HD Comcast Box and an HDTV

You can't watch HDTV without a HD cable box...
I know that already. I was wondering about regular channels, such as Fox or Lifetime or CNN and such.

Your cable box won't magically project HD channels and all that without being a proper HD box. You need to call your cable company and see if your current box will do the trick, most likely not.
Nope, this box will not do the trick. Need to get an upgraded one.

An HDTV simply allows for HD channels to be displayed, but regular channels will still look shi;tty as ever.
Sorta answers my question??

Answers in bold. But it doesn't exactly address my concerns.

Say for instance I switched the old box from the older TV and plug it into the new TV. Will it show a TV channel? Or will it simply show no channel at all?