Lattices of Intermediate Theories via Ruitenburg's Theorem
aa r X i v : . [ m a t h . L O ] A p r Lattices of Intermediate Theories viaRuitenburg’s Theorem ∗ Gianluca Grilletti and Davide Emilio Quadrellaro
Institute for Logic, Language and Computation (ILLC), Amsterdam, The NetherlandsDepartment of Mathematics and Statistics, University of Helsinki, Finland [email protected] [email protected]
Abstract
For every univariate formula χ we introduce a lattices of intermediate theories: the latticeof χ -logics. The key idea to define χ -logics is to interpret atomic propositions as fixpoints ofthe formula χ , which can be characterised syntactically using Ruitenburg’s theorem. Wedevelop an algebraic duality between the lattice of χ -logics and a special class of varieties ofHeyting algebras. This approach allows us to build five distinct lattices—corresponding tothe possible fixpoints of univariate formulas—among which the lattice of negative variantsof intermediate logics. We describe these lattices in more detail. This paper introduces a family of lattices of intermediate theories, building on three resultsfrom the literature: the duality between intermediate logics and varieties of Heyting algebras, anovel algebraic semantics for inquisitive logic and negative variants, and Ruitenburg’s Theorem.Intermediate logics ([5, 11]) are classes of formulas closed under uniform substitution andmodus ponens, lying between intuitionistic logic (
IPC ) and classical logic (
CPC ). This familyof logics has been studied using several semantics, as for example Kripke semantics, Bethsemantics, topological semantics and algebraic semantics (for an overview see [1]). Among these,the algebraic semantics based on Heyting algebras plays a special role: every intermediate logicis sound and complete with respect to some class of Heyting algebras. This connection between intermediate logics and Heyting algebras has been studied usingtools from universal algebra. As a consequence of Birkhoff’s Theorem ([3, 4]), the lattice ofvarieties of Heyting algebras HA is dually isomorphic to the lattice of intermediate logics IL .This result allows to characterise properties of intermediate logics in terms of properties of thecorresponding variety, and viceversa.Inquisitive logic InqB ([9, 18, 7, 8]) is an extension of classical logic that encompasses logicalrelation between questions in addition to statements. The logic was originally defined throughthe support semantics , a generalisation of the standard truth-based semantics of
CPC . Ciardelliet al. gave an axiomatisation of the logic, showing that it sits between
IPC and
CPC , andhighlighting connections with other intermediate logics such as Maksimova’s logic ND , Kreisel-Putnam logic KP and Medvedev’s logic ML ([6]). However, InqB itself is not an intermediatelogic, since it is not closed under uniform substitution. ∗ The authors would like to thank Nick Bezhanishvili for comments and discussions on this work. The firstauthor was supported by the European Research Council (ERC, grant agreement number 680220). The secondauthor was supported by Research Funds of the University of Helsinki. Kripke semantics is known to be incomplete for some intermediate logics. However, it is still an openproblem whether this hold for Beth and topological semantics ([13, 1]).attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro
An algebraic semantics for
InqB has been defined in [2], based on the corresponding algebraicsemantics for intermediate logics. The key idea is to restrict the interpretation of atomicpropositions to range over regular elements of a Heyting algebra, that is, over fixpoints ofthe operator ¬¬ . This restriction allows to have a sound and complete algebraic semantics,despite the failure of the uniform substitution principle. As shown in [16], this approach canbe extended to the class of DNA -logics, also known as negative variants of intermediate logics ([14, 12]). Moreover, this leads naturally to a duality between
