the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. These are the three fundamental laws of logic and while they seem to be true all the time, I am wondering if there are certain fields where one of the three laws doesn't quite apply, especially in physics and mathematics. Is this the case?

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    There are some logics that deny non-contradiction or excluded middle. See plato.stanford.edu/entries/dialetheism and plato.stanford.edu/entries/logic-intuitionistic. – Eliran May 18 '19 at 6:17
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    Keep in mind that these laws apply to our use of language, not to nature. They are "true all the time" because we choose to use sentences (for the most part) in a way that complies with them. So it is more accurate to say not that they are "true" but that they are pragmatically useful to adopt. There are alternative ways to use language, as Eliran mentioned, that are better for certain purposes: constructive mathematics (no excluded middle), non-ideal reasoning (no non-contradiction), quantum events (no identity), etc. – Conifold May 18 '19 at 6:25
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    Maybe (see Kant) the laws of logic are the way our brain is "programmed" to understand the world; if so, necessarily they applies to every field of human knowledge. – Mauro ALLEGRANZA May 18 '19 at 10:10
  • see also quantum logic, whose rules of inference consider the principles of quantum mechanics for their justifications. – Graham Kemp May 23 '19 at 0:28

You will find that there is no "law" of logic that is held universally. And to speak "of logic" confuses the issue because there isn't a single logic but rather numerous logics.

Technically all you need to have a logic is a language (syntax or rules for what counts as a well formed formula), definitions of interpretation (or semantics), operators and their interpretations, and rules of inference.

There is incredible flexibility here and you can be arbitrary although the systems are typically judged by their usefulness in solving problems. This is much like mathematics, which has systems where parallel lines intersect, or where division by zero is allowed.

Graham Priest did a great interview with 3am magazine a few years back on non-traditional logics, but 3am does not seem to keep an archive of their publications. Fortunately a copy can be found at a site maintained by Richard Marshall the journalist who did the interview. The interview is an accessible overview of what paraconsistent logics and dialetheism are and why people care about them.

Francesco Berto wrote a journal article "Meaning, Metaphysics and Contradiction" in American Philosophical Quarterly, Vol 43, no. 4 (October 2006) looking at paraconsistent logics and dialetheism which have both challenged the law of non-contradiction Article available on Jstor.

Quantum physics inspired the need for logics that handle more than binary values or a scale from 0 to 1 (such as various fuzzy logics). This lead to eight valued logics. Here is a journal article on work done on 8-value logics The focus in this article is from a computer science perspective.

Studia Logica: An International Journal for Symbolic Logic Vol. 92, No. 2, Jul., 2009 was an entire volume you may find of use on truth, semantics, and some work on non-traditional logic. Here is a link to that volume/issue

And finally Hegel used the dialectic to challenge Aristotalean assumptions on logic. Here is an article on Hegel's logic

Jumping into professional-level logic is not easy. What I found is only an example of scholarship published in journals on these topics, and it may serve as an entry for you. Good luck!

  • Quantum physics inspired the need for logics that handle more than binary values or a scale from 0 to 1 (such as various fuzzy logics). This lead to eight valued logics. Multi-valued logics (3, 4, .. whatever, nothing special with 8) have naught to do with QM. They were initially studied by Lukasiewicz from a purely mathematical perspective. Indeed, von Neumann tried to define a QM logic but this didn't go anywhere fast; nowadays it seems that Girard's Linear Logic, invented in the 70s, would be a good fit to describe QM processes but I know nothing precise on that. – David Tonhofer May 23 '19 at 23:03
  • As for Computer Science, well, fuzzy logic is used for expressing fuzzy set membership and (with some epistemological handwaving) "approximate values" and lives in control systems mainly. Some instances of Paraconsistent Logic, i.e. logic able to handle contradictions, uses Truth Value Lattices. Very cool. – David Tonhofer May 23 '19 at 23:06

I am just going to add this : these laws might best be considered axioms, an axiom is not true in the acceptance you're using, rather, it's self-refering; you can say it's truth is out of question, hence you cannot speak of its truth (since truth is, most usally, considered a compliance-relation between a "refering object" and a "refered object" (this is a very weak analogy, it's just to convey the point).

In many ways, your are free to drop an axiom, but in doing so : you'll have then construct "logical coherence" without it, something that has been done is mathematics and other fields.

Good day

  • All three of your mentioned concepts share one thing, they are all artificial constructs which attempt to pre-determine what is acceptable discourse and search for truth value. They should, all three, be ignored and discarded. CMS – user37981 May 24 '19 at 12:01
  • @CharlesMSaunders: I think Berson's take on these matters (although maybe not explicit in form) is very relevant and joins your suggestion : a construct is a dissociation from the "élan vital ", one that is however usefull. I'd also agree that they are usefull however restrictive they might be. Overcoming them will come at a price of forging inhanced constructions. However, something bothers me in equiting axioms with mere constructions. I'd not say they are constructed, I'd say they are stated, much like a picture is stated mentally. – Gloserio May 24 '19 at 12:09
  • @CharlesMSaunders: from a linguistic point of view, axioms operate as a pivot of the grammar. There is not point defying grammar IMO, you can however suggest another. – Gloserio May 24 '19 at 12:11

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