Is pure relativism impossible?
I use to think that relativism was a logically inconsistent position. But maybe it is not about truth, but about consciousness and its subtle focus. If truth is empty of objectivity, is this a fact ad therefore objective? "It's all empty". But "is that claim substantial?". It just descends in an endless spiril, more often being on the side of objectivity but forever sliding away from it. It's easier at that point to to say, instead of "there is no truth", that "everything and anything is true".
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From one of Tom Waits' songs, as best I recall: “Everything you can think of is true [...]” This can only make sense, thought, from the roundabout perspective that whatever is believed to be true will, for that same span of time, be true to the given believer. The topic is complex for me since it gets into complex epistemological dichotomies between, what I will for brevity term, “believed truths” and “true beliefs”.
Still, being in accord with your observation that to claim no objectivity exists is to claim an objective/impartial stipulation of what is objective/ontically real (in which case, the statement becomes a logical contradiction: objectivity both exists and does not exist at the same time and in the same respect):
Because the only noncontradictory conclusion obtainable is that some non-relative ontic reality exists (so to phrase), some believed truths can only be untrue. More specifically, wherever there is logical contradiction between believed truths, one, some, or else all of these believed truths will need to be false. This, at least, wherever the law of noncontradiction is granted.
"Is that true?"
"Is there truth in that definition?"
"No, there is no substance"
"Any substance in that response"
"No"
The problem of philosophy
"Fact" and "truth" aren't the same thing in my usage.
So objective facts don't imply anything about objectivity for truth.
The first is a principle from early Mahayana Buddhist philosophy - the 'doctrine of two truths'. This is the principle that there is a domain of conventional reality (sa?v?ti-satya) which roughly encompasses what we understand as the phenomenal domain, the realm of appearance, empirical facts.
But there is also the domain of ultimate truth (param?rtha-satya) which is 'known by the Buddha, beyond word and thought and can be positively grasped only by intuition. Empirical truth, on the other hand, is based upon knowledge of the external world by means of verbal designation. In the final analysis, however, phenomenal existence has no independent substantiality corresponding to the words used to describe it. Such existence, as asserted by realists, is merely fictitious or conventional 1. Not seeing this is to 'absolutize the relative' which is very close in meaning to 'scientism'.
The second heuristic is Kant's assertion that he is at once an empirical realist and a transcendental idealist (summary here.) This allows Kant to recognise the validity of empirical truth claims, whilst also asserting the fundamental role of the categories of the intellect (etc) in our cognition of reality.
I think most of the confusion around this topic arises because of the insertion of science into the domain previously occupied by religion and philosophy. In the latter, there is the ability to acknowledge the reality of a philosophical absolute (typically but always conceived of as 'God') which is felt or intuited but can't be directly known by the mundane intellect. It is precisely that which is rejected by modern naturalism which invariably wishes to locate the absolute in the domain of the relative, or to 'absolutize' scientific knowledge. You see a lot of that in popular discourse.
A Pope would be obliged to say something like that, but there is actually plenty of cross-cultural dialogue between Buddhists and Catholics, and even a popular movement called [url=https://www.google.com/search?ei=ENhpXaSsAvzXz7sP7Li4qAs&q=Zen+Catholicism&oq=Zen+Catholicism&gs_l=psy-ab.3..0i71l8.0.0..14657...0.3..0.0.0.......0......gws-wiz.7Ot6a8fxS0M&ved=0ahUKEwik9OrohKzkAhX863MBHWwcDrUQ4dUDCAo&uact=5] Zen Catholicism.
[/url]Quoting Gregory
Caution is needed with the word 'substance' in the context of philosophy. It was originally used to translate the Arisotelian 'ouisia' which is nearer in meaning to 'being' than to 'stuff'. Substance is 'the bearer of predicates' - a rough way of putting it is that the substantial being is what a being really is, whereas the accidents (blue eyes, for example) are incidental to it. So if you look at passages in philosophy where the term 'substance' is used, and mentally interpolate the word 'being', while it's not quite correct, it does provide a very different perspective on the usual way of understanding 'substance' as 'a truly existing thing'.
Buddhism famously denies substance but again caution is needed, because it developed in a very different cultural milieu. The Buddha's interlocutors were Brahmins who had their own doctrine of 'being' which was also quite different from the Aristotelian. It's an interesting question to study but requires careful interpretation. (Incidentally, a lot of what I said in my above post was derived from a 1955 book by a British-trained Indian scholar called Murti, called The Central Philosophy of Buddhism.)
There are prohibitions on unrestricted comprehension, e.g. the expression, "the set of all sets", is not allowed. Bertrand Russell famously asked, "Does the set of all sets that do not contain themselves, contain itself?" That was the beginning of the gigantic foundational crisis in mathematics at the very end of the 19th century.
In fact, it is still ongoing. The lowermost foundations of (classical) mathematics have turned out to be impredicative (circular). There is no hope for a ramified foundation.