It looks a lot like my General Sikorski, which is blooming for me now in Wa state. Mine has a red bar along the midribs when the flower first opens, and then fades to blue. Those large flowered blue ones are hard to ID. There are so many of them and the color doesn't photograph well. You might check out clematis on the web's search feature. You can go to clematis search, put in blue as the tepal color, white as another color(for the anthers, stamens) pruning type 2 and it will give you a lot of photo's to look at. That might help you ID it.

Here is a link that might be useful: Clematis on the web

It was bought at Wal Mart and the tag says Josephine TM. I don't know if I can trust this. It doesn't look like the pictures of Josephine TM. If I can't ID it, how do I care for it?

Josephine has a double pink flower. Your plant looks like a large flowered hybrid pruning group 2. It blooms on old wood and new wood. You don't have to prune it and it will bloom in the spring on the old wood. If you water and fertilize it after the spring bloom you can get a rebloom on the new wood later in the summer or fall. You can also hard prune it every spring if you prefer. You will miss blooms in the spring but it will bloom for you on new wood in the summer. Clematis like moisture during the growing season and fertilizer. A tomato or rose fertilizer is fine. Type 2 clematis grow well here in the PNW. With mine I did very little pruning for the first 10 years. They started getting bare legs and very few blossoms at the bottom of the plant. This year I hard pruned my type 2 to rejuvinate them. They produced a lot of new vines at the base and have a very full look now. So if they get too bare at the bottom you can always hard prune them in the spring. Good luck with your clematis, whatever it is.