There has been a great deal of effort over the centuries to understand truth, and a number of models have been created as a result, one of them being the bimodal model in which every proposition is either true or false. There are obvious advantages to this model, and many domains where it is sufficient. However, it doesn’t mesh well with how human cognition views truth. I’ll give two examples: propositions that presuppose something false, and relative truth.
The first one is something like this: John found out that Hillary Clinton has been elected president. Obviously, this isn’t true. However, it isn’t false, because its negation isn’t true either: John didn’t find out that Hillary Clinton has been elected president. (Compare these with John found out/didn’t find out that Donald Trump has been elected president.) A fundamental principle of bivalent logic is that “P or not P” is always true, so the existence of propositions for which that principle doesn’t hold messes the whole thing up. One way to account for such infelicitous natural language propositions is with a trivalent system, in which a proposition is either true, false, or sans truth value; another is to propose a system of types of propositions such that some are bivalent (so ordinary logic prevails) and others are trivalent (and a more complicated logic is required). (A related issue is that some sentences are designed not to have truth values at all: questions, commands, and speech acts such as “Hello” and “I divorce you, I divorce you, I divorce you”; but I digress.)
The second has to do with propositions like Hillary Clinton is a progressive Democrat and Bernie Sanders is a progressive Democrat. Notice that those propositions encompass a wide range of subjective and objective criteria, including how the speaker/listener define progressive and Democrat, and the relative importance of the elements of those definitions. (Here on Daily Kos, I read more than one debate about both of those propositions during the campaign.) It is of course possible to declare that one or the other of them is either true or false, but as human beings we know that that is not adequate. Our intuitions about such propositions tell us that some are more true than others. On this site, people disagree about which of the above propositions is truer than the other; this means that truth, at least for certain propositions, is a gradation. The best-known model for dealing with this kind of truth is known as fuzzy logic, and it’s greatest proponent is of course Lotfi Zadeh. A common special case of propositions that tend to have fuzzy truth values is that of hyperbole: “Hillary Clinton/George Bush/Donald Trump is a criminal”. Here, how one feels emotionally about the proposition becomes highly relevant.
As I mentioned, how the speaker and listener define terms and rank criteria for determining the truth of propositions is central to all this. When all speakers and listeners in a given sphere of discourse agree on definitions and criteria, then these quibbles tend to fade away, and discourse appears almost completely populated by propositions with clearly discernible bivalent truth values. There may be a few cases where disagreement exists even within such an homogenous group, and those cases will tend to be the locus of a lot of discussion within the group. However, what we are seeing now is a clash among subgroups of Americans that disagree sharply on both definitions and criteria. Not only that, there are many highly emotional hyperbolic propositions that have polluted our sphere of discourse.
Some people believe (or as they might put it, “know”) that there are completely objective logical or empirical criteria that can be used to resolve the differences among groups of definitions and truth criteria. In fact, whether there are or not is probably impossible to determine and in any case largely irrelevant, because human beings do not live in a completely objective logical and empirical universe. For one thing, we do not have a direct, 100% reliable connection to [what most of us believe is an underlying] objective reality. We have the problems of our senses, which are imperfect, our attention, which tends to wander, our memories, which are leaky and constructive, and our frame of reference, which tends to have difficulty incorporating things that differ from what we’re used to. For another thing, we have psychological and sociological factors—individual priorities, things we’ve learned, religious faith or the secular equivalent, loyalty to and preference for one’s own group (or the opposite)—that cause us to have systematic biases toward or away from this or that set of facts, definitions, or truth criteria. I do not think it likely that objective logical or empirical criteria will resolve the disagreements we are seeing between the right and left wings of American politics.
I don’t know how this will end. In the past, these sharp differences in how we assign truth values to the world of discourse have never been resolved. Instead, some kind of force has caused one set of definitions and truth criteria to become dominant over the other. Frequently, this occurs when one subgroup is much larger and/or more powerful than the other(s). But this apparent resolution of the underlying fundamental differences in definitions and truth criteria has been shown time after time to be illusory: another shift in power inevitably occurs and what was dominant becomes once again subordinate.
But even in a subordinate posture, we continue to cling to our previously acquired definitions and truth criteria, biding our time.