Michael Lynch's book True to Life (2004) is an outstanding book. It's chock-full-of arguments concerning the value of truth and how it's linked to important and timely topics like politics.
There's one argument appearing early in the text that he uses to show that truth is objective (p. 10):
1. If I know something, then it's that I know that I don't know everything and neither does anyone else. (premise)
2. There are some things we won't ever know and there are things we think we know but we don't. (premise)
3. So, truth is objective. (1,2)
First, let's admit that the argument is abductive or an inference to the best explanation rather than what it may appear to be, like a modus ponens or modus tollens argument. From our best evidence that I don't know everything and others don't know everything, as well as the assumption that we won't ever collectively know some things we may infer that truth is objective.
Second, something that troubled me upon reading that argument in 2004 was the difficulty I had moving from my or our ignorance to truth's objectivity. There are two reasons why I think the move is an illicit one: (1) our ignorance only proves our fallibility not truth's objectivity and (2) the argument presumes a special understanding of objectivity.
I agree that there are things that we won't ever know, and I agree that I (or we) don't know everything. For example, we will never know what the last day on Earth will be like, whether there are impossible objects, or how many licks it takes to get the center of a Tootsie Roll Pop (well, maybe we'll know that eventually). And, there are things we think we know but don't. For instance, we think we know that COVID-19 originated in a wet market in Wuhan, China, but do we really know that? Anyway, each of these claims show that we are ignorant. Ignorance, at least here, is not knowing what the facts are. When we are ignorant of x, we fail to know what x is, why it has come about, or what explains x as a phenomenon. Ignorance could also include not being able to ever know something. We may never know the number of stars that ever existed in the universe. We don't have the time or the energy to count all of them. In each of these cases ignorance brings to the fore our epistemic liabilities.
Our epistemic liabilities, however, seem to float free from whether truth is objective. Sure, there is a fact of the matter that there are y number of stars that have existed in the universe. But the fact of the matter doesn't make truth objective. The limits of epistemic accessibility has to do with our own liabilities, and these liabilities are unconnected with whether there is a fact of the matter. That there is a fact of the matter is a metaphysical thesis. Epistemic theses are a far cry from something being objective. The most we can infer from our own ignorance is that we don't know a lot. Objectivity is surely more than expressing something about our limited cognitive abilities.
This brings us to the second problem. If the objectivity of truth were to follow from the first two premises, then we might think that truth's objectivity is presumed in the premises of the argument (particularly 2). The presumption is that we're ignorant. Let's take objectivity to be that a proposition is true, whether there were agents here to know them or not. Truth's objectivity, then, is a matter of being true independently of knowers. That would make our own ignorance incompatible with truth being objective. If we tweak our understanding of truth's objectivity slightly and admit that objectivity is a matter of any number of people agreeing to the truth of some proposition, then we understand objectivity to be intersubjective. It wouldn't be a matter of consensus, per se, but the truth of a proposition would be determined by its agreement with the way the world is. For these two reasons, I see that such a definition of truth's objectivity may be deficient.
Let's consider matters from a different direction. Instead of worrying about what objectivity is and how it is connected with truth, let's consider the claim that epistemic access gives rise to truth's objectivity. Just what would it take for our epistemic failure to access worldly or contingent truths that imply something about the objectivity of truth? The argument must go something like this. If we don't know everything and our knowledge of the world is limited, then that doesn't rule out that there are truths beyond our capacity for knowing them. Anything beyond our capacity for knowing them suggests something like objectivity. Therefore, our own ignorance implies that truth is objective, and so truth is objective. It is a part of our ignorance that shows truth to be objective.
I have already recommended against such a view by showing how our epistemic access to the truth of some proposition may not entail objectivity in a metaphysical sense, i.e., in the sense that we usually mean when we talk about moral objectivity. Murder is morally wrong insofar as the unjustified intentional killing of someone is morally wrong. Even if we lived in a world where murder was acceptable, broadly speaking, it wouldn't overcome the criticism that murder is objectively morally reprehensible. That's the main reason I think we may want to resist the above argument. My main complaint is that truth's being objective doesn't follow from the premises. And , even if it did follow, the argument seems to presume truth's objectivity in the premises.