Comments on approach 1a:
To clarify what I meant by approach 1a, it's as if the writer of a criterion failure example preceded his statement of his scenario or example with the following introductory paragraph:
"If what is said in the following paragraph (I'll call it the scenario paragraph) were a true statement, then, as you can judge for yourself, the premise of the criterion would be met, and, as I'll show, if [some particular voting system] is used, the requirement of the criterion is not met."
(here would follow the scenario paragraph)
What I've just said in the introductory paragraph could be a true statement, even if part of what is said in the scenario paragraph that follows it has imprecise meaning, unknown meaning, or no meaning. That's because in the quoted paragraph I said "_If_ what is said in the following paragraph were a true statement." So, whether everything in the scenario paragreaph means anything and could be true, is irrelevant. The quoted paragraph merely talks about _if_ what follows were true.
So, if the scenario paragraph says that a majority of the voters prefer X to Y, and since WDSC says that if a majority prefer X to Y then WDSC's requirement must be met, and if, when that voting system is used in the situation described in the scenario paragraph, the majority said in that paragraph to prefer X to Y have no way to meet WDSC's requirement, then it can be said that the method fails the criterion, even without a definition for "prefer", for the reasons described above. Whatever "prefer" means, the scenario paragraph says those voters prefer X to Y, and WDSC says that if a majorilty prefer X to Y, then they must be able to meet WDSC's requirement.
Of course the writer of an example could actually word it by starting with a paragraph like that introductory paragraph, to clarify the point that I'm making here. But of course the introductory paragraph isn't needed, because the things that it asserts, when true, are just as true without being asserted in that paragraph.
The preceding is intended to clarify approach 1a, which argues that well-defined criteria could use the word "prefer" even if it had imprecise meaning, unknown meaning, or no meaning.
Comments on approach 1b:
To show that "prefer" has a meaning, whether or not that meaning is precisely-defined, I'll use that word here:
Markus and some others may prefer that "prefer" not be used in definitions of criteria, but expressing that preference isn't the same as showing that such criteria are ambiguous, unclearly-defined, or not well-defined.
I define criteria in terms of preference because that avoids the problems that I've described that some criteria would otherwise have. But it would be great if someone posted definitions equivalent to my criteria, without using the word "prefer". The only legimate objections to the use of "prefer" in criterion definitions are aesthetic objections. I prefer the aesthetic objections to the problems that some criteria would have without the use of "prefer".
Markus, you keep saying that definitions equivalent to mine, but which don't use "prefer" are or should be possible. Can you or can you not post one?
Mike Ossipoff
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