DNA -logics and a special class ofvarieties, analogous to the one for intermediate logics.Ruitenburg’s theorem ([19]) concerns sequences of formulas of the following kind: α := p α n +1 := α [ α n / p ]where α is a formula and p is a fixed atomic proposition. In particular, Ruitenburg’s theoremstates that this sequence is definitively periodic with period 2—modulo logical equivalence. Forexample, if we take α := ¬ p we can see that ¬ p ≡ ¬¬¬ p , showing that ¬ p is a fixpoint of theoperator ¬¬ . Ghilardi and Santocanale give an alternative proof of this result in [20], studyingendomorphisms of finitely generated Heyting algebras. This proof makes use of the dualityintroduced above and it highlights the relevance of the algebraic interpretation of Ruitenburg’sTheorem.In this paper we use Ruitenburg’s theorem and its algebraic interpretation to define a latticeof intermediate theories in the same spirit as the negative variants. Fixed a univariate formula χ ,we define an algebraic semantics by restricting valuations to range over fixpoints of the formula χ —which can be characterised using Ruitenburg’s Theorem. This allows us to build the latticeof χ -logics, intermediate theories characterised in terms of the fixpoint-axiom χ ( p ) ↔ p . Weshow that the algebraic semantics is sound and complete for these logics; and we developed aduality theory for these logics analogous to the one for negative variants. We also show thatthere are only six possible fixpoints for univariate formulas: ⊤ , p, ¬ p, ¬¬ p, p ∨¬ p, ⊥ . This allowsus to characterise and describe all the possible lattices of χ -logics built using this approach.In Section 2 we introduce some preliminary notions on intermediate logics and their algebraicsemantics, the theory of algebraic duality for such logics and Ruitenburg’s Theorem. In Section 3we define χ -logics and give a brief overview of their main properties that can be derived in purelysyntactic terms. In Section 4, fixed a formula χ , we introduce a novel algebraic semantics for χ -logics based on Ruitenburg’s Theorem and we define a notion of variety of Heyting algebrassuitable to study χ -logics, namely χ -varieties. In Section 5 we develop an algebraic dualitytheory for these logics, showing that the lattice of χ -logics is dually isomorphic to the lattice of χ -varieties. Finally, in Section 6 we show there are only 5 distinct lattices of χ -logics for anyunivariate formula χ , we describe their properties in more detail and we study the relationsbetween them. Conclusions and possible directions for future work are presented in Section 7. In this Section we summarise the theory from the literature used throughout the paper.
Fix an infinite set AP of atomic propositions and consider the set of formulas L generated bythe following grammar: φ ::= p | ⊥ | φ ∧ φ | φ ∨ φ | φ → φ attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro where p ∈ AP. As usual, we will introduce the shorthand ¬ φ := φ → ⊥ for negation . Hence-forth we will leave the sets AP and L implicit, referring to atomic propositions from AP andto formulas from L simply as atomic propositions and formulas respectively. To indicate a se-quence of propositions h p , . . . , p n i we will often use the notation p , and similarly for sequencesof formulas ( φ = h φ , . . . , φ n i ) and sequences of other objects.Consider formulas φ, ψ and an atomic proposition p . We will indicate with φ [ ψ / p ] the formulaobtained by substituting every occurrence of p in φ with the formula ψ . More generally, given ψ = h ψ , . . . , ψ n i a sequence of formulas and p = h p , . . . , p n i a sequence of distinct atomicpropositions, we will indicate with φ [ ψ / p ] the formula obtained by substituting simultaneously each p i with ψ i . With abuse of notation, when we take χ a univariate formula—that is, aformula with only one free variable—we will indicate the sequence h χ ( p ) , . . . , χ ( p n ) i with thenotation χ ( p ); for example, the notations φ [ ¬ p / p ] and φ [ h¬ p , . . . , ¬ p n i / h p , . . . , p n i ] indicate thesame formula.We will refer to the intuitionistic propositional calculus (see for example [5, 11]) as IPC .With slight abuse of notation, we will write
IPC also to refer to the set of validities of thiscalculus.An intermediate logic ([5, 11]) is a set of formulas L with the following properties:1. IPC ⊆ L ⊆ CPC ;2. L is closed under modus ponens : If φ ∈ L and φ → ψ ∈ L , then ψ ∈ L ;3. L is closed under uniform substitution : If φ ∈ L and ρ is a substitution, then φ [ ρ ] ∈ L .Given Γ a set of formulas, we indicate with MP(Γ) the smallest set of formulas extending Γ andclosed under modus ponens; and with US(Γ) the smallest set of formulas extending Γ and closedunder uniform substitution. It is immediate to prove such sets always exist and that if Γ ⊆ CPC ,then MP(US(Γ)) is the smallest intermediate logic extending Γ: we will call MP(US(Γ)) theintermediate logic generated by Γ.Intermediate logics form a structure of bounded lattice under the set-theoretic inclusion. Inparticular, the meet and join operations are L ∧ L := L ∩ L and L ∨ L := MP( L ∪ L ),and IPC and
