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Is it - must there be - possible to track all logical statements back to the fundamental laws of logic ( the law of identity, the law of non-contradiction, etc.) when it comes to "classical logic"? Are all logic derived from these fundamental laws?

### The problem here, I think, is

The problem here, I think, is that there's no one answer to the question "What are the fundamental laws of logic?" We can do things in different ways, and things which are fundamental on some accountings will be derived on others.
Let's assume that there is a definite answer to the question "What are the logical truths of classical logic?" (I'm using this as a proxy for "logical statements." If we want to expand it to include principles of inference, like modus ponens, that's okay too.) Note that the set of all such truths will be infinite, but that's okay. And to make "classical logic" well-defined, let's assume we mean truth-functional and first-order predicate logic, in which our first assumption is indeed correct. Then there are sets of rules and/or axiom schemes that provably allow the derivation of every logical truth thus understood. As just noted, there is no one way to so this, and the different ways won't contain the same axioms and/or rules. Even "the law of non-contradiction" will show...

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