What do we use to check the validity of a logic system?

Question

The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which is argued to be self-defeating.[25] Since the early 20th century, certain logicians have proposed logics that deny the validity of the law.

https://en.wikipedia.org/wiki/Law_of_noncontradiction#Alleged_impossibility_of_its_proof_or_denial

So reading this article on Wikipedia we are told that we can't disprove or prove this law, so what are the means to know if a logic system, alternative logic system that add or removes logical laws, is valid or not and what are the different validity criterias and norms used to verify if a logic system is true, valid or useful?

Answer 1

The object of valid reasoning is to proceed from assumptions to conclusions without introducing error in the reasoning process itself. The objection to fallacies in rhetorical logic is that they may introduce error in the reasoning process. It is entirely possible to reach true conclusions with fallacious reasoning; it is the possibility of reaching false ones as well using the same kind of argument that makes fallacies invalid.

This is much simpler to do with classical two-valued propositional logic. In that case, the truth of the material conditional is necessary and sufficient to establish logical validity. With variants where one or other of the laws of classical logic is deemed not to hold, the problem of establishing whether an argument is valid becomes much trickier. In those cases, it is customary to establish formal rules of inference in an attempt to assure that an argument is valid.

The law of non-contradiction is one of the tools for evaluating logical arguments, but it really only applies in two-valued systems. In multi-valued logics, its utility is severely limited.

Answer 2

The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which is argued to be self-defeating.

This is no quite true.

The law of contradiction, which says that a contradiction is false, is one of the fundamental assumptions which defines logical reasoning. We can choose to ignore it and pretend that we can still reason logically but this is just equivocation. Together with the law of identity and the law of excluded middle, the law of contradiction defines logical reasoning. This is so because these assumption are fundamental. Fundamental here means that we recognise that we cannot choose to ignore these laws and still reason logically. We can always pretend otherwise, but this would be equivocation. This is playing with toy logic while pretending to do real logic.

Since the early 20th century, certain logicians have proposed logics that deny the validity of the law.

Words are cheap and anyone can pretend to reason logically while ignoring the law of contradiction. Too bad. If you ignore the fundamental laws of logic, how are you going to reason logically? Do you really play chess if you choose to ignore the rules of chess? We cannot stop anyone pretending they have a logic based on denying the validity of the law of contradiction, but this is just equivocation, and equivocation is bad logic.

So reading this article on Wikipedia we are told that we can't disprove or prove this law, so what are the means to know if a logic system, alternative logic system that add or removes logical laws, is valid or not and what are the different validity criterias and norms used to verify if a logic system is true, valid or useful?

Any system which ignores any of the fundamental laws of logic is not a logic system, and therefore irrelevant to human logic.

Such systems should be regarded as mathematical systems, not logical ones. They can only be logical systems in name only. Good mathematics, possibly, not good formal logic. Not at all.