My question may be pedantic, but I find the syntax used to describe the "power" of a hypothesis test really annoying, and I just want to either correct my understanding of the concept, or understand why it is, seemingly, mislabeled.
My understanding of "power" is the probability that a test will reject $H_0$, when $H_a$ is true. Just from this wording, the power of a test seems to be the power in favor of $H_a$, not the power against it. In other words, if $H_a$ is true, you want your test to reject $H_a$. However, my text, Introduction to the Practice of Statistics, used as a text book at our community college, states that this probability is "the power against $H_a$".
Why do they call the probability that $H_0$ is rejected when $H_a$ is true the power against $H_0$ instead of the power in favor of $H_a$, or simply the power of $H_a$?