Confusion between true negatives and false positives (double negation?!) Given the below term definitions

..my question is: why is the logical negation (opposite) of a false negative not a true negative? This would imply 1-α=β, which does not have to be the case.
And same for positives: I'd have thought that what is left after counting the false positives is the true positives. In other words, that one minus the false positive rate, i.e. 1-α, is is the true positive rate, which is 1-β. This would imply α=β, which is (again) not necessarily true!
Can anyone help clear the confusion?
 A: The terms 'postive' and 'negative' are used with respect to underlying null ($H_0$) and alternate ($H_1$) hypotheses. Both hypotheses specify how your test statistic should be distributed, given that they are correct.
If the $H_0$ is correct, the test statistic will be distributed according to the null distribution, and it will have a probability of $\alpha$ of falling in the rejection region. I.e. with probability $\alpha$ you will reject the null hypothesis even if it is true - this is the False Positive Rate. Conversely, there is $1-\alpha$ probability of failing to reject the null hypothesis; this is the True Negative Rate (since the null hypothesis is true here, you should not reject it).
Now consider that $H_0$ is false, and $H_1$ is true. Now the test statistic is distributed according to the alternate distribution. With probability $\beta$, the test statistic will fall outside the rejection region. This is the False Negative Rate, since you failed to reject $H_0$ even when it was false. Conversely, with probability $1-\beta$ you correctly reject the null hypothesis, hence it is called the True Positive Rate.
I found the following image which might be helpful in understanding (source: probabilitycourse.com, chapter 8).

Here $X$ is the test statistic. The null hypothesis is rejected if $X>c$. Following from the above discussion, we can express $\alpha$ and $\beta$ as these conditional probabilities.
$$\alpha = p(X>c|H_0) \implies 1-\alpha = p(X<c|H_0)$$
$$\beta = p(X<c|H_1) \implies 1-\beta = p(X>c|H_1)$$
A: Let me try to explain some more: Suppose the patient is sick and given a test - the test can result either in a true positive outcome or in a false negative outcome, and so these two probabilities (1-beta and beta, respectively) add up to 1. Suppose the patient is healthy and given a test - the test can result in a true negative outcome or a false positive outcome and these two probabilities (1-alpha and alpha, respectively) add up to one. Each set of probabilities is conditioned on the health status of the patient.
A: I see some confusion here, maybe I can help :


*

*A true positive is something your method identifies as a positive, and actually is. If you run a HIV test and it's positive, and if the patient is sick, then it's a true positive.

*A false positive is something your method identifies as a positive, but is not. In the same example, your patient is not sick but the test is positive.

