Where does the Logistic Distribution get its name? Having read around on the topic I understand its application as a close approximation of the normal distribution with a nicer mathematical form, but where does its name come from?
Is it associated with computer logic and its binary nature?

The cumulative distribution function of the logistic distribution is the logistic function
$$
F(x) = \frac{1}{1+e^{-(x-\mu)/s}}
$$
For an explanation of where the logistic function got its name, check the What does the name "Logistic Regression" mean? thread.
