[famouswaffles]

The second. Mathematical logic thrives on precision, clear definitions, and unambiguous axioms, but real-world systems are often marked by vagueness, uncertainty, and dynamic change.

Gödel’s Incompleteness Theorems also demonstrates that in any sufficiently powerful mathematical system, there are true statements that cannot be proven within the system. This implies that no matter how refined a logical system you devise, it will invariably be incomplete or inconsistent when grappling with real-world phenomena.

[YeGoblynQueenne]

Gödel didn't say anything about real world phenomena. He was talking about formal languages and mathematics.

[famouswaffles]

Of course. But if you truly cannot model every true statement in any formally devised system, then you are by definition going to have to reject valid rules that your logic cannot verify if you intend your system to perfectly logical.

[YeGoblynQueenne]

I believe that's right but only in a deductive setting, and as long as there's a requirement for soundness. Inductive and abductive logical inference are not sound and they are very useful for real-world decision-making. But that is a developing field and there are still many unknowns there.