Potholing that last sentence was mostly for fun.
By virtue of the definition of a logarithm, exp(log(x)) = x, we can derive that since the exponential function is well-defined for complex numbers, so is the logarithm. Taking the logarithm of a negative number nets you the logarithm of the absolute plus imaginary pi. The real part of any logarithm is a symmetric function, and there are probably a few other interesting properties of logarithms in complex analysis that I don't know of.
log(0) is undefined, as you note, but that does not mean the limit x -> 0 log(x) is undefined. It is in fact a pole singularity (if I have understood my singularity analysis correct). No matter how you approach it, you get negative infinity. So given your "logarithmic velocities," I counter with the fact that using a limit it is still a perfect isomorphism. Limit of x -> -inf exp(x) is indeed 0, so when using limits (which is practically what Real Numbers are for) your argument that log isn't an isomophism from reals to reals is hereby proven invalid (if you want a step by step proof, just ask, it'll be a fun exercise).
Given that logarithms are a category-theoretic isomorphism from the reals onto the reals (from the multiplication group onto the addition group) there is no reason why log-odds isn't as valid as odds, which is as valid as ]0,1[ probabilities. Infinities are not valid Reals, 0 and 1 are not valid probablities QED.
As I said. Do not challenge anyone (including me) on the abstract algebra of the matter.
[I do apologize if the argument is poorly formulated, I am as of writing mildly intoxicated]
Yes, log(x) is an isomorphism from the positive reals as a multiplicative group to the real numbers as an additive group. As a result, it is only an isomorphism from multiplicative probabilities to additive log-probabilities if you assume that 0 is not a probability to begin with, which is circular logic.
I was very interested in the discussions and opinions that grew out of the last time this was played, but find digging through 800+ comments for a new game to start on the same thread annoying. I also don't want this game ruined by a potential sock puppet (whom ever it may be). So here's a non-sockpuppetiered Irrationality Game, if there's still interest. If there isn't, downvote to oblivion!
The original rules:
Enjoy!