Every rule in chess is arbitrary.
Arbitrary is only an issue if it creates an inconsistency. The moves assigned to all of the pieces are abitrary for example, but that's part of the premise of the game. With the stalemate = draw rule, the concept of a forfeit is applied inconsistently. On the one hand, someone who can't move loses, and on the other hand, someone who can't move draws.
Logic is irrelevant here.
Obviously not, given that the logic of the stalemate rule is the point of contention.
Set of rules has to be consistent and unambiguous and this is the case with chess rules
You could say that the stalemate rule is consistently inconsistent, so from that perspective, the rule works. It is like if I watch a movie with a plot hole. The plot hole represents an internal inconsistency, but no matter how many times I watch it, the plot hole is consistently there.
The creators of the game most likely implimented the rule to give the game more complexity. Its no fun when you're down a pawn and you have no drawing chances. A skilled player should be able to draw the game, you should have to prove you can win, not just get a pawn up advantage, trade pieces, and win automatically due to no stalemate.
How do you decide if an anology is valid? To decide that, you already need to know that the argument in question holds in both the context of your original proposition and the context of the anology.
Anologies are not, have never been and will never be valid reasoning. They can just help you understand a point, not proof it.
Your notion of logic is too far away from mine (and many others, including mathematical and philosophical logicians) to have a fruitful discussion. We are talking about two totally different things using the same name for both (I don't think it's a foreign language issue). Let's stop now
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That being said: I like the concept of stalemate.