Logistic regression output and probability What is the interpretation of the number that is the output of the logistic regression function? 
The logistic function 
$$f(\vec{x}) = \frac{1}{1+e^{-g(\vec{x})}}$$
(where $g$ is a linear function) is supposed to map a continuous variable (or more generally a whole bunch of totally ordered variables) to between 0 and 1.
I've always assumed it is the probability of inclusion in one set or the other. The range is $[0,1]$ (well, maybe not 0 and 1), which is what a probability is. And frankly anything between 0 and 1, what else could it be other than a probability. 
But looking at the curve, I started to doubt. I wondered if it is necessarily to be interpreted as a probability. It looks like a probability but is it really? Just because they share the same range doesn't mean they are the same. If $f(\vec{x}) = .75$, does that really mean that $75\%$ of $f$ is less than $f(\vec{x})$?
This could go in two directions:


*

*Suppose it is a probability, or more exactly the probability of a 'true', '1', or 'positive' classification of a point in the domain. How is this justified?

*Suppose not. Then what is it exactly and why? How far a way is it from a probability (numerically and conceptually)?
Another way to say this is what is so special about $1/(1+e^{-g(x)})$? Why not any monotonically increasing odd (about $y=1/2$) function with the same range, like 
$$f(x) = \frac{\tan^{-1}(g(x))+\pi/2}{\pi}$$ or
$$f(x) = {\rm erf}(g(x)) = \frac{2}{\sqrt{\pi}} \int_{-\infty}^x e^{-t^2} \ {\rm d}t$$ (very close but not equal to the logistic function) 

or frankly 
$$f(x) = 
\begin{array}{ll}
0,& {\rm if\  } g(x) < 0\\
1,& {\rm if\  } g(x) >= 0
\end{array}$$ ?
 A: 
What is the interpretation of the number that is the output of the logistic regression function?

Logistic regression as understood in recent decades is explicitly used as a model for Bernoulli or binomial data (with extensions into other cases such as multinomial), where the model if for the parameter, $p$, which is indeed a probability.
However, logistic regression has its origins in modelling the growth of a proportion over time[1] (which may be continuous), so in origins it bears a close link to nonlinear models that fit logistic growth curves

And frankly anything between 0 and 1, what else could it be other than a probability.

Well, something between 0 and 1 could be a model a continuous fraction such as the proportion of substance A in a mix of things. Can logistic regression model such a thing? The model for the mean makes sense, but the model for the variance doesn't necessarily make sense; in logistic regression the variance function is of the form $\mu(1-\mu)$. This is directly related to the variance of a Bernoulli. 
However (for example) one could consider approximating something like a beta (which has variance function proportional to $\mu(1-\mu)$) by a quasi-binomial model; then we wouldn't necessarily be modelling a probability as such, but we'd still arguably be using logistic regression to do it.
So while it's nearly always conceived as a model for a probability, it doesn't necessarily have to be.

Suppose it is a probability, or more exactly the probability of a 'true', '1', or 'positive' classification of a point in the domain. How is this justified?

I don't understand the question here. If it's explicitly a model for $p$ in a Bernoulli, what additional sort of justification do you seek? Of course the link function may be wrong (while that's no great difficulty - since other links could be used - we would no longer be doing logistic regression). 
[1]: Cramer, J.S.  (2002),
"The Origins of Logistic Regression,"
Tinbergen Institute, December
http://papers.tinbergen.nl/02119.pdf
