Semisimplicity, Glivenko theorems, and the excluded middle
aa r X i v : . [ m a t h . L O ] J a n SEMISIMPLICITY, GLIVENKO THEOREMS, AND THEEXCLUDED MIDDLE
TOM ´AˇS L ´AVIˇCKA AND ADAM PˇRENOSIL
Abstract.
We formulate a general, signature-independent form of the lawof the excluded middle and prove that a logic is semisimple if and only if itenjoys this law, provided that it satisfies a weak form of the so-called incon-sistency lemma of Raftery. We then show that this equivalence can be usedto provide simple syntactic proofs of the theorems of Kowalski and Krachtcharacterizing the semisimple varieties of FL ew -algebras and Boolean algebraswith operators, and to extend them to FL e -algebras and Heyting algebras withoperators. Moreover, under stronger assumptions this correspondence worksat the level of individual models: the semisimple models of such a logic areprecisely those which satisfy an axiomatic form of the law of the excludedmiddle, and a Glivenko-like connection obtains between the logic and its ex-tension by the axiom of the excluded middle. This in particular subsumes thewell-known Glivenko theorems relating intuitionistic and classical logic andthe modal logics S4 and S5. As a consequence, we also obtain a descriptionof the subclassical substructural logics which are Glivenko related to classicallogic. Introduction
This paper investigates the relationship between three fundamental notions oflogic and algebra, namely semisimplicity, the law of the excluded middle (LEM),and Glivenko theorems, which provide double negation translations between logics.These are all shown to be part of a single coherent circle of ideas: in the firstpart of the paper we prove that the semisimplicity of a logic is equivalent to someform of the LEM, while in the second part we establish a Glivenko-like connectionbetween a logic and what we call its semisimple companion, which is in many casesaxiomatized by the LEM.Both the LEM and Glivenko theorems presuppose some notion of negation. Forthe purposes of this paper, we take the defining feature of negation to be a weakform of the so-called inconsistency lemma (IL) of Raftery [27], which in its strongform states that ϕ is inconsistent with Γ if and only if ¬ ϕ is provable from Γ. If sucha negation is available in a logic L, then an axiomatic extension of L is semisimpleif and only if it enjoys the LEM. This provides a useful strategy for describingthe semisimple axiomatic extensions of L, which for algebraizable logics such asthe substructural logic FL ew or the global modal logic K amounts to describingthe semisimple subvarieties of the corresponding varieties of algebras. We shall see Mathematics Subject Classification.
Primary 03G27, 03C05, 06F05.
Key words and phrases.
Semisimplicity, Glivenko’s theorem, law of the excluded middle, alge-braic logic, inconsistency lemma, structural completeness.The research of the first author was supported by project no. GA17-04630S of the Czech ScienceFoundation. that such non-trivial algebraic questions concerning semisimplicity can be answeredusing a purely syntactic approach via the LEM, and more importantly that doingso will result in brief and transparent proofs.The study of the semisimple extensions of a given logic L naturally leads oneto consider the semisimple companion of L, which we define as the logic α Th (L)determined by the maximal consistent theories of L. The semisimple companion isoften axiomatized relative to L by an axiomatic form of the LEM. It can be thoughtof as a dual counterpart to the so-called structural (or admissible) completion of L,which is the logic determined by the smallest theory of L, i.e. the set of theoremsof L. Just like the rules valid in the structural completion may be described interms of the theorems of L as the so-called admissible rules of L, we describe therules valid in the semisimple companion in terms of the antitheorems of L as whatwe call the antiadmissible rules of L. The inconsistency lemma then allows us toreformulate antiadmissibility as a Glivenko-like theorem connecting L and α Th (L)by means of a double negation translation. Examples of this phenomenon aboundamong substructural and modal logics.The present paper therefore brings together several strands of research withinnon-classical logic: the study of inconsistency lemmas initiated by Raftery [27], whoestablished the equivalence between semisimplicity and the LEM (more precisely,what he called the classical IL) assuming a strong form of the IL, the descriptionsof the semisimple varieties of FL ew -algebras and modal algebras (Boolean alge-bras with operators) due to Kowalski and Kracht [18, 19], and the investigation ofGlivenko theorems due to Cignoli & Torrens [8, 9] in the context of extensions ofintegral residuated structures, Galatos & Ono [16] in the context of substructurallogics, and Bezhanishvili [3] in the context of intuitionistic modal logics. We extendvarious results from these papers and put them into a uniform perspective. Thetheory of structural completions and admissible rules [24, 1, 28] also finds a dualcounterpart in our study of semisimple companions and antiadmissible rules.The general approach to Glivenko theorems due to Torrens [30], on the otherhand, is not directly comparable to the approach we take in the present paper.Torrens understands Glivenko theorems to be relative to a unary formula whichrepresents a homomorphism onto an algebra of “regular” elements, while we under-stand Glivenko theorems to be relative to a family of sets of unary formulas relatedto inconsistency by means of an IL. Both approaches have their merits: ours doesnot directly yield the known Glivenko theorem relating H´ajek’s basic fuzzy logicBL and the infinite-valued Lukasiewicz logic Ł (because BL does not satisfy astrong enough IL), while the approach of Torrens does not directly yield the knownGlivenko theorem relating the global modal logics S4 and S5 (because ¬ (cid:3) ¬ (cid:3) x does not represent a homomorphism onto an algebra of regular elements). We do,however, obtain a “local” Glivenko theorem relating BL to the infinitary version of Lukasiewicz logic Ł ∞ .It is important to observe that although we choose to formulate our results interms of Hilbert-style consequence relations (consequence relations on formulas),they can be extended to the so-called k -deductive systems of Blok & Pigozzi [5](consequence relations on k -tuples of formulas), and even further to universal Hornlogic without equality [13]. These in particular subsume the equational consequencerelations of (generalized) quasivarieties, which are classes of algebras axiomatizedby (possibly infinitary) implications between equalities in a given set of variables. EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 3
The extension to the k -deductive setting does not involve any substantial mathe-matical work, however, it does come with an increase in the complexity of notation.We thus opt for formulating our results in logical terms and using the notion ofalgebraizability to transfer them to the algebraic realm. The reader familiar with k -deductive systems will see that our results can instead immediately be translatedinto results concerning equational consequence relations and (generalized) quasi-varieties without any need for an intermediate step in the form of a Hilbert-styleconsequence relation.1.1. Semisimple algebras and logics.
Having outlined the main themes of thepresent paper, let us now consider them in more detail, starting with semisimplicity.Recall that an algebra is called simple if it has precisely two congruences, namely theequality relation and the total relation, and it is called semisimple if it is a subdirectproduct of simple algebras, or equivalently if the equality relation is the intersectionof all maximal non-trivial congruences. A variety (equational class) of algebras K is semisimple if each algebra in K is semisimple. Boolean algebras, distributivelattices, and De Morgan algebras are all semisimple varieties, while Heyting algebrasand modal algebras (Boolean algebras with operators) are examples of varietieswhich are not semisimple. If we replace congruences by K -congruences (congruences θ on A such that A /θ ∈ K ), this definition of semisimplicity extends to all classesof algebras in a given signature closed under subdirect products.Our notion of a semisimple logic, by contrast, will be syntactic. The maximalnon-trivial theories of a logic are called simple , a theory is semisimple if it is anintersection of simple theories, and a logic is semisimple if all of its theories are.A theory here is a set of formulas closed under the consequence relation of the logic.Under some mild assumptions, this extends to a semantic form of semisimplicity:each model of the logic is a subdirect product of simple models.If a logic L enjoys a tight connection to a class of algebras K , then the semanticsemisimplicity of L is equivalent to the semisimplicity of K in the algebraist’s sense.In technical terms, this holds if L is (at least weakly) algebraizable and K formsthe algebraic counterpart of L. We can thus use the results proved here to answeralgebraic questions concerning semisimplicity. We focus on cases where K forms avariety, but the same reasoning applies to (generalized) quasivarieties.We consider two kinds of problems involving semisimplicity. Firstly:given a variety K , describe its semisimple algebras.For example, a Heyting algebra is semisimple if and only if it is a Boolean algebra.Similarly, an S4 modal algebra (i.e. a Boolean algebra with a topological interioroperator) is semisimple if and only if it is an S5 modal algebra (an S4 modal algebrasatisfying x ≤ (cid:3)♦ x ). A related but distinct problem is:given a variety K , describe its semisimple subvarieties.For example, a result of Kowalski [18] states that a variety of FL ew -algebras (i.e.bounded commutative integral residuated lattices) is semisimple if and only if itvalidates the equation x ∨ ¬ ( x n ) ≈ n , while an analogous result ofKowalski & Kracht [19] states that a variety of modal algebras is semisimple if andonly if it is validates the equational axioms of weak n -transitivity and n -cyclicityfor some n . However, an individual FL ew -algebra or modal algebra may well besemisimple without validating any of these equations. T. L´AVIˇCKA AND A. P ˇRENOSIL
In case the variety K forms the algebraic counterpart of an (at least weakly)algebraizable logic L, the first of these problems is equivalent to:describe the semisimple models of L . Similarly, the second one is then equivalent to:describe the semisimple axiomatic extensions of L . We shall see that, using the equivalence between semisimplicity and the LEM,both types of problems can be attacked by entirely syntactic methods. To see thisstrategy in action, the interested reader may skip ahead to Subsection 3.7, wherewe extend the results of Kowalski and Kracht to FL e -algebras and modal Heytingalgebras. Note that the original proof of Kowalski for FL ew -algebras involves severalpages of algebraic computations, which are far from straightforward, while our prooftakes about two paragraphs (given the equivalence between semisimplicity and theLEM).1.2. The law of the excluded middle.
Let us now explain what we mean bythe LEM. We take it to be a syntactic principle which, roughly speaking, statesthat if in a given context Γ a formula is derivable from both ϕ and the negation of ϕ , then it is already derivable from Γ. For example, classical logic enjoys the LEMin the form Γ , ϕ ⊢ CL ψ and Γ , ¬ ϕ ⊢ CL ψ = ⇒ Γ ⊢ CL ψ, while the global modal logic S5 enjoys the LEM in the formΓ , ϕ ⊢ S5 ψ and Γ , ¬ (cid:3) ϕ ⊢ S5 ψ = ⇒ Γ ⊢ CL ψ. More generally, a whole family of formulas may play the role of a negation. Suchprinciples will be called local
LEMs, as opposed to the global
LEMs shown above.For example, the infinitary Lukasiewicz logic Ł ∞ enjoys the following local LEM:Γ , ϕ ⊢ Ł ∞ ψ and Γ , ¬ ( ϕ n ) ⊢ Ł ∞ ψ for each n ∈ ω = ⇒ Γ ⊢ Ł ∞ ψ. Although the LEM is formulated as an implication between valid rules, under suit-able assumptions it can be expressed in axiomatic form.The connection between semisimplicity and the LEM presupposes that the nega-tion satisfies a so-called inconsistency lemma (IL). This is an equivalence which re-lates inconsistency and validity in a manner analogous to the following equivalencefor classical logic: Γ , ϕ ⊢ CL ∅ ⇐⇒ Γ ⊢ CL ¬ ϕ. Here Γ , ϕ ⊢ CL ∅ represents the classical inconsistency of the set of formulas Γ , ϕ .For example, the global modal logic S4 enjoys the ILΓ , ϕ ⊢ S4 ∅ ⇐⇒ Γ ⊢ S4 ¬ (cid:3) ϕ, which indicates that the connective ¬ (cid:3) x (rather than merely ¬ x ) functions as anegation in S4, while the ( n + 1)-valued Lukasiewicz logic Ł n +1 enjoys the ILΓ , ϕ ⊢ Ł n +1 ∅ ⇐⇒ Γ ⊢ Ł n +1 ¬ ( ϕ n ) . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 5
As with LEMs, we call such equivalences global
ILs, in contrast to local
ILs suchas the one enjoyed by the substructural logic FL ew and its axiomatic extensions, aswell as the infinitary Lukasiewicz logic Ł ∞ :Γ , ϕ ⊢ FL ew ∅ ⇐⇒ Γ ⊢ FL ew ¬ ( ϕ n ) for some n ∈ ω. Parametrized local ILs, enjoyed e.g. by FL, are more general still.Such equivalences (in their global form) were first studied as conditions in theirown right by Raftery [27]. In addition to ordinary global ILs, Raftery also consid-ered classical global ILs. These combine an ordinary global IL with what we call a dual
IL, generalizing the classical equivalenceΓ ⊢ CL ϕ ⇐⇒ Γ , ¬ ϕ ⊢ CL ∅ . Equivalently, a classical IL combines an ordinary IL and with a LEM. Rafterythen proved that for each algebraizable logic L whose algebraic counterpart is aquasivariety K , the classical global IL for the logic L corresponds precisely to thefiltrality of the quasivariety K , i.e. to relative semisimplicity plus the equationaldefinability of principal relative congruences in K .We are now ready to state our main results relating the semisimplicity of a logicand the LEM (Theorems 3.35 and 3.43), which can be thought of as extendingRaftery’s results to the (parametrized) local form of the classical IL: if L enjoys aparametrized local IL, thenall theories of L are semisimple if and only if L satisfies the LEM . If L enjoys a local IL, then this can be upgraded to:all models of L are semisimple if and only if L satisfies the LEM . Since the (parametrized) local IL is inherited by all axiomatic extensions of L, thesetheorems will enable us to characterize the semisimple axiomatic extensions of thesubstructural logic FL e and the intuitionistic modal logic IK.1.3. Semisimple companions.
To tackle the second problem involving semi-simplicity, namely describing the semisimple models of a given logic, we studythe (syntactic) semisimple companion α Th (L) of a logic L, defined as the logic de-termined by the simple theories of L. That is, α Th (L) is the logic of all matricesof the form h Fm L , T i where T ranges over simple theories of L. Our next result(Theorem 4.16) states that if L enjoys a local IL, thena model of L is semisimple if and only if it is a model of α Th (L) . The problem of describing the semisimple models of L thus reduces to the problemof axiomatizing α Th (L).For instance, the semisimple companion of H´ajek’s basic fuzzy logic BL (whichinherits a local IL from FL ew ) is the infinitary Lukasiewicz logic Ł ∞ . This isbecause the algebraic counterpart of BL is the variety of BL-algebras, semisimpleBL-algebras coincide with semisimple MV-algebras (see [31]), and these in turnform the algebraic counterpart of Ł ∞ (as opposed to the finitary logic Ł , whosealgebraic counterpart is the variety of MV-algebras). In this case, we used ourability to identify the semisimple algebras in the given variety to describe α Th (BL).However, we can also proceed in the opposite direction.The problem of axiomatizing the semisimple companion α Th (L) can be solved inan entirely mechanical manner whenever L enjoys a global IL and the LEM can be T. L´AVIˇCKA AND A. P ˇRENOSIL expressed in axiomatic form. This occurs if L has either a well-behaved disjunctionor a well-behaved implication. More precisely, we assume either a global deduction–detachment theorem (DDT), generalizing the equivalenceΓ , ϕ ⊢ CL ψ ⇐⇒ Γ ⊢ CL ϕ → ψ, or a global proof by cases property (PCP), generalizing the equivalenceΓ , ϕ ⊢ CL χ and Γ , ψ ⊢ CL χ ⇐⇒ Γ , ϕ ∨ ψ ⊢ CL χ. For example, the global modal logic S4 enjoys a global DDT in the formΓ , ϕ ⊢ S4 ψ ⇐⇒ Γ ⊢ S4 (cid:3) ϕ → ψ, as well as a global PCP in the formΓ , ϕ ⊢ S4 χ and Γ , ψ ⊢ S4 χ ⇐⇒ Γ , (cid:3) ϕ ∨ (cid:3) ψ ⊢ S4 χ. These indicate that the connectives (cid:3) x → y and (cid:3) x ∨ (cid:3) y (rather than merely x → y and x ∨ y ) function respecitvely as an implication and a disjunction in S4.In logics which enjoy the global IL and either the global DDT or the global PCP,the LEM may be expressed by an axiom of one of the two forms( p → q ) → (( ¬ p → q ) → q ) or p ∨ ¬ p, where the negation, implication, and disjunction are the connectives which occurin the IL, DDT, and PCP. These need not coincide with the connectives whichare customarily denoted ¬ x , x → y , and x ∨ y in the given logic: in S4 these arerespectively ¬ (cid:3) x , (cid:3) x → y , and (cid:3) x ∨ (cid:3) y .Whenever such an axiomatic form of the LEM is available, the semisimple com-panion of L is the extension of L by the axiom of the excluded middle (Proposi-tion 4.22). It immediately follows (Theorem 4.23) thata model of L is semisimple if and only if it is satisfies the (axiomatic) LEM . This subsumes the facts that semisimple Heyting algebras are precisely Booleanalgebras, and semisimple S4 modal algebras are precisely S5 modal algebras. (Recallthat S5 is the extension of S4 by the axiom p → (cid:3)♦ p , which is equivalent in S4 tothe LEM in the form (cid:3) p ∨ (cid:3) ¬ (cid:3) p , thus α Th (S5) = S4.) These two particular factsare of course not difficult to prove directly, but the merit of our approach lies inproviding a uniform proof of such facts in any logic with a well-behaved negationand implication or disjunction.1.4. Glivenko theorems.
The framework outlined above now allows us to spellout a precise connection between semisimplicity and Glivenko theorems. The semi-simple companion can be thought of as a dual counterpart to the so-called structuralcompletion of a logic: the semisimple companion of L is the logic determined bythe largest non-trivial theories of L, while the structural completion of L is thelogic determined by the smallest theory of L, i.e. by the matrix h Fm L , Thm L i .Just like the structural completion is the strongest logic with the same theorems,the antistructural completion of a compact logic is the strongest logic with the sameantitheorems. This relates the study of semisimple companions to the establishedtopic of structural completeness (see e.g. [24, 1, 28]). Indeed, we may call α Th (L)the antistructural completion of L. EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 7
In particular, the notion of an admissible rule (see [28]) has a natural counterpartin the theory of semisimple companions. Recall that a rule Γ ⊢ ϕ is valid in thestructural completion of L if it is admissible : ∅ ⊢ L σ [Γ] = ⇒ ∅ ⊢ L σ ( ϕ ) for each substitution σ .Similarly (Proposition 4.6), a rule Γ ⊢ ϕ is valid in the antistructural completionof a compact logic L, i.e. in α Th (L), if it is antiadmissible : σ ( ϕ ) , ∆ ⊢ L ∅ = ⇒ σ [Γ] , ∆ ⊢ L ∅ for each ∆ ⊆ Fm L and each substitution σ .For compact logics which enjoy a local IL (Proposition 4.11), this simplifies to: ϕ, ∆ ⊢ L ∅ = ⇒ Γ , ∆ ⊢ L ∅ for each ∆ ⊆ Fm L.A Glivenko theorem relating L and α Th (L) now results if we apply the IL tothis simplified definition of antiadmissibility and recall that Γ ⊢ α Th (L) ϕ if andonly if Γ ⊢ ϕ is antiadmissible in L. This subsumes as special cases the familiarGlivenko theorems for intuitionistic logic and S4 (see [17, 22]), or more preciselytheir extensions concerning valid rules rather than theorems:Γ ⊢ CL ϕ ⇐⇒ Γ ⊢ IL ¬¬ ϕ, Γ ⊢ S5 ϕ ⇐⇒ Γ ⊢ S4 ¬ (cid:3) ¬ (cid:3) ϕ. For a logic such as FL ew which only enjoys a local IL with respect to the family offormulas ¬ ( x n ), we obtain what we might call a local Glivenko theorem:Γ ⊢ α Th (FL ew ) ϕ ⇐⇒ Γ ⊢ FL ew {¬ ( ¬ ϕ n ) f ( n ) | n ∈ ω } for some f : ω → ω. In particular, a local Glivenko theorem connects BL and α Th (BL) = Ł ∞ :Γ ⊢ Ł ∞ ϕ ⇐⇒ Γ ⊢ BL {¬ ( ¬ ϕ n ) f ( n ) | n ∈ ω } for some f : ω → ω. This Glivenko theorem can perhaps more naturally be expressed as:Γ ⊢ Ł ∞ ϕ ⇐⇒ for each n ∈ ω there is some k ∈ ω such that Γ ⊢ BL ¬ ( ¬ ϕ n ) k . In case Γ is finite, this is superseded by the global Glivenko theorem due to Cignoliand Torrens [8]: Γ ⊢ Ł ∞ ϕ ⇐⇒ Γ ⊢ Ł ϕ ⇐⇒ Γ ⊢ BL ¬¬ ϕ. Note that this approach to Glivenko theorems, which relies on the relationshipbetween negation and inconsistency, differs substantially from the approach of Cig-noli and Torrens [8, 30] and of Galatos & Ono [16], which relies on a suitableterm-definable projection onto an algebra of regular elements.2.
