Gödelian Logic refers to logic, math, and arguments in the style of Kurt Gödel. Specifically - his two incompleteness theorems, and one completeness theorem. Due to their tricky and subtle nature, his incompleteness theorems are possibly the most misunderstood theorems of all time.
“All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described.” - Douglas Hofstadter, Gödel, Escher, Bach
This theorem is less well known than the other two, which came after it, but also less misunderstood.
One way you might think to get around Gödel's Incompleteness, is to leave behind logical certainty, and instead assign probabilities to logical statements.