6

It seems that the answer would be yes, especially when we think about the example of the preface paradox (authors stating in prefaces "the errors that are found herein are mine alone", i.e. believing they made some errors, even though they presumably proofread their own book, and also believe to have eliminated them). What do philosophers say on this issue?

7

It can. Ramsey put it nicely in his "last papers" written around 1929 under the influence of Peirce's pragmatism (quoted from Marion, Wittgenstein, Ramsey and British Pragmatism):

"We want our beliefs to be consistent not only with one another but also with the facts: nor is it even clear that consistency is always advantageous; it may well be better to be sometimes right than never right. Nor when we wish to be consistent are we always able to be: there are mathematical propositions whose truth or falsity cannot as yet be decided. Yet it may humanly speaking be right to entertain a certain degree of belief in them on inductive or other grounds: a logic which proposes to justify such a degree of belief must be prepared actually to go against formal logic; for to a formal truth formal logic can only assign a belief of degree 1.[...] This point seems to me to show particularly clearly that human logic or the logic of truth, which tells men how they should think, is not merely independent of but sometimes actually incompatible with formal logic."

What Ramsey calls "human logic" was to be the basis for his theory of subjective probability, degrees of belief, and is now called epistemic logic, to which he was a precursor. His first point is that our beliefs are unlikely to be always in line with the truth, so insisting on consistency may lead to spreading falsehoods around more than is otherwise necessary.

A good example of this was the then recent Bohr's model of the atom. When proposed it was inconsistent with classical electrodynamics, the only electrodynamics then available and amply confirmed by multiple experiments, because according to it rotating electrons had to radiate light and quickly fall upon the nucleus, contrary to what Bohr postulated. Yet it made good predictions about spectral lines, so there were reasons to hold on to both, despite their inconsistency. The inconsistent conglomerate of classical electrodynamics and "old" quantum mechanics was more "consistent with the facts" than available consistent alternatives. It is similar with the preface paradox. Echoing Peirce, Ramsey continued:

"This is a kind of pragmatism: we judge mental habits by whether they work, i.e., whether the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to".

Ramsey's second point is related to what Hintikka later called "logical omniscience", the unrealistic assumption often made that knowledge of the premises implies knowledge of all of their consequences, see Is the problem of logical omniscience intractable? It is plainly contradicted by the fact that we are yet to know the truth value of the Riemann hypothesis despite knowing all the axioms of set theory. Many mathematicians believe it to be true "on inductive or other grounds" (aesthetic, perhaps), and their beliefs may turn out to be inconsistent, if not on this then on something else.

The point is that our logical clairvoyance is limited, and we may not even know if the beliefs we hold are, in fact, consistent. What we use in practice is only "surface information" extracted from our beliefs, not all of the "depth information" of their consequences, see What is the difference between depth and surface information?

"A belief […] is a map of neighbouring space by which we steer. It remains such a map however much we complicate it or fill in details. But if we professedly extend it to infinity, it is no longer a map; we cannot take it in or steer by it. Our journey is over before we need its remoter parts."

Even consistency of arithmetic has not been (and can not be, without making disputable assumptions) proven, but that hardly reflects on its utility to us. In his late period Wittgenstein, who worked with Ramsey for a year in 1929, came to similar conclusions. As he wrote in Remarks on the Foundations of Mathematics:

"If a contradiction were now actually found in arithmetic - that would only prove that an arithmetic with such a contradiction in it could render very good service; and it would be better for us to modify our concept of the certainty required, than to say that it really not yet have been a proper arithmetic."

See The later Wittgenstein’s guide to contradictions by Persichetti for a commentary.

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