Preliminaries
We now introduce the necessary preliminaries. We recall some basic notions ofabstract algebraic logic concerning logics and their models. The reader unfamiliarwith this framework may find the recent textbook [14] helpful. We then introduceour running examples: substructural and normal modal logics. Finally, we reviewsome notational conventions used in this paper.
T. L´AVIˇCKA AND A. P ˇRENOSIL
Logics. A logic L is a structural closure operator on the absolutely free algebra(the algebra of formulas) Fm L on a given set of variables Var L, where structuralitymeans that if T is a closed set of formulas, i.e. a theory of L, then so is σ − [ T ] forany substitution (homomorphism) σ : Fm L → Fm L. Equivalently, we can thinkof a logic as a consequence relation, denoted Γ ⊢ L ϕ , between sets of formulas Γand formulas ϕ such that • ϕ ⊢ L ϕ , • if Γ ⊢ L ϕ , then Γ , ∆ ⊢ L ϕ , • if Γ ⊢ L Φ and Φ ⊢ L ϕ , then Γ ⊢ L ϕ , and • if Γ ⊢ L ϕ , then σ [Γ] ⊢ L σ ( ϕ ) for each substitution σ .Here Γ ⊢ L Φ abbreviates the claim that Γ ⊢ L ϕ for each ϕ ∈ Φ. Given L as astructural closure operator, we define Γ ⊢ L ϕ to hold if and only if ϕ is in theclosure of Γ. Conversely, given L as a consequence relation, we define the closedsets (theories) of L to be the sets of formulas T such that T ⊢ L ϕ implies ϕ ∈ T .The set of all theories of L forms a complete lattice Th L.A logic L is finitary , or more generally κ -ary, if Γ ⊢ L ϕ implies Γ ′ ⊢ L ϕ for somefinite Γ ′ ⊆ Γ, or more generally for some Γ ′ ⊆ Γ such that | Γ ′ | < κ .A pair h A , F i consisting of an algebra A and a set F ⊆ A is called a matrix .A matrix is a model of a logic L if for each homomorphism h : Fm L → A the set h − [ F ] is a theory of L, in other words if h designates ϕ whenever it designates Γ.Here we say that h designates a set of formulas Φ in h A , F i if h [Φ] ⊆ F . If h A , F i is a model of L, we also say that F is a filter of L on A . For each algebra A theset of all filters of L on A forms a closure system. The filter generated by X ⊆ A isthen denoted Fg A L X . Mimicking the notation for formulas, we also write X ⊢ A L a for a ∈ Fg A L X .A matrix h A , F i , and by extension the filter F , is called trivial if F = A .In particular, the trivial theory of L is the theory Fm L. A simple theory of Lis a maximal non-trivial theory, while a semisimple theory of L is an intersectionof simple theories of L. A logic L is then called semisimple if each theory of L issemisimple. Every semisimple logic is coatomic : each non-trivial theory is includedin a simple theory.An antitheorem of a logic L is a set of formulas Γ which cannot be jointlydesignated in any non-trivial model. In other words, for each non-trivial model h A , F i of L and each homomorphism h : Fm L → A it is not the case that h [Γ] ⊆ F . The claim that Γ is an antitheorem of L shall be abbreviated by Γ ⊢ L ∅ . It is straightforward to prove the following analogues of monotonicity, cut, andstructurality for antitheorems: • if Γ ⊢ L ∅ , then Γ ⊢ L ϕ , • if Γ ⊢ L ∅ , then Γ , ∆ ⊢ L ∅ , • if Γ ⊢ L Φ and Φ ⊢ L ∅ , then Γ ⊢ L ∅ , and • if Γ ⊢ L ∅ , then σ [Γ] ⊢ L ∅ for each substitution σ .A logic L is compact , or more generally κ -compact , if Γ ⊢ L Fm L implies Γ ′ ⊢ L Fm L for some finite Γ ′ ⊆ Γ, or more generaly for some Γ ′ ⊆ Γ such that | Γ ′ | < κ . We take ⊢ ∅ to be a single indivisible symbol, rather than analyzing it as ⊢ Φ for Φ = ∅ .(The latter would be appropriate if sets of formulas Φ to the right of the turnstile were given adisjunctive rather than conjunctive reading.) While this notation clashes with the convention thatΓ ⊢ L Φ means Γ ⊢ L ϕ for each ϕ ∈ Φ, no confusion is likely to occur. The motivation behind thisnotation stems from the duality between theorems and antitheorems.
EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 9
In particular, each compact theory has a finite antitheorem, and conversely eachfinitary logic with an antitheorem is compact. For finitary logics the compactnessof L coincides with the lattice-theoretic compactness of Fm L in the lattice Th L.Observe that each compact logic is coatomic.A logic need not have antitheorems: for example, in the positive fragment ofclassical logic (whose signature consists of ∧ , ∨ , ⊤ , → ) every formula can be desig-nated in the two-element Boolean matrix h B , { }i by the homomorphism sendingeach variable to 1. Thus Γ ⊢ L ∅ is a stronger claim than Γ ⊢ L Fm L.
Fact 2.1.
If a variable p does not occur in Γ , then Γ ⊢ L ∅ if and only if Γ ⊢ L p .Proof. If h : Fm L → A designates Γ in a model h A , F i of L, then for each a ∈ A there is a homomorphism g a : Fm L → A such that g a [Γ] ⊆ F and g a ( p ) = a . Thus F = A whenever Γ can be designated in h A , F i . (cid:3) Fact 2.2. Γ ⊢ L ∅ if and only if σ [Γ] ⊢ L Fm L for each substitution σ .Proof. Right to left, let σ and τ be substitutions such that no formula in σ [Fm L]contains the variable p and τ ( σ ( ϕ )) = ϕ for each ϕ . Then σ [Γ] ⊢ L p , so σ [Γ] ⊢ L ∅ by the previous fact. Applying τ now yields that Γ = τ [ σ [Γ]] ⊢ L ∅ . (cid:3) Running examples: substructural logics.
Let us now introduce the twofamilies of logics which will serve as running examples throughout the paper. Wefirst consider substructural logics, which we take to be the axiomatic extensions ofthe Full Lambek calculus FL. A standard reference text for substructural logicsis [15]. The signature of FL consists of: • the lattice meet and join ∧ and ∨ , • the top and bottom constants ⊤ and ⊥ , • the monoidal multiplication and unit · and 1, • the residuals of multiplication \ and / , and • the constant 0.The traditional inclusion of the constant 0 in the signature will be of little conse-quence to us (in fact, of no consequence except for the very last theorem of thispaper). The reader may well decide to drop this constant from the signature. Thepresence of ⊥ , on the other hand, will be crucial.We define the consequence relation of FL in terms of its algebraic semantics givenby FL-algebras. These are algebras in the above signature such that h A, ∧ , ∨ , ⊤ , ⊥i is a bounded lattice, h A, · , i is a monoid, and moreover the residuation law holds: x ≤ z/y ⇐⇒ x · y ≤ z ⇐⇒ y ≤ x \ z. FL-algebras form a variety. The subvariety of FL e -algebras imposes the axiom ofcommutativity (also known as exchange) x · y ≈ y · x , or equivalently x \ y ≈ y/x .The subvariety of FL ew -algebras moreover imposes the axiom of integrality 1 ≈ ⊤ (also known as weakening), or equivalently x · y ≤ x ∧ y . Imposing the axiom ofcontraction x ≤ x · x on FL ew -algebras yields the variety of Heyting algebras.Throughout the paper we use the standard notation x := 1 and x n +1 := x n · x .The varieties of FL n e and FL n ew algebras are then defined as the subvarieties of FL e and FL ew which impose the axiom of n -contraction (1 ∧ x ) n +1 ≈ (1 ∧ x ) n . Thenotation ¬ x abbreviates x \⊥ , with ¬ x n interpreted as ¬ ( x n ). The consequence relation of the logic FL is defined in terms of the equationalconsequence relation (cid:15) FL of FL-algebras as follows:Γ ⊢ FL ϕ ⇐⇒ { ≤ γ | γ ∈ Γ } (cid:15) FL ≤ ϕ. Analogous equivalences relate the logics FL e , FL ew , FL n e , and FL n ew with the cor-responding classes algebras. In the terminology of abstract algebraic logic, theselogics are algebraizable , and their algebraic counterparts are the varieties of FL-algebras, FL e -algebras, and FL ew -algebras. We shall not need to define these termsprecisely for the purposes of this paper.In FL e -algebras we use x → y to denote the element x \ y = y/x . In FL e andits extensions, we thus in effect replace the two residuals of multiplication in thesignature of FL by a single residual → .Among the best-known extensions of FL ew are the finite-valued Lukasiewiczlogics Ł n +1 , the finitary Lukasiewicz logic Ł , and its infinitary counterpart Ł ∞ .These may all be defined in terms of a single matrix and its finite submatrices,namely the standard Lukasiewicz chain [0 , Ł with the filter { } . This is the realinterval [0 ,
1] equipped with the standard lattice structure and the operations x · y := max( x + y − , ,x → y := min(1 − x + y, . In terms of these, we may define some further operations: ¬ x := x → ⊥ = 1 − x, x := 0 ,x ⊕ y := ¬ ( ¬ y · ¬ x ) = min( x + y, , ( n + 1) x := x ⊕ nx. The matrix h [0 , Ł , { }i determines the infinitary Lukasiewicz logic Ł ∞ , of whichthe finitary Lukasiewicz logic Ł is the finitary companion. That is,Γ ⊢ Ł ϕ ⇐⇒ Γ ′ ⊢ Ł ∞ ϕ for some finite Γ ′ ⊆ Γ . The logics Ł n +1 are then determined by the submatrices over { n , n , . . . , n − n , nn } .The failure of finitarity in Ł ∞ is witnessed by the rule {¬ p → q n | n ∈ ω } ⊢ Ł ∞ p ∨ q, which in fact axiomatizes Ł ∞ relative to Ł . Nevertheless, it is known that Ł ∞ iscompact [6], along with other related logics such as the infinitary version of theproduct logic Π ∞ [10] or the infinitary version BL ∞ of H´ajek’s basic fuzzy logic. Such logics provide a good motivation for taking care to formulate our theory forcompact rather than just finitary logics.2.3.
Running examples: normal modal logics.
Just like substructural logicsare axiomatic extensions of FL, classical normal modal logics are axiomatic exten-sions of the basic global modal logic K, and intuitionistic modal logics are axiomaticextensions of its intuitionistic counterpart IK. Although many such logics enjoy arelational (Kripke) semantics, we shall define the consequence relations of K and The logic Π ∞ is determined by the standard product chain [0 , Π , while BL ∞ is determinedby the standard BL-chains, i.e. BL-algebras over the unit interval [0 , ∞ follows from the compactness of Ł ∞ and the decomposition of standard BL-chains as ordinal sumsof Lukasiewicz, product, and G¨odel components [23, Theorem 2.16]. The key observation is thatif a finite set Γ is satisfiable in the standard product chain [0 , Π or the standard G¨odel chain[0 , G , then it is satisfiable in the standard Lukasiewicz chain [0 , Ł . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 11
IK in terms of their algebraic semantics given by what we call modal algebras andmodal Heyting algebras.An introduction to classical modal logic may be found in [7, 4]. For an overviewof different approaches to intuitionistic modal logic, the reader may consult [29].Our motivation for considering intuitionistic modal logics is to show how easy itis to extend the existing results for classical modal logics given the machineryintroduced here. However, no familiarity with intuitionistic modal logic will berequired: throughout the paper the reader may easily substitute a correspondingclassical modal logic whenever we discuss an intuitionistic one.We define modal algebras as Boolean algebras with a unary operator (cid:3) whichsatisfies the equations (cid:3) ( x ∧ y ) ≈ (cid:3) x ∧ (cid:3) y and (cid:3) ⊤ ≈ ⊤ . Modal Heyting algebrasare Heyting algebras equipped unary operators (cid:3) and ♦ which satisfy the followingequations: (cid:3) ( x ∧ y ) ≈ (cid:3) x ∧ (cid:3) y (cid:3) ⊤ ≈ ⊤ (cid:3) ( x → y ) ≤ ♦ a → ♦ b ♦ ( x ∨ y ) ≈ ♦ x ∨ ♦ y ♦ ⊥ ≈ ⊥ ♦ x → (cid:3) y ≤ (cid:3) ( x → y )Each modal algebra is a modal Heyting algebra if we take ♦ x := ¬ (cid:3) ¬ x . However,the two operators are in general not interdefinable in modal Heyting algebras.Let us remark here that although the axiom ( ♦ p → (cid:3) p ) → (cid:3) ( p → q ) is usuallytaken to be part of the basic intuitionistic modal logic, we do not use this axiomanywhere. In other words, we might even take our basic intuitionistic modal logicto be slightly weaker and drop the inequality ♦ x → (cid:3) y ≤ (cid:3) ( x → y ) from the abovedefinition of a modal Heyting algebra.Throughout the paper we use the notation (cid:3) n x := x ∧ (cid:3) x ∧ · · · ∧ (cid:3) n x, ♦ n x := x ∨ ♦ x ∨ · · · ∨ ♦ n x, where (cid:3) n and ♦ n denote a string of n boxes or diamonds. The varieties of IKn . . n ∈ ω : • IKn . (cid:3) n x ≤ (cid:3) n +1 x (weak n -transitivity), • IKn . . ≈ x ∨ (cid:3) ¬ (cid:3) n x ( n -cyclicity), • IS4-algebras: (cid:3) x ≤ x (reflexivity) and (cid:3) x ≤ (cid:3)(cid:3) x (transitivity).Kn . . x ≤ (cid:3)♦ x . The dual equations ♦ n +1 x ≤ ♦ n x , x ≤ ♦ x ,and ♦♦ x ≤ ♦ x may be added to the above definitions of IKn . (cid:15) IK of modal Heyting algebras as follows:Γ ⊢ IK ϕ ⇐⇒ { ≈ γ | γ ∈ Γ } (cid:15) IK ≈ ϕ. Analogous equivalences relate the other intuitionistic modal logics introduced abovewith the corresponding classes of modal Heyting algebras. These logics are, in theterminology of abstract algebraic logic, algebraizable with their algebraic counter-parts being the varieties of the same name.Although we shall not need to use Kripke semantics in this paper, for the benefitof the reader let us recall that • S4 is the global logic of reflexive and transitive Kripke frames, and • S5 is the global logic of reflexive, transitive, and symmetric Kripke frames.To define the Kripke semantics for Kn .
4, let us introduce the notation R n for the n -fold relational composition of a binary relation R on a set W : R is the equalityrelation on W and R n +1 = R n ◦ R . We then use R ≤ n to denote the relation S ≤ i ≤ n R i , i.e. the relation of being accessible in at most n steps. The relation R will be called weakly n -transitive if R ≤ ( n +1) ⊆ R ≤ n , and n -cyclic if uRw implies wR ≤ n u . Then • Kn . n -transitive Kripke frames. • Kn .
45 is the logic of n -cyclic weakly n -transitive Kripke frames.Similar claims could be made for the intuitonistic versions of these logics.2.4. Notational conventions.
We now review some notational conventions em-ployed throughout the paper. For a given logic L, κ ranges over all cardinals suchthat 1 < κ ≤ | Var L | + , possibly finite, while α ranges over all ordinals 0 < α < κ ,and ϕ and p range over α -tuples of formulas and variables. A set of formulas inthe variables p, q, r is denoted e.g. I( p, q, r ), in which case I( ϕ, ψ, χ ) denotes the setof formulas obtained by substituting ϕ for p , ψ for q , and χ for r . If Φ is a set offormulas of cardinality at most equal to | α | , then I(Φ , ψ, χ ) abbreviates I( ϕ, ψ, χ )for some α -tuple of formulas ϕ consisting precisely of the elements of Φ, some ofthem possibly repeated. For each α -tuple of formulas ϕ and each substitution σ ,the tuple obtained by applying σ to each element of ϕ is denoted σ ( ϕ ). Analogousconventions hold for α -tuples a of elements of any given algebra. We generally avoidexplicitly stating the lengths of tuples to avoid cluttering the text with technicalitieswhich can be easily inferred from the context.3. Semisimplicity and the law of the excluded middle
The aim of this section is to show that under suitable conditions, semisimplicityand the law of the excluded middle (LEM) are equivalent. Theorem 3.35 providesthe syntactic version of this equivalence, while Theorem 3.43 provides a semanticversion, which is then used in Subsection 3.7 to describe the semisimple varietiesof FL e -algebras and modal Heyting algebras.We first introduce the syntactic principles which will play a crucial role in ourframework: deduction–detachment theorems (DDTs), inconsistency lemmas (ILs),dual inconsistency lemmas (dual ILs), and simple inconsistency lemmas (simpleILs). Dual ILs in fact turn out to be nothing but LEMs in disguise. As the readerwill readily observe, these principles are preserved under axiomatic extensions aswell as linguistic reducts, provided that the reducts contain all the syntactic ma-terial required to express the principle in question. Our methods will be entirelysyntactic, with the exception of results which relate syntactic and semantic semi-simplicity (Subsection 3.6).Although we only investigate DDTs, ILs, and dual ILs from a syntactic point ofview, the reader should be aware that these principles also have natural semanticcorrelates. For DDTs such correlates are well known, as the theory of DDTs is aclassical part of abstract algebraic logic (see e.g. [12, 14]). By contrast, the theoryof ILs has not been systematically investigated so far, with the exception of theglobal IL studied by Raftery [27]. Since developing this theory is outside the scopeof the present paper, let us merely remark here that a theory of ILs parallel to the EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 13 existing theory of DDTs can be built where the known semantic characterizationsof so-called global, local, and parametrized local DDTs have analogues usuallyobtained by restricting to simple theories or models at the appropriate places.The facts about DDTs stated here are well known and their proofs can be foundin the literature (see again [12, 14]), although we extend the standard frameworkslightly by allowing for infinitary DDTs. The facts about ILs are not present inthe literature. However, many of them are extensions of the results of Raftery [27]on global ILs in finitary protoalgebraic logics to a more general setting. In suchcases we provide a reference to the corresponding proposition in [27]. Some ofthese extensions are straightforward, while in other cases some subtleties arise.Moreover, while our proofs are invariable syntactic in nature, some of Raftery’sproofs are semantic in nature and involve protoalgebraicity.3.1.
Deduction theorems.
A deduction–detachment theorem (DDT) for a givenlogic is, informally speaking, an equivalence which allows us to move formulasfrom the left of the turnstile to the right in a manner analogous to the followingequivalence for classical logic:Γ , ϕ ⊢ CL ψ ⇐⇒ Γ ⊢ CL ϕ → ψ. In finitary logics, such an equivalence allows us to reduce questions of validity toquestions of theoremhood.Many familiar logics enjoy a DDT. For example, the above equivalence holds forintuitionistic logic, while a more general one holds for the n -contractive Full Lambekcalculus with exchange FL n e and its axiomatic extensions:Γ , ϕ ⊢ FL n e ψ ⇐⇒ Γ ⊢ FL n e (1 ∧ ϕ ) n → ψ. A similar equivalence holds for the global modal logic IKn . , ϕ ⊢ IKn . ψ ⇐⇒ Γ ⊢ IKn . (cid:3) n ϕ → ψ. The above equivalences all involve a single formula schema to the right of theturnstile, no matter what Γ, ϕ , or ψ are. Such DDTs are called global DDTs.A more complicated DDT holds for the Full Lambek calculus with exchange FL e and its axiomatic extensions:Γ , ϕ ⊢ FL e ψ ⇐⇒ Γ ⊢ FL e (1 ∧ ϕ ) n → ψ for some n ∈ ω. For FL ew and its axiomatic extensions this simplifies to:Γ , ϕ ⊢ FL ew ψ ⇐⇒ Γ ⊢ FL ew ϕ n → ψ for some n ∈ ω. Similarly, the global modal logic IK and its axiomatic extensions, including theclassical modal logic K, enjoy the following equivalence:Γ , ϕ ⊢ IK ψ ⇐⇒ Γ ⊢ IK (cid:3) n ϕ → ψ for some n ∈ ω. Recall that (cid:3) n x stands for x ∧ (cid:3) x ∧ · · · ∧ (cid:3) n x , where (cid:3) n abbreviates a string of n boxes. Such equivalences, which feature a whole family of formulas (or moregenerally sets of formulas) to the right of the turnstile, are called local DDTs.Some logics, such as the Full Lambek calculus FL do not enjoy even a localform of the DDT, but they satisfy a still more general form of the DDT, called aparametrized local DDT (see [15, Chapter 2.4]). Other variants of the DDT notconsidered in the present paper also exist, such as so-called contextual deduction theorems [26]. Having sketched some of the most important examples of DDTs, letus now state the relevant definitions properly. Definition 3.1.
