This is a bit of a circular question. What do we actually mean by 'truth', especially when referring to using notions of truth in reasoning about the world?
An instructive exercise is to ask why we should think that boolean logic as we use it today is 'true'. When we get down to it, it seems to me that when we say that it is 'true', we actually mean that it is useful: we find it to be a reliable tool with which to do reasoning, and we find that whenever we make a mistake, the error can be traced to the ideas to which we applied logic, and not to the logic itself. That is to say: it seems a mode of reasoning which is as good as the information upon which it acts — it is subject to the maxim of Garbage In, Garbage Out, but it is not so brittle or sensitive to error that it is useless. This prompts us to trust it.
If this is true of boolean logic, could another more subtle logic also prove similarly useful, or even more useful? Of course, boolean logic forces us to make precise distinctions when asking questions, which is potentially useful. But when considering the world around us, we run into famous problems such as the Sorites paradox. These paradoxes can be described as paradoxes of language rather than logic, but the distinction is perhaps illusuory: just as we are encouraged to ask whether a pile of sand is "a heap" or is not "a heap", we ask ourselves whether the Sorites itself indicates "a weakness of language" or "a weakness of logic". Because logic is only as useful as the language which is fed to it, and language only as useful for reasoning as the logic applied to it, the real question is how best to repair the flaw in the Language/Logic system, if we decide that it is worthwhile.
But the real teaser is not the Sorites, and artificial distinctions in colloquial language, but advances in science. What ought we to say about Newtonian Mechanics — is it true? Well, it's useful enough for most purposes. It's not badly wrong. It's not correct enough to high enough precision for all purposes, but you can build bridges, skyscrapers, and lunar expeditions relying on just Newtonian mechanics and a bit of crude material science. And what about the Earth being round? If by 'round' you mean "a sphere", you should account for the bulging at the equator due to the centrifugal force; and if you mean "an oblate spheroid", you're still neglecting the variation in elevation. These descriptions are practical approximations to a very complicated 'truth', that is, a complicated state of affairs; and yet they're not what we would describe as "horribly wrong". They're just not 'perfectly' right; you can get closer to the truth by a more detailed description.
So, if you're principally interested in ordinary language (which I'd be careful about — is cold a thing that can seep into your bones, does the Sun really rise and fall in the sky?), I would say that we already implicitly recognise that multivalued logic is more useful — 'truer' — than boolean logic. It's just that when we demand precision, we don't know how to move forward except to replace the complicated messy notion of truth with the idealised boolean one, and hope that degrees of error are numerically measurable, so that we can capture the truth (and our uncertainty about it) in quantity rather than verity.
The only question is whether this is the best way of doing things. I rather hope that mathematics yields a multivalued logic which has greater practical power than boolean logic. And although it would be a formidable task to do so, I don't believe that it is a priori impossible.