Three challenges to truth
Part 1: The Truth™
Part 2: No truth
Part 3: Truthiness
The foundation of non-bullshit metaphysics is the discussion, construction and defense of the notion of "truth". What is this notion?
We can first look at the cognitive/linguistic job for which we employ the notion of "truth". Regardless of of the specific word we use, we still want some word to denote this job, and "truth" is at least as good a word as any other, with the advantage of historical use.
The first job we want to do is discuss distinctions. Thus the notion of truth defines the notion of falsity: If we discuss something as true then it follows that we discuss something else as false—we are making a distinction. The second job we want to do is discuss universal distinctions: If we discuss something as true for someone, somewhere, at some time, we discuss it as true for everyone, everywhere, always; likewise for falsity.
These are not the only cognitive/linguistic jobs we want to do, but we have perfectly good words, sanctified by historical use, to label those other jobs. The most important of these jobs is to discuss non-universal distinctions; we label that job as "opinion".
Since truth refers to a specifically cognitive/linguistic job, it follows that truth and falsity are properties of statements or sets of statements, in the extended sense that we can discuss particular states of mind or neural states in the same sort of language we use to discuss written or spoken statements in a natural language such as English. Mind/brain states are physical, symbolic representation with a vocabulary and grammar, just like statements in natural language.
There are three challenges to this job, hence three challenges to truth.
The first challenge to truth is the "modernist" notion of The Truth™. The Truth™ is a challenge to ordinary small-t truth because it attempts to over-determine truth by privileging a particular context evaluating statements.
We know from Quine et al. that no statement even has meaning, much less truth, by itself. To understand a statement, we need a context. There are two important contexts: the intensional and the extensional. The intensional context governs how we interpret the statement, how the words or concepts or mental states hook up with other mental states. Examples of components of this intensional context applied to natural language statements are dictionaries, thesauri, and grammars. Another example of an intensional context is an axiom set, such as Peano's arithmetic, coupled with the grammar of propositional calculus. The extensional context is just the real world.
The Truth™ is a challenge to truth because it privileges a specific intensional context as true (and thus implies that alternative intensional contexts are false), and therefore the evaluations made under the privileged context become not just true (i.e. true in that context) but The Truth™, as all other contexts are false.
For instance, it is an ordinary truth that "2+2=4" is a theorem of the intensional context of ordinary integer arithmetic. When this intensional context is used to evaluate extensional reality, it is true in the sense that if you put two stones in a jar, and then put two more in, and then count the stones in the jar, you will count to four.
To construct an obviously absurd example, The Truth™ would declare ordinary integer arithmetic a privileged "true" context. Thus "2+2=4" is not just a truth about arithmetic, or a truth about some particular set of stones in a jar, but The Truth™. Of course, once we start applying the underlying concepts of our arithmetical context to other extensional realities, such as the motion of the hour hand of an ordinary 12 hour clock, we start running into silly contradictions: "7+8=15" is a valid statement of ordinary arithmetic and a true statement about stones in jars, but "7+8=3" is a valid statement of modulo-12 arithmetic and a true statement about the position of an hour hand.
Obviously, the statement "7+8=15" means different things in different intensional and extensional contexts; we are entirely justified in considering the same set of symbols to represent different statements in different contexts.
I constructed the above example specifically to highlight the absurdity, but the concept of privileging an intensional context crops up time and again in more subtle ways. For instance modernist meta-mathematics privileged the abstract conjecture-proof intensional context, making alternative contexts false (or at least marginalized as non-mathematics). Postmodernist mathematicians are now exploring a rich vein of alternative contexts, including computational, representational, and probabilistic mathematics.
The most historically egregious example of privileging an intensional context to establish The Truth™ is privileging specific cultural and religious contexts to establish The Truth™ in the realm of ethics.
It is a straightforward truth that people in Western societies approve of democracy, a truth in the sense that it is unproblematic that even an alien species observing us from galaxy NGC 6745—regardless of their particular characteristics, assuming only they were able to rationally discuss the subject at all—would conclude that common approval of democracy is a true statement about the people in Western societies.
However, if we arbitrarily privilege this particular social/cultural context as the true context, then "democracy is good" becomes not just a truth but The Truth™; if a society or culture does not embrace democracy, then their own context is false. Likewise for Christianity or Islam, sexual morality and driving on the right-hand side of the road.
Any time you hear someone talking in terms of The Truth™ (you can actually hear the capital letters and the ™ symbol), look carefully and you'll see they're always privileging some particular intensional context for establishing The Truth™.
The Truth™ is fundamentally a challenge to truth because small-t truth is always dependent on intensional context. (The context dependence does not entail subjectivism, though, because context is a particular subjective state, it is not established by particular subjective states.) There is no way of establishing the truth of an intensional context, in the sense of applying to contexts the job we expect of "truth". Specifically we have no basis other than the arbitrary (non-universal) imposition of a context to distinguish a true context from a false one. Thus in making the distinction we just push the arbitrary non-universalism around; we don't eliminate it.
The Truth™ fundamentally, then, is revealed as bullshit in the philosophical sense: Using truth-language while being fundamentally indifferent to truth.
 For the purpose of this discussion, we will presuppose the notion of scientific realism, that a real world exists independent of our minds and—usually directly but often indirectly—causes our perceptual experiences. All the same sorts of conclusions, however, apply mutatis mutandis to Phenomenalism, with perception standing in for extension, from which we can construct the notion of scientific realism.
 Postmodernism will get its share of criticism in the next installment: "No truth".
 See A Glance at Postmodern Pedagogy of Mathematics (h/t to philosophical bits).
 My apologies: This comment is the soul of (bad) obscurity. I promise to clarify in the future.