A logic L enjoys the κ -ary parametrized local DDT , where κ ≤| Var L | + , if for each ordinal < α < κ there is a family of sets Φ α ( p, q, r ) , where p is an α -tuple of variables, such that for each set of formulas Γ and each α -tuple offormulas ϕ : Γ , ϕ ⊢ L ψ ⇐⇒ Γ ⊢ L I( ϕ, ψ, π ) for some I( p, q, r ) ∈ Φ α ( p, q, r ) and some π. It enjoys the κ -ary local DDT if each Φ α can be chosen of the form Φ α ( p, q ) .It enjoys the κ -ary global DDT if each Φ α can be chosen of the form { I( p, q ) } . The whole collection of families Φ α will be referred to as the DDT family Φ.Moreover, • by the unary DDT we mean the 2-ary DDT, • by the finitary DDT we mean the ω -ary DDT, and • by the unrestricted DDT we mean the κ -ary DDT for κ = | Var L | + .The technical condition κ ≤ | Var L | + , which accounts for the fact that there are notuples of distinct variables of length | Var L | + , will be always be assumed when wetalk about κ -ary DDTs, or indeed any other κ -ary syntactic principles. For a finitary ( κ -ary) logic we may assume without loss of generality that eachof the families Φ α consists of finite sets (sets of cardinality less than κ ) only. Alogic which enjoys a unary global (local, parametrized local) DDT in fact enjoysthe finitary global (local, parametrized local) local DDT. Moreover, a κ -ary logicwhich enjoys a κ -ary (parametrized) local DDT in fact enjoys the unrestricted(parametrized) local DDT.Only unary (or equivalently, finitary) DDTs are usually considered in the litera-ture. This omission is entirely justified when it comes to global DDTs in a settingwhere each connective is finitary: the reader may easily observe that even clas-sical logic fails to enjoy the infinitary global DDT. However, as a consequence offinitarity, classical logic does enjoy the unrestricted local DDT in the formΓ , ϕ ⊢ CL ψ ⇐⇒ Γ ⊢ CL (cid:16)^ ϕ ′ (cid:17) → ψ for some finite subtuple ϕ ′ of ϕ. More interestingly, the infinitary Lukasiewicz logic Ł ∞ enjoys a more complicatedform of the unrestricted local DDT. Since ILs and dual ILs are instrumental inobtaining the infinitary local DDT of Ł ∞ , we postpone the formulation of thisunrestricted DDT until after these have been introduced (see Example 3.24).Our reasons for considering infinitary DDTs are three-fold. Firstly, we gain somegenerality with no corresponding increase in the complexity of our proofs. Secondly,building an arity into the definition of a DDT makes the parallels between thetheory of DDTs and the theory of ILs more explicit. A unary IL need not induce In other words, the unary DDT allows us to transport a single formula from the left of theturnstile to the right. One might alternatively tweak the definition of the κ -ary DDT in case κ isfinite to make sure that it is the 1-ary DDT rather than the 2-ary DDT which transports a singleformula across the turnstile. We opt against doing so as it would be rather tedious to split thedefinition of each syntactic principle into two cases depending on whether κ is finite. If | Fm L | = | Var L | , which is the case for most “everyday” logics, then there are no tuples ofdistinct formulas of length greater than | Var L | anyway. This is trivial for global and local DDTs. For parametrized local DDTs the presence ofparameters complicates things, but the implication holds by Fact 3.2.
EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 15 a finitary IL, therefore one is led to consider ILs of different arities anyway, and nosubstantial simplicity is gained by restricting to finitary ILs. Finally, for infinitaryprotoalgebraic logics it is the unrestricted DDT rather than the finitary DDT whichcorresponds to a natural semantic condition, namely the Filter Extension Property(see [20, Proposition 6.95]).Although we did not provide an example of a parametrized local DDT in action,let us recall here that logics with such a DDT admit a more practical character-ization in terms of the existence of a set of formulas which satisfies the axiom ofReflexivity and the rule of Modus Ponens. Most “everyday” logics with an impli-cation connective therefore enjoy at least this form of the DDT.
Fact 3.2.
The following are equivalent for each logic L : (1) L enjoys the unary parametrized local DDT. (2) L enjoys the unrestricted parametrized local DDT. (3) L has a protoimplication set, i.e. a set of formulas ∆( p, q ) in two variablessuch that ∅ ⊢ L ∆( p, p ) and p, ∆( p, q ) ⊢ L q .Proof. The equivalence between (1) and (3) is well known, see e.g. Theorems 6.7and 6.22 of [14]. The equivalence between conditions (2) and (3) is not statedexplicitly there, but it holds by a trivial modification of the proof of Theorem 6.22of [14]. (cid:3)
If a logic enjoys a parametrized local DDT w.r.t. two families Φ and Φ ′ , then theseare equivalent in the following sense: for each I( p, q, r ) ∈ Φ α there are I ′ ( p, q, r ′ ) ∈ Φ ′ and π ′ such that I( p, q, r ) ⊢ L I ′ ( p, q, π ′ ), and conversely for each I ′ ( p, q, r ′ ) ∈ Φ ′ α there are I( p, q, r ) ∈ Φ and π such that I ′ ( p, q, r ′ ) ⊢ L I( p, q, π ). Fact 3.3.
Let L be a logic which enjoys the κ -ary parametrized local DDT w.r.t. afamily Φ . Then L enjoys the κ -ary parameterized local DDT w.r.t. a family Φ ′ ifand only if Φ and Φ ′ are equivalent.Proof. We omit the proof of this known fact, since an analogous fact will be provedfor ILs. (cid:3)
An equivalent formulation of the (parametrized) local DDT is what we call the(surjective) substitution swapping property.
Definition 3.4.
A logic L enjoys κ -ary (surjective) substitution swapping , where κ ≤ | Var L | + , if for each ordinal < α < κ , each (surjective) substitution σ , each α -tuple of formulas ϕ , each ψ , and each L -theory Tσ ( ϕ ) , T ⊢ L σ ( ψ ) = ⇒ ϕ, σ − [ T ] ⊢ L ψ. Proposition 3.5.
A logic L enjoys the κ -ary (parametrized) local DDT if and onlyif it enjoys κ -ary (surjective) substitution swapping, for κ ≤ | Var L | + .Proof. We again omit the proof of this known fact (see e.g. [14, Section 6.2]), sincean analogous fact will be proved for ILs. (cid:3)
Although in this paper we shall not be concerned with the semantic correlatesof the various forms of the DDT, let us briefly state what they are for the benefitof the algebraically minded reader. If L is an algebraizable logic and K is itsequivalent algebraic semantics, then L always enjoys the parametrized local DDT.Moreover, modulo some finitarity and finiteness assumptions, it enjoys the local DDT if and only if K enjoys the Relative Congruence Extension Property (RCEP),and it enjoys the global DDT if and only if K has Equationally Definable RelativePrincipal Congruences (EDPRC).3.2. Inconsistency lemmas.
While DDTs allow us to move formulas from theleft of the turnstile to the right in any context, inconsistency lemmas (ILs) have amore limited scope: they allow us to do so if the premises are inconsistent. Thatis, they generalize the following equivalence for classical logic:Γ , ϕ , . . . , ϕ k ⊢ CL ∅ ⇐⇒ Γ ⊢ L ¬ ( ϕ ∧ · · · ∧ ϕ k ) . This is essentially the DDT for CL with the falsum constant in place of ψ . In com-pact logics, such an equivalence allows us to reduce questions of inconsistency toquestions of validity. It will come as no surprise to the reader that such equivalencesalso come in global, local, and parametrized local forms.Each logic with a DDT and a falsum constant representing an inconsistent propo-sition enjoys an IL. Fragments of such logics which drop the implication but retainthe negation provide examples of logics which enjoy the IL but not a DDT. For ex-ample, (the conjunction–negation–product–unit fragment of) FL n e and its axiomaticextensions enjoy the IL in the formΓ , ϕ , . . . , ϕ k ⊢ FL n e ∅ ⇐⇒ Γ ⊢ FL n e ¬ (1 ∧ ϕ ∧ · · · ∧ ϕ k ) n , while (the negation–conjunction–box fragment of) IKn . , ϕ , . . . , ϕ k ⊢ IKn . ∅ ⇐⇒ Γ ⊢ IKn . ¬ (cid:3) n ( ϕ ∧ · · · ∧ ϕ k ) . The local form of the IL is analogous to the local form of the DDT. For example,in FL e and its axiomatic extensions, it takes the formΓ , ϕ , . . . , ϕ k ⊢ FL e ∅ ⇐⇒ Γ ⊢ FL e ¬ (1 ∧ ϕ ∧ · · · ∧ ϕ k ) n for some n ∈ ω. Similarly, in IK and its axiomatic extensions, it takes the formΓ , ϕ , . . . , ϕ k ⊢ IK ∅ ⇐⇒ Γ ⊢ IK ¬ (cid:3) n ( ϕ ∧ · · · ∧ ϕ k ) for some n ∈ ω. A substructural logic may well enjoy a “better” IL than the one which comesfrom its DDT. For instance, the product fuzzy logic Π only enjoys the local DDTinherited from FL ew , but it enjoys the global IL of classical logic:Γ , ϕ , . . . , ϕ k ⊢ Π ∅ ⇐⇒ Γ ⊢ Π ¬ ( ϕ ∧ · · · ∧ ϕ k ) . A different example is the infinitary Lukasiewicz logic Ł ∞ , which enjoys a com-plicated local DDT (see Example 3.23), but by compactness it inherits the fairlysimple unrestricted local IL of the finitary Lukasiewicz logic Ł (i.e. of FL ew ). Theseare both local principles, but the local IL enjoyed by Ł ∞ is substantially simplerthan the local DDT enjoyed by Ł ∞ . Definition 3.6.
A logic L enjoys the κ -ary parametrized local IL , where κ ≤| Var L | + , if for each ordinal < α < κ there is a family of sets Ψ α ( p, r ) , where p is an α -tuple of variables, such that for each set of formulas Γ and each α -tuple offormulas ϕ : Γ , ϕ ⊢ L Fm L ⇐⇒ Γ ⊢ L I α ( ϕ, π ) for some I( p, r ) ∈ Ψ α and some π. It enjoys the κ -ary local IL if each Ψ α can be chosen of the form Ψ α ( p ) . It enjoysthe κ -ary global IL if each Ψ α can be chosen of the form { I( p ) } . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 17
Fact 3.7.
Each logic with the parametrized local IL has an antitheorem.Proof.
Consider any I ∈ Ψ . Then I( p, p ) ⊢ L I( p, p ), where p is a tuple consistingentirely of copies of the variable p , therefore p, I( p, p ) ⊢ L Fm L. It follows that p, I( p, p ) is an antitheorem. (cid:3) The remarks which follow the definition of DDTs (Definition 3.1) also apply, mutatis mutandis , to ILs. In particular, • by the unary IL we mean the 2-ary IL, • by the finitary IL we mean the ω -ary IL, and • by the unrestricted IL we mean the κ -ary IL for κ = | Var L | + .The condition κ ≤ | Var L | + will again be assumed when talking about κ -ary ILs.If a logic is κ -compact, then we may assume without loss of generality that thefamilies Ψ α consist of sets of cardinality less than κ . Each unary IL induces afinitary IL provided that the logic in question has a conjunction: a set of formulas V ( p, q ) in two variables such that V ( p, q ) ⊢ L p and V ( p, q ) ⊢ L q and p, q ⊢ L V ( p, q ).Moreover, a κ -compact logic with a κ -ary (parametrized) local IL in fact enjoys theunrestricted (parametrized) local IL.If a logic enjoys the IL w.r.t. two families, then the two families are in a suitablesense equivalent (see [27, Remark 3.2]). Namely, we say that two families of sets offormulas Ψ α ( p, q ) and Ψ ′ α ( p, q ) for 0 < α < κ are equivalent if for each I ∈ Ψ α thereare I ′ ∈ Ψ ′ α and π ′ such that I( p, q ) ⊢ L I ′ ( p, π ′ ), and conversely for each I ′ ∈ Ψ ′ α there are I ∈ Ψ α and π such that I ′ ( p, q ) ⊢ L I( p, π ).Observe that if Ψ is a local inconsistency family, i.e. it only consists of sets ofthe form I( p ), and Ψ ′ is equivalent to Ψ, then Ψ ′ can be transformed into a localinconsistency family replacing each I ′ ( p, q ) ∈ Ψ ′ α by I ′ ( p, σ ( π ′ )) where π ′ is thetuple whose existence is asserted by the equivalence between Ψ and Ψ ′ and σ isany substitution such that σ ( p ) = p and σ ( q ) ⊆ p . Moreover, if Ψ is a globalinconsistency family, i.e. it consists of a single set of the form I( p ), then Ψ ′ mayalso be restricted to a single set of the form I( p ). Fact 3.8.
Let L be a logic which enjoys the κ -ary parametrized local IL w.r.t. afamily Ψ . Then L enjoys the κ -ary parametrized local IL w.r.t. a family Ψ ′ if andonly if Ψ and Ψ ′ are equivalent.Proof. Right to left, if Γ ⊢ L I( ϕ, π ) for I ∈ Ψ α , then by structurality I( ϕ, π ) ⊢ L I ′ ( ϕ, σ ( π ′ )) for some I ′ ∈ Ψ ′ α where σ is a substitution such that σ ( p ) = ϕ and σ ( q ) = π . Thus Γ ⊢ L I ′ ( ϕ, σ ( π ′ )) by cut. Conversely, there is a substitution σ ′ such that Γ ⊢ L I ′ ( ϕ, π ′ ) for I ′ ∈ Ψ α implies Γ ⊢ L I( ϕ, σ ′ ( π )) for some I ∈ Ψ α . Theconditions Γ ⊢ L I( ϕ, π ) for some π and Γ ⊢ L I ′ ( ϕ, π ′ ) for some π ′ are thereforeequivalent.Left to right, by the IL I( p, q ) ⊢ L I( p, q ) for I ∈ Ψ α implies p, Ψ( p, q ) ⊢ L ∅ ,hence by the IL there are I ′ ∈ Ψ ′ α and π ′ such that Ψ( p, q ) ⊢ L I ′ ( p, π ′ ). The otherimplication is proved by switching the roles of Ψ and Ψ ′ . (cid:3) Each local DDT induces a corresponding local IL, provided that the logic inquestion has an antitheorem (see [27, Corollary 3.9]). In the parametrized case,this holds if the antitheorem is small enough.
Fact 3.9.
Each logic which enjoys the κ -ary global (local, parametrized local) DDTalso enjoys the κ -ary global (local, parametrized local) IL, provided that it has anantitheorem (in the parametrized local case, of cardinality at most | Var L | ). Proof.
If a logic has an antitheorem, then by structurality for antitheorems it hasan antitheorem of cardinality at most | Var L | in one variable only, say Π( p ) for p ∈ p . Suppose that L enjoys the parametrized local DDT w.r.t. Φ. We define thefamily Ψ α as follows: Ψ α := { I f | f : Π → Φ α } , whereI f ( p, r ) := [ π ∈ Π f ( π )( p, π ( p ) , r π ) , where r obtained by collecting the tuples r π into a single tuple, and r π are tuples ofdistinct variables such that r π and r π ′ are disjoint for distinct π and π ′ , and theydo not share any variables with p . In the parametrized local case, such tuples existby the assumption on the cardinality of Π.Then ϕ, I f ( π )( ϕ, π ( ϕ ) , r π ) ⊢ L π ( ϕ ) for each π ∈ Π by the DDT, therefore ϕ, I f ( ϕ, π ) ⊢ L Π( ϕ ). But by structurality for antitheorems Π( ϕ ) ⊢ L ∅ , hence ϕ, I f ( ϕ, π ) ⊢ L ∅ . Conversely, if Γ , ϕ ⊢ L ∅ , then Γ , ϕ ⊢ L Π( ϕ ) and by the DDTfor each π ∈ Π there are I ∈ Φ and ρ π such that Γ ⊢ L I( ϕ, π ( ϕ ) , ρ π ). ThereforeΓ ⊢ L I f ( ϕ, ρ ), where f : Π → Φ α is the function which assigns the set I to π and ρ is a tuple obtained by collecting all the tuples ρ π into a single tuple. Here we usethe assumption about the disjointness of the tuples r π .If no parameters occur in Φ, then the assumption on the cardinality of Φ α for 0 < α < κ is not needed, and the resulting family Ψ also does not containparameters. Moreover, if Φ α is a singleton, so is Ψ α . (cid:3) The technical assumption about the cardinality of an antitheorem is satisfiedwhenever | Fm L | = | Var L | , which is true for most “everyday” logics. However, theassumption will fail e.g. in a logic which has countable many variables but eachantitheorem contains uncountably many constants.The (surjective) substitution swapping property, which is equivalent to the (para-metrized) local DDT, has a direct analogue for (parametrized) local ILs. Definition 3.10.
A logic L with an antitheorem enjoys κ -ary (surjective) sub-stitution swapping for antitheorems , where κ ≤ | Var L | + , if for each ordinal <α < κ , each (surjective) substitution σ , each α -tuple of formulas ϕ , and each theory T of L σ ( ϕ ) , T ⊢ L ∅ = ⇒ ϕ, σ − [ T ] ⊢ L ∅ . Proposition 3.11.
A logic L enjoys the κ -ary (parametrized) local IL if and onlyif it has an antitheorem and it enjoys κ -ary (surjective) substitution swapping forantitheorems, provided that κ ≤ | Var L | + .Proof. Suppose that L enjoys the parametrized local IL. Then L has an anti-theorem. Now suppose that σ ( ϕ ) , T ⊢ L ∅ for some L-theory T and some surjectivesubstitution σ . By the IL w.r.t. a family Ψ there are I ∈ Ψ and π such that T ⊢ L I( σ ( ϕ ) , π ). Because σ is surjective, there are formulas ρ such that π = σ ( ρ ),therefore T ⊢ L I( σ ( ϕ ) , σ ( ρ )) and T ⊢ L σ [I( ϕ, ρ )]. But then σ − [ T ] ⊢ L I( ϕ, ρ ) andby the IL ϕ, σ − [ T ] ⊢ L ∅ .Conversely, suppose that L has an antitheorem and enjoys surjective substitutionswapping for antitheorems. We define the family Ψ so thatI( p, q ) ∈ Ψ α ⇐⇒ p, I( p, q ) ⊢ L ∅ . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 19
If Γ ⊢ L I( ϕ, π ), then Γ , ϕ ⊢ L ∅ because ϕ, I( ϕ, π ) ⊢ L ∅ by structurality for anti-theorems. Conversely, suppose that Γ , ϕ ⊢ L ∅ and let σ be a surjective substi-tution such that σ ( p ) = ϕ . Such a substitution exists because p is an α -tuplefor some 0 < α < κ ≤ | Var L | + , i.e. α has cardinality at most | Var L | . Thenby surjective substitution swapping for antitheorems σ − [ T ] , p ⊢ L ∅ where T isthe L-theory generated by Γ, hence σ − [ T ] ∈ I( p, q ). Applying σ yields that T = σ [ σ − [ T ]] ∈ I( ϕ, σ ( q )), hence Γ ⊢ L I( ϕ, π ) for π := σ ( q ).Now suppose that L enjoys the local IL and σ ( ϕ ) , T ⊢ L ∅ for some L-theory T .By the IL w.r.t. Ψ there is some I ∈ Ψ such that T ⊢ L I( σ ( ϕ )), i.e. T ⊢ L σ [I( ϕ )].But then σ − [ T ] ⊢ L I( ϕ ), hence ϕ, σ − [ T ] ⊢ L ∅ .Conversely, suppose that L enjoys substitution swapping for antitheorems. Wedefine the family Ψ so thatI( p ) ∈ Ψ α ⇐⇒ p, I( p ) ⊢ L ∅ . As before, Γ ⊢ L I( ϕ ) implies that Γ , ϕ ⊢ L ∅ . Conversely, suppose that Γ , ϕ ⊢ L ∅ and let σ be a substitution such that σ ( p ) = ϕ and σ ( q ) = ϕ ∈ ϕ . Let τ be asubstitution such that τ ( p ) = p and τ ( q ) = p otherwise. Then by substitutionswapping for antitheorems σ − [ T ] , p ⊢ L ∅ where T is the L-theory generated by Γ.By structurality for antitheorems, we may apply τ to obtain τ [ σ − [ T ]] , p ⊢ L ∅ . Thus τ [ σ − [ T ]] ∈ Ψ α ( p ). Applying σ again yields σ [ τ [ σ − [ T ]]] , ϕ ⊢ L ∅ . Since Γ ⊢ L T , itnow suffices to prove that σ [ τ [ σ − [ T ]]] ⊆ T . Consider therefore a formula α ( p, q ) ∈ σ [ τ [ σ − [ T ]]]. Then there is a formula β ( p, q ) such that β ( ϕ, ϕ ) = σ ( β ( p, q )) ∈ T and α ( p, q ) = σ ( τ ( β ( p, q ))) = σ ( β ( p, p )) = β ( ϕ, ϕ ) where ϕ = σ ( p ). Thus α ( p, q ) ∈ T . (cid:3) Dual and classical inconsistency lemmas.