CPC are the minimum and maximum respectively. We will refer to this lattice withthe notation IL .In the literature, several semantics have been proposed to study these logics: Kripke se-mantics, Beth semantics and topological semantics are some famous examples (see [1] for anoverview of some well-known semantics). In this paper we will focus on the so-called algebraicsemantics : given an Heyting algebra H and a function V : AP → H —which we will refer to asa valuation —we define recursively by the following clauses the interpretation J φ K HV of a formula φ in H under V . J p K HV = V ( p ) J ⊤ K HV = 1 H J ⊥ K HV = 0 H J φ ∧ ψ K HV = J φ K HV ∧ H J ψ K HV J φ ∨ ψ K HV = J φ K HV ∨ H J ψ K HV J φ → ψ K HV = J φ K HV → H J ψ K HV where 1 H , H , ∧ H , ∨ H , → H indicate the constants and operations of the algebra H . We saythat φ is true in H under V and we write ( H, V ) (cid:15) φ if J φ K HV = 1. We say that φ is valid in H and we write H (cid:15) φ if it is true in H under any valuation V .As a shorthand, we will indicate with [ p a , . . . , p n a n ] a generic valuation V suchthat V ( p i ) = a i —without specifying its value on the atomic formulas different from p , . . . , p n .3 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro Given a function f : H n → H we will call it a polynomial if it is obtained by composing thefunctions 1 H , H , ∧ H , ∨ H and → H (where we identify the constants 1 H and 0 h with the corre-sponding 0-ary functions). Given a formula φ ( p , . . . , p n ), we can associate to it the polynomial φ (indicated with the bold font) defined as: φ : H n → Ha J φ K H [ p a ,...,p n a n ] Moreover, it is immediate to show that for every polynomial f , there exists a (non-unique)formula φ such that f = φ . In the study of intermediate logics, a special role is played by the collection of those algebras defined by a certain intermediate logic. Given L ∈ IL , define the variety generated by L as theset Var ( L ) := { H ∈ HA | ∀ φ ∈ L. H (cid:15) φ } where HA indicates the class of all Heyting algebras. We will call a class V ⊆ HA a variety if V is closed under the operations H , S , P defined over subclasses of HA as follows:H( C ) := { H ∈ HA | ∃ A ∈ C . A ։ H } (homomorphic images)S( C ) := { H ∈ HA | ∃ A ∈ C . H ֒ → A } (subalgebras)P( C ) := ( Y i ∈ I A i ∈ HA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∀ i ∈ I. A i ∈ C ) (products)It is easy to prove that Var ( L ) is indeed a variety; moreover the following well-known resultsgive us an alternative characterisation of varieties: Theorem 1 (Tarski’s theorem; [21], Theorem 9.5 in [4]) . Given
C ⊆ HA a class of algebras, V ( C ) := HSP( C ) is the smallest variety containing C . Theorem 2 (Birkhoff’s theorem; [3], Theorem 11.9 in [4]) . A class of algebras
C ⊆ HA is avariety iff it is equationally definable, that is, there exists a set of formulas F ⊆ L such that C = { H ∈ HA | ∀ φ ∈ F. H (cid:15) φ } . Given these results, it is easy to show that varieties form a bounded distributive lattice underthe inclusion order. In particular, the meet and join operations are V ∧ V := V ∩ V and V ∨ V := V ( V ∪ V ). We will refer to this lattice with the notation HA . Moreover, given a variety V we can define a set of formulas that characterises it: Log ( V ) := { φ ∈ L | ∀ H ∈ V . H (cid:15) φ } It is easy to prove that
Log ( V ) is an intermediate logic; and that—using Birkhoff’s theorem— L = Log ( Var ( L )) and V = Var ( Log ( V )) for every intermediate logic L and variety V . Moreover,since Var and
Log are antitone maps, these maps are dual isomorphisms between the lattice IL and the lattice HA : Theorem 3 (Duality; Theorem 7.54 in [5]) . The lattice of intermediate logics is dually isomor-phic to the lattice of varieties of Heyting algebras, i.e. IL ∼ = op HA . Notice the difference between HA (the class of all Heyting algebras) and HA (the lattice of varieties ofHeyting algebras). In particular HA ∈ HA . attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro For the remainder of this Section, we will indicate with p a fixed atomic proposition. Let φ ( p, q )be a formula, where p, q contain all the atomic propositions appearing in φ . A folklore resultsays that the formulas φ ( p, q ) φ ( p, q ) := φ ( φ ( φ ( p, q ) , q ) , q )are equivalent in classical logic. Ruitenburg extended this result to intuitionistic logic ([19]). Definition 4.
