Editing Provability logic (section)

==Examples==
There are a number of provability logics, some of which are covered in the literature mentioned in {{sectionlink||References}}. The basic system is generally referred to as '''GL''' (for [[Kurt Gödel|Gödel]]–[[Martin Hugo Löb|Löb]]) or '''L''' or '''K4W''' ('''W''' stands for [[Well-founded relation|well-foundedness]]). It can be obtained by adding the modal version of [[Löb's theorem]] to the [[normal modal logic|logic '''K''']] (or '''K4''').

Namely, the '''axioms''' of '''GL''' are all [[tautology (logic)|tautologies]] of classical [[propositional logic]] plus all formulas of one of the following forms: 
* '''Distribution axiom''': {{math|□(''p'' → ''q'') → (□''p'' → □''q'');}}
* '''Löb's axiom''': {{math|□(□''p'' → ''p'') → □''p''.}}
And the '''rules of inference''' are:
* '''''Modus ponens''''': From ''p'' → ''q'' and ''p'' conclude ''q'';
* '''Necessitation''': From <math>\vdash</math> ''p'' conclude <math>\vdash</math> {{math|□''p''}}.