So Bertrand has made a logical fallacy because "The inability, or disinclination, to disprove a claim does not render that claim valid, nor give it any credence whatsoever".
Yes, he was making a logical fallacy, on purpose: he mimicked the logic behind god-worship.
"There is a god; I can't prove there is one, but you (an atheist, for instance, being "you") can't prove that there is not one. Therefore there is one."
This is the logic Bertrand Russell paralleled in his famous tea-pot-in-orbit thought experiment, and his reasoning is false, perfectly pointing out how the god-worshippers' reasoning is logically faulty when they arrive at an absolute and invincible belief in god.
-- Updated 2017 August 29th, 3:04 am to add the following --
I don't believe there's a tea pot orbiting Saturn, therefore, there is a tea pot orbiting Saturn. And the burden of proof is on you to demonstrate that there isn't one.
1. I believe there is NO tea pot orbiting Saturn.
2. Therefore there is a tea pot orbiting Saturn.
There is no logical connection here. There is the word "therefore" but it connects two incongruous claims.
The whole thing is bizarre. It is not a product of logic, and therefore you can't measure it with the yardstick that measures logic (such as "is this a fallacy or not").
This is not a fallacy. There is no logic connecting the two claims.
1. "I believe X. Therefore X does not exist."
2. This can only be true if whatever I believe in does not exist.
3. If someone finds an instance of what I believe in exists, then the general rule of 2 is falsified.
4. And after that the point in 1. is falsified.
The falsification proof hinges on the fact that there are some things I believe in, yet they exist.
In other words, the proof hinges on an empirical finding, and that is exactly what the tea-pot experiment hinges on. No find, then theoretically possible, but practically useless. Yes, find, then proof is not necessary, because it's a fact.