Editing Admissible rule (section)

==Examples==

*[[classical logic|Classical propositional calculus]] (''CPC'') is structurally complete.<ref>Chagrov & Zakharyaschev (1997), Thm. 1.25</ref> Indeed, assume that ''A''/''B'' is a 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'' = <math>\top</math> if ''v''(''p'') = 1, and ''σp'' = <math>\bot</math> 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]] all of whose elements have a name in the language of ''L''.)
*The [[Georg Kreisel|Kreisel]]–[[Hilary Putnam|Putnam]] rule (also known as [[Ronald Harrop|Harrop]]'s rule, or [[independence of premise]] rule)
::<math>(\mathit{KPR})\qquad\frac{\neg p\to q\lor r}{(\neg p\to q)\lor(\neg p\to r)}</math>
:is admissible in the [[intuitionistic logic|intuitionistic propositional calculus]] (''IPC''). In fact, it is admissible in every superintuitionistic logic.<ref>Prucnal (1979), cf. Iemhoff (2006)</ref> On the other hand, the formula
::<math>(\neg p\to q\lor r)\to ((\neg p\to q)\lor(\neg p\to r))</math>
:is not an intuitionistic theorem; hence ''KPR'' is not derivable in ''IPC''. In particular, ''IPC'' is not structurally complete.
*The rule
::<math>\frac{\Box p}p</math>
:is admissible in many modal logics, such as ''K'', ''D'', ''K''4, ''S''4, ''GL'' (see [[Kripke semantics#Correspondence and completeness|this table]] for names of modal logics). It is derivable in ''S''4, but it is not derivable in ''K'', ''D'', ''K''4, or ''GL''.
*The rule
::<math>\frac{\Diamond p\land\Diamond\neg p}\bot</math>
:is admissible in normal logic <math>L \supseteq S4.3</math>.<ref>Rybakov (1997), p. 439</ref> It is derivable in ''GL'' and ''S''4.1, but it is not derivable in ''K'', ''D'', ''K''4, ''S''4, or ''S''5.
*[[Löb's theorem|Löb's rule]]
::<math>(\mathit{LR})\qquad\frac{\Box p\to p}p</math>
:is admissible (but not derivable) in the basic modal logic ''K'', and it is derivable in ''GL''. However, ''LR'' is not admissible in ''K''4. In particular, it is ''not'' true in general that a rule admissible in a logic ''L'' must be admissible in its extensions.
*The [[intermediate logic|Gödel–Dummett logic]] (''LC''), and the modal logic ''Grz''.3 are structurally complete.<ref name="hsc">Rybakov (1997), Thms. 5.4.4, 5.4.8</ref> The [[t-norm fuzzy logics|product fuzzy logic]] is also structurally complete.<ref>Cintula & Metcalfe (2009)</ref>