What is the system of logic which models reality and, furthermore, which models human reason?
Of course, objective reality (that is, reality as it is before it's perceived) may operate under a logic which is not conceivable or understandable by humans, or even fit our understanding of what a "logic" is, but I'm simply concerned with the logic that underpins our perceived reality. Also, I think it's important to note that the logic which describes reality may be fundamentally different, and even incompatible, to that which describes human reason (although I will argue that that's not the case). Furthermore, for the sake of a roadmap, I want to say that while I outline the problem in the first 3 paragraphs, somewhere in the 4th paragraph is where I will start talking about and arguing my own ideas. I'm not horribly confident about the solutions I give, but I still want to offer my own thoughts on the problem and am eager to hear what people have to think about them and the problem as a whole.
Any pursuit of truth (i.e. an argument) is necessarily described by a system of logic which describes how truth may be derived and understood. This is true of philosophy, science, math, etc. But, if these pursuits of truth are to be valid for our reality, they must abide by the logic which describes our reality. Otherwise, the logic used is either incompatible with that of reality, or outside of it entirely, in which cases the argument's conclusion is either wrong or indeterminate, respectively.
For example, consider the argument: "water is wet implies hot dogs are tacos", "water is wet", therefore "hot dogs are tacos". This argument contains 3 assumptions, the two initial statements and that modus ponens is a valid rule of inference. If the negation of any of these 3 is true (either as axioms or provable statements), then the argument is false within our reality. If all 3 are true within the logic of reality, then the argument's conclusion is true. But, if any of the 3 assumptions is not axiomatically nor provably true within reality, then the argument's conclusion is indeterminate (conceptually similar to a Godel statement in math and computer science).
It follows then to ask: What is the system of logic which models reality? This system of logic, if an accurate model, would provide undeniable truth and validity to all argumentation which concerns reality. The problem is, how would this model be justified? If it is argued for, the argument itself must rely on a logic. This logic, if not equivalent to that which it is arguing for, will then provide a conclusion which is either wrong or indeterminate. So, for the argument to have any merit, the logic which underpins it must be identical to that which it is arguing models reality. This, however, is circular reasoning. It seems then that the logic which underpins reality cannot be logically determined. What does this mean then for philosophy and science? Is it all useless? How can physics, for example, determine the foundations from which all of reality is derived, if it seems that doing so is illogical? This leads me to my question about the logical modeling of human reason.
*This is where I begin to transition into talking about my own ideas concerning a solution to these issues.
You may say, for example, that if this is true then we can arbitrarily assert a logic to model reality, and this assertion, no matter how absurd, is equally valid as any other assertion. But, if you were to claim, say, that reality is described by a classical logic (laws of identity, excluded middle, and non-contradiction) and has the two axioms: "god is good" and "god is evil", then by the principle of explosion this becomes a trivialized logic in which every conceivable statement is a theorem. This however can be observed as not true, in that I am not on Mars right now so the statement "I am on Mars" must be false. The key realization here is that my observation is required, which leads to two things: one, our perception is necessary to have any information about reality, and two, the principles of induction are inherent to how humans process our perception. Reality can not be understood unless we take, axiomatically, at least a subset of our observations as true information, and the principles of induction as valid rules of inference. Therefore, an inductive logic (e.g. science) must be the logic which accurately models human reason. This lens of inductive reasoning is unavoidable in understanding reality as humans, and therefore, we must consider it a subset of the system of logic which models reality (to avoid the aforementioned problems). In a kind of poetic sense, this means that we model our own reality by virtue of perceiving it, and that this model is subjective to our individual perception.
From here, we may argue a logic to model reality by use of inference. First off, I would like to talk about Aristotle's formalism of classical logic which underlies all of western civilization. Aristotle proposed that, for any statement P, the following is true: "P is P", "P is either true or false, not anything else or in between", and "the statement 'P and not P' must be false". This is the foundation behind most civilized thought throughout history. The second statement (law of excluded middle) wasn't questioned and its negation formalized until the early 20th century. The third statement (law of noncontradiction) didn't have its negation formalized until the past ~3 decades. These formalisms have led to many-valued and paraconsistent logics. Please note that the ideas of these logics have been talked about for centuries (particularly many-valued logics in ancient China), but they weren't properly understood and formalized until the times listed above as far as I'm aware.
