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Are there impossible non-classical logics? For every combination of logic rules we can reject, is there a non-classical logic that's associated with it?

I found these:

Computability logic is a semantically constructed formal theory of computability—as opposed to classical logic, which is a formal theory of truth—that integrates and extends classical, linear and intuitionistic logics.

Dynamic semantics interprets formulas as update functions, opening the door to a variety of nonclassical behaviours

Many-valued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic, which permit any real number between 0 and 1 as a truth value.

Intuitionistic logic rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws;

Linear logic rejects idempotency of entailment as well;

Modal logic extends classical logic with non-truth-functional ("modal") operators.

Paraconsistent logic (e.g., relevance logic) rejects the principle of explosion, and has a close relation to dialetheism;

Quantum logic

Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;

Non-reflexive logic (also known as "Schrödinger logics") rejects or restricts the law of identity;[3]

But since there are 5 laws.

Each logical system in this class shares characteristic properties:[4]

Law of excluded middle and double negation elimination
Law of noncontradiction, and the principle of explosion
Monotonicity of entailment and idempotency of entailment
Commutativity of conjunction
De Morgan duality: every logical operator is dual to another

But since there are 5 laws, there are at the very least 10 different non-classical logic laws, but the above list don't include every combinations of logic rules we can reject. So I am wondering if it's true or not.

Sayaman
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    What does "impossible" mean? People usually study and give names to logics (and theories generally) that have some supporting intuitions/applications behind them, not just some random combinations of axioms that can be put together. But there are studies of independence of Boolean axioms from each other that give models where all but some hold, see e.g. [Ninomiya-Mukaidono, Independence of each axiom in a set of axioms and complete sets of axioms of Boolean algebra](https://ieeexplore.ieee.org/document/1011088). – Conifold Nov 02 '21 at 00:51
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    The five characteristic properties you mention are not sufficient to distinguish classical logic from all non-classical ones. There are other distinctions, e.g. over valency, order, variable and quantifier type, etc. I classified the main non-classical logics in my answer to this question: https://philosophy.stackexchange.com/questions/77449/in-how-many-and-which-ways-can-a-logic-be-non-classical-are-there-systems-for-o/77453#77453 – Bumble Nov 02 '21 at 12:34
  • All of the logics you described are all under the MATHEMATICAL LOGIC category. Are you aware all logic is not based in Mathematics? Seems you are making a huge assumption with the question.Classical logic is a synonym for Mathematical logic.Classical logic does NOT refer to Aristotelian logic for instance.Aristotelian logic has three famous so called laws not 5. You question would not make any sense if we speak of Aristotelian logic. You would need to restate the question or add more specific details to be evaluated correctly. Only Aristotle's logic is independent of humans beings creating it. – Logikal Nov 02 '21 at 22:05
  • @Logikal Aristotalean bivalent logic *is* the basis for the vast majority of mathematical logic. – Bertrand Wittgenstein's Ghost Nov 29 '21 at 03:44

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