"Is Logic Empirical?" strongly suggests a question that I would like very much to get a handle on.
That phrase is a title of an article by Hilary Putnam, and, according to synopses/reviews, the paper deals very narrowly with the possibility that laws of logic may need to be revised in view of new empirical knowledge about quantum mechanics. While that idea may be an intriguing idea, it seems very much beside the main point of my question, except for a tangential connection.
So maybe I should ask the question in this way: How on earth did humankind ever get the idea, at genesis, that deductive logic is useful for obtaining new knowledge? (Incidentally, by "logic" I mean deductive logic unless I indicate otherwise.) It seems inescapable that deductive logic must have developed in prehistoric times in conjoint parallel with the linguistic structures of logic. So in that sense, it may be said that the idea that deductive logic is indubitably valid has an empirical foundation.
Supposedly, something about human experience led humans to think they were on to something in developing a tradition of logic. But it also seems to me that belief in the indubitable validity of logic is very much a dogma; so it seems to me that, in the spirit of Descartes's questioning of philosophical foundations, one ought to examine by empirical, scientific studies if possible whether logic does indeed lead to indubitable new knowledge.
Also, although human history may be highly relevant empirically, human history in no way constitutes a double-blind, random, unbiased statistical test of hypotheses. (Such statistical tests seem to be considered the gold standard for empirical testing -- at least for medical research questions.) There is simply too much cultural preselection going on for things to be otherwise.
So how do we grapple with the essentially Cartesian question of empirical foundations for deductive logic?
By way of context, I am especially motivated in posing this question in view of the seemingly extremely towering theories of algebraic topology, seemingly towering all the way to the moon, to use hyperbole. That seems like a very incredible dependence on the idea that deductive logic leads to indubitable new knowledge. Very highly abstract, set-theoretic developments in theories of probability theory and stochastic processes also provide similar motivation for these questions.
Now, I realize that there seems to be a strain of mathematicians who simply regard such theories as merely a formalistic game in the axiomatic tradition of Euclidean geometry and that empirical relevance to the "real world" outside of mathematical theory is simply a non-issue in developing such theories.
There is another perspective that I should also mention here. One sometimes sees physical theories whose only empirical tests are relatively remote empirical verifications of empirical consequences. But of course, if logic is indubitable, there is a strong preference for directly verifying the theories themselves -- that is, to reach the chain of logic much earlier in the chain. It seems such verifications could lead to much stronger, more powerful, much more universally useful knowledge about the theories in question -- that is, provided logic is shown to be indubitably valid, say based on overwhelming empirical verification.
Finally, in regard to the Putnam discussion about the relevance of quantum mechanics: Beyond grandiose ideas of human capabilities, there seems to be no reason to presume that humans can ever develop a perfect theory of the physical universe; that idea of achieving theoretical perfection seems to be an empirically unconfirmable idea.
So it seems from the outset that issues about consistency with quantum mechanics are not at all compelling for a relevance to the question of empirical foundations of logic.
Perhaps I should make more explicit the implicit main question I have in mind. Namely, are there any published, comprehensive studies that thoroughly explore the empirical foundations for the idea that deductive logic is a reliable tool for obtaining new knowledge of the "natural" world outside the formalistic framework of logic? Of course, that a proposition follows from premises according to formalistic rules is often a matter of empirical verification by grinding through the rules. But my question focuses on whether logic adds any new knowledge of an empirical nature outside the formalism. Bertrand Russell said that the rules of logic are a priori knowledge. I think he was probably just recapping general rhetoric of the times about the idea. But that does not seem good enough. I find it hard to think that the rules of logic have no firm, scientifically empirical foundation in order for them, in a scientifically compelling fashion, to be considered useful outside of entertainment purposes. For example, I'm thinking that the Pythagorean theorem might form part of the fabric of such new knowledge of the natural world. The theorem does indeed seem very relevant and empirically verifiable in the natural world, up to small errors of measurement. And so it seems the theorem might be considered a partial proof by inductive logic that deductive logic has practical relevance to the natural world. But in the spirit of empiricism, it seems that much more proof of an empirical nature is needed.
Another example might be the uses of logic for Newtonian mechanics and Newtonian gravity and the consequences (of sorts) in celestial mechanics even though these applications may not have the relative "perfection" of Einsteinian relativity.