Hilary Putnam argues that formal systems cannot even be used to perform hypothesis selection. Whether you wish to characterize this as "prescription" is up to you, but one might suspect that if formal systems cannot serve here exclusively, they also cannot serve exclusively in other domains with a prescriptive element (e.g. ethics).
Epistemic Values are Values Too
The classical pragmatists, Peirce, James, Dewey, and Mead, all held that value and normativity permeate all of experience. In the philosophy of science, what this point of view implied is that normative judgments are essential to the practice of science itself. These pragmatist philosophers did not refer only to the kind of normative judgments that we call "moral" or "ethical"; judgments of "coherence," "plausibility," "reasonableness," "simplicity," and of what Dirac famously called the beauty of a hypothesis, are all normative judgments in Charles Peirce's sense, judgments of "what ought to be" in the case of reasoning.
Carnap tried to avoid admitting this by seeking to reduce hypothesis-selection to an algorithm—a project to which he devoted most of his energies beginning in the early 1950s, but without success. In Chapter 7, I shall look in detail at this and other unsuccessful attempts by various logical positivists (as well as Karl Popper) to avoid conceding that theory selection always presupposes values, and we shall see that they were, one and all, failures. But just as these empiricist philosophers were determined to shut their eyes to the fact that judgment of coherence, simplicity (which is itself a whole bundle of different values, not just one "parameter"), beauty, naturalness, and so on, are presupposed by physical science, likewise many today who refer to values as purely "subjective" and science as purely "objective" continue to shut their eyes to this same fact. Yet coherence and simplicity and the like are values. (The Collapse of the Fact/Value Dichotomy, 30–31)
When Carnap failed to find "an algorithm", he failed to find a formal system. When Putnam talks about "values", he means areas of human judgment which cannot be encapsulated in formal systems.
This being said, it's not clear that the right thing to say is "human judgement trumps logic". There are many kinds of logic. Instead, what I think you really want to say is that humans can always find a given formal system to be too restrictive for current circumstances, and thus "grow" it in order to handle the new situation. One might view this as "rationalizing", but that would presume that our minds cannot grasp on to higher orders of logic than they can formalize. This seems contentious, given concepts such as tacit knowledge and the more general unarticulated background. Any stance of scientific realism must suppose that the new rationality of a scientific revolution somehow continues the old understanding; otherwise, you end up calling it completely wrong. There is, however, also a discontinuity in scientific revolutions, with many treatments. One of my favorites is Alasdair MacIntyre's 1977 Epistemological Crises, Dramatic Narrative and the Philosophy Of Science, in The Monist.