I have been reading a little about realism and anti-realism which has left me thinking that my metaphysical beliefs put me in both camps? Let me explain. I'm inclined to accept the correspondence theory of truth which, I think, puts me in the realism camp as to my ontology. However, while I believe there exists a world external to mind, I do not think we come to know that world directly. Our experience and knowledge of the world is mediated by the brain which uses conceptual frameworks to make sense of all the raw data we are bombarded with daily. So it would seem, ontologically I'm a realist but epistemologically I'm an anti-realist. Does this make any sense?

Let's make two initial comments to muddy the waters! 1) Accepting some version of a correspondence theory of truth -- e.g. accepting that a true proposition is made true by the existence of a corresponding fact -- doesn't ipso fact make you a realist in your ontology. It will obviously depend what you think about facts ! (You could still be an idealist like Berkeley, and suppose the only facts are ultimately those involving God, other spirits, and their ideas.) 2) Accepting that our knowledge of the world depends on a lot of processing of data by the brain using built-in cognitive mechanisms doesn't make you an anti-realist in epistemology. You could still hold that when those processes are working reliably, they successfully give you epistemic access to facts that obtain independently of you and your cognitive mechanisms. I'd say that talk of a "correspondence theory of truth", "realism about ontology", "conceptual frameworks", and "epistemological anti-realism" is all far too slippery...

Is it correct to say that square circles (and other incoherent ideas) do not exist? Or would it be more accurate to say they neither exist, nor don't exist?

You need to be careful to distinguish things and ideas here. Is the question about square circles or about the idea of a square circle ? Compare: there are no such things as unicorns . It would plainly be wrong to say that they "neither exist nor don't exist": unicorns definitely don't exist! But the idea of a unicorn exists and seems coherent enough. Indeed, we are tempted to suppose that there could have been things that fitted the idea. Likewise, there are no such things as round squares . Like unicorns, round squares definitely do not exist. But this time, though we can frame the idea of a round square -- we grasp that something counts as a round square if it is round and a square! -- the idea is a self-contradictory one in the sense that nothing can possibly count as fitting our idea here.

Over a year ago, I read Quine's Two Dogmas for a philosophy class. One part in it makes the step from talking about meanings to abolishing meanings and talking only about synonymy. I never quite got that. I mean, if there are two things similar (or the same) about something, don't they each have to have those things? If two pieces of string have the same length, they have each have a length, and they happen to be the same. Likewise for any other properties I could think of, such as color, volume, mass, etc. I don't see how sameness could not imply those "intermediary entities" which are the same. Thanks.

Consider an example from Frege: the direction of the line L is identical to the direction of the line M if and only if L is parallel to M. That's true. But how should we read it? Do we read it as explaining the notion of being parallel in terms of the identity of two abstract objects, i.e. two directions? Or do we take it the other way about, as partially explaining talk about two abstract objects, directions, in terms of the already-understood notion of lines being parallel? There's lots to be said for taking it the second way, as introducing reference to certain abstract objects in terms of something more familiar. Likewise: the meaning of "gorse" is identical to the meaning of "furze" if and only if "gorse" and "furze" are synonymous. That looks true too. But how should we read it? Do we read it as explaining the notion to synonymy in terms of the identity of two abstract objects, meanings? Or do we take it the other way about, as (hopefully) partially explaining talk about two...