Wavejumper:
The whole concept of Truth becomes as fuzzy, undefined and mysterious as the concept "reality".
Only meaningful declarative statements can be true or false. In non-bivalent logic systems one may also allow values of indeterminate. Notice here that the idea of a 'perfect' truth is really no longer applicable. Given a statement, it is either true, false, or indeterminate, maybe some fuzzy degree thereof, (or if it is a sort of statement that does not have truth-value, then it is neither).
The philosopher Ludwig Wittgenstein (1889 - 1951) provided the following model for thinking about the world. The world is the totality of facts. Facts consist of states of affairs.
A fact (perhaps not as scientists use the term) is simply whatever is the case - however the world is. Note, this is a statement about the world, not a statement about our knowledge or capacity for knowledge about the world. Facts hold whether there is actual 'vagueness' in the world or not. This is a presupposition, but does not require an argument because otherwise, there is no world (even if there is no 'mind independent' world, the presupposition is still admissible).
The most basic (non-inferential) statement that one can make about the world is an observation report - a statement that has as its semantic content one's immediate sensory experience (in an earlier post I have provided a model for thinking about how these statements can be true or false).
Such statements cannot be truly asserted independent of the experience that prompts them and they are never epistemically 'certain' - in the sense that such a statement could not possibly be false. They are always probable and prone to error. So we require justification. Justification for empirical claims never logically entail the truth of such claims (otherwise they would be necessary truths, which would mean such claims are certain, which they are not -- notice this is a claim about knowledge acquisition, not what might be the case for states of affairs). The strength of a justifier is: if the justifier is true, the more likely the claim being justified is true. The relationship is only probable, and probability is determined by 'relevant alternatives', i.e. for an empirical claim, empirical alternatives are more relevant than logical alternatives, similar empirical alternatives are more relevant than non-similar empirical alternatives, and so forth. This doesn't always hold, but for exceptions some account must be offered. This does assume that states of affairs (at least as experienced, which is the denotation of observation reports) are 'law-like' (which is an admissible assumption 1. experience would be incomprehensible otherwise, and experience is comprehensible, 2. it makes no assumption to what the world might be like independent of experience (my model of how observation statements are true or false allows for the possibility that such statements can be be true and the world (in itself) be very different or even incomprehensible for us).
Subjectivity and objectivity are (I suggest) not best thought of as 'mental' and 'non-mental' (this distinction, as connoting a real distinction, is under fire. Many philosophers (at least the ones concerned with what is going on in empirical science) already consider such concepts to be outdated, and relegated to our 'folk' ways of speaking about ourselves. Subjective / objective distinction is relevant in thinking about statements - it is an epistemic distinction. A subjective statement is one that either intrinsically depends on some person for its truth-value, or is a statement that cannot be true or false, i.e. 'I like apples', 'Picasso is the best painter ever'. An objective statement is one that is not contingent on some particular person or persons for its truth-value, i.e. 'There is a tree in my front yard'. This may be true, it may be false, but whichever it is, it doesn't depend on my mere belief about it that it is so. Notice that objective claims have this quality even if one believes that trees are not there when we are not observing them -- objectivity is a quality of statements...not of 'objects'. If I say, 'Trees only exist when we are observing them' (whatever that is supposed to mean)...while trees may depend on our observation, the truth of the statement does not depend on our observation.
Objective claims are only probable, their justification requires inter-subjectivity...this should all be pretty familiar for you scientific minded thinkers.