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| In [[logic]], a '''modal companion''' of a [[intermediate logic|superintuitionistic]] (intermediate) logic ''L'' is a [[normal modal logic|normal]] [[modal logic]] which interprets ''L'' by a certain canonical translation, described below. Modal companions share various properties of the original [[intermediate logic]], which enables to study intermediate logics using tools developed for modal logic.
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| ==Gödel–McKinsey–Tarski translation==
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| Let ''A'' be a [[Propositional calculus|propositional]] [[intuitionistic logic|intuitionistic]] formula. A modal formula ''T''(''A'') is defined by induction on the complexity of ''A'':
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| :<math>T(p)=\Box p</math> for any [[propositional variable]] <math>p</math>,
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| :<math>T(\bot)=\bot,</math>
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| :<math>T(A\land B)=T(A)\land T(B),</math>
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| :<math>T(A\lor B)=T(A)\lor T(B),</math>
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| :<math>T(A\to B)=\Box(T(A)\to T(B)).</math>
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| As negation is in intuitionistic logic defined by <math>A\to\bot</math>, we also have
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| :<math>T(\neg A)=\Box\neg T(A).</math>
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| ''T'' is called the '''Gödel translation''' or '''[[Kurt Gödel|Gödel]]–[[J. C. C. McKinsey|McKinsey]]–[[Alfred Tarski|Tarski]] translation'''. The translation is sometimes presented in slightly different ways: for example, one may insert <math>\Box</math> before every subformula. All such variants are provably equivalent in [[Kripke semantics |S4]].
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| ==Modal companions==
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| For any normal modal logic ''M'' which extends '''S4''', we define its '''si-fragment''' ρ''M'' as
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| :<math>\rho M=\{A\mid M\vdash T(A)\}.</math>
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| The si-fragment of any normal extension of '''S4''' is a superintuitionistic logic. A modal logic ''M'' is a '''modal companion''' of a superintuitionistic logic ''L'' if <math>L=\rho M</math>.
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| Every superintuitionistic logic has modal companions. The '''smallest modal companion''' of ''L'' is
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| :<math>\tau L=\mathbf{S4}\oplus\{T(A)\mid L\vdash A\},</math>
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| where <math>\oplus</math> denotes normal closure. It can be shown that every superintuitionistic logic also has the '''largest modal companion''', which is denoted by σ''L''. A modal logic ''M'' is a companion of ''L'' if and only if <math>\tau L\subseteq M\subseteq\sigma L</math>.
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| For example, '''S4''' itself is the smallest modal companion of the intuitionistic logic ('''IPC'''). The largest modal companion of '''IPC''' is the [[Andrzej Grzegorczyk|Grzegorczyk]] logic '''Grz''', axiomatized by the axiom
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| :<math>\Box(\Box(A\to\Box A)\to A)\to A</math>
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| over '''K'''. The smallest modal companion of the classical logic ('''CPC''') is Lewis' '''S5''', whereas its largest modal companion is the logic
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| :<math>\mathbf{Triv}=\mathbf K\oplus(A\leftrightarrow\Box A).</math>
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| More examples:
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| {| border="1" cellspacing="0" cellpadding="5"
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| |''L''
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| |τ''L''
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| |σ''L''
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| |other companions of ''L''
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| |-
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| |'''IPC'''
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| |'''S4'''
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| |'''Grz'''
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| |'''S4.1''', '''Dum''', ...
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| |-
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| |'''KC'''
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| |'''S4.2'''
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| |'''Grz.2'''
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| |'''S4.1.2''', ...
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| |-
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| |'''LC'''
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| |'''S4.3'''
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| |'''Grz.3'''
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| |'''S4.1.3''', '''S4.3Dum''', ...
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| |-
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| |'''CPC'''
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| |'''S5'''
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| |'''Triv'''
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| |see below
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| |-
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| |}
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| ==Blok–Esakia isomorphism==
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| The set of extensions of a superintuitionistic logic ''L'' ordered by inclusion forms a [[complete lattice]], denoted Ext''L''. Similarly, the set of normal extensions of a modal logic ''M'' is a complete lattice NExt''M''. The companion operators ρ''M'', τ''L'', and σ''L'' can be considered as mappings between the lattices Ext'''IPC''' and NExt'''S4''':
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| :<math>\rho\colon\mathrm{NExt}\,\mathbf{S4}\to\mathrm{Ext}\,\mathbf{IPC},</math>
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| :<math>\tau,\sigma\colon\mathrm{Ext}\,\mathbf{IPC}\to\mathrm{NExt}\,\mathbf{S4}.</math>
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| It is easy to see that all three are [[monotonic function|monotone]], and <math>\rho\circ\tau=\rho\circ\sigma</math> is the identity function on Ext'''IPC'''. [[Larisa Maksimova|L. Maksimova]] and [[Vladimir V. Rybakov|V. Rybakov]] have shown that ρ, τ, and σ are actually [[complete lattice#Morphisms of complete lattices|complete lattice homomorphism]]s. The cornerstone of the theory of modal companions is the '''Blok–Esakia theorem''', proved independently by [[Willem Blok|Wim Blok]] and [[Leo Esakia]]. It states