I want to propose that a classical logic is a very poor and inaccurate model of reality, and that a many-valued, perhaps even paraconsistent logic, is a much better one.
Firstly, there's a lot of justification in modern physics for a many-valued model. Quantum mechanics is the most successful and accurate theory of reality we have ever developed, but it is fundamentally probabilistic. That is, whether or not a particle is at a specific location, has a certain amount of momentum, or that it even exists, etc., can only be calculated as a probability. We can't know these things certainly, but only can calculate the probability that they are the case. For a long time, it was posited that there were "hidden variables", i.e. factors at play that we don't understand which determine these probabilities and are themselves certain. However, the Nobel prize in physics was recently awarded for Bell's theorem, which mathematically proves that hidden-variable theories are incompatible with quantum mechanics. The fact that these hidden-variable theories also don't stand up nearly as well to experimentation would justify that quantum mechanics is our most accurate theory of reality. This seems to justify using a probabilistic logic to model reality. That is, a logic where the truth value of a given statement is a number n, where n ranges from 0 to 1 and is the probability that the given statement is true. But, from the law of large numbers and observed properties of physics, these probabilities "cancel out" on a larger scale (physically, more mass/energy) such that the accuracy of this model could theoretically be approached asymptotically by that of a non-probabilistic logic (not necessarily classical), wherein the n for a statement gets infinitesimally close to 0 or 1, but never reaches it.
And, while I'm less confident about this, I think a paraconsistent logic would more accurately model reality than a non-contradictory or trivial logic. Paraconsistent logics allow for contradictions, and I feel contradictions are inherent to the human condition. Think absurdism in that we crave meaning but objective meaning can't be obtained, or sociologically in that we wish to self actualize but at the same time desire social connections which requires conformity which hinders actualization, or in game development in that the same company that created Fallout 76 also created Skyrim (possibly the most earth shattering contradiction of them all). These contradictions necessarily exist within the human condition, but we, as humans, must persist and reason how to make choices and live our lives anyways. This seems to suggest that paraconsistent reasoning is inherent to the logic of reality, or, more fundamentally, to the logic of human reason (which as aforementioned is a subset of the logic of reality). I'm only less confident about this argument because it's much more nuanced and philosophical than the quantum mechanics justification for probabilistic logic and feels less fundamental or necessary, although I do think it's right.
Therefore, I posit that the logic which most accurately models our reality is a probabilistic, paraconsistent logic.
The reason all of this is important, especially if you're to agree with me that classical logics are poor models, is that much of our pursuit of truth is built on classical logic. Mathematics is a classical logic for example, yet it's used to describe physics which seems to rely on a non-classical logic. Does this prompt the development of a "new math" which can be more logically compatible with physics? Furthermore, much of Western civilization is built on non-contradictory logic. How would it make sense, both systematically and culturally, to introduce paraconsistent logic so that our civilization is more logically compatible with the human condition? I believe the way we reason about things is often a much bigger, much more telling issue than the particular things we reason about. However, I'm quite worried that the ways we reason are largely non-standardized and incompatible with reality, which can lead to very fundamental problems in the ways we decide to lead our lives and our societies.
If you've managed to read all of this, I greatly appreciate your time. I know I wrote a lot, and I'm worried that I may have been overly wordy or unclear at points, although it's hard to tell since it's such a nuanced topic to begin with, but I am grateful if I've even only been marginally understood. These questions and their implications have been on my mind a lot recently and I've only now been able to really put a finger on it and put it into words, and I am very excited to hear what others have to say, as this is a topic I have not been able to find anything about online. I'm not sure if the concept of logically modeling reality and human reason is new, if it's equivalent to another idea and I'm just thinking about it differently, or if I simply haven't been looking hard enough to find anything about it. Either way, if anybody has any resources regarding this I would greatly appreciate them, and if you have any questions please don't hesitate to comment asking for clarifications. Also, I'm aware my format is kind of unorthodox, especially since aspects of a proposed answer are mixed in with my asking of the question (although I feel like this is often typical in philosophy), so if anybody has any suggestions on reformatting or anything of that nature, please let me know but I ask that you simply flag me with a suggestion or send me a notification rather than editing it yourself. Thank you!