Wash the apples.
Quarter and core the apples, but don't peel them. Toss them, peels and all, into the biggest pot you have. I use an 8 or 10 quart stockpot.
Pour an inch or two of water or apple juice in the pot, and cook the apples. I have stainless steel pots, so I cook them for a half hour or so on high. Depending on your cookware, you may have to cook the apples at a lower temperature. It will take a bit longer, but is otherwise not a big deal.
Take the cooked apples out of the pan and run them through a vegetable grinder/strainer, food mill, or blender. It will turn into applesauce. (So you could stop here if that's what you wanted.)
Put the applesauce back into the pot. Add the sugar, Splenda, and cinnamon to taste.
Cook the applesauce on low, stirring often (though not continuously). It will take another half hour or longer until it is a rich brown and a nice, thick consistency. Be careful that it doesn't burn; that will ruin the whole batch.
If your apple butter gets too thick, you can either put it in the blender or thin it with a wee bit of water or apple juice.
If you can it in sterile jars, the apple butter will last almost forever. In the freezer, you can keep apple butter for six months to a year. In the refrigerator, it will stay good for about a month (If you don't eat it sooner).