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 ---- Election-methods mailing list - see http://electorama.com/em for list info