Given φ ( p, q ) a formula, define the formulas { φ n ( p, q ) } n ∈ N recursively as follows: φ ( p, q ) := p φ n ( p, q ) := φ ( φ n − ( p, q ) , q )That is, φ n is obtained by substituting φ n − for p in φ . Theorem 5 (Ruitenburg’s theorem; [19]) . For every formula φ ( p, q ) , the sequence φ , φ , φ , . . . is—modulo logical equivalence—definitely periodic with period . That is, there exists a naturalnumber n such that: φ n ↔ φ n +2 ∈ IPC (1)We will call the smallest n for which Equation 1 holds the Ruitenburg index (or simply the index ) of φ . Moreover, we will call φ n the fixpoint of the formula φ .We can also see Ruitenburg’s result as an algebraic fixpoint theorem. Let A be a Heytingalgebra, a a sequence of elements in A and f ( x, y ) a polynomial. Then a consequence ofRuitenburg’s theorem is that the operator f ( x, a ) = f ( f ( x, a ) , a ) admits a fixpoint. Andindeed, this is an equivalent formulation of Theorem 5, as can be easily shown by applying itto the Lindebaum-Tarski algebra of IPC .As proven by Ruitenburg (Example 2.5 in [19]), there is no uniform bound for all theformulas φ , but each formula admits an index. However, for some classes of formulas we canfind a uniform bound: Lemma 6 (Proposition 2.3 in [19]) . If χ ( p ) is a univariate formula, then χ ↔ χ ∈ IPC . Moreover, the fixpoint of χ ( p ) is equivalent to one of the following formulas: ⊥ , p , ¬ p , ¬¬ p , p ∨ ¬ p , ⊤ . We give an elementary proof of this result in Appendix A, different from the original one givenby Ruitenburg in [19]. χ -logics In analogy with the case of negation, given a formula χ we are interested in logics arising byinterpreting atoms as fixpoints for the operator χ : we will call these logics χ -logics .In this paper, we start the study of this family of logics by considering only univariateformulas for two reasons: the presence of additional atoms requires a generalisation of theduality results presented in Section 2 to Heyting algebras with constants; and restricting ourattention to univariate formulas allows us to give a more in-depth description of all the latticesof χ -logics generated through this procedure—which are finitely many, as shown at the end ofthis section. Definition 7 ( χ -logic) . Let χ ( p ) be a univariate formula and Γ a set of formulas. We definethe χ -logic generated by Γ as the smallest set of formulas Γ χ with the following properties: 5 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro IPC ⊆ Γ χ ;2. If φ ∈ Γ and σ is a substitution, then φ [ σ ] ∈ Γ χ ;3. χ ( p ) ↔ p ∈ Γ χ for every atomic proposition p ;4. Γ χ is closed under modus ponens : if φ ∈ Γ χ and φ → ψ ∈ Γ χ , then ψ ∈ Γ χ .Condition 3 requires atoms to behave as fixpoints of the operator χ . This condition togetherwith uniform substitution would impose that all formulas behave like fixpoints, which is arequirement too strong for our purposes (we will see later that χ -logics are generally not closedunder uniform substitution). That is why we require the uniform substitution principle onlyfor formulas in Γ, that is, Condition 2.Notice that we can interpret Γ χ as the set of valid formulas of a Hilbert-style deductive sys-tem where Conditions 1, 2 and 3 define the axioms—to be more precise, the axiom schemata—while Conditions 4 specifies modus ponens as the only rule of the system. This suggests thefollowing characterisation of χ -logics. Lemma 8.
