Admissible Rule - Examples

Examples

• Classical propositional calculus (CPC) is structurally complete. Indeed, assume that A/B is non-derivable rule, and fix an assignment v such that v(A) = 1, and v(B) = 0. Define a substitution σ such that for every variable p, σp = if v(p) = 1, and σp = if v(p) = 0. Then σA is a theorem, but σB is not (in fact, ¬σB is a theorem). Thus the rule A/B is not admissible either. (The same argument applies to any multi-valued logic L complete with respect to a logical matrix whose all elements have a name in the language of L.)
• The Kreisel–Putnam rule (aka Harrop's rule, or independence of premise rule)

is admissible in the intuitionistic propositional calculus (IPC). In fact, it is admissible in every superintuitionistic logic. On the other hand, the formula

is not an intuitionistic tautology, hence KPR is not derivable in IPC. In particular, IPC is not structurally complete.

• The rule

is admissible in many modal logics, such as K, D, K4, S4, GL (see this table for names of modal logics). It is derivable in S4, but it is not derivable in K, D, K4, or GL.

• The rule

is admissible in every normal modal logic. It is derivable in GL and S4.1, but it is not derivable in K, D, K4, S4, S5.

• Löb's rule

is admissible (but not derivable) in the basic modal logic K, and it is derivable in GL. However, LR is not admissible in K4. In particular, it is not true in general that a rule admissible in a logic L must be admissible in its extensions.

• The Gödel–Dummett logic (LC), and the modal logic Grz.3 are structurally complete. The product fuzzy logic is also structurally complete.