respective indistinguishable samples of some substance is greater than $0^\circ C$ or not. John uses $T_1$ to measure his sample and finds that his result is just statistically significantly different from $0^\circ C$. John celebrates and begins making plans to publish his result. You use $T_2$ to measure your sample and happen to measure exactly the same value as John. You celebrate as well and begin to think about how you can beat John to publication. "Not so fast", John says. "Your experiment was different from mine. I was bound to use $T_1$ all along, whereas you had only a 50% chance of using $T_2$. You need to include that fact in your calculations. When you do, you'll find that your result is no longer significant."
The book I'm using then states: "According to radically behaviouristic forms of frequentism, John is correct."
Question: On what base would a frequentist say that John is correct? The book I'm using provides this example to illustrate the Conditionality Principle (according to the CP John would be incorrect), but it doesn't explain why frequentists would even say otherwise.
Thanks in advance!
This article gives some philosophical treatment of this idea, and the introduction briefly talks about the various principles and how they fit together. The key of the article is that, even though Birnbaum's proof that the Principle of Sufficiency ($\mathcal{S}$) and Conditionality Principle ($\mathcal{C}$) imply the Likelihood Principle ($\mathcal{L}$), they say:
The general idea they present is that if one wants to agree with $\mathcal{S}$ , but is opposed (to some degree) to $\mathcal{L}$ , then "one must attach great importance to the possibility of choosing randomly between $E$ and $E'$" (where $E$ and $E'$ are experiments to be analyzed.
As in your question, if someone was a radically behaviouristic frequentist, they may be opposed to the inconsistencies of the likelihood principle with classical hypothesis testing, but still like the concept of sufficiency. The article examines utilizing random "mixtures" of experiments to establish a domain where $\mathcal{S}\land \neg\mathcal{L}$.