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| :''The mappings ρ and σ are mutually [[inverse function|inverse]] lattice [[isomorphism]]s of'' Ext'''IPC''' ''and'' NExt'''Grz'''. | |
| Accordingly, σ and the [[function (mathematics)#Restrictions and extensions|restriction]] of ρ to NExt'''Grz''' are called the '''Blok–Esakia isomorphism'''. An important corollary to the Blok–Esakia theorem is a simple syntactic description of largest modal companions: for every superintuitionistic logic ''L'',
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| :<math>\sigma L=\tau L+\mathbf{Grz}.</math>
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| ==Semantic description==
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| The Gödel translation has a frame-theoretic counterpart. Let <math>\mathbf F=\langle F,R,V\rangle</math> be a [[transitive relation|transitive]] and [[reflexive relation|reflexive]] modal [[general frame]]. The [[preorder]] ''R'' induces the [[equivalence relation]]
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| :<math>x\sim y \iff x\,R\,y \land y\,R\,x</math>
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| <!-- x \mathrel{R} y is broken by texvc -->
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| on ''F'', which identifies points belonging to the same cluster. Let <math>\langle\rho F,\le\rangle=\langle F,R\rangle/{\sim}</math> be the induced [[quotient set|quotient]] [[partial order]] (i.e., ρ''F'' is the set of [[equivalence class]]es of <math>\sim</math>), and put
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| :<math>\rho V=\{A/{\sim}\mid A\in V,A=\Box A\}.</math>
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| Then <math>\rho\mathbf F=\langle\rho F,\le,\rho V\rangle</math> is an intuitionistic general frame, called the '''skeleton''' of '''F'''. The point of the skeleton construction is that it preserves validity modulo Gödel translation: for any intuitionistic formula ''A'',
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| :''A'' is valid in ρ'''F''' if and only if ''T''(''A'') is valid in '''F'''.
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| Therefore the si-fragment of a modal logic ''M'' can be defined semantically: if ''M'' is complete with respect to a class ''C'' of transitive reflexive general frames, then ρ''M'' is complete with respect to the class <math>\{\rho\mathbf F;\,\mathbf F\in C\}</math>.
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| The largest modal companions also have a semantic description. For any intuitionistic general frame <math>\mathbf F=\langle F,\le,V\rangle</math>, let σ''V'' be the closure of ''V'' under Boolean operations (binary [[intersection (set theory)|intersection]] and [[complement (set theory)|complement]]). It can be shown that σ''V'' is closed under <math>\Box</math>, thus <math>\sigma\mathbf F=\langle F,\le,\sigma V\rangle</math> is a general modal frame. The skeleton of σ'''F''' is isomorphic to '''F'''. If ''L'' is a superintuitionistic logic complete with respect to a class ''C'' of general frames, then its largest modal companion σ''L'' is complete with respect to <math>\{\sigma\mathbf F;\,\mathbf F\in C\}</math>.
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| The skeleton of a [[Kripke frame]] is itself a Kripke frame. On the other hand, σ'''F''' is never a Kripke frame if '''F''' is a Kripke frame of infinite depth.
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| ==Preservation theorems==
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| The value of modal companions and the Blok–Esakia theorem as a tool for investigation of intermediate logics comes from the fact that many interesting properties of logics are preserved by some or all of the mappings ρ, σ, and τ. For example,
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| *[[decidability (logic)|decidability]] is preserved by ρ, τ, and σ,
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| *[[Kripke semantics#Finite model property|finite model property]] is preserved by ρ, τ, and σ,
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| *[[tabular logic|tabularity]] is preserved by ρ and σ,
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| *[[Kripke semantics#Correspondence and completeness|Kripke completeness]] is preserved by ρ and τ,
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| *[[first-order logic|first-order]] definability on Kripke frames is preserved by ρ and τ.
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| ==Other properties==
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| Every intermediate logic ''L'' has an [[infinite set|infinite]] number of modal companions, and moreover, the set <math>\rho^{-1}(L)</math> of modal companions of ''L'' contains an [[infinite descending chain]]. For example, <math>\rho^{-1}(\mathbf{CPC})</math> consists of '''S5''', and the logics <math>L(C_n)</math> for every positive integer ''n'', where <math>C_n</math> is the ''n''-element cluster. The set of modal companions of any ''L'' is either [[countable set|countable]], or it has the [[cardinality of the continuum]]. Rybakov has shown that the lattice Ext''L'' can be [[embedding|embedded]] in <math>\rho^{-1}(L)</math>; in particular, a logic has a continuum of modal companions if it has a continuum of extensions (this holds, for instance, for all intermediate logics below '''KC'''). It is unknown whether the converse is also true.
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| The Gödel translation can be applied to [[rule of inference|rule]]s as well as formulas: the translation of a rule
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| :<math>R=\frac{A_1,\dots,A_n}{B}</math>
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| is the rule
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| :<math>T(R)=\frac{T(A_1),\dots,T(A_n)}{T(B)}.</math>
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| A rule ''R'' is [[admissible rule|admissible]] in a logic ''L'' if the set of theorems of ''L'' is closed under ''R''. It is easy to see that ''R'' is admissible in a superintuitionistic logic ''L'' whenever ''T''(''R'') is admissible in a modal companion of ''L''. The converse is not true in general, but it holds for the largest modal companion of ''L''.
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| ==References==
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| *Alexander Chagrov and Michael Zakharyaschev, ''Modal Logic'', vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
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| *Vladimir V. Rybakov, ''Admissibility of Logical Inference Rules'', vol. 136 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1997.
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| [[Category:Modal logic]]
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