Just like ordinary ILs allowus to reduce questions of inconsistency in a compact logic to questions of validity,dual ILs are syntactic principles which allow for the converse reduction, in a manneranalogous to the following equivalence for classical logic:Γ ⊢ CL ϕ ⇐⇒ Γ , ¬ ϕ ⊢ L ∅ . More generally, the finite-valued Lukasiewicz logics Ł n satisfy the dual ILΓ ⊢ Ł n +1 ϕ ⇐⇒ Γ , ¬ ϕ n ⊢ Ł n +1 ∅ , while in S5 the dual IL takes the formΓ ⊢ S5 ϕ ⇐⇒ Γ , ¬ (cid:3) ϕ ⊢ S5 ∅ , Most of our examples of logics with the IL, however, do not enjoy the dual IL:intuitionistic logic does not, neither do FL e , FL n e , K, or S4.The dual local IL follows a quantifier pattern dual to that of ordinary local ILs:a universal quantifier takes the place of the existential one. This corresponds tothe fact that while in both cases the IL family informally has a disjunctive reading,the IL family occurs to the right of the turnstile, while the dual IL family occursto the left of the turnstile.For example, the infinitary Lukasiewicz logic Ł ∞ enjoys the dual local ILΓ ⊢ Ł ∞ ϕ ⇐⇒ Γ , ¬ ϕ n ⊢ Ł ∞ ∅ for each n ∈ ω. This is an immediate consequence of the fact that for each element a of the standard Lukasiewicz chain on the real interval [0 ,
1] either a = 1 or a n = 0 for some n . Thefinitary Lukasiewicz logic Ł , on the other hand, only enjoys this dual local IL for finite sets of formulas Γ: the above equivalence fails for Ł if we take Γ to be {¬ ϕ → ϕ n | n ∈ ω } . We shall see that this failure of the dual local IL correspondsto the fact that Ł ∞ is semisimple while Ł is not. Definition 3.12.
A logic L enjoys the κ -ary dual parametrized local IL , where κ ≤ | Var L | + , if for each ordinal < α < κ there is a family of sets of formulas Ψ α ( p, q ) , where p is an α -tuple of variables, such that for each set of formulas Γ and each α -tuple of formulas ϕ Γ ⊢ L ϕ ⇐⇒ Γ , I( ϕ, π ) ⊢ L Fm L for each set I ∈ Ψ and each tuple π. It enjoys the κ -ary dual local IL if each Ψ α can be chosen of the form Ψ α ( p ) . Itenjoys the κ -ary dual global IL if each Ψ α can be chosen of the form { I( p ) } . Fact 3.13.
Each logic with the dual parametrized local IL has an antitheorem.Proof.
The dual IL implies that p, I( p, p ) ⊢ L Fm L, where p is a tuple consistingentirely of copies of the variable p . Thus p, I( p, p ) is an antitheorem. (cid:3) For dual (parametrized) local ILs there is no need to take note of the arity ofthe dual IL, unlike in the case of the dual global IL.
Fact 3.14.
Each logic with the unary dual (parametrized) local IL has the unre-stricted dual (parametrized) local IL.Proof.
It suffices to take Ψ α ( p, q ) := S p i ∈ p Ψ α ( p i , q ). (cid:3) A suitable dual local IL allows us to strengthen the κ -compactness of a logic to κ -arity, provided that the cardinal κ is a regular. Fact 3.15.
Let κ be a regular infinite cardinal. If a κ -compact logic L enjoys thedual local IL w.r.t. Ψ such that | Ψ | < κ , then L is a κ -ary logic.Proof. If Γ ⊢ L ϕ , then Γ , I( ϕ ) ⊢ L ∅ for all I ∈ Ψ . For each I ∈ Ψ , then istherefore Γ I ⊆ Γ of cardinality less than κ such that Γ , I( ϕ ) ⊢ L ∅ . It follows fromthe regularity of κ that Γ ′ := S I ∈ Ψ Γ I is a set of cardinality less than κ such thatΓ ′ , I( ϕ ) ⊢ L ∅ . Thus Γ ′ ⊢ L ϕ by the dual local IL. (cid:3) The dual IL becomes an especially powerful principle in conjunction with theordinary IL. Following Raftery [27], we call such a combination a classical IL.
Definition 3.16.
A logic enjoys the κ -ary classical IL w.r.t. a family Ψ if it enjoysboth the ordinary and the dual κ -ary IL w.r.t. Ψ . We now show that if a logic enjoys an ordinary IL with respect to a family Ψas well as some form of the dual IL w.r.t. some family, then it in fact it enjoys aclassical IL w.r.t. the family Ψ. This simple observation will turn out to be crucialwhen applying the framework developed here to specific logics.
Proposition 3.17.
Let L be a logic which enjoys the κ -ary parametrized local ILw.r.t. a family Ψ . If L enjoys the dual parametrized local IL, then it enjoys the κ -ary classical parametrized local IL w.r.t. Ψ .Proof. The left-to-right direction of the dual IL follows immediately from the ILapplied to I( ϕ, π ) ⊢ L I( ϕ, π ). Conversely, suppose that Γ L ψ for some ψ ∈ ϕ .Since L enjoys the unary dual IL w.r.t. some family Ψ ′ , then there are I ′ ∈ Ψ ′ and π such that Γ , I ′ ( ψ, π ) L ∅ . But the dual IL applied to ψ ⊢ L ψ yields that EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 21 I ′ ( ψ, π ) , ψ ⊢ L ∅ . By monotonicity I ′ ( ψ, π ) , ϕ ⊢ L ∅ , hence by the IL there areI ∈ Ψ α and ρ such that I ′ ( ψ, π ) ⊢ L I( ϕ, ρ ). Thus Γ , I ′ ( ψ, π ) L ∅ implies thatΓ , I( ϕ, ρ ) L ∅ . (cid:3) The mere fact that a logic enjoys of a global IL therefore immediately upgradesa dual parametrized local IL to a dual global IL. Similarly, a finitary parametrizedlocal IL immediately upgrades a unary dual IL to a finitary one.
Corollary 3.18.
If a logic enjoys the κ -ary parametrized local IL and the unarydual local (global) IL, then it enjoys the κ -ary classical local (global) IL. Recalling the notion of equivalence of IL families (Fact 3.8), it follows that theclassical IL is preserved under equivalence of IL families (see [27, p. 400]).
Fact 3.19. If L enjoys a classical IL w.r.t. Ψ , then it enjoys a classical IL w.r.t. Ψ ′ if and only if Ψ and Ψ ′ are equivalent. The power of a dual parametrized local IL, a seemingly weak principle on itsown, therefore lies in upgrading ordinary ILs to classical ILs, which then implyfull-blooded DDTs (see [27, Corollary 4.4]).
Proposition 3.20.
Let L be a logic which enjoys the classical ( κ + λ ) -ary local(global) IL w.r.t. Ψ such that each set in Ψ has cardinality less than λ + 1 . Then L enjoys the κ -ary local (global) DDT.Proof. Fix some α such that 0 < α < κ . Let us define the family of functions F := { f : Ψ → [ n ∈ ω Ψ n | f (J) ∈ Ψ | J | + | α | for each J ∈ Ψ } . For each f ∈ F we define the set of formulas I f ( p, q ) as follows:I f ( p, q ) := [ J ∈ Ψ f (J)( p ∪ J( q )) . We claim that L enjoys the local DDT with respect to Φ := { I f | f ∈ F} . The claimfor the classical global IL then follows immediately from the fact that F and there-fore Φ is a singleton if each Ψ n is a singleton.If Γ , ϕ ⊢ L ψ , then Γ , ϕ, J( ψ ) ⊢ L ∅ for each J ∈ Ψ by the dual IL. Considersome J ∈ Ψ . By assumption | J | < κ . Then by the IL there is K ∈ Ψ | J | + | α | suchthat Γ ⊢ L K( ϕ ∪ J( ψ )). Let f ∈ F be the function which to each J ∈ Ψ assignsthis K ∈ Ψ | J | + | α | . Then Γ ⊢ L I f ( ϕ, ψ ) for some f ∈ F . Conversely, suppose thatΓ ⊢ L I f ( ϕ, ψ ). Then Γ , ϕ, J( ψ ) ⊢ L ∅ for each J ∈ Ψ by the IL, therefore Γ , ϕ ⊢ L ψ by the dual IL. (cid:3) Proposition 3.21.
Let L be a logic which enjoys the classical ( κ + 1) -ary global(local, parametrized local) IL w.r.t. Ψ such that each set in Ψ has cardinality lessthan κ . Then L has a protoimplication set.Proof. (Recall the definition of a protoimplication set from Fact 3.2.) By the dualIL p ⊢ L p implies p, I( p, π ) ⊢ L ∅ for each I ∈ Ψ and each tuple π . Thus by theassumption concerning the cardinality of sets in Ψ , for each I ∈ Ψ and each tuple π there are J ∈ Ψ | I | +1 and ρ such that ∅ ⊢ L J( { p } ∪ I( p, π ) , ρ ) . Let Γ be the union of the sets of formulas J( { p } ∪ I( q, π ) , ρ ) where q is distinctfrom p and J( { p } ∪ I( p, π ) , ρ ) ranges over all the sets obtained above. Then bystructurality ∅ ⊢ L σ [Γ] where σ is the substitution such that σ ( p ) = p and σ ( r ) = q for every variable r other than p . On the other hand, p, I( q, π ) , Γ ⊢ L ∅ for eachI ∈ Ψ and each π by the IL, therefore p, Γ ⊢ L q by the dual IL. Structurality nowimplies that p, σ [Γ] ⊢ L q , thus σ [Γ] is the required protoimplication set. (cid:3) Corollary 3.22.
Each κ -compact logic with the κ -ary classical global (local, para-metrized local) IL enjoys the κ -ary global (local, parametrized local) DDT, where κ ≤ | Var L | + is an infinite cardinal. In the local and global cases, our proof that a classical IL yields a DDT wasconstructive. This recipe for constructing DDTs from classical ILs is in fact emi-nently practical. To illustrate this, let us apply it to obtain a local DDT for theinfinitary Lukasiewicz logic Ł ∞ , which to the best of our knowledge was previouslyunknown. This also yields the only example known to us of a local DDT where theDDT family necessarily consists of infinite sets. Example 3.23. Ł ∞ enjoys the unary local DDT: Γ , ϕ ⊢ Ł ∞ ψ ⇐⇒ Γ ⊢ Ł ∞ { f ( n )( ϕ → ψ n ) | n ∈ ω } for some f : ω → ω, or equivalently: Γ , ϕ ⊢ Ł ∞ ψ ⇐⇒ for each n ∈ ω there is k ∈ ω such that Γ ⊢ Ł ∞ k ( ϕ → ψ n ) . However, it does not enjoy the unary LDDT w.r.t. any family of finite sets.Proof.
By compactness, Ł ∞ enjoys the same local IL as its finitary counterpart Ł .The above recipe for constructing the local DDT now yields the DDT family con-sisting of the sets {¬ ( p · ¬ q n ) f ( n ) | n ∈ ω } for f : ω → ω, and each such set is equivalent, formula by formula, to { f ( n )( p → q n ) | n ∈ ω } .If Ł ∞ satisfied the local DDT w.r.t. a family Φ of finite sets, then we claim that itsatisfies the local DDT w.r.t. the family { p n → q | n > } . Recall that this is a DDTfamily for Ł , and Ł is precisely the finitary companion of Ł ∞ . If Γ ⊢ Ł ∞ ϕ n → ψ ,then Γ , ϕ ⊢ Ł ∞ ψ because ϕ, ϕ n → ψ ⊢ Ł ∞ ψ . Conversely, ϕ, I( ϕ, ψ ) ⊢ Ł ∞ ψ for eachI ∈ Φ, therefore ϕ, I( ϕ, ψ ) ⊢ Ł ψ by the finiteness assumption and I( ϕ, ψ ) ⊢ Ł ϕ n → ψ for some n > Ł . Thus I( ϕ, ψ ) ⊢ Ł ∞ ϕ n → ψ for some n > ϕ n → ψ ⊢ Ł ∞ I( ϕ, ψ ) for some I ∈ Φ by the local DDT w.r.t. Φapplied to ϕ, ϕ n → ψ ⊢ Ł ∞ ψ . Therefore Φ and { p n → q | n > } are equivalentfamilies and Ł ∞ enjoys the local DDT w.r.t. the latter.We now show that Ł ∞ does not satisfy the local DDT w.r.t. this family. LetΓ := { p i +1 → ( q i +1 → q i ) , ¬ q → q ii | i ∈ ω } . Then Γ , p ⊢ Ł ∞ q . This is because Γ , p ⊢ Ł ∞ q i +1 → q i for each i ∈ ω , henceΓ , p ⊢ Ł ∞ ¬ q → q i for each i ∈ ω , but {¬ q → q i | i ∈ ω } ⊢ Ł ∞ q .We prove that Γ Ł ∞ p n → q for each n >
0. Given an ε ∈ (0 , v such that v ( p ) := 1 − εn + 1 ,v ( q i ) := min (cid:18) , − ε + 1 + · · · + in + 1 ε (cid:19) . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 23
Then v ( p n ) = 1 − nεn +1 (cid:2) − ε = v ( q ), hence v ( p n − q ) <
1. Moreover, v ( q i +1 − q i ) ≤ i +1 n +1 ε = 1 − v ( p i +1 ), thus v ( p i +1 ) ≤ v ( q i +1 → q i ). It follows that v ( p i +1 → ( q i +1 → q i )) = 1. Finally, observe that v ( q i ) < · · · + i = i ( i +1)2 ≤ n , i.e. onlyif i ≤ √ n . But then v ( q ii ) ≥ − iε ≥ − √ nε . Taking any ε ≤ − √ nε ,e.g. ε := √ n , now yields v ( ¬ q ) = ε ≤ − √ nε ≤ v ( q ii ) for each i ∈ ω . Thus v ( ¬ q → q ii ) = 1 for each i ∈ ω . (cid:3) The logic Ł ∞ in fact enjoys an unrestricted local DDT. Example 3.24.
The infinitary Lukasiewicz logic Ł ∞ enjoys the unrestricted localDDT: Γ , ϕ ⊢ Ł ∞ ψ ⇐⇒ for each n there is k and a finite subtuple ϕ ′ of ϕ such that Γ ⊢ Ł ∞ k (cid:16)^ ϕ ′ → ψ n (cid:17) . Proof.
By compactness, Ł ∞ has the same finitary local IL as Ł , which again bycompactness extends to an unrestricted local IL as follows:Γ , ϕ ⊢ Ł ∞ ∅ ⇐⇒ Γ ⊢ Ł ∞ ¬ (cid:16)^ ϕ ′ (cid:17) n for some finite subtuple ϕ ′ of ϕ. (The conjunction V ϕ ′ may equivalently be replaced by a product.) The recipe forconstructing DDTs from classical ILs now yields the stated DDT. (cid:3) Simple inconsistency lemmas.
The ordinary IL and the dual IL in factcoincide when restricted to simple theories T , thanks to the equivalences T ⊢ L ϕ ⇐⇒ T, ϕ L ∅ ,T ⊢ L I( ϕ ) ⇐⇒ T, I( ϕ ) L ∅ . This common restriction of the IL and dual IL to simple theories will be called thesimple IL, although it might equally well be called the simple dual IL.
Definition 3.25.
A logic L enjoys the κ -ary simple parametrized local IL , where κ ≤ | Var L | + , if for each ordinal < α < κ there is a family of sets of formulas Ψ α ( p, q ) , where p is an α -tuple of variables, such that for each simple theory T of L and each α -tuple ϕ we have ϕ, I( ϕ, π ) ⊢ L ∅ for each π and T, ϕ ⊢ L ∅ = ⇒ T ⊢ L I( ϕ, π ) ⊢ L ∅ for some set I ∈ Ψ and some tuple π, or equivalently T, I( ϕ, π ) ⊢ L ∅ for each set I ∈ Ψ and each tuple π = ⇒ T ⊢ L ϕ. It enjoys the κ -ary simple local IL if each Ψ α can be chosen of the form Ψ α ( p ) . Itenjoys the κ -ary simple global IL if each Ψ α can be chosen of the form { I( p ) } . Just like with dual (parametrized) local ILs, there is again no need to take note ofthe arity of a simple (parametrized) local IL, since each unary simple (parametrized)local IL extends to an unrestricted one.While the ordinary (parametrized) local IL is equivalent to the (surjective) sub-stitution swapping property for antitheorems, the simple (parametrized) local IL isequivalent to a weaker condition, namely the closure of the family of simple theoriesunder preimages with respect to (surjective) substitutions. For many of our proofsthis weaker condition will be sufficient.
Proposition 3.26.
The following are equivalent for each coatomic logic L with anantitheorem: (1) L enjoys the simple parametrized local IL. (2) If σ is surjective and T is a simple theory of L , then so is σ − [ T ] .Proof. (1) ⇒ (2): if σ − [ T ] L ϕ for some simple theory T , then T L σ ( ϕ ), hence T, σ ( ϕ ) ⊢ L ∅ by the simplicity of T and T ⊢ L I( σ ( ϕ ) , π ) for some I ∈ Ψ and π by the simple local IL. By surjectivity π = σ ( ρ ) for some ρ , so T ⊢ L σ [I( ϕ, ρ )]. Itfollows that σ − [ T ] ⊢ L I( ϕ, ρ ) and σ − [ T ] , ϕ ⊢ L ∅ . Thus σ − [ T ] is simple.(2) ⇒ (1): take Ψ := { I( p, q ) | p, I( p, q ) ⊢ L ∅} and asume that T, ϕ ⊢ L ∅ forsome simple theory T . Consider a surjective substitution such that σ ( p ) = ϕ . Then T L ϕ = σ ( p ), hence σ − [ T ] L p . But the theory σ − [ T ] is simple by assumption,hence σ − [ T ] , p ⊢ L ∅ and σ − [ T ] ∈ Ψ . That is, σ − [ T ] = I( p, q ) for some I ∈ Ψ .Then T ⊢ L σ [ σ − [ T ]] = I( ϕ, σ ( q )). (cid:3) Proposition 3.27.
