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Can. Or you can not share it.

By and large, now it makes sense to talk only about symbolic (mathematical, formal) logic, which is just as scientific as we consider scientific mathematics, of which symbolic logic is a part. Leibniz's logic is essentially informal, while Boole's “logic” is just one approach to constructing formal systems. Zadeh's theory fits into the definition of non-classical logic.

Formal logic has a very simple classification:

By order (i.e., by the entities under consideration): logic of propositions, predicates of the first, second, etc. orders

According to compliance with the requirements for classical logic: strict ambiguity, context extensionality, corresponding understanding of truth, the assumption of incompleteness of the subject area, the absence of empty names; if logic meets all the requirements, it is classical, if not-non-classical (for non-classical logic, there is a separate classification, which is not given here: those who wish can read, for example, the textbook “Introduction to Logic” by Bocharov and Markin: everything is more or less described there).

You can divide anything you want into anything you want. And you can not share it. You can not do logic at all, but bake pies-sometimes the effect is much cooler.

If we talk about logic and how to divide it, then it is already wrong to start with some personal references. It's like dividing philosophy into the philosophies of Aristotle (unexpectedly), Kant, Wittgenstein, and Foucault. Logic, like any other science, has its own historical stages. For example, we most often divide all logic into traditional (that is, the Aristotelian tradition based on the 3 laws of logic (identity, non-contradiction, and the excluded third) and modern (which began with Leibniz, but is not limited to him). At the same time, modern logic is divided into classical and non-classical. Classical logic is a formalized science that largely reflects the Aristotelian tradition mixed with mathematics. Non-classical logic is an interesting concept that considers various logics based on the principle “let's abandon some law and see what happens”.

That is, if we consider the periodization of logic as a science, we will get several periods.

If you separate out individual logics as sets of methods/tools blah-blah-blah, then you can't do with just Aristotle/Boolean and so on. Moreover, in my humble opinion, it is precisely Aristotle's logic that does not exist, since his apparatus is too scattered and unstructured. For example, non-classical logic includes a cloud of different modal (judgment evaluation logics) and not so much logics: temporal, alethic, epistemic, deontic, and about thirty more.

In addition, logic can also be divided into formal and informal logic. And here in general you are drowning, because for informal logic they have not yet come up with a clear definition.

If you are too lazy to read all the above, then you can simply answer: no, you can't )

I don't pretend to be a classifier of logic or at least a specialist in the history of logic, but for me personally, the term “George Boole logic” sounds strange in the context of logic classification for at least two reasons. There is such a term as Boolean algebra – this is a related concept to logic, but still it is not the same thing. I imagine that the main merit of Boole was precisely the allocation of Boolean algebra, and this direction received a certain development. For example, Boolean algebras are used in set theory, but unlike logic, they are not a fundamental concept. In the context of enriching logic itself, I think it is more common to talk about Frege's contribution. Although, of course, such things as, for example, relations appeared before Frege, and perhaps now it is generally accepted to consider “Frege's logic” as Boole's logic. But I doubt it.