At bottom, logic operators like 'if' 'and' 'or' 'not' can be reduced to one operator:
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| (which is called "the sheffer stroke" or "nand" operator)
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and all basic logical theorems can be derived from one fundamental axiom like:
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(a|(b|c))|((d|(d|d))|((e|b)|((a|e)|(a|e))))
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For example from this, you can define all the basic logical operators, and propositional logic theorems, such as DeMorgan's laws.
Of course this system does not include quantifiers like 'all', or modal operators like 'possible', ... this would require a more complex system. But still, spelled out like this, a few questions become apparent:
1. what is logic?
2. is this the only system of logic that can be defined?
3. what is the foundation for this logic?
4. if something is "proven," how do we know the proof is itself valid?