In logic, there is a bridge. It connects two shores: admissible rules (those that don't violate truth) and derivable rules (those obtainable from axioms). In ordinary, finite cases, the bridge is sturdy. But if there are infinitely many rules — the bridge collapses.

This property is called structural completeness of a logical system. Until now, it was known that many well-known logics, such as Dummett's or Medvedev's logic, are complete under finite rules. But new research has shown: once you allow infinite sets of premises, they lose this property. The reason is a cunning infinite construction: an algebra of order ω+2 (a nearly countable chain of elements with two extra ones on top). It's like a recipe: if you add infinitely many ingredients, the dish no longer works by the book.

The only systems where the bridge withstands infinite rules are those generated by a single finite algebra. Such systems are called tabular. They can be described by a simple list of all possible situations. All others, even slightly more complex, fail the test. The study also clarifies the inner workings of logical systems and shows how to build counterexamples. The discovery links κ-structural completeness (where κ is the allowed size of infinity) to deep questions about whether a logic can be axiomatized in a finite way at all. The bridge of logic, alas, is rated only for finite weight.