The following are equivalent for each coatomic logic L with anantitheorem: (1) L enjoys the simple local IL. (2) If σ is a substitution and T a simple theory of L , then so is σ − [ T ] .Proof. (1) ⇒ (2): if σ − [ T ] L ϕ for some simple theory T , then T L σ ( ϕ ), hence T, σ ( ϕ ) ⊢ L ∅ by the simplicity of T and T ⊢ L I( σ ( ϕ )) = σ [I( ϕ )] for some I ∈ Ψ by the simple local IL. It follows that σ − [ T ] ⊢ L I( ϕ ) and σ − [ T ] , ϕ ⊢ L ∅ . Thus σ − [ T ] is simple.(2) ⇒ (1): take Ψ := { I( p ) | p, I( p ) ⊢ L ∅} and asume that T, ϕ ⊢ L ∅ forsome simple theory T . Consider the substitutions σ ( q ) = ϕ and τ ( q ) = p for eachvariable q . Then T L ϕ = σ ( p ), so σ − [ T ] L p , but the theory σ − [ T ] is simple byassumption, hence σ − [ T ] , p ⊢ L ∅ and τ [ σ − [ T ]] ∈ Ψ . Moreover, T ⊢ L σ [ τ [ σ − [ T ]]]by the same reasoning as in the proof of Proposition 3.11. (cid:3) The strong three-valued Kleene logic K is an example of a logic with the simpleglobal IL but not even the parametrized local IL. We provide a sketch of the prooffor readers familiar with K (see e.g. [25, Ch. 7]). Fact 3.28.
The strong three-valued Kleene logic K enjoys the simple global ILw.r.t. ¬ p but not the parametrized local IL.Proof. Recall that a literal is a formula of the form p or ¬ p where p is a variable.If Γ is a set of literals and ϕ is a literal, then Γ , ϕ ⊢ K ∅ if and only if Γ ⊢ K ¬ ϕ .Using the disjunctive normal form and the proof by cases propertyΓ , ϕ ⊢ K χ and Γ , ψ ⊢ K χ ⇐⇒ Γ , ϕ ∨ ψ ⊢ K χ, this equivalence extends to arbitrary sets of formulas Γ. This now implies that eachsimple theory T contains either p or ¬ p for each variable p . By induction over thecomplexity of ϕ , it follows that T contains either ϕ or ¬ ϕ for each ϕ . The simpleglobal IL follows: T, ϕ ⊢ K ∅ = ⇒ T K ϕ = ⇒ T ⊢ K ¬ ϕ .On the other hand, if K enjoys the parametrized local IL w.r.t. a family Ψ, then p, I( p, q ) ⊢ K ∅ for each I ∈ Ψ and each q . It follows that I( p, q ) ⊢ K ¬ p , e.g. byusing the disjunctive normal form. But p ∧ ¬ p ⊢ K ∅ and ∅ K ¬ ( p ∧ ¬ p ). (cid:3) EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 25
The law of the excluded middle.
The dual IL can also be restated inthe form of a law of the excluded middle (LEM). Here we take the LEM to be aprinciple which states that if in some context Γ we can deduce ψ from both ϕ andthe negation of ϕ , then we may deduce ψ from Γ itself. Although we take the LEMto be a meta-rule (an implication between valid rules) rather than an axiom or arule, we shall see later that the LEM can often be reduced to the validity of someset of axioms or rules.The canonical example of the LEM is of course the one for classical logic:Γ , ϕ ⊢ CL ψ Γ , ¬ ϕ ⊢ CL ψ Γ ⊢ CL ψ (The horizontal line above represents an implication from the premises to the con-clusion.) More generally, the finite-valued Lukasiewicz logics Ł n +1 enjoy the LEMin the form: Γ , ϕ ⊢ Ł n +1 ψ Γ , ¬ ϕ n ⊢ Ł n +1 ψ Γ ⊢ Ł n +1 ψ The global modal logic S5 enjoys the LEM in the form:Γ , ϕ ⊢ S5 ψ Γ , ¬ (cid:3) ϕ ⊢ S5 ψ Γ ⊢ S5 ψ The above implications are all examples of global LEMs. The local form ofthe LEM follows the same quantifier pattern as the local form of the dual IL. Forexample, the infinitary Lukasiewicz logic Ł ∞ enjoys the local LEMΓ , ϕ ⊢ Ł ∞ ψ Γ , ¬ ϕ n ⊢ Ł ∞ ψ for each n ∈ ω Γ ⊢ Ł ∞ ψ Let us now provide a precise definition of the LEM in our sense of the term.
Definition 3.29.
A logic L enjoys the κ -ary parametrized local LEM , where κ ≤| Var L | + , if for each ordinal < α < κ there is a family of sets of formulas Ψ α ( p, q ) ,where p is an α -tuple of variables, such that ϕ, I( ϕ, π ) ⊢ L Fm L for each I ∈ Ψ α and each ϕ and π, and moreover Γ ⊢ L ψ whenever Γ , ϕ ⊢ L ψ and Γ , I( ϕ, π ) ⊢ L ψ for each set I ∈ Ψ and each tuple π. It enjoys the local LEM if each Ψ α can be chosen of the form Ψ α ( p ) . It enjoys the global LEM if each Ψ α can be chosen of the form { I( p ) } . It is no accident that the LEMs above correspond precisely to the dual ILsenjoyed by these logics.
Proposition 3.30.
A logic enjoys the κ -ary dual parametrized local IL w.r.t. thefamily Ψ if and only if it enjoys the κ -ary parametrized local LEM w.r.t. Ψ .Proof. If L has the LEM w.r.t. Ψ and Γ , I( ϕ, π ) ⊢ L Fm L for each I ∈ Ψ and each π , then for each ϕ i ∈ ϕ the LEM for ψ := ϕ i yields Γ ⊢ L ϕ i . The other implicationof the dual IL follows from ϕ, I( ϕ, π ) ⊢ L Fm L for each I ∈ Ψ.Conversely, suppose L has the dual IL w.r.t. Ψ and Γ , ϕ ⊢ L ψ and Γ , I( ϕ, π ) ⊢ L ψ for each I ∈ Ψ and each π . We need to show that Γ ⊢ L ψ .By the dual IL Γ , I( ϕ, π ) ⊢ L ψ yields that Γ , I( ϕ, π ) , J( ψ, ρ ) ⊢ L ∅ for each J ∈ Ψ and each ρ . By the dual IL this implies Γ , J( ψ, ρ ) ⊢ L ϕ . Applying the assumption that Γ , ϕ ⊢ L ψ yields Γ , J( ψ, ρ ) ⊢ L ψ , but ψ, J( ψ, ρ ) ⊢ L ∅ , therefore Γ , J( ψ, ρ ) ⊢ L ∅ .But by the dual IL this implies that Γ ⊢ L ψ .The equivalences for dual local and global ILs and the corresponding formsof the LEM are an immediate consequence of the equivalence between the dualparametrized local IL and the parametrized local LEM. (cid:3) Corollary 3.31.
A logic enjoys the κ -ary dual global (local, parametrized local) ILif and only if it enjoys the κ -ary global (local, parametrized local) LEM. In particular, for the (parametrized) local form of the LEM there is no need totake note of the arity, since the unary (parametrized) local LEM extends to anunrestricted one by Fact 3.14.3.6.
Syntactic characterization of semisimplicity.
We now show that, assum-ing the simple IL, semisimplicity is in fact equivalent to the law of the excludedmiddle (LEM), or equivalently the dual IL. Recall that the simple IL follows bothfrom the ordinary IL and from the dual IL. We first show that the dual IL thenextends from simple to semisimple theories.
Proposition 3.32. If L enjoys the κ -ary simple parametrized local IL w.r.t. afamily Ψ , then for each semisimple theory T of L T, I( ϕ, π ) ⊢ L ∅ for each set I ∈ Ψ and each tuple π = ⇒ T ⊢ L ϕ. Proof. If T L ϕ and T is semisimple, then there is some semisimple theory T ′ ⊇ T such that T ′ L ϕ . By the simple IL there are I ∈ Ψ α and π such that T ′ , I( ϕ, π ) L ∅ , hence also T, I( ϕ, π ) L ∅ . (cid:3) Proposition 3.33.
Each semisimple logic with the κ -ary parametrized local ILw.r.t. Ψ enjoys the κ -ary classical parametrized local IL w.r.t. Ψ .Proof. The IL implies the simple IL, which implies the dual IL for semisimpletheories. But by assumption each theory of the logic is semisimple. (cid:3)
Conversely, the dual parametrized local IL implies semisimplicity.
Proposition 3.34.
Each coatomic logic with the dual parametrized local IL is semi-simple.Proof.
Suppose that Γ L ϕ . Then by the dual IL there are I ∈ Ψ and π suchthat Γ , I( ϕ, π ) L ∅ . By coatomicity Γ , I( ϕ, π ) extends to a simple L-theory ∆. But ϕ, I( ϕ, π ) ⊢ L ∅ by the dual IL applied to ϕ ⊢ L ϕ , therefore ∆ L ϕ . It follows thateach L-theory Γ is semisimple. (cid:3) Combining these simple observations with the equivalence between the LEM, thedual IL, and the classical IL now yields the following theorem, which is the mainsyntactic result of the first half of this paper.
Theorem 3.35.
The following are equivalent for each coatomic logic L which enjoysthe simple (parametrized) local IL w.r.t. Ψ : (1) L is semisimple. (2) L enjoys the (parametrized) local LEM. (3) L enjoys the dual (parametrized) local IL. (4) L enjoys the (parametrized) local LEM w.r.t. Ψ . (5) L enjoys the dual (parametrized) local IL w.r.t. Ψ . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 27
An analogous equivalence for the global LEM and the dual global IL immediatelyfollows.
Theorem 3.36.
The following are equivalent for each coatomic logic L which enjoysthe κ -ary simple global local IL w.r.t. Ψ : (1) L is semisimple. (2) L enjoys the unary global LEM. (3) L enjoys the unary dual global IL. (4) L enjoys the κ -ary global LEM w.r.t. Ψ . (5) L enjoys the κ -ary dual global IL w.r.t. Ψ . Let us remark here that, while the equivalence between the originally algebraicnotion of semisimplicity and the logical notion of the LEM is certainly of interestconceptually, in practical applications it is crucial that we know the specific formof the LEM. It would be quite difficult to apply this theorem to describe, say, thesemisimple extensions of a given logic L, as these could enjoy the LEM with respectto widely varying families Ψ. However, as we shall see, the theorem can be veryhelpful when describing the semisimple axiomatic extensions of a logic L whichenjoys the IL w.r.t. Ψ, since each of them inherits this specific IL.It will not have escaped the reader looking to apply the previous theorem toestablish the semisimplicity of some class of algebras that there is a catch to it: wehave only defined a syntactic notion of semisimplicity (each theory is an intersec-tion of simple theories), whereas the algebraist’s notion of semisimplicity is semanticand relates to algebras. Let us therefore show how to get from the syntactic semi-simplicity of an (at least weakly) algebraizable logic to the semisimplicity of thecorresponding class of algebras. This will require somewhat stronger assumptions.We call a logic L semantically semisimple if each L-filter on each algebra A is an intersection of simple (i.e. maximal non-trivial) L-filters on A . We call it semantically coatomic if the lattice of L-filters on each algebra is coatomic, i.e. ifeach L-filter on each algebra A is included in some simple L-filter on A . Clearlyeach semantically semisimple or coatomic logic is also syntactically so. Fact 3.37.
The following implications hold: (1)
Each finitary logic with an antitheorem is semantically compact. (2)
Each semantically compact logic is semantically coatomic.Proof.
Item (1) holds because L-filter generation yields an algebraic closure opera-tor on each algebra if L is finitary, and the existence of a finite antitheorem ensuresthat the trivial filter on each algebra is finitely generated. Item (2) is a generallattice theoretic fact. (cid:3)
The semantic counterparts of the syntactic principles defined in the previoussubsections are obtained by replacing the quantification over Γ, ϕ , π and ψ withquantification over all algebras A (instead of the algebra Fm L) and all subsets X ⊆ A , all tuples a, b ∈ A , and all elements c ∈ A . The semantic κ -ary dualparametrized local IL , for example, states that for each algebra A , each X ⊆ A ,and each α -tuple a ∈ A for 0 < α < κX ⊢ A L a ⇐⇒ X, I( a, b ) ⊢ A L A for each set I ∈ Ψ α and each tuple b. Recall that X ⊢ A L a is our notation for the claim that each element of a belongs tothe L-filter of A generated by X . Proposition 3.38.
Each semantically coatomic logic which enjoys the semanticsimple dual parametrized local IL is semantically semisimple.Proof.
This is a straightforward semantic analogue of Proposition 3.34. (cid:3)
Given the above implications, in order to prove that a finitary semisimple logicwith an antitheorem is semantically semisimple, it suffices to show that the syntacticdual parametrized local IL extends to a semantic one. If the logic in question is notfinitary but it is at least compact, then one moreover has to show that syntacticcompactness extends to semantic compactness.Such theorems, which extend a syntactic property to a corresponding semanticone, are called transfer theorems in abstract algebraic logic. The proof of thetransfer theorem for the classical local IL will involve several technical lemmas,which are in fact nothing but straightforward modifications of existing proofs founde.g. in [14]. The reader unfamiliar with the techniques involved in such proofs isadvised to skip the details and jump to Theorem 3.43 and Fact 3.44.In the following pair of lemmas, recall that the existence of a protoimplicationset is equivalent to the parametrized local DDT (Fact 3.2). Moreover, each logic Lsuch that | Fm L | = | Var L | is λ -ary for λ = | Var L | + . Lemma 3.39.
Let L be λ -ary logic for λ = | Var L | + with a protoimplication set.Then X ⊢ A L a if and only if there are Γ , ϕ , and a homomorphism h : Fm L → A such that h [Γ] ⊆ X ∪ Fg A L ∅ , h ( ϕ ) = a , and Γ ⊢ L ϕ .Proof. This is a standard lemma about filter generation for finitary logics (see [14,Proposition 6.12]). The only subtlety involved in generalizing the proof to λ -arylogics is that given a set of rules { Γ i ⊢ ϕ i | i ∈ I } such that | I | < λ for each i ∈ I ,we need to be able to rename the variables in these rules so that distinct rulesuse disjoint sets of variables. The assumption that λ = | Var L | + takes care of thisproblem. (cid:3) Lemma 3.40.
Let L be λ -ary logic for λ = | Var L | + which has an antitheorem anda protoimplication set. Then X ⊢ A L A if and only if there is Γ and a homomorphism h : Fm L → A such that h [Γ] ⊆ X ∪ Fg A L ∅ and Γ ⊢ L ∅ .Proof. The right-to-left implication is straightforward. Conversely, because L is λ -ary and has an antitheorem, it has an antitheorem Π of cardinality at most | Var L | ,w.l.o.g. in a single variable p . Then any homomorphism g : Fm L → A yields aset P := g [Π] such that P ⊢ A L A and | P | ≤ | Var L | . By Lemma 3.39, for each a ∈ P there are formulas Γ a , ϕ a and a homomorphism h a : Fm L → A such that h a [Γ a ] ⊆ X ∪ Fg A L ∅ , h a ( ϕ a ) = a , and Γ a ⊢ L ϕ a . Because | P | ≤ | Var L | , we mayassume that the formulas Γ a , ϕ a and the formulas Γ a ′ , ϕ a ′ do not share any variablesif a and a ′ are distinct. We thus obtain a set Γ := S a ∈ P Γ a , a tuple of formulas ϕ which consists of the formulas ϕ a , and a homomorphism h : Fm L → A such that h [Γ] ⊆ X ∪ Fg A L ∅ , h ( ϕ ) = a , and Γ ⊢ L ϕ . We may assume w.l.o.g. that Γ does notcontain the variable p .If ∆ is a protoimplication set for L, let Γ ′ := Γ ∪ S π ∈ P ∆( ϕ g ( π ) , π ) and take h ′ : Fm L → A to be the homomorphism which agrees with h on every variableexcept h ′ ( p ) = g ( p ), then h ′ [Γ ′ ] ⊆ X ∪ Fg A L ∅ , since h ′ ( ϕ g ( π ) ) = g ( π ) = h ′ ( π ).Moreover, now Γ ′ ⊢ L ∅ because Γ ⊢ L ϕ and ϕ, S π ∈ Π ∆( ϕ g ( π ) , π ) ⊢ L Π. (cid:3) EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 29
The equivalence of syntactic and semantic compactness for logics which satisfythe assumptions of the previous lemma now follows immediately.
Proposition 3.41.
Let L be a λ -ary logic for λ = | Var L | + which has an anti-theorem and a protoimplication set. If L is (syntactically) κ -compact, then it issemantically κ -compact. We are now in a position to extend the syntactic classical IL to a semantic one.
Proposition 3.42.
Let L be a λ -ary logic for λ = | Var L | + . If L enjoys thesyntactic κ -ary classical local IL w.r.t. a family Ψ such that | Ψ α | ≤ | Var L | for each Ψ α and each set in Ψ has cardinality less than κ , then it enjoys the semantic κ -aryclassical local IL w.r.t. Ψ .Proof. Recall that the classical IL implies the DDT (Proposition 3.20), thereforethe previous two lemmas apply. We first show the transfer of the IL. One direction isclear: X ⊢ A L I( a, b ) for I ∈ Ψ α and b ⊆ A implies X, a ⊢ A L A because a, I( a, b ) ⊢ A L A by virtue of p, I( p, q ) being an antitheorem. Conversely, suppose that X, a ⊢ A L A where a ∈ A has length 0 < α < κ . Then by the previous lemma there are Γ, ϕ , and a homomorphism h : Fm L → A such that h [Γ] ⊆ Fg A L ∅ , h ( ϕ ) = a , andΓ , ϕ ⊢ L ∅ . By the IL there are I ∈ Ψ α and π such that Γ ⊢ L I( ϕ, π ), thus for b := h ( π ) we have I( a, b ) = h [I( ϕ, π )] ⊆ Fg A L h [Γ] ⊆ Fg A L X .For the dual IL, one direction is again clear: X ⊢ A L a implies X, I( a ) ⊢ A L A foreach I ∈ Ψ α because a, I( a ) ⊢ A L A . Conversely, suppose that X, I( a ) ⊢ A L A for eachI ∈ Ψ α . Then by the previous fact for each I ∈ Ψ α there are Γ I , Λ I ⊆ Fm L, and h I : Fm L → A such that h I [Γ I ] ⊆ X ∪ Fg A L ∅ and h I [Λ I ] ⊆ I( a ) and Γ I , Λ I ⊢ L ∅ .More precisely, for each λ I ∈ Λ I there is ι λ I ∈ I such that h ( λ I ) = ι λ I ( a ). We mayassume w.l.o.g. that there is an α -tuple of variables p which do not occur in anyΓ I , Λ I . Moreover, since | Ψ α | ≤ | Var L | we may assume that the sets Γ I , Λ I andΓ I ′ , Λ I ′ do not share any variables for distinct I and I ′ . This allows us to definea homomorphism h : Fm L → A such that h [Γ I ] = h I [Γ I ], h [Λ I ] = h I [Λ I ], and h ( a ) = a .Now consider Γ := S I ∈ Ψ α Γ I and Γ ′ := Γ ∪ S I ∈ Ψ α S λ I ∈ Λ I ∆( ι λ I ( p ) , λ I ). Then h [Γ ′ ] ⊆ X ∪ Fg A L ∅ because h ( λ I ) = ( ι λ I ( a )) = h ( ι λ I ( p )). Moreover, Γ ′ , I( p ) ⊢ L Λ I for each I ∈ Ψ α , hence Γ ′ , I( p ) ⊢ L ∅ for each p . By the dual local IL it follows thatΓ ′ ⊢ L p . Finally, h ( p ) = a implies that a ∈ Fg A L X . (cid:3) Combining the semantic results of this subsection with the characterization ofsyntactic semisimplicity (Theorem 3.35) now yields the following theorem, whichis the main semantic result of the first half of the paper. It upgrades our pre-vious characterization of syntactic semisimplicity to a characterization of seman-tic semisimplicity, provided that the logic enjoys a local IL rather than merely aparametrized local one and certain very mild technical conditions are met.
Theorem 3.43.
The following are equivalent for a semantically coatomic logic L which enjoys the κ -ary local IL w.r.t. Ψ , provided that | Ψ α | ≤ | Var L | for all Ψ α ,each set in Ψ has cardinality less than κ , and L is λ -ary for λ = | Var L | + : (1) L is semantically semisimple. (2) L is syntactically semisimple. (3) L enjoys the unary local LEM. (4) L enjoys the unary dual local IL. (5) L enjoys the unary classical local IL. (6) L enjoys the semantic κ -ary local LEM w.r.t. Ψ . (7) L enjoys the semantic κ -ary dual local IL w.r.t. Ψ . (8) L enjoys the semantic κ -ary classical local IL w.r.t. Ψ .In particular, the above equivalence holds for each compact L with the κ -ary local ILw.r.t. Ψ , provided that | Ψ α | ≤ | Var L | for each Ψ α and each set in Ψ has cardinalityless than κ .Proof. The implication from semantic semisimplicity to the syntactic semisimplicityis trivial, and the equivalence between the syntactic conditions has already beenestablished in Theorem 3.35. The proof of the equivalence between the syntacticLEM and the syntactic dual IL (Proposition 3.30) immediately extends to theirsemantic counterparts, as does the proof that the syntactic κ -ary IL and unarydual IL imply the syntactic κ -ary classical IL (Proposition 3.17). The implica-tion from the semantic classical IL to semantic semisimplicity is Proposition 3.38.Finally, the implication from the syntactic unary classical IL (which implies thesyntactic κ -ary classical IL by Proposition 3.17) to the semantic κ -ary classical ILis Proposition 3.42.Each of the conditions implies the unary classical local IL without the use of se-mantic coatomicity and λ -arity. But compactness together with the unary classicallocal IL imply that L is λ -ary for λ = | Var L | + by Fact 3.15, using the assump-tion that | Ψ | < λ . Moreover, compactness together with λ -arity and the unaryparametrized local DDT (which follows from the unary classical local IL by Proposi-tion 3.21) imply the semantic compactness of L by Proposition 3.41. The hypothesesabout λ -arity and semantic coatomicity may thus be omitted if L is compact. (cid:3) Recall again that each logic with | Fm L | = | Var L | , in particular each logic withat most countably many connectives and constants, is λ -ary for λ = | Var L | + .In order to use the above theorem to prove the semisimplicity of a class of alge-bras, it remains to connect the semantic semisimplicity of a (weakly) algebraizablelogic with the semisimplicity of its algebraic counterpart. Fact 3.44.
Let L be a (weakly) algebraizable logic and let K be its algebraic coun-terpart. Then K is semisimple if and only if L is semantically semisimple.Proof. By [14, Proposition 6.117] the lattice of K -congruences and the lattice ofL-filters on each algebra are isomorphic. (cid:3) Applications.
We now show how the equivalence between semisimplicity andthe LEM (Theorem 3.35) can be used to describe the semisimple axiomatic exten-sions of a given logic. We consider two cases: the semisimple axiomatic extensionsof the logics FL e and IK. By Theorem 3.43 and Fact 3.44, this problem is equivalentto the problem of describing the semisimple varieties of FL e -algebras and modalHeyting algebras.The semisimple varieties of FL ew -algebras and modal algebras were already de-scribed by Kowalski [18] and by Kowalski & Kracht [19] respectively. The proofs ofKowalski and Kracht are algebraic in nature and, although their overall structure issimilar, they involve fairly complicated computations specific to the two varieties.In contrast, having isolated the common core of the two proofs into Theorem 3.35,our proofs are relatively brief and purely syntactic. EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 31
Theorem 3.45.
An axiomatic extension of FL e is semisimple if and only if theformula (1 ∧ p ) ∨ (1 ∧ ¬ (1 ∧ p ) n ) is a theorem of L for some n ∈ ω .Proof. Let L be an axiomatic extension of FL e . Then L is finitary and it inheritsthe local DDT and local IL of FL e . By Theorem 3.35, the semisimplicity of L istherefore equivalent to the LEM w.r.t. {¬ (1 ∧ p ) n | n ∈ ω } :Γ , ϕ ⊢ L ψ and Γ , ¬ (1 ∧ ϕ ) n ⊢ L ψ for each n ∈ ω = ⇒ Γ ⊢ L ψ. Suppose first that L is semisimple and take ϕ := p , ψ := q , andΓ := { (1 ∧ p ) → q } ∪ { (1 ∧ ¬ (1 ∧ p ) n ) → q | n ∈ ω } . The LEM yields that Γ ⊢ L q . Since (1 ∧ ¬ (1 ∧ p ) i ) → q ⊢ L (1 ∧ ¬ (1 ∧ p ) j ) → q holdsfor j ≤ i , finitarity implies that for some k ∈ ω (1 ∧ p ) → q, (1 ∧ ¬ (1 ∧ p ) k ) → q ⊢ L q. Substituting (1 ∧ p ) ∨ (1 ∧ ¬ (1 ∧ p ) k ) for q now yields the desired theorem.Conversely, recall that L inherits the proof by cases property (PCP) of FL e :Γ , ϕ ⊢ L χ and Γ , ψ ⊢ L χ ⇐⇒ Γ , (1 ∧ ϕ ) ∨ (1 ∧ ψ ) ⊢ L χ. It immediately follows that if (1 ∧ p ) ∨ (1 ∧ ¬ (1 ∧ p ) n ) is a theorem of L, then Lenjoys the LEM w.r.t. {¬ (1 ∧ p ) n | n ∈ ω } . (cid:3) We have in fact proved the above claim for any extension of FL e (or more gen-erally of any reduct of FL e ) which shares the same local DDT. Corollary 3.46.
A variety of FL e -algebras is semisimple if and only if it satisfiesthe equation (1 ∧ x ) ∨ (1 ∧ ¬ (1 ∧ x ) n ) ≈ for some n . Specializing the above results to FL ew and FL ew -algebras allows us to simplifytheir statement somewhat. Theorem 3.47.
An axiomatic extension of FL ew is semisimple if and only if p ∨¬ p n is a theorem of L for some n ∈ ω . Corollary 3.48 ([18]) . A variety of FL ew -algebras is semisimple if and only if itsatisfies the equation x ∨ ¬ x n ≈ for some n . The proof of the analogous theorem for IK will be slightly more complicated.Let us first show that the axiom of n -cyclicity can be given a slightly different butequivalent form in IKn . Fact 3.49.
Let L be an extension of IK and k ≥ . If p ∨ (cid:3) ¬ (cid:3) n p is a theorem of L , then so is p ∨ (cid:3) k ¬ (cid:3) kn p .Proof. Suppose that p ∨ (cid:3) ¬ (cid:3) n p and p ∨ (cid:3) k ¬ (cid:3) kn p are theorems. Then substi-tuting (cid:3) kn p for p in the former yields the theorem (cid:3) kn p ∨ (cid:3) ¬ (cid:3) n (cid:3) kn p , whichcombined with p ∨ (cid:3) k ¬ (cid:3) kn p yields the theorem p ∨ (cid:3) k (cid:3) ¬ (cid:3) n (cid:3) kn p , or equivalently p ∨ (cid:3) k +1 ¬ (cid:3) ( k +1) n p . (cid:3) Fact 3.50.
Let L be an extension of IKn . for n ≥ . Then p ∨ (cid:3) ¬ (cid:3) n p is atheorem of L if and only if (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p is. Proof.
The right-to-left direction is immediate given the definition of (cid:3) n . Con-versely, if p ∨ (cid:3) ¬ (cid:3) n p is a theorem, then so is p ∨ (cid:3) n ¬ (cid:3) n p by the previous fact.Substituting (cid:3) n p for p yields that (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n + n p . Finally, (cid:3) n p → (cid:3) n +1 p isa theorem because L extends IKn .
4, therefore (cid:3) n p → (cid:3) k p is a theorem for each k ≥ n . Taking k := n + n now yields the theorem (cid:3) n p → (cid:3) n + n p , which combinedwith (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n + n p yields the theorem (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p . (cid:3) Theorem 3.51.
An axiomatic extension of IK is semisimple if and only if theformulas p ∨ (cid:3) ¬ (cid:3) n p and (cid:3) n p → (cid:3) n +1 p are theorems of L for some n ∈ ω .Proof. Let L be an axiomatic extension of IK. Then L is finitary and it inheritsthe local DDT and local IL of IK. By Theorem 3.35, the semisimplicity of L istherefore equivalent to the LEM w.r.t. {¬ (cid:3) n p | n ∈ ω } .Suppose first that (cid:3) n p → (cid:3) n +1 p and p ∨ (cid:3) ¬ (cid:3) n p are theorems of L. Recallthat the latter is equivalent to (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p by Fact 3.50. Then the local DDTinherited by L from IK reduces to the global DDT w.r.t. (cid:3) n p → q . It now sufficesto show that L enjoys the global LEM w.r.t. ¬ (cid:3) n p . Using the global DDT of L,this global LEM is equivalent to the following implication:Γ ⊢ L (cid:3) n ϕ → ψ Γ ⊢ L (cid:3) n ¬ (cid:3) n ϕ → ψ Γ ⊢ L ψ This implication holds if (and only if) (cid:3) n p → q, (cid:3) n ¬ (cid:3) n p → q ⊢ L q , i.e. if (andonly if) (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p is a theorem of L.Conversely, suppose that L is semisimple. We first prove that p ∨ (cid:3) ¬ (cid:3) n p is atheorem of L for some n ∈ ω . To this end, take Γ := { q → (cid:3) k p | k ∈ ω } . Then p, Γ ⊢ L p ∨ (cid:3) ¬ q and ¬ (cid:3) k p, Γ ⊢ L ¬ q ⊢ L (cid:3) ¬ q ⊢ L p ∨ (cid:3) ¬ q for each k ∈ ω , thereforeΓ ⊢ L p ∨ (cid:3) q by the LEM. Because q → (cid:3) k +1 p ⊢ L q → (cid:3) k p , by finitarity there issome n ∈ ω such that q → (cid:3) n p ⊢ L p ∨ (cid:3) ¬ q . Substituting (cid:3) n p for q now yieldsthat (cid:3) n p ∨ (cid:3) ¬ (cid:3) n p , hence also p ∨ (cid:3) ¬ (cid:3) n p , is a theorem of L.Consider now the function f ( k ) = kn + 1 and takeΓ := { p → q } ∪ { (cid:3) f ( k ) ¬ (cid:3) k p → q | k ∈ ω } . Then p, Γ ⊢ L q and ¬ (cid:3) k p, Γ ⊢ L q for each k ∈ ω , therefore Γ ⊢ L q by the LEM. Byfinitarity there is some m ≥ p → q, { (cid:3) f ( k ) ¬ (cid:3) k p → q | k ≤ m } ⊢ L q. Substituting the formula ψ := p ∨ W k ≤ m (cid:3) f ( k ) ¬ (cid:3) k p for q yields that ψ is a theoremof L. By Fact 3.49 the formula p ∨ (cid:3) k ¬ (cid:3) kn p is a theorem of L for each k ≤ m .Substituting ¬ (cid:3) mn p for p in ψ yields the theorem ¬ (cid:3) mn p ∨ _ k ≤ m (cid:3) f ( k ) ¬ (cid:3) k ¬ (cid:3) kn (cid:3) ( m − k ) n p, since (cid:3) mn p is equivalent (in every context) to (cid:3) kn (cid:3) ( m − k ) n p for each k ≤ m . Ap-plying the substitution instances (cid:3) ( m − k ) n p ∨ (cid:3) k ¬ (cid:3) kn (cid:3) ( m − k ) n p of the theorem p ∨ (cid:3) k ¬ (cid:3) kn p to the theorem displayed above yields ¬ (cid:3) mn p ∨ _ k ≤ m (cid:3) f ( k ) (cid:3) ( m − k ) n p. For our function f ( k ) := kn +1 this is equivalent to ¬ (cid:3) mn p ∨ (cid:3) mn +1 p , which implies (cid:3) mn p → (cid:3) mn +1 p . Of course, if p ∨ (cid:3) ¬ (cid:3) n p is a theorem, so is p ∨ (cid:3) ¬ (cid:3) mn p . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 33 (Let us also provide a semantic proof that n -cyclicity plus the complicatedtheorem p ∨ W k ≤ m (cid:3) f ( k ) ¬ (cid:3) k p imply weak n -transitivity. We only state the prooffor classical modal logic for the sake of simplicity, but the argument can be ex-tended to the intuitionistic setting. These axiom of n -cyclicity and the theorem p ∨ W k ≤ m (cid:3) f ( k ) ¬ (cid:3) k p are equivalent to so-called Sahlqvist formulas, therefore theycorrespond certain first-order conditions on Kripke frames and the logic L is com-plete with respect to the class of all Kripke frames satisfying these conditions (seee.g. [4, Sections 3.6 and 4.3]). Using the notation introduced in Subsection 2.3, n -cyclicity corresponds to the condition that uRw implies wR ≤ k u . The more com-plicated theorem corresponds to the following condition:for each u ∈ W there is some n ≤ m such that uR ≤ nk +1 w implies wR ≤ n u. But by the first condition wR ≤ n u implies uR ≤ nk w , therefore for each u ∈ W there issome n ≤ m such that uR ≤ nk +1 w implies that uR ≤ nk w . It follows that uR ≤ mk +1 w implies that uR ≤ mk w . Informally, all of the alternatives imply that each for eachpath of length mk + 1 there is a shortcut of length at most mk . The formula (cid:3) mk p → (cid:3) mk +1 p is therefore a theorem of L.) (cid:3) Corollary 3.52.
A variety of modal Heyting algebras is semisimple if and only ifit satisfies the equations ≈ x ∨ (cid:3) ¬ (cid:3) n x and (cid:3) n x ≤ (cid:3) n +1 x for some n ∈ ω . Corollary 3.53 ([19]) . A variety of modal algebras is semisimple if and only if itsatisfies the equations x ≤ (cid:3)♦ n x and (cid:3) n x ≤ (cid:3) n +1 x for some n ∈ ω . Glivenko theorems and semisimple companions
In this section we study the (syntactic) semisimple companion of a logic, whichfor compact logics is the largest extension with the same inconsistent sets of for-mulas as the original logic. The rules valid in the semisimple companion can bedescribed in terms of the antitheorems of the original logic. This leads to what wecall antiadmissible rules, by analogy with the admissible rules of a logic. In fact, thesemisimple companion can be thought of as a construction dual to the structuralcompletion, which is the largest extension with the same theorems as the originallogics.Under favourable circumstances, the models of the semisimple companion areprecisely the semisimple models of the original logic. This fact can be applied todescribe the semisimple algebras in a given variety, provided that we know howto axiomatize the semisimple companion of the corresponding logic. This strat-egy works for logics with a well-behaved implication or disjunction (which satisfythe global DDT or the global PCP). For such logics the semisimple companion isaxiomatized by an axiomatic form of the LEM.The description of antiadmissible will then allow us to establish a Glivenko-likeconnection between a logic and its semisimple companion, subsuming several exist-ing Glivenko theorems in the literature. In particular, we describe all extensions ofFL e which enjoy a Glivenko-like connection to classical logic.4.1. The semisimple companion of a logic.
Let us start by recalling the rele-vant definitions from the previous section. A simple theory of L is a largest non-trivial theory of L, while a semisimple theory is an intersection of simple theories.A logic L is called (syntactically) semisimple if each theory of L is semisimple. The (syntactic) semisimple companion of L, denoted α Th (L), is then defined as the logic determined by the simple, or equivalently semisimple, theories of L. That is, it isthe logic determined by the matrices h Fm L , T i where T ranges over the simple, orequivalently semisimple, theories of L. Note that it is not obvious from the defini-tion whether this companion is a semisimple logic, although it is always completewith respect to its simple theories.The semisimple companion can only be expected to have a significant relation tothe original logic if each non-trivial theory may be extended to a simple theory, acondition that we call (syntactic) coatomicity . Otherwise, much information aboutthe theories of the original logic may be lost. This condition will be assumed inalmost all of the results of this section. Recall that in particular each compact logicis coatomic. Fact 4.1.
Let L be a coatomic logic. Then α Th (L) is the largest logic with the samesimple theories as L . In particular, α Th (L) is also coatomic.Proof. If L ′ has the same simple theories as L, then each matrix of the form h Fm L , T i for some simple L-theory T is a model of L ′ , hence L ′ ≤ α Th (L). Clearlyeach simple theory of L is a simple theory of α Th (L). Conversely, each simple theory T of α Th (L) is a theory of L, so by coatomicity it extends to a simple theory T ′ ofL, which is by definition a theory of α Th (L). But T is a simple theory of α Th (L),therefore T ′ = T and T is L-simple. (cid:3) Fact 4.2.
Let L be a coatomic logic. An extension L ′ of L has the same simpletheories as L if and only if it satisfies the equivalence Γ ⊢ L Fm L ⇐⇒ Γ ⊢ L ′ Fm L . Proof.
If L and L ′ have the same simple theories and Γ L Fm L, then by thecoatomicity of L there is a simple L-theory, hence also a simple L ′ -theory, T ⊇ Γ.But T L ′ Fm L, hence also Γ L ′ Fm L. Of course Γ ⊢ L Fm L implies Γ ⊢ L ′ Fm Lfor each extension L ′ of L.Conversely, if L ′ satisfies the equivalence and T is a simple theory of L, thenby the equivalence T L ′ Fm L. Let T ′ be the L ′ -theory generated by T . Then T ′ is a non-trivial L-theory extending T , hence T ′ = T by the L-simplicity of T .Thus T is an L ′ -theory, and in fact a simple L ′ -theory because L ′ is an extensionof L. On the other hand, if T is a simple theory of L ′ , then it is a theory of L.If T ′ is a non-trivial theory of L which properly extends T , then by the assumedequivalence T ′ L ′ Fm L, therefore T ′ = T by the L ′ -simplicity of T . Thus T is asimple L-theory. (cid:3) Corollary 4.3.
Let L be a coatomic logic. Then the semisimple companion α Th (L) is the largest extension of L such that Γ ⊢ L Fm L ⇐⇒ Γ ⊢ α Th (L) Fm L . If a logic fails to be coatomic, then pathological behaviour can occur. In theextreme case, it may lack simple theories altogether. For example, consider thelogic with a single binary operation x · y axiomatized by the rules p · q ⊢ p, p · ( q · r ) ⊢ ( p · q ) · r, ( p · q ) · r ⊢ p · ( q · r ) . The lattice of theories of this logic is isomorphic to the lattice of sets of wordsover the given variables closed under taking initial segments. This lattice has no
EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 35 coatoms, so the semisimple companion of this logic is the trivial logic. It maytherefore even happen that ∅ ∈
Th L while ∅ ⊢ α Th (L) Fm L.Among logics which have an antitheorem, such pathological behaviour can onlyoccur if the logic is not finitary, since each non-trivial finitary logic with an anti-theorem is compact, hence coatomic. One fairly natural example is the followingexpansion of G¨odel’s well-known super-intuitionistic logic. Consider the real interval [0 ,
1] with the natural order viewed as a Heyting algebra(i.e. as a bounded distributive lattice with the binary operation → such that x → y = y if y < x and x → y = 1 otherwise) expanded by a constant c q interpreted by q for each rational q ∈ [0 , G Q and define the G¨odel logicof order with rational constants GO Q as the logic (over some given set of variables)in the signature of the algebra G Q such thatΓ ⊢ GO Q ϕ ⇐⇒ ^ h [Γ] ≤ h ( ϕ ) for each homomorphism h : Fm GO Q → G Q . The presence of infinitely many constants is not an essential feature of this example.We could make do with the unary function x x instead. Example 4.4.
G¨odel logic of order with rational constants has no simple theoriesand no largest extension with the same antitheorems.Proof.
Consider the logics GO Q q for q ∈ [0 , ∩ Q defined (over the same algebra offormulas) asΓ ⊢ GO Q q ϕ ⇐⇒ ^ h [Γ] ∧ q ≤ h ( ϕ ) for each hom. h : Fm GO Q → G Q . Clearly GO Q = GO Q . Moreover, GO Q r is a proper extension of GO Q q if r < q because ∅ ⊢ GO Q r c q if and only if r < q . Finally, Γ is an antitheorem of GO Q q for q > V h [Γ] = 0 for each homomorphism h : Fm GO Q → G Q .It follows that adding any finite set of axioms ∅ ⊢ c q for q > Q does notchange the set of antitheorems, since it results in a logic below some GO Q q with q >
0. However, adding all of these axioms results in the logic becoming trivialthanks to the antitheorem { c q | q ∈ (0 , ∩ Q } .If T is a non-trivial theory of GO Q , then T GO Q c q for some constant c q with q >
0. Because T is not an antitheorem of GO Q , it is also not an antitheoremof GO Q q , hence the theory of GO Q q generated by T is a non-trivial theory of GO Q properly extending T by virtue of containing c q . (cid:3) Contrast this with the situation for theorems rather than antitheorems: eachlogic L, finitary or not, is guaranteed to have a largest extension with the sameset of theorems Thm L, called the structural completion of L, which is defined bythe single matrix h Fm L , Thm L i . The reader may observe a natural duality be-tween the semisimple companion defined the maximal (non-trivial) theories andthe structural completion defined by the unique minimal theory. Indeed, the semi-simple companion may well be called the antistructural completion : if a coatomiclogic L has an antitheorem, then α Th (L) is precisely the largest extension of L withthe same set of antitheorems. This logic in fact lacks subdirectly irreducible models altogether (see [21, Section 3.2]).
Antiadmissible rules.
The rules valid in the structural completion of alogic L can be described in terms of the theorems of L: they are precisely the admissible rules of L, that is, rules Γ ⊢ ϕ such that ∅ ⊢ L σ [Γ] = ⇒ ∅ ⊢ L σ ( ϕ ) for each substitution σ. Given the duality between the structural completion and the semisimple companion,we now wish to provide a similar description of the rules valid in the semisimplecompanion α Th (L) in terms of the antitheorems of L.Throughout this section only, we shall abuse notation and talk about the ruleΓ ⊢ ϕ , i.e. the set of rules Γ ⊢ ϕ i for ϕ i ∈ ϕ , where ϕ is an α -tuple for 0 < α < κ .This is not an essential point: the reader is free to substitute ϕ for ϕ throughoutthis chapter. Definition 4.5.
A rule Γ ⊢ ϕ will be called antiadmissible in L if σ ( ϕ ) , ∆ ⊢ Fm L = ⇒ σ [Γ] , ∆ ⊢ Fm L for each substitution σ and ∆ ⊆ Fm L , or equivalently if σ ( ϕ ) , T ⊢ Fm L = ⇒ σ [Γ] , T ⊢ Fm L for each substitution σ and T ∈ Th L . Proposition 4.6.
A rule Γ ⊢ ϕ is antiadmissible in a coatomic logic L if and onlyif Γ ⊢ α Th (L) ϕ .Proof. If the rule Γ ⊢ ϕ is valid in α Th (L), then σ ( ϕ ) , ∆ ⊢ α Th (L) Fm L implies σ [Γ] , ∆ ⊢ α Th (L) Fm L by structurality and cut. Applying the equivalence Γ ⊢ L Fm L ⇐⇒ Γ ⊢ α Th (L) Fm L now yields the antiadmissibility of Γ ⊢ ϕ . Conversely,suppose that Γ ⊢ ϕ is not valid in α Th (L). Then there is a simple L-theory T suchthat Γ ⊢ ϕ fails in the matrix h Fm L , T i , i.e. there is a substitution σ such that σ [Γ] ⊆ T and σ ( ϕ ) * T , hence σ ( ϕ ) , T ⊢ L Fm L but σ [Γ] , T L Fm L. (cid:3)
A rule Γ ⊢ ϕ is admissible in a logic L if and only if the extension of L by thisrule has the same theorems as L. An analogous observation can be made aboutantiadmissible rules in coatomic logics with an antitheorem. Corollary 4.7.
A rule Γ ⊢ ϕ is antiadmissible in a coatomic logic L with anantitheorem if and only if extending L by this rule does not add any antitheorems. Without the assumption of coatomicity, we can still show that at least the finitaryantiadmissible rules form a finitary logic.
Fact 4.8.
The finitary antiadmissible rules of a logic form a finitary logic. Moreprecisely, the set of all rules Γ ⊢ ϕ such that Γ ′ ⊢ ϕ is antiadmissible in L for somefinite Γ ′ ⊆ Γ is a finitary logic.Proof. Reflexivity, monotonicity, structurality, and finitarity hold by definition.Moreover, antiadmissible rules are closed under finitary cut: if the rules Γ ⊢ ϕ and ϕ, ∆ ⊢ ψ are antiadmissible, then σ ( ψ ) , Φ ⊢ Fm L = ⇒ σ ( ϕ ) , σ [∆] , Φ ⊢ Fm L = ⇒ σ [Γ] , σ [∆] , Φ ⊢ Fm L , therefore the rule Γ , ∆ ⊢ ψ is antiadmissible in L. Full cut now follows fromfinitarity and finitary cut. (cid:3) EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 37
The above fact fails if we look at all antiadmissible rules. These are closedunder reflexivity, monotonicity, structurality, and finitary cut, but not under fullcut. For example, in the G¨odel logic of order with rational constants GO Q (recallExample 4.4) each of the rules ∅ ⊢ c q for q ∈ (0 , ∩ Q is antiadmissible, as is thevalid rule { c q | q ∈ (0 , ∩ Q } ⊢ c , but the rule ∅ ⊢ c is not antiadmissible, sincethis would imply that GO Q is the trivial logic.It may be instructive to contrast this with the situation of admissible rules.Essentially, a rule is admissible if all of its instances lead only from theorems totheorems. If the terminal nodes of a proof (which is a well-founded tree) are theo-rems and all rules applied in the proof preserve theoremhood, then by induction onwell-founded trees the conclusion must be a theorem too. However, the analogousargument for antiadmisible rules fails: if the conclusion of a proof is an antitheorem,and all rules reflect antitheoremhood in each context (if Γ ⊢ ϕ is an instance ofthe rule and ϕ, ∆ is an antitheorem, then so is Γ , ∆), then we cannot infer thatthe set of all terminal nodes of the proof forms an antitheorem: induction on well-founded trees does not work in this direction. Such an inference only works for finiteproofs, hence the above fact. Note that a finitary logic may well have infinitaryantiadmissible rules, therefore restricting to antiadmissible rules in finitary logicsdoes not help.In the rest of this subsection, we show that in many cases the definition of anantiadmissible rule may be simplified. Indeed, the double quantification over con-texts ∆ and substitutions σ makes the definition of an antiadmissible rule morecomplicated than the analogous definition of an admissible rule. Remarkably, itturns out that we can often do without the quantification over substitutions. Thishas no analogue in the case of admissible rules, where quantification over substitu-tions is of course essential.The proof will rely on the (parametrized) local ILs introduced in Section 3.Recall that in Proposition 3.11 we proved that the κ -ary (parametrized) local IL isequivalent in a logic with an antitheorem to (surjective) substitution swapping forantitheorems: for each (surjective) substitution σ , each theory T of a logic L withan antitheorem, and each α -tuple of formulas ϕ for 0 < α < κσ ( ϕ ) , T ⊢ L ∅ = ⇒ ϕ, σ − [ T ] ⊢ L ∅ . In the statement of the following proposition, an invertible substitution is asubstitution ι such that τ ◦ ι is the identity map on Fm L for some substitution τ .Invertible substitutions are precisely those injective substitutions τ such that τ ( p ) ∈ Var L for each variable p . Proposition 4.9.
Let L be a coatomic logic with the unary parametrized local IL.Then Γ ⊢ ϕ is antiadmissible in L if and only if ι ( ϕ ) , ∆ ⊢ L ∅ = ⇒ ι [Γ] , ∆ ⊢ L ∅ for each invertible substition ι and each ∆ . Proof.
Suppose that Γ ⊢ ϕ satisfies the implication and σ ( ϕ ) , T ⊢ L ∅ for sometheory T of L. Let us first construct an invertible substitution ι and a surjectivesubstitution τ such that σ = τ ◦ ι .To define ι and τ , pick X ⊆ Var L such that | X | = | Var L | = | Var L \ X | , and aninjective map f : Var L → X and a surjective map g : X → Var L such that g ◦ f isthe identity map on Var L. The map f extends to an invertible substitution ι withan inverse λ such that λ ◦ ι is the identity map on Fm L. We can then define τ such that the restriction τ : X → Fm L is defined as τ ( p ) = ( σ ◦ λ )( p ) for p ∈ X and therestriction τ : Var L \ X → Var L is surjective. Then τ is a surjective substitutionsuch that σ = τ ◦ ι .The assumption σ ( ϕ ) , T ⊢ L ∅ , i.e. ( τ ◦ ι )( ϕ ) , T ⊢ L ∅ , implies ι ( ϕ ) , τ − [ T ] ⊢ L ∅ by surjective substitution swapping for antitheorems. The assumed implicationnow yields ι [Γ] , τ − [ T ] ⊢ L ∅ . Finally, we may apply the substitution τ to obtain( τ ◦ ι )[Γ] , τ [ τ − [ T ]] ⊢ L ∅ , hence σ [Γ] , T ⊢ L ∅ . (cid:3) For the purposes of the following proposition, a rule Γ ⊢ ϕ is said to omitenough variables if | Var L \ Var(Γ , ϕ ) | = | Var L | , where Var ∆ denotes the set of allvariables which occur in ∆. In other words, it leaves | Var L | variables unused. Thisin particular covers all finitary rules Γ ⊢ ϕ . Proposition 4.10.
Let L be a coatomic logic with a κ -ary parametrized local IL.Then a rule Γ ⊢ ϕ which omits enough variables is antiadmissible in L if and only if ϕ, ∆ ⊢ L ∅ = ⇒ Γ , ∆ ⊢ L ∅ for each ∆ ⊆ Fm L . Proof.
Let Γ ⊢ ϕ be a rule satisfying the implication, let ι be an invertible substi-tution, and let ι ( ϕ ) , T ⊢ L ∅ for some theory T of L. By the previous proposition itsuffices to show that ι [Γ] , T ⊢ L ∅ .The substitution ι is restricts to an injective map f : Var L → Var L. Since | Var L \ Var(Γ , ϕ ) | = | Var L | = κ , there is a bijective map g : Var L → Var L whichagrees with f on Var(Γ , ϕ ). The bijection g extends to a bijective substitution σ of Fm L such that σ agrees with ι on Γ and ϕ . But ι ( ϕ ) , T ⊢ L ∅ implies σ ( ϕ ) , T ⊢ L ∅ ,hence ϕ, σ − [ T ] ⊢ L ∅ by surjective substitution swapping for antitheorems. Theassumed implication yields Γ , τ − [ T ] ⊢ L ∅ . We may apply the substitution τ toobtain τ [Γ] , T ⊢ L ∅ , i.e. σ [Γ] , T ⊢ L ∅ . (cid:3) In other words, for all but the most pathological antiadmissible rules which useup almost all variables, we may omit the quantification over substitutions σ from thedefinition of antiadmissibility. If the logic enjoys a local IL, then no such technicalcondition is necessary. Proposition 4.11.
Let L be a coatomic logic with a simple local IL. Then a rule Γ ⊢ ϕ is antiadmissible in L if and only if ϕ, ∆ ⊢ L ∅ = ⇒ Γ , ∆ ⊢ L ∅ for each ∆ ⊆ Fm L . Moreover, we may restrict ∆ to range over the simple theories of L .Proof. If σ [Γ] , ∆ L ∅ , then by coatomicity σ [Γ] , ∆ extends to a simple theory T , hence T ⊢ L σ [Γ] and σ − [ T ] ⊢ L Γ. By assumption Γ , σ − [ T ] L ∅ implies ϕ, σ − [ T ] L ∅ . But σ − [ T ] is simple by the simple local IL, therefore σ − [ T ] ⊢ L ϕ and T ⊢ L σ ( ϕ ). It follows that σ ( ϕ ) , ∆ L ∅ . (cid:3) This last result gives us a convenient syntactic handle on the valid rules of α Th (L)in terms of the antitheorems of L, which will later be leveraged to yield Glivenkotheorems connecting L and α Th (L).4.3. Semisimple theories and models of L . Having linked the valid rules of thesemisimple companion of L with the antitheorems of L, we now wish to relate thetheories and models of the semisimple companion of L with the semisimple theoriesand models of L. More precisely, we want to identify sufficient conditions underwhich these classses of theories and models coincide.
EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 39
Proposition 4.12.
Let L be coatomic with the simple local IL. Then the theoriesof α Th (L) are precisely the semisimple theories of L . In particular, α Th (L) is semi-simple.Proof. Each semisimple theory of L is a theory of α Th (L) because each simpletheory of L is by definition a theory of α Th (L) and the theories of α Th (L) are closedunder intersections. Conversely, let T be a theory of α Th (L) such that T α Th (L) ϕ .To prove the proposition it suffices to find a simple L-theory U extending T suchthat U L ϕ . But T α Th (L) ϕ implies by Proposition 4.11 that there is an L-theory V such that ϕ, V ⊢ L ∅ and T, V L ∅ . By coatomicity the set T, V extends to asimple L-theory U . Then ϕ, V ⊢ L ∅ and U, V L ∅ , therefore U L ϕ . (cid:3) In fact, α Th (L) inherits the local IL of L and upgrades it to a classical one. Proposition 4.13.
Let L be coatomic with the κ -ary par. local IL w.r.t. Ψ . Then α Th (L) enjoys the κ -ary par. local IL w.r.t. Ψ . If L moreover has the simple localIL, then α Th (L) enjoys the κ -ary classical par. local IL w.r.t. Ψ .Proof. Suppose that Γ , ϕ ⊢ α Th (L) ∅ . Then Γ , ϕ ⊢ L ∅ , therefore by the IL Γ ⊢ L I( ϕ, π )for some I ∈ Ψ α and some π . It follows that Γ ⊢ α Th (L) I( ϕ, π ). On the other hand,if Γ ⊢ α Th (L) I( ϕ, π ) for some I ∈ Ψ α and some π , then Γ , ϕ ⊢ α Th (L) ∅ because ϕ, I( ϕ, π ) ⊢ L ∅ implies ϕ, I( ϕ, π ) ⊢ α Th (L) ∅ .Suppose now that L enjoys a simple local IL and Γ α Th (L) ϕ . Then by Propo-sition 4.11 there is some simple theory T such that T, ϕ ⊢ L ∅ but T, Γ L ∅ . Thesimple local IL w.r.t. Ψ ′ yields T ⊢ L I ′ ( ϕ ) for some I ′ ∈ Ψ ′ , therefore Γ , I ′ ( ϕ ) L ∅ and Γ , I ′ ( ϕ ) α Th (L) ∅ . (cid:3) We now show that, under suitable assumptions, the models of α Th (L) are thesemisimple models of L. Lemma 4.14.
Let L be a logic with the semantic simple local IL. If h : A → B isa homomorphism and h B , G i is a simple model of L , then so is h A , h − [ G ] i .Proof. If a / ∈ h − [ G ], then h − [ G ] , a ⊢ A L A : a / ∈ h − [ G ] = ⇒ h ( a ) / ∈ G = ⇒ G, h ( a ) ⊢ A L B (by the simplicity of h B , G i )= ⇒ G ⊢ L I( h ( a )) for some I ∈ Ψ (by the semantic simple IL)= ⇒ h [I( a )] ⊆ G for some I ∈ Ψ (because h is a homomorphism)= ⇒ I( a ) ⊆ h − [ G ] for some I ∈ Ψ = ⇒ h − [ G ] , a ⊢ A L A (by the semantic IL) . It follows that h − [ F ] is a simple L-filter on A . (cid:3) Lemma 4.15.
Let L be a semantically coatomic logic with the semantic simplelocal IL. Then the simple models of α Th (L) are precisely the simple models of L .Proof. By the previous lemma, if h A , F i is a simple model of L, then for eachhomomorphism h : Fm L → A the set of formulas h − [ F ] is a simple theory of L,and thus a theory of α Th (L). It follows that h A , F i is a (simple) model of α Th (L).Conversely, if h A , F i is a simple model of α Th (L), then by the semantic coatomic-ity of L the filter F extends to a simple filter G of L, and by the implication proved in the previous paragraph G is a non-trivial filter of α Th (L). Because F was asimple filter of α Th (L), it follows that G = F , therefore F is a simple filter of L. (cid:3) Theorem 4.16.
Let L be a compact logic which enjoys the κ -ary local IL w.r.t. afamily Ψ such that | Ψ α | ≤ | Var L | for each Ψ α and each set in Ψ has cardinalityless than κ . Then the models of α Th (L) are precisely the semisimple models of L .In particular, α Th (L) is semantically semisimple.Proof. By Lemma 4.15 it suffices to show that α Th (L) is semantically semisimple.We know that it enjoys the unary classical local IL w.r.t. Ψ by Proposition 4.13.Semantic semisimplicity follows by Theorem 3.35. (cid:3) In other words, whenever the above theorem applies, to describe the semisimplemodels of a logic L it suffices to axiomatize α Th (L). We now show that undercertain conditions this can be done in an entirely mechanical manner, since α Th (L)turns out to be the extension of L by an axiomatic form of the LEM. We illustratethis on FL n e and IKn . DDT–axiomatic form of the LEM and the
PCP–axiomatic form of the LEM .In the following, p ⇒ q and ⊥ shall denote sets of formulas in the same way thatΓ( p, q ) and ∆ do. Given these two sets of formulas, we define ∼ ϕ := ϕ ⇒ ⊥ = [ π ∈ ⊥ ϕ ⇒ π. If ϕ is a finite tuple of formulas, we define the set of formulas ϕ ⇒ ψ as follows: ∅ ⇒ ψ := { ψ } , h ϕ, ϕ n +1 i ⇒ ψ := { ϕ n +1 ⇒ χ | χ ∈ ϕ ⇒ ψ } . Fact 4.17. If L enjoys the unary global DDT w.r.t. p ⇒ q , then it enjoys the finitaryglobal DDT w.r.t the family { p , . . . , p n } ⇒ q . Proposition 4.18.
Let L be a logic with a finite antitheorem ⊥ and a unaryglobal DDT w.r.t. a finite set p ⇒ q . Then L enjoys the unary LEM if and only if ( p ⇒ q ) ⇒ (( ∼ p ⇒ q ) ⇒ q ) is a theorem of L .Proof. We know that L enjoys the unary LEM if and only if it enjoys the unaryglobal LEM w.r.t. ¬ p (Propositions 3.30 and 3.17). By the global DDT this LEMamounts to the implicationΓ ⊢ L ϕ ⇒ ψ Γ ⊢ L ∼ ϕ ⇒ ψ Γ ⊢ L ψ This is equivalent to ϕ ⇒ ψ, ∼ ϕ ⇒ ψ ⊢ L ψ for all ϕ and ψ , i.e. p ⇒ q, ∼ p ⇒ q ⊢ L q .By the global DDT, this is equivalent to ∅ ⊢ L ( p ⇒ q ) ⇒ (( ∼ p ⇒ q ) ⇒ q ). (cid:3) We call ( p ⇒ q ) ⇒ (( ∼ p ⇒ q ) ⇒ q ) the DDT–axiomatic form of the LEM for L.For example, the DDT-axiomatic form of the LEM for FL n e and IKn . p → n q ) → n (( ¬ n p → n q ) → n q ) , EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 41 where x → n y := (1 ∧ x ) n → y and ¬ n x := ¬ (1 ∧ x ) n in the case of FL n e and x → n y := (cid:3) n x → y and ¬ n x := ¬ (cid:3) n x in the case of IKn . e w.r.t. the family {{ p → n q } | n ∈ ω } , perhaps on accountof being an axiomatic extension of FL e . Then, using this local DDT, L enjoys thelocal LEM w.r.t. the family {{¬ n p } | n ∈ ω } if and only if for each m ∈ ω and eachfunction f : ω → ω Γ ⊢ L ϕ → m ψ Γ ⊢ L ¬ n ϕ → f ( n ) ψ Γ ⊢ L ψ This is equivalent to the fact that for each m ∈ ω and f : ω → ω and all ϕ and ψϕ → m ψ, {¬ n ϕ → f ( n ) ψ | n ∈ ω } ⊢ L ψ. By structurality this is equivalent to the fact that for each m ∈ ω and f : ω → ωp → m q, {¬ n p → f ( n ) q | n ∈ ω } ⊢ L q. One could go a step further and use finitarity and the local DDT to express thisrule as the validity of some set of theorems. However, the resulting family of setsof theorems would be rather complicated.A more familiar form of the LEM can be obtained for logics which enjoy whatCintula & Noguera [11] call the proof by cases property (PCP). For example, thelogic FL e and its axiomatic extensions satisfy the equivalenceΓ , ϕ ⊢ FL e χ and Γ , ψ ⊢ FL e χ ⇐⇒ Γ , (1 ∧ ϕ ) ∨ (1 ∧ ψ ) ⊢ FL e χ, while IKn . , ϕ ⊢ IKn . χ and Γ , ψ ⊢ IKn . χ ⇐⇒ Γ , (cid:3) n ϕ ∨ (cid:3) n ψ ⊢ IKn . χ. The logic IK only satisfies a local form of this condition, namelyΓ , ϕ ⊢ IK χ and Γ , ψ ⊢ IK χ ⇐⇒ Γ , (cid:3) n ϕ ∨ (cid:3) n ψ ⊢ IK χ for some n ∈ ω. Definition 4.19.
