The only positive thing about minimum wage logic is that it is easily defeated. Common sense economics rejects it, but non-economic arguments cause it problems, too.
Here is a non-economic challenge to minimum wage proponents I have yet to hear convincingly answered (perhaps a commenter or my counterpart Tim can provide an explanation): If someone—say, John—wants to work for six dollars an hour at Company X, and Company X wants to hire John for that wage, why should that be illegal?