Let L be the intermediate logic generated by Γ . Then Γ χ = L χ .Proof. The left-to-right containment is immediate, since the operator ( − ) χ is monotone. As forthe other containment, notice that Conditions 1, 2 and 4 impose that L ⊆ Γ χ , from which theresult follows.So we can think of χ -logics as always generated by a corresponding intermediate logic insteadof a generic set of formulas. To stress this point, given an intermediate logic L we will call L χ the χ -variant of L . Notice that a direct consequence of Lemma 8 is that any set satisfyingConditions 2, 3 and 4 is the χ -variant of some intermediate logic L .Restricting our attention to intermediate logics allows us to give an alternative characteri-sation of χ -logics. Lemma 9.
Let χ ( p ) be a univariate formula and n be its index. Given L an intermediate logic,we have L χ = { φ ( p ) | φ [ χ n ( p ) / p ] ∈ L } . Proof.
Call the set on the right-hand side M .Firstly, we will show that M satisfies the conditions in Definition 7. Since L contains IPC and is closed under modus ponens and uniform substitution, we easily obtain Conditions 1, 2and 4. As for Condition 3, since n is the index of χ , we have χ n +2 ( p ) ↔ χ n ( p ) ∈ L , from whichit follows χ ( p ) ↔ p ∈ M for every atomic proposition p .Secondly, we need to show that M is the smallest set satisfying these conditions. To do so,we will use the following fact: given formulas α = h α , . . . , α l i , β = h β , . . . , β l i , γ formulas anddistinct atomic propositions q = h q , . . . , q l i , we have ^ i ≤ l ( α i ↔ β i ) → ( γ [ α / q ] ↔ γ [ β / q ] ) ∈ IPC ⊆ X Consider now a set X satisfying the conditions of Definition 7.Since χ ( q ) ↔ q ∈ X for every q and X is closed under uniform substitution, it follows thatalso χ ( q ) ↔ χ ( q ) ∈ X . Moreover, since( α ↔ β ) → ( ( β ↔ γ ) → ( α ↔ γ ) ) ∈ IPC ⊆ X attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro by closure under modus ponens we obtain that χ ( q ) ↔ q ∈ X . Iterating this reasoning, weobtain that χ n ( q ) ↔ q ∈ X for every q ; or χ n +1 ( q ) ↔ q for every q —depending on the parityof n . Assume the former is the case; the treatment of the other case is analogous.Consider now a formula φ ( p ) with p = h p , . . . , p l i . Combining the previous facts we get: χ n ( p i ) ↔ p i ∈ X for every i ≤ l and V i ≤ l ( χ n ( p i ) ↔ p i ) → ( φ [ χ n ( p ) / p ] ↔ φ ( p ) ) ∈ X implies φ [ χ n ( p ) / p ] ↔ φ ( p ) ∈ X Suppose now φ [ χ n ( p ) / p ] ∈ L . Since α → ( ( α ↔ β ) → β ) ∈ IPC ⊆ X and X is closed under modus ponens, it follows that φ ( p ) ∈ X ; and consequently M ⊆ X .So M is the smallest set satisfying the Conditions in Definition 7, thus proving L χ = M aswanted.It is interesting to notice that instances of χ -logics have already been studied in the literature:an example is inquisitive logic InqB . In fact, reinterpreting Theorem 3.4.9 in [6], we obtain thefollowing:
Theorem 10 (Ciardelli; Theorem 3.4.9 in [6]) . KP ¬ p = ND ¬ p = ML ¬ p = InqB
One can easily show that the set L χ in general is not closed under uniform substitution . Nonethe-less, the set L χ is closed under a weaker notions of substitution, that is, atomic substitution :give σ ∈ AP → AP a permutation of the atomic propositions, if φ ( p ) ∈ L χ then φ ( σ ( p )). Thatis to say, even though atomic propositions play a special role—fix-points of χ —they are stillconsidered as generic entities, in that they are indistinguishable from one another. Moreover, asnoted by Iemