View source for Admissible rule

In [[Logic|logic]], a [[Rule of inference|rule of inference]] is '''admissible''' in a [[Formal system|formal system]] if the set of [[Theorem|theorem]]s of the system does not change when that rule is added to the existing rules of the system. In other words, every [[Well-formed formula|formula]] that can be [[Formal proof|derived]] using that rule is already derivable without that rule, so, in a sense, it is redundant. The concept of an admissible rule was introduced by [[Biography:Paul Lorenzen|Paul Lorenzen]] (1955).

==Definitions==

Admissibility has been systematically studied only in the case of structural (i.e. [[Philosophy:Substitution (logic)|substitution]]-closed) rules in propositional [[Philosophy:Non-classical logic|non-classical logic]]s, which we will describe next.

Let a set of basic [[Logical connective|propositional connective]]s be fixed (for instance, <math>\{\to,\land,\lor,\bot\}</math> in the case of superintuitionistic logics, or <math>\{\to,\bot,\Box\}</math> in the case of [[Modal logic|monomodal logic]]s). Well-formed formulas are built freely using these connectives from a [[Countable set|countably infinite]] set of [[Propositional variable|propositional variable]]s ''p''<sub>0</sub>, ''p''<sub>1</sub>, .... A [[Philosophy:Substitution (logic)|substitution]] ''σ'' is a function from formulas to formulas that commutes with applications of the connectives, i.e.,
:<math>\sigma f(A_1,\dots,A_n)=f(\sigma A_1,\dots,\sigma A_n)</math>
for every connective ''f'', and formulas ''A''<sub>1</sub>, ... , ''A''<sub>''n''</sub>. (We may also apply substitutions to sets Γ of formulas, making {{nowrap|1=''σ''Γ = {''σA'': ''A'' &isin; Γ}.}}) A Tarski-style consequence relation<ref>Blok & Pigozzi (1989), Kracht (2007)</ref> is a relation <math>\vdash</math> between sets of formulas, and formulas, such that
{{ordered list|start=1
|<math>A\vdash A,</math>
|if <math>\Gamma\vdash A</math> then <math>\Gamma,\Delta\vdash A,</math> ("weakening")
|if <math>\Gamma\vdash A</math> and <math>\Delta,A\vdash B</math> then <math>\Gamma,\Delta\vdash B,</math> ("composition")
}} 
for all formulas ''A'', ''B'', and sets of formulas Γ, Δ. A consequence relation such that
{{ordered list|start=4
|if <math>\Gamma\vdash A</math> then <math>\sigma\Gamma\vdash\sigma A</math>
}}
for all substitutions ''σ'' is called '''structural'''. (Note that the term "structural" as used here and below is unrelated to the notion of [[Structural rule|structural rule]]s in [[Sequent calculus|sequent calculi]].) A structural consequence relation is called a '''propositional logic'''. A formula ''A'' is a theorem of a logic <math>\vdash</math> if <math>\varnothing\vdash A</math>.

For example, we identify a superintuitionistic logic ''L'' with its standard consequence relation <math>\vdash_L</math> generated by [[Modus ponens|modus ponens]] and axioms, and we identify a [[Philosophy:Normal modal logic|normal modal logic]] with its global consequence relation <math>\vdash_L</math> generated by modus ponens, necessitation, and (as axioms) the theorems of the logic.

A '''structural inference rule'''<ref>Rybakov (1997), Def. 1.1.3</ref> (or just '''rule''' for short) is given by a pair (Γ, ''B''), usually written as
:<math>\frac{A_1,\dots,A_n}B\qquad\text{or}\qquad A_1,\dots,A_n/B,</math>
where Γ&nbsp;=&nbsp;{''A''<sub>1</sub>, ... , ''A''<sub>''n''</sub>} is a finite set of formulas, and ''B'' is a formula. An '''instance''' of the rule is
:<math>\sigma A_1,\dots,\sigma A_n/\sigma B</math>
for a substitution ''σ''. The rule Γ/''B'' is '''derivable''' in <math>\vdash</math>, if <math>\Gamma\vdash B</math>. It is '''admissible''' if for every instance of the rule, ''σB'' is a theorem whenever all formulas from ''σ''Γ are theorems.<ref>Rybakov (1997), Def. 1.7.2</ref> In other words, a rule is admissible if it, when added to the logic, does not lead to new theorems.<ref>[http://www.illc.uva.nl/D65/artemov.pdf From de Jongh’s theorem to intuitionistic logic of proofs]</ref> We also write <math>\Gamma\mathrel{|\!\!\!\sim} B</math> if Γ/''B'' is admissible. (Note that <math>\phantom{.}\!{|\!\!\!\sim}</math> is a structural consequence relation on its own.)

Every derivable rule is admissible, but not vice versa in general. A logic is '''structurally complete''' if every admissible rule is derivable, i.e., <math>{\vdash}={\,|\!\!\!\sim}</math>.<ref>Rybakov (1997), Def. 1.7.7</ref>

In logics with a well-behaved [[Logical conjunction|conjunction]] connective (such as superintuitionistic or modal logics), a rule <math>A_1,\dots,A_n/B</math> is equivalent to <math>A_1\land\dots\land A_n/B</math> with respect to admissibility and derivability. It is therefore customary to only deal with [[Unary operation|unary]] rules ''A''/''B''.

==Examples==

*[[Philosophy: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 Kreisel–[[Biography:Hilary Putnam|Putnam]] rule (also known as Harrop's rule, or [[Philosophy:Independence of premise|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 every normal modal logic.<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>

==Decidability and reduced rules==

The basic question about admissible rules of a given logic is whether the set of all admissible rules is decidable. Note that the problem is nontrivial even if the logic itself (i.e., its set of theorems) is [[Philosophy:Decidability (logic)|decidable]]: the definition of admissibility of a rule ''A''/''B'' involves an unbounded universal quantifier over all propositional substitutions. Hence ''a priori'' we only know that admissibility of rule in a decidable logic is <math>\Pi^0_1</math> (i.e., its complement is recursively enumerable). For instance, it is known that admissibility in the bimodal logics ''K''<sub>''u''</sub> and ''K''4<sub>''u''</sub> (the extensions of ''K'' or ''K''4 with the universal modality) is undecidable.<ref>Wolter & Zakharyaschev (2008)</ref> Remarkably, decidability of admissibility in the basic modal logic ''K'' is a major open problem.

Nevertheless, admissibility of rules is known to be decidable in many modal and superintuitionistic logics. The first decision procedures for admissible rules in basic [[Transitive relation|transitive]] modal logics were constructed by Rybakov, using the '''reduced form of rules'''.<ref>Rybakov (1997), §3.9</ref> A modal rule in variables ''p''<sub>0</sub>, ... , ''p''<sub>''k''</sub> is called reduced if it has the form
:<math>\frac{\bigvee_{i=0}^n\bigl(\bigwedge_{j=0}^k\neg_{i,j}^0p_j\land\bigwedge_{j=0}^k\neg_{i,j}^1\Box p_j\bigr)}{p_0},</math>
where each <math>\neg_{i,j}^u</math> is either blank, or negation <math>\neg</math>. For each rule ''r'', we can effectively construct a reduced rule ''s'' (called the reduced form of ''r'') such that any logic admits (or derives) ''r'' if and only if it admits (or derives) ''s'', by introducing extension variables for all subformulas in ''A'', and expressing the result in the full [[Disjunctive normal form|disjunctive normal form]]. It is thus sufficient to construct a decision algorithm for admissibility of reduced rules.

Let <math>\textstyle\bigvee_{i=0}^n\varphi_i/p_0</math> be a reduced rule as above. We identify every conjunction <math>\varphi_i</math> with the set <math>\{\neg_{i,j}^0p_j,\neg_{i,j}^1\Box p_j\mid j\le k\}</math> of its conjuncts. For any subset ''W'' of the set <math>\{\varphi_i\mid i\le n\}</math> of all conjunctions, let us define a Kripke model <math>M=\langle W,R,{\Vdash}\rangle</math> by
:<math>\varphi_i\Vdash p_j\iff p_j\in\varphi_i,</math>
:<math>\varphi_i\,R\,\varphi_{i'}\iff\forall j\le k\,(\Box p_j\in\varphi_i\Rightarrow\{p_j,\Box p_j\}\subseteq\varphi_{i'}).</math><!-- \mathrel{R} is parsed but deleted by texvc. bloody hell -->
Then the following provides an algorithmic criterion for admissibility in ''K''4:<ref>Rybakov (1997), Thm. 3.9.3</ref>

'''Theorem'''. The rule <math>\textstyle\bigvee_{i=0}^n\varphi_i/p_0</math> is ''not'' admissible in ''K''4 if and only if there exists a set <math>W\subseteq\{\varphi_i\mid i\le n\}</math> such that
#<math>\varphi_i\nVdash p_0</math> for some <math>i\le n,</math>
#<math>\varphi_i\Vdash\varphi_i</math> for every <math>i\le n,</math>
#for every subset ''D'' of ''W'' there exist elements <math>\alpha,\beta\in W</math> such that the equivalences
::<math>\alpha\Vdash\Box p_j</math> if and only if <math>\varphi\Vdash p_j\land\Box p_j</math> for every <math>\varphi\in D</math>
::<math>\alpha\Vdash\Box p_j</math> if and only if <math>\alpha\Vdash p_j</math> and <math>\varphi\Vdash p_j\land\Box p_j</math> for every <math>\varphi\in D</math>
:hold for all ''j''.

Similar criteria can be found for the logics ''S''4, ''GL'', and ''Grz''.<ref>Rybakov (1997), Thms. 3.9.6, 3.9.9, 3.9.12; cf.&nbsp;Chagrov&nbsp;&amp;&nbsp;Zakharyaschev&nbsp;(1997), §16.7</ref> Furthermore, admissibility in intuitionistic logic can be reduced to admissibility in ''Grz'' using the [[Philosophy:Modal companion|Gödel–McKinsey–Tarski translation]]:<ref>Rybakov (1997), Thm. 3.2.2</ref>
:<math>A\,|\!\!\!\sim_{IPC}B</math> if and only if <math>T(A)\,|\!\!\!\sim_{Grz}T(B).</math>

Rybakov (1997) developed much more sophisticated techniques for showing decidability of admissibility, which apply to a robust (infinite) class of transitive (i.e., extending ''K''4 or ''IPC'') modal and superintuitionistic logics, including e.g. ''S''4.1, ''S''4.2, ''S''4.3, ''KC'', ''T''<sub>''k''</sub> (as well as the above-mentioned logics ''IPC'', ''K''4, ''S''4, ''GL'', ''Grz'').<ref>Rybakov (1997), §3.5</ref>

Despite being decidable, the admissibility problem has relatively high [[Computational complexity theory|computational complexity]], even in simple logics: admissibility of rules in the basic transitive logics ''IPC'', ''K''4, ''S''4, ''GL'', ''Grz'' is coNEXP-complete.<ref>Jeřábek (2007)</ref> This should be contrasted with the derivability problem (for rules or formulas) in these logics, which is [[PSPACE]]-complete.<ref>Chagrov & Zakharyaschev (1997), §18.5</ref>

==Projectivity and unification==

Admissibility in propositional logics is closely related to unification in the equational theory of [[Modal algebra|modal]] or [[Heyting algebra]]s. The connection was developed by Ghilardi (1999, 2000). In the logical setup, a '''unifier''' of a formula ''A'' in the language of a logic ''L'' (an ''L''-unifier for short) is a substitution ''σ'' such that ''σA'' is a theorem of ''L''. (Using this notion, we can rephrase admissibility of a rule ''A''/''B'' in ''L'' as "every ''L''-unifier of ''A'' is an ''L''-unifier of ''B''".) An ''L''-unifier ''σ'' is '''less general''' than an ''L''-unifier ''τ'', written as ''σ'' ≤ ''τ'', if there exists a substitution ''υ'' such that
:<math>\vdash_L\sigma p\leftrightarrow \upsilon\tau p</math>
for every variable ''p''. A '''complete set of unifiers''' of a formula ''A'' is a set ''S'' of ''L''-unifiers of ''A'' such that every ''L''-unifier of ''A'' is less general than some unifier from ''S''. A most general unifier (MGU) of ''A'' is a unifier ''σ'' such that {''σ''} is a complete set of unifiers of ''A''. It follows that if ''S'' is a complete set of unifiers of ''A'', then a rule ''A''/''B'' is ''L''-admissible if and only if every ''σ'' in ''S'' is an ''L''-unifier of ''B''. Thus we can characterize admissible rules if we can find well-behaved complete sets of unifiers.

An important class of formulas that have a most general unifier are the '''projective formulas''': these are formulas ''A'' such that there exists a unifier ''σ'' of ''A'' such that
:<math>A\vdash_L B\leftrightarrow\sigma B</math>
for every formula ''B''. Note that ''σ'' is an MGU of ''A''. In transitive modal and superintuitionistic logics with the [[Kripke semantics#Finite model property|finite model property]], one can characterize projective formulas semantically as those whose set of finite ''L''-models has the '''extension property''':<ref>Ghilardi (2000), Thm. 2.2</ref> if ''M'' is a finite Kripke ''L''-model with a root ''r'' whose cluster is a [[Singleton (mathematics)|singleton]], and the formula ''A'' holds at all points of ''M'' except for ''r'', then we can change the valuation of variables in ''r'' so as to make ''A'' true at ''r'' as well. Moreover, the proof provides an explicit construction of an MGU for a given projective formula ''A''.

In the basic transitive logics ''IPC'', ''K''4, ''S''4, ''GL'', ''Grz'' (and more generally in any transitive logic with the finite model property whose set of finite frame satisfies another kind of extension property), we can effectively construct for any formula ''A'' its '''projective approximation''' Π(''A''):<ref>Ghilardi (2000), p. 196</ref> a finite set of projective formulas such that
#<math>P\vdash_L A</math> for every <math>P\in\Pi(A),</math>
#every unifier of ''A'' is a unifier of a formula from Π(''A'').
It follows that the set of MGUs of elements of Π(''A'') is a complete set of unifiers of ''A''. Furthermore, if ''P'' is a projective formula, then
:<math>P\,|\!\!\!\sim_L B</math> if and only if <math>P\vdash_L B</math>
for any formula ''B''. Thus we obtain the following effective characterization of admissible rules:<ref>Ghilardi (2000), Thm. 3.6</ref>
:<math>A\,|\!\!\!\sim_L B</math> if and only if <math>\forall P\in\Pi(A)\,(P\vdash_L B).</math>

==Bases of admissible rules==

Let ''L'' be a logic. A set ''R'' of ''L''-admissible rules is called a '''basis'''<ref>Rybakov (1997), Def. 1.4.13</ref> of admissible rules, if every admissible rule Γ/''B'' can be derived from ''R'' and the derivable rules of ''L'', using substitution, composition, and weakening. In other words, ''R'' is a basis if and only if <math>\phantom{.}\!{|\!\!\!\sim_L}</math> is the smallest structural consequence relation that includes <math>\vdash_L</math> and ''R''.

Notice that decidability of admissible rules of a decidable logic is equivalent to the existence of recursive (or recursively enumerable) bases: on the one hand, the set of ''all'' admissible rules is a recursive basis if admissibility is decidable. On the other hand, the set of admissible rules is always co-recursively enumerable, and if we further have a recursively enumerable basis, set of admissible rules is also recursively enumerable; hence it is decidable. (In other words, we can decide admissibility of ''A''/''B'' by the following [[Algorithm|algorithm]]: we start in parallel two exhaustive searches, one for a substitution ''σ'' that unifies ''A'' but not ''B'', and one for a derivation of ''A''/''B'' from ''R'' and <math>\vdash_L</math>. One of the searches has to eventually come up with an answer.) Apart from decidability, explicit bases of admissible rules are useful for some applications, e.g. in [[Proof complexity|proof complexity]].<ref>Mints & Kojevnikov (2004)</ref>

For a given logic, we can ask whether it has a recursive or [[Finite set|finite]] basis of admissible rules, and to provide an explicit basis. If a logic has no finite basis, it can nevertheless have an '''independent basis''': a basis ''R'' such that no proper subset of ''R'' is a basis.

In general, very little can be said about existence of bases with desirable properties. For example, while tabular logics are generally well-behaved, and always finitely axiomatizable, there exist tabular modal logics without a finite or independent basis of rules.<ref>Rybakov (1997), Thm. 4.5.5</ref> Finite bases are relatively rare: even the basic transitive logics ''IPC'', ''K''4, ''S''4, ''GL'', ''Grz'' do not have a finite basis of admissible rules,<ref>Rybakov (1997), §4.2</ref> though they have independent bases.<ref>Jeřábek (2008)</ref>

===Examples of bases===
*The empty set is a basis of ''L''-admissible rules if and only if ''L'' is structurally complete.
*Every extension of the modal logic ''S''4.3 (including, notably, ''S''5) has a finite basis consisting of the single rule<ref>Rybakov (1997), Cor. 4.3.20</ref>
::<math>\frac{\Diamond p\land\Diamond\neg p}\bot.</math>
*{{ill|Albert Visser|lt=Visser|nl}}'s rules
::<math>\frac{\displaystyle\Bigl(\bigwedge_{i=1}^n(p_i\to q_i)\to p_{n+1}\lor p_{n+2}\Bigr)\lor r}{\displaystyle\bigvee_{j=1}^{n+2}\Bigl(\bigwedge_{i=1}^{n}(p_i\to q_i)\to p_j\Bigr)\lor r},\qquad n\ge 1</math>
:are a basis of admissible rules in ''IPC'' or ''KC''.<ref>Iemhoff (2001, 2005), Rozière (1992)</ref>
*The rules
::<math>\frac{\displaystyle\Box\Bigl(\Box q\to\bigvee_{i=1}^n\Box p_i\Bigr)\lor\Box r}{\displaystyle\bigvee_{i=1}^n\Box(q\land\Box q\to p_i)\lor r},\qquad n\ge0</math>
:are a basis of admissible rules of ''GL''.<ref>Jeřábek (2005)</ref> (Note that the empty disjunction is defined as <math>\bot</math>.)
*The rules
::<math>\frac{\displaystyle\Box\Bigl(\Box(q\to\Box q)\to\bigvee_{i=1}^n\Box p_i\Bigr)\lor\Box r}{\displaystyle\bigvee_{i=1}^n\Box(\Box q\to p_i)\lor r},\qquad n\ge0</math>
:are a basis of admissible rules of ''S''4 or ''Grz''.<ref>Jeřábek (2005, 2008)</ref>

==Semantics for admissible rules==

A rule Γ/''B'' is '''valid''' in a modal or intuitionistic Kripke frame <math>F=\langle W,R\rangle</math>, if the following is true for every valuation <math>\Vdash</math> in ''F'':
:if for all <math>A\in\Gamma</math> <math>\forall x\in W\,(x\Vdash A)</math>, then <math>\forall x\in W\,(x\Vdash B)</math>.
(The definition readily generalizes to [[General frame|general frame]]s, if needed.)

Let ''X'' be a subset of ''W'', and ''t'' a point in ''W''. We say that ''t'' is
*a '''reflexive tight predecessor''' of ''X'', if for every ''y'' in ''W'': ''t R y'' if and only if ''t'' = ''y'' or for some ''x'' in ''X'': ''x'' = ''y'' or ''x R y'' ,
*an '''irreflexive tight predecessor''' of ''X'', if for every ''y'' in ''W'': ''t R y'' if and only if for some ''x'' in ''X'': ''x'' = ''y'' or  ''x R y'' .
We say that a frame ''F'' has reflexive (irreflexive) tight predecessors, if for every ''finite'' subset ''X'' of ''W'', there exists a reflexive (irreflexive) tight predecessor of ''X'' in ''W''.

We have:<ref>Iemhoff (2001), Jeřábek (2005)</ref>
*a rule is admissible in ''IPC'' if and only if it is valid in all intuitionistic frames that have reflexive tight predecessors,
*a rule is admissible in ''K''4 if and only if it is valid in all [[Transitive relation|transitive]] frames that have reflexive and irreflexive tight predecessors,
*a rule is admissible in ''S''4 if and only if it is valid in all transitive [[Reflexive relation|reflexive]] frames that have reflexive tight predecessors,
*a rule is admissible in ''GL'' if and only if it is valid in all transitive converse [[Well-founded relation|well-founded]] frames that have irreflexive tight predecessors.

Note that apart from a few trivial cases, frames with tight predecessors must be infinite. Hence admissible rules in basic transitive logics do not enjoy the finite model property.

==Structural completeness==

While a general classification of structurally complete logics is not an easy task, we have a good understanding of some special cases.

Intuitionistic logic itself is not structurally complete, but its ''fragments'' may behave differently. Namely, any disjunction-free rule or implication-free rule admissible in a superintuitionistic logic is derivable.<ref>Rybakov (1997), Thms. 5.5.6, 5.5.9</ref> On the other hand, the [[Biography:Grigori Mints|Mints]] rule
:<math>\frac{(p\to q)\to p\lor r}{((p\to q)\to p)\lor((p\to q)\to r)}</math>
is admissible in intuitionistic logic but not derivable, and contains only implications and disjunctions.

We know the ''maximal'' structurally incomplete transitive logics. A logic is called '''hereditarily structurally complete''', if any extension is structurally complete. For example, classical logic, as well as the logics ''LC'' and ''Grz''.3 mentioned above, are hereditarily structurally complete. A complete description of hereditarily structurally complete superintuitionistic and transitive modal logics was given respectively by Citkin and Rybakov. Namely, a superintuitionistic logic is hereditarily structurally complete if and only if it is not valid in any of the five Kripke frames<ref name="hsc"/>
::[[File:Tsitkin frames.svg]]
Similarly, an extension of ''K''4 is hereditarily structurally complete if and only if it is not valid in any of certain twenty Kripke frames (including the five intuitionistic frames above).<ref name="hsc"/>

There exist structurally complete logics that are not hereditarily structurally complete: for example, [[Intermediate logic|Medvedev's logic]] is structurally complete,<ref>Prucnal (1976)</ref> but it is included in the structurally incomplete logic ''KC''.

==Variants==

A '''rule with parameters''' is a rule of the form
:<math>\frac{A(p_1,\dots,p_n,s_1,\dots,s_k)}{B(p_1,\dots,p_n,s_1,\dots,s_k)},</math>
whose variables are divided into the "regular" variables ''p''<sub>''i''</sub>, and the parameters ''s''<sub>''i''</sub>. The rule is ''L''-admissible if every ''L''-unifier ''σ'' of ''A'' such that ''σs''<sub>''i''</sub>&nbsp;=&nbsp;''s''<sub>''i''</sub> for each ''i'' is also a unifier of ''B''. The basic decidability results for admissible rules also carry to rules with parameters.<ref>Rybakov (1997), §6.1</ref>

A '''multiple-conclusion rule''' is a pair (Γ,Δ) of two finite sets of formulas, written as
:<math>\frac{A_1,\dots,A_n}{B_1,\dots,B_m}\qquad\text{or}\qquad A_1,\dots,A_n/B_1,\dots,B_m.</math>
Such a rule is admissible if every unifier of Γ is also a unifier of some formula from Δ.<ref>Jeřábek (2005); cf. Kracht (2007), §7</ref> For example, a logic ''L'' is consistent iff it admits the rule
:<math>\frac{\;\bot\;}{},</math>
and a superintuitionistic logic has the disjunction property iff it admits the rule
:<math>\frac{p\lor q}{p,q}.</math>
Again, basic results on admissible rules generalize smoothly to multiple-conclusion rules.<ref>Jeřábek (2005, 2007, 2008)</ref> In logics with a variant of the disjunction property, the multiple-conclusion rules have the same expressive power as single-conclusion rules: for example, in ''S''4 the rule above is equivalent to
:<math>\frac{A_1,\dots,A_n}{\Box B_1\lor\dots\lor\Box B_m}.</math>
Nevertheless, multiple-conclusion rules can often be employed to simplify arguments.

In [[Proof theory|proof theory]], admissibility is often considered in the context of [[Sequent calculus|sequent calculi]], where the basic objects are sequents rather than formulas. For example, one can rephrase the [[Cut-elimination theorem|cut-elimination theorem]] as saying that the cut-free sequent calculus admits the [[Cut rule|cut rule]]
:<math>\frac{\Gamma\vdash A,\Delta\qquad\Pi,A\vdash\Lambda}{\Gamma,\Pi\vdash\Delta,\Lambda}.</math>
(By abuse of language, it is also sometimes said that the (full) sequent calculus admits cut, meaning its cut-free version does.) However, admissibility in sequent calculi is usually only a notational variant for admissibility in the corresponding logic: any complete calculus for (say) intuitionistic logic admits a sequent rule if and only if ''IPC'' admits the formula rule that we obtain by translating each sequent <math>\Gamma\vdash\Delta</math> to its characteristic formula <math>\bigwedge\Gamma\to\bigvee\Delta</math>.