A logic L enjoys the binary global proof by cases property (PCP) if there is a set of formulas ⊔ ( p, q ) such that Γ , ϕ ⊢ L χ and Γ , ψ ⊢ L χ ⇐⇒ Γ , ⊔ ( ϕ, ψ ) ⊢ L χ for some n ∈ ω. Fact 4.20 ([11, Lemma 3.10]) . Let ⊔ (Φ , Ψ) := S ϕ ∈ Φ S ψ ∈ Ψ ⊔ ( ϕ, ψ ) . If L enjoysthe binary global PCP w.r.t. ⊔ ( p, q ) , then for all finite sets Φ and ΨΓ , Φ ⊢ L χ and Γ , Ψ ⊢ L χ ⇐⇒ Γ , ⊔ (Φ , Ψ) ⊢ L χ. Proposition 4.21.
Let L be a logic which enjoys the unary global IL w.r.t. a finiteset ∼ p and the binary global PCP w.r.t. ⊔ ( p, q ) . Then L enjoys the unary LEM ifand only if ⊔ ( p, ∼ p ) is a theorem of L .Proof. We know that L enjoys the unary LEM if and only if it enjoys the unaryglobal LEM w.r.t. ¬ p (Propositions 3.30 and 3.17). By the global binary PCP thisamounts to the implication Γ , ⊔ ( ϕ, ∼ ϕ ) ⊢ L ψ Γ ⊢ L ψ This implication is equivalent to ∅ ⊢ L ⊔ ( ϕ, ∼ ϕ ). (cid:3) We call ⊔ ( p, ∼ p ) the PCP–axiomatic form of the LEM for L. For example, thePCP–axiomatic form of the LEM for FL n e is(1 ∧ p ) ∨ (1 ∧ ¬ (1 ∧ p ) n ) , while the PCP–axiomatic form of the LEM for IKn . (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p, which we have already shown to be equivalent to p ∨ (cid:3) ¬ (cid:3) n p in Fact 3.50.We now show that the DDT–axiomatic and PCP–axiomatic form of the LEMprecisely describes the semisimple models of L. Proposition 4.22.
Let L be a coatomic logic with a finite antitheorem and a unaryglobal DDT w.r.t. a finite set (alternatively, with a unary global IL w.r.t. a finiteset and a binary global PCP). Then α Th (L) is precisely the extension of L by theDDT–axiomatic (alternatively, the PCP–axiomatic) form of the LEM.Proof. For the IL set, DDT set, and PCP set we use the notation ∼ ϕ , ϕ ⇒ ψ ,and ⊔ (Φ , Ψ) introduced in Subsection 3.5. Let L ′ be the extension of L by theDDT–axiomatic form of the LEM. We first prove that L ′ ≤ α Th (L). Since α Th (L)is complete w.r.t. the simple theories of L, it suffices to prove that each simpletheory T of L is a theory of L ′ . But for each simple theory T of L either T ⊢ L ϕ or T, ϕ ⊢ L ∅ , in which case T ⊢ L ∼ ϕ . In either case T, ϕ ⇒ ψ, ∼ ϕ ⇒ ψ ⊢ L ψ by theDDT, therefore T proves the DDT–axiomatic form of the LEM again by the DDT.Thus T is a theory of L ′ .Conversely, the logic L ′ is an axiomatic extension of L, therefore it inherits theglobal DDT of L. By Proposition 4.18 the DDT–axiomatic form of the LEM impliesthat L ′ is semisimple. It thus suffices to prove that each simple theory T of L ′ isa theory of α Th (L). But because L is coatomic, T extends to a simple theory T ′ of L, which is a theory of L ′ by the previous paragraph. The simplicity of T withrespect to L ′ now implies that T ′ = T , hence T is a simple theory of L. As such, itis by definition a theory of α Th (L).The proof in the PCP case is analogous. If T is a simple theory of L, then either T ⊢ L ϕ or T ⊢ L ∼ ϕ , so T ⊢ L ⊔ ( ϕ, ∼ ϕ ). Conversely, the logic L ′ which extendsL by the PCP–axiomatic form of the LEM inherits the binary global PCP as wellas the unary global IL of L. By Proposition 4.21 the PCP–axiomatic of the LEMimplies that L ′ is semisimple. The same argument as in the DDT case now showsthat each simple theory T of L ′ is a theory of α Th (L). (cid:3) Theorem 4.23.
Let L be a compact logic which enjoys a unary global DDT w.r.t. afinite set (alternatively, a unary global IL w.r.t. a finite set and a binary globalPCP). Then a model of L is semisimple if and only if it validates the DDT–axiomatic (alternatively, the PCP–axiomatic) form of the LEM.Proof. This follows from the previous proposition and Theorem 4.16. (cid:3)
If L is a (weakly) algebraizable and K is its algebraic counterpart, then on eachalgebra the lattice of L-filters is isomorphic to the lattice of K -congruences, thereforethe above theorem tells us that the semisimple algebras of K are precisely thosewhich satisfy the equational translation of (either of the two forms of) the LEM.For FL n e algebras and IKn . EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 43
Fact 4.24. An FL n e -algebra is semisimple if and only if it satisfies the equation ≤ x ⊔ ¬ n x , or equivalently the equation x → n y ≤ ( ¬ n x → n y ) → n y . Fact 4.25. An IKn . -algebra is semisimple if and only if it satisfies the equation ≈ x ⊔ ¬ n x , or equivalently the equation x → n y ≤ ( ¬ n x → n y ) → n y . The simpler form of the axiom of n -cyclicity (Fact 3.50) yields an alternativeformulation of the last fact. Fact 4.26. A Kn . -algebra is semisimple if and only if it satisfies x ≤ (cid:3)♦ n x . An IKn . -algebra is semisimple if and only if it satisfies ≈ x ∨ (cid:3) ¬ (cid:3) n x . These facts may not be particularly difficult to prove directly. Our point,however, is that the above theorem enables us to prove such results in a uniform wayfor any logic which satisfies certain syntactic prerequisites. Moreover, it immedi-ately extends to all (weakly) algebraizable fragments of these logics which containsufficient syntactic material to express the IL and the DDT or the PCP. (Althoughthe two forms of the LEM are equivalent for FL n e and IKn .
4, in such fragments onlyone of them may be expressible.)4.4.
Glivenko theorems.
In this final section, we show that a Glivenko-like con-nection obtains between a logic and its semisimple companion, subsuming severalknown Glivenko theorems under a single umbrella.
Proposition 4.27.
Let L be a coatomic logic with the κ -ary local IL w.r.t. a family Ψ such that each I ∈ Ψ has cardinality less than κ . Then Γ ⊢ α Th (L) ϕ if and onlyif for each I ∈ Ψ there is some J ∈ Ψ | I | such that Γ ⊢ L J(I( ϕ )) .Proof. By Proposition 4.6 the valid rules of α Th (L) coincide with the antiadmissiblerules of L. Using Proposition 4.11 and the local IL, the antiadmissible rules of Lare precisely those rules Γ ⊢ ϕ such that ∆ ⊢ L I( ϕ ) implies Γ , ∆ ⊢ L ∅ for each ∆and each I ∈ Ψ . In other words, Γ , I( ϕ ) ⊢ L ∅ for each I ∈ Ψ . Another applicationof the local IL yields the desired conclusion. (cid:3) The above equivalence may be called a local
Glivenko theorem connecting L and α Th (L): it states thatΓ ⊢ α Th (L) ϕ ⇐⇒ Γ ⊢ L { f (I)(I( ϕ )) | I ∈ Ψ } for some f : Ψ → [ <α<κ Ψ α such that f (I) ∈ Ψ | I | . For example, for FL ew we get the local Glivenko theoremΓ ⊢ α Th (FL ew ) ϕ ⇐⇒ Γ ⊢ FL ew {¬ ( ¬ ϕ n ) f ( n ) | n ∈ ω } for some f : ω → ω, while for IK we get the local Glivenko theoremΓ ⊢ α Th (IK) ϕ ⇐⇒ Γ ⊢ IK {¬ (cid:3) f ( n ) ( ¬ (cid:3) n ϕ | n ∈ ω } for some f : ω → ω. Of course, for such Glivenko theorems to be of interest we need to be able to identifythe semisimple companions of these logics. For example, for BL we get the localGlivenko theoremΓ ⊢ Ł ∞ ϕ ⇐⇒ Γ ⊢ BL {¬ ( ¬ ϕ n ) f ( n ) | n ∈ ω } for some f : ω → ω. If Γ is finite, this theorem is superseded by the much simpler equivalenceΓ ⊢ Ł ϕ ⇐⇒ Γ ⊢ BL ¬¬ ϕ, which was proved by Cignoli & Torrens [8, Theorem 2.1]. The most satisfactory results hold for logics which enjoys the global IL w.r.t.some set ∼ x , and moreover the LEM can be expressed in axiomatic form. For suchlogics, α Th (L) is the extension of L by the DDT–axiomatic or PCP–axiomatic formof the LEM, as shown in the previous subsection, and moreover we obtain a globalrather than a local form of the Glivenko theorem:Γ ⊢ α Th (L) ϕ ⇐⇒ Γ ⊢ L ∼∼ ϕ. This immediately yields the following Glivenko theorems in the substructural andintuitionistic modal cases.
Fact 4.28.
Let L be an axiomatic extension of FL n e and let L+ lem be the extensionof L by the axiom (1 ∧ p ) ∨ (1 ∧ ¬ (1 ∧ p ) n ) . Then Γ ⊢ L+ lem ϕ ⇐⇒ Γ ⊢ L ¬ (1 ∧ ¬ (1 ∧ ϕ ) n ) n . Fact 4.29.
Let L be an axiomatic extension of IKn . and let L + lem be theextension of L by the axiom (cid:3) n p ∨ (cid:3) n ¬ (cid:3) n p (or equivalently, p ∨ (cid:3) ¬ (cid:3) n p ). Then Γ ⊢ L+ lem ϕ ⇐⇒ Γ ⊢ L ¬ (cid:3) n ¬ (cid:3) n ϕ. The familiar Glivenko theorem for intuitionistic logicΓ ⊢ CL ϕ ⇐⇒ Γ ⊢ IL ¬¬ ϕ, as well as the Glivenko theorem for S4Γ ⊢ S5 ϕ ⇐⇒ Γ ⊢ S4 ♦(cid:3) ϕ, are immediate special cases (see [17, 22]). We also obtain the Glivenko theorem ofBezhanishvili [3, Thm 10] relating two different intuitionistic versions of the modallogic S5, namely the logics known as MIPC and WS5. MIPC is the logic of so-called monadic Heyting algebras, i.e. modal Heyting algebras which satisfy (cid:3)(cid:3) x ≈ (cid:3) x ≤ x , x ≤ ♦ x ≈ ♦♦ x and x ≤ (cid:3)♦ x and ♦(cid:3) x ≤ x , and WS5 adds the axiom ¬ (cid:3) ¬ x ≈ ♦ x , or equivalently 1 ≈ (cid:3) x ∨ ¬ (cid:3) x , to MIPC [2, Thm 23]. Fact 4.30.
WS5 = MIPC + lem .Proof.
If 1 ≈ (cid:3) x ∨ (cid:3) ¬ (cid:3) x , then 1 ≈ (cid:3) x ∨¬ (cid:3) x , using the axiom (cid:3) x ≤ x . Conversely,if 1 ≈ (cid:3) x ∨ ¬ (cid:3) x , then 1 ≈ x ∨ (cid:3) ¬ (cid:3) x is equivalent to ¬ (cid:3) ¬ (cid:3) x ≤ x , but indeed ¬ (cid:3) ¬ (cid:3) x = ♦(cid:3) x ≤ x . (cid:3) Theorem 4.23 now immediately yields the known fact that WS5-algebras areprecisely the semisimple monadic Heyting algebras [2, Thm 24].
Fact 4.31 ([3]) . The following Glivenko theorem connects the intuitionistic modallogics
MIPC and
WS5 : Γ ⊢ WS5 ϕ ⇐⇒ Γ ⊢ MIPC ¬¬ (cid:3) ϕ. Cignoli and Torrens prove the equivalence for theorems, but their proof extends to consequencerelations. In fact, the equivalence extends to a connection between Ł ∞ and the infinitary logicBL ∞ , defined as the logic of all BL-algebras over the unit interval [0 , ⊢ Ł ∞ ϕ ⇐⇒ Γ ⊢ BL ∞ ¬¬ ϕ , from which Γ ⊢ Ł ϕ ⇐⇒ Γ ⊢ BL ¬¬ ϕ follows. EMISIMPLICITY, GLIVENKO THEOREMS, AND THE EXCLUDED MIDDLE 45
Proof.
In view of the general Glivenko theorem for axiomatic extensions of IKn . ¬¬ (cid:3) ϕ and ¬ (cid:3) ¬ (cid:3) ϕ are equivalent in MIPC.In monadic Heyting algebras we have (cid:3) ¬ (cid:3) x ≤ ¬ (cid:3) x , thus ¬¬ (cid:3) x ≤ ¬ (cid:3) ¬ (cid:3) x . Con-versely ♦ z ≤ ¬ (cid:3) ¬ z implies that ¬ (cid:3) ¬ (cid:3) ¬ z ≤ ¬ z , so taking z := ¬ y yields ¬ (cid:3) ¬ (cid:3) y ≤¬ (cid:3) ¬ (cid:3) ¬¬ y ≤ ¬¬ y , and taking y := (cid:3) x yields ¬ (cid:3) ¬ (cid:3) x = ¬ (cid:3) ¬ (cid:3)(cid:3) x ≤ ¬¬ (cid:3) x . (cid:3) Finaly, let us describe the subclassical substructural logics which are Glivenkorelated to classical logic. This extends the Glivenko equivalence between the fuzzylogic SBL (strict BL) and classical logic CL due to Cignoli & Torrens [8, Theo-rem 2.2]. The following theorem is the only place in the paper where we use theconstant 0 in the signature of FL. Let us use the notation ¬ x := x \ ¬ x := 0 /x (the shape of the symbol indicates whether x occurs to the left or to the right ofthe slash). Theorem 4.32.
Let L be a compact logic such that FL ≤ L ≤ CL . Then thefollowing are equivalent: (1) Γ ⊢ CL ϕ ⇐⇒ Γ ⊢ L ¬¬ ϕ for all Γ and ϕ , (2) Γ ⊢ CL ϕ ⇐⇒ Γ ⊢ L ¬¬ ϕ for all Γ and ϕ , (3) Γ ⊢ CL ¬ ϕ ⇐⇒ Γ ⊢ L ¬ ϕ for all Γ and ϕ , (4) Γ ⊢ CL ¬ ϕ ⇐⇒ Γ ⊢ L ¬ ϕ for all Γ and ϕ , (5) Γ , ϕ ⊢ L ⇐⇒ Γ ⊢ L ¬ ϕ for all Γ and ϕ , and ¬ ( p ∧ ¬ q ) ⊢ L ¬¬ ( p \ q ) . (6) Γ , ϕ ⊢ L ⇐⇒ Γ ⊢ L ¬ ϕ for all Γ and ϕ , and ¬ ( p ∧ ¬ q ) ⊢ L ¬¬ ( q/p ) .Proof. We prove the equivalence of (1), (3), and (5). The equivalence of (2), (4),and (6) is proved in an entirely analogous manner, and (3) and (4) are equivalentbecause ¬ p ⊢ FL ¬ p and ¬ p ⊢ FL ¬ p .(1) ⇒ (3): if Γ ⊢ CL ¬ ϕ , then Γ ⊢ CL ¬ ϕ , therefore Γ ⊢ L ¬¬¬ ϕ by (1). But ¬ p ⊢ FL ¬ p and ¬¬¬ p ⊢ FL ¬ p , therefore Γ ⊢ L ¬¬¬ ϕ and Γ ⊢ L ¬ ϕ . The rule ¬ p ⊢ FL ¬ p now yields Γ ⊢ L ¬ ϕ . Conversely, Γ ⊢ L ¬ ϕ implies Γ ⊢ CL ¬ ϕ because Lis subclassical.(3) ⇒ (5): the rule in question is valid in CL, thus it is valid in L by (3).If Γ ⊢ L ¬ ϕ , then Γ , ϕ ⊢ L p, ¬ p ⊢ FL
0. Conversely, if Γ , ϕ ⊢ L
0, thenΓ , ϕ ⊢ CL
0, hence Γ ⊢ CL ¬ ϕ , and Γ ⊢ L ¬ ϕ by (3).(5) ⇒ (3): let L be the extension of L by the rule 0 ⊢ ⊥ . We claim thatΓ ⊢ L ⇐⇒ Γ ⊢ L ∅ . The left-to-right direction is trivial, and conversely either the proof of Γ ⊢ L ⊥ doesnot use the rule 0 ⊢ ⊥ , in which case Γ ⊢ L ⊥ ⊢ L
0, or it does use the rule, in whichcase Γ ⊢ L
0. Now if α Th (L ) = CL, thenΓ ⊢ CL ϕ ⇐⇒ Γ , ¬ ϕ ⊢ CL ∅ by the IL of CL ⇐⇒ Γ , ¬ ϕ ⊢ L ∅ because α Th (L ) = CL ⇐⇒ Γ ¬ ϕ ⊢ L ⇐⇒ Γ ⊢ L ¬¬ ϕ by (5) . It will therefore suffice to prove that α Th (L ) = CL. The logic L enjoys the global IL of intuitionistic logic: in one direction, Γ ⊢ L ¬ ( ϕ ∧ · · · ∧ ϕ k ) implies Γ , ϕ , . . . , ϕ k ⊢ L ∅ , since 0 ⊢ L ⊥ . Conversely,Γ , ϕ , . . . , ϕ k ⊢ L ∅ = ⇒ Γ , ϕ , . . . , ϕ k ⊢ L ⇒ Γ , ϕ ∧ · · · ∧ ϕ k ⊢ L ⇒ Γ ⊢ L ¬ ( ϕ ∧ · · · ∧ ϕ k ) by (5)= ⇒ Γ ⊢ L ¬ ( ϕ ∧ · · · ∧ ϕ k ) . The logic α Th (L ) enjoys the classical global IL with respect to the same IL familyby Proposition 4.13, therefore it enjoys the global DDT in the formΓ , ϕ ⊢ α Th (L ) ψ ⇐⇒ Γ ⊢ α Th (L ) ¬ ( ϕ ∧ ¬ ψ ) . Since L is compact and enjoys a finitary global IL w.r.t. a family of finite sets,the assumed rule ¬ ( ¬ p ∧ q ) ⊢ L ¬¬ ( p \ q ) yields that ¬ ( p ∧ ¬ q ) ⊢ α Th (L ) p \ q by theGlivenko theorem connecting L and α Th (L ), i.e. by Proposition 4.27. Due to thisrule, α Th (L ) enjoys the DDT of intuitionistic logic:Γ , ϕ ⊢ α Th (L ) ψ ⇐⇒ Γ ⊢ α Th (L ) ϕ \ ψ. This DDT immediately implies that the axioms of Exchange, Contraction, andWeakening, i.e. the formulas ( p \ ( q \ r )) \ ( q \ ( p \ r ), ( p \ ( p \ q )) \ ( p \ q ) , and p \ ( q \ p ), aretheorems of α Th (L ). Since adding these axioms to FL yields intuitionistic logic,IL ≤ L ≤ CL. But each non-trivial theory of IL extends to a simple theory of IL,which is a theory of CL and therefore also a theory of L . It follows that the simpletheories of IL and L coincide, therefore α Th (L ) = α Th (IL) = CL. (cid:3) The associativity of FL was not used in the above proof, therefore the sametheorem in fact holds for extensions of the non-associative Full Lambek calculus.
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Institute of Information Theory and Automation, Czech Academy ofSciences, Pod Vod´arenskou vˇeˇz´ı 4, Praha 8, 182 00, Czechia
Email address : lavicka.thomas@@gmail.com (Adam Pˇrenosil) Department of Mathematics, Vanderbilt University, 1326 StevensonCenter, Nashville, TN 37240, USA
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