Errant beliefs are not simply about factual matters, they are about concepts, including what it means to be wrong.
In the television show, The Big Bang Theory, in the episode "The Hofstadter Isotope," Sheldon Cooper discusses the nature of "wrong" with comic book store owner Stuart:
Stuart: Ooh, Sheldon, I'm afraid you couldn't be more wrong.
Sheldon: More wrong? Wrong is an absolute state and not subject to gradation.
Stuart: Of course it is. It's a little wrong to say a tomato is a vegetable. It's very wrong to say it's a suspension bridge.
If you agree with Sheldon, then you believe that there cannot be degrees of falsehood. Sheldon claims that "wrong is an absolute state," but fails to follow up that claim with anything new. He simply rephrases his position.
Stuart's view, on the other hand, relates to scientific exploration as science gets closer and closer to uncovering new natural truths (i.e., whether loop quantum gravity or string theory bests explains the physical world). Thus, when we talk about a tomato being a vegetable, we are closer to the truth than when we talk about a tomato being a suspension bridge. Again, it is all a matter of 'closeness.'
Often times we hear the expression "There's some truth in that," or "That's closer to the truth."
If truth can be measured in degrees, so too can falsity. In all reality, "A tomato is a suspension bridge" is no more wrong than "A tomato is a vegetable," but the second claim seems closer to the truth.
For example, there are no possible worlds in which a tomato is and is not a suspension bridge. In some possible worlds, however, a tomato may not be a vegetable. Of course in our world - in the real world - where a tomato is a vegetable, a tomato being a vegetable and not a suspension bridge represents actuality and not fiction (although, admittedly, the idea of a tomato being a suspension bridge is quite entertaining... it reminds me of James and the Giant Peach, in which a peach is a ship).
In reality, Stuart's refutation shows Sheldon that being wrong is subject to gradation. This graduation is exactly what Stuart offers with his suspension bridge example. If you can give even one (true) refutation, then logically speaking, you've just won the argument.
Being right about being wrong is right up a philosopher's alley, even if the concept, itself, is wrong.