==Notes==
{{reflist|3}}

==References==

*W. Blok, D. Pigozzi, ''Algebraizable logics'', [[Organization:Memoirs of the American Mathematical Society|Memoirs of the American Mathematical Society]] 77 (1989), no. 396, 1989.
*A. Chagrov and M. Zakharyaschev, ''Modal Logic'', Oxford Logic Guides vol. 35, Oxford University Press, 1997. {{ISBN|0-19-853779-4}}
*P. Cintula and G. Metcalfe, ''Structural completeness in fuzzy logics'', [[Notre Dame Journal of Formal Logic]] 50 (2009), no.&nbsp;2, pp.&nbsp;153–182. {{doi|10.1215/00294527-2009-004}}
*A. I. Citkin, ''On structurally complete superintuitionistic logics'', Soviet Mathematics - Doklady, vol. 19 (1978), pp.&nbsp;816&ndash;819.
*S. Ghilardi, ''Unification in intuitionistic logic'', [[Journal of Symbolic Logic]] 64 (1999), no. 2, pp.&nbsp;859–880. [http://projecteuclid.org/euclid.jsl/1183745815 Project Euclid] [https://www.jstor.org/stable/view/2586506 JSTOR]
*S. Ghilardi, ''Best solving modal equations'', [[Annals of Pure and Applied Logic]] 102 (2000), no. 3, pp.&nbsp;183–198. {{doi|10.1016/S0168-0072(99)00032-9}}
*R. Iemhoff, ''On the admissible rules of intuitionistic propositional logic'', Journal of Symbolic Logic 66 (2001), no. 1, pp.&nbsp;281–294. [http://projecteuclid.org/euclid.jsl/1183746371 Project Euclid] [https://www.jstor.org/stable/view/2694922 JSTOR]
*R. Iemhoff, ''Intermediate logics and Visser's rules'', Notre Dame Journal of Formal Logic 46 (2005), no. 1, pp.&nbsp;65–81. {{doi|10.1305/ndjfl/1107220674}}
*R. Iemhoff, ''On the rules of intermediate logics'', [[Archive for Mathematical Logic]], 45 (2006), no. 5, pp.&nbsp;581–599. {{doi|10.1007/s00153-006-0320-8}}
*E. Jeřábek, ''Admissible rules of modal logics'', [[Journal of Logic and Computation]] 15 (2005), no. 4, pp.&nbsp;411–431. {{doi|10.1093/logcom/exi029}}
*E. Jeřábek, ''Complexity of admissible rules'', Archive for Mathematical Logic 46 (2007), no. 2, pp.&nbsp;73–92. {{doi|10.1007/s00153-006-0028-9}}
*E. Jeřábek, ''Independent bases of admissible rules'', Logic Journal of the IGPL 16 (2008), no. 3, pp.&nbsp;249–267. {{doi|10.1093/jigpal/jzn004}}
*M. Kracht, ''Modal Consequence Relations'', in: Handbook of Modal Logic (P. Blackburn, J. van Benthem, and F. Wolter, eds.), Studies of Logic and Practical Reasoning vol. 3, Elsevier, 2007, pp.&nbsp;492–545. {{ISBN|978-0-444-51690-9}}
*P. Lorenzen, ''Einführung in die operative Logik und Mathematik'', Grundlehren der mathematischen Wissenschaften vol. 78, Springer–Verlag, 1955.
*G. Mints and A. Kojevnikov, ''Intuitionistic Frege systems are polynomially equivalent'', Zapiski Nauchnyh Seminarov POMI 316 (2004), pp.&nbsp;129–146. [http://www.emis.de/journals/ZPOMI/v316/p129.ps.gz gzipped PS]
*T. Prucnal, ''Structural completeness of Medvedev's propositional calculus'', Reports on Mathematical Logic 6 (1976), pp.&nbsp;103–105.
*T. Prucnal, ''On two problems of Harvey Friedman'', Studia Logica 38 (1979), no. 3, pp.&nbsp;247–262. {{doi|10.1007/BF00405383}}
*P. Rozière, ''Règles admissibles en calcul propositionnel intuitionniste'', Ph.D. thesis, Université de Paris VII, 1992. [http://www.pps.jussieu.fr/~roziere/admiss/these.pdf PDF]
*V. V. Rybakov, ''Admissibility of Logical Inference Rules'', Studies in Logic and the Foundations of Mathematics vol. 136, Elsevier, 1997. {{ISBN|0-444-89505-1}}
*F. Wolter, M. Zakharyaschev, ''Undecidability of the unification and admissibility problems for modal and description logics'', [[ACM Transactions on Computational Logic]] 9 (2008), no. 4, article no. 25. {{doi|10.1145/1380572.1380574}} [https://web.archive.org/web/20110718020147/http://tocl.acm.org/accepted/318wolter.pdf PDF]

{{DEFAULTSORT:Admissible Rule}}
[[Category:Rules of inference]]
[[Category:Modal logic]]

{{Sourceattribution|Admissible rule}}