I suppose it is possible to say that truth doesn't exist, so long as you don't mean it in a logical way; rather, only as an intuition or hunch. But then we're back to faith, and what you're really saying is that faith is a reliable guide to know things we cannot otherwise know.
A paradox enters here, because it would imply that what is determined is not strictly true (since there can be neither truth nor falsehood). What I mean is, imagine a closed system; or better, imagine that the system of logic really were complete and self-sufficient. If that were the case, then everything within the system would be as predetermined as 2 + 2 = 4.
But in reality, no human system is complete, and every system contains assumptions that cannot be proved by the system. Rather, they come from outside the system, via anything from spontaneous hunches, to tacit foreknowledge of an impending solution, to God reaching down vertically via revelation. But ultimately the human circle can only be completed by and with God.
That latter is another Unavoidable Truth. You can of course say that God doesn't exist, but be mindful of what you are implying with that statement.
First of all, if it is true, then it is false, because you have eliminated any ground or basis for truth. But let that pass. What is really happening is that you are truly subjecting yourself to the Ultimate Con, because you are pretending to close a circle that in principle can never be closed by human effort.
In the absence of God, man is like a gaping wound on the surface of being. What I mean is that human beings are uniquely "opened up" to knowledge and experience, but without any reason for being so. Prior to the emergence of man, animals are indeed self-contained within their own neurology and instincts. Animals are not capable of exiting that closed circle, or of even knowing that there is anything outside their immediate experience.
I was attempting to explain this principle to a fellow over at Instapundit the other day. I see from subsequent comments that he is a conservative. But he's obviously not conservative all the way down -- and up. For he was simultaneously insisting that Darwinism is true, and that man may discover his first principles in logic.
I ask you: how can this be the case? Even if you stipulate the Darwinism part, how did randomly evolved animals escape the closed circle of Darwinism in order to know it from the outside? How do definitionally immanent creatures attain transcendence? I attempted to express this in an aphoristic way that was perhaps too aphoristic: "If Darwinism is true, then it is false."
That is not an argument, it's just a logical entailment. It is a necessary conclusion. Conversely, if you are going to affirm that Darwinism is literally true, then you can only do so by stepping outside Darwinism and making an appeal to faith, as per the above. But then you've opened a whole can of cosmic wormholes, such that Darwinism becomes a rather meager thing. It's just one wormhole among many.
When we get right down to it, the ultimate question is the relationship between epistemology and ontology, of knowing and being. As we said the other day, Kant is the one who officially presided over the divorce, ironically, in 1776, when the Critique was published.
I say "ironic" because that was the same year a bunch of dead white guys opened up their political manifesto (after a brief paragraph of throat-clearing) with the line, "We hold these truths to be self-evident..."
We can only thank Providence that Kant wasn't there, because he would have immediately taken issue with the gag about self-evident truths. How would they have expressed it in a manner acceptable to Kant?
"We hold this truth to be self-evident, that truth is unknowable by man. Rather, we can only know forms of our own sensibility. In other words, we have only access to appearances, never reality.
"Having said that, we infer that there must be a noumenal reality behind the phenomenal appearances. It's just that we can't know anything about it.
"Anyhoo, that's the bad news. The good nous (lol!) is that religion is all about the noumemal. I know what you're thinking -- 'so, religion is a lot of nonsense about the unknowable?' Not so fast! It might be true. You never know. That's why religion involves faith.
"Therefore, I propose this rewording: 'We hold this irrational faith to be entirely self-serving...'"
As it is unable to explain that consciousness which creates it, science, when it finishes explaining everything, will not have explained anything. --Nicolás Gómez Dávila