[EM] Re: Approval doesn't meaningfully meet Majority or Mutual Majority.

[James Green-Armytage]

Mike,
	I am not making errors, it's just that you are not grasping my meaning. I
will try to reword my argument in a painstakingly clear and precise way
below.

Main point: The way you word your PMC definition is not meaningful. You
should word it more like your MMC and CC definitions. 
Note: In this posting I will refer to your MMC and CC definitions as PMMC
and PCC, because they are preference criteria.

Point 1. Your PMC definition should be more like your PMMC definition.
	1a. Your PMMC definition is this:
"If a set of voters consisting of more than half of the voters prefer all
the candidates in set S to all the other candidates, and vote sincerely,
the winner should be from set S."
	1b. In order to be meaningful, your PMC definition should be more like
your PMMC definition, that is, like this:
"If a set of voters consisting of more than half of the voters prefer
candidate X to all the other candidates, and vote sincerely, the winner
should be candidate X."

Point 2: If you worded your PMMC like your PMC, approval would pass it.
	2a. Your current PMC definition is this: 
"If a set of voters consisting of more than half of the voters prefer X to
each one of the other candidates, then they should have a sincere way to
elect X."
	2b. If you worded your PMMC definition in an analogous way, it would be
like this:
"If a set of voters consisting of more than half of the voters prefer all
members of set S to each one of the other candidates, then they should
have a sincere way to elect a member of set S."
	2c. Approval passes PMMC if worded as above (in 2b). This is not a
meaningful definition of PMMC.

Point 3: If you worded your PCC like your PMC, approval would pass it.
	3a. Your current definition of PCC is this:
"If there's a CW, and if everyone votes sincerely, then the CW should win."
	3b. If you worded your PCC like your PMC definition in part 3, it would
be like this:
"If there is a CW, then there should be a sincere way to elect the CW."
	3c. Approval passes PCC if worded as above (in 3b). This is not a
meaningful definition of PCC.

Summary: Points 2 and 3 lend support to the argument that your wording of
PMC is not a meaningful one, i.e. not in accord with the intent of the
majority criterion. Point 1 shows how you can reword it to be meaningful,
i.e. by making it more like your PMMC definition.
	Take your PMMC definition, your PCC definition, and your current PMC
definition. It's kind of like Sesame Street: 
"One of these things is not like the others / One of these things just
doesn't belong / Can you tell which thing is not like the others / By the
time I finish my song?"
	The one that doesn't belong is your current PMC definition.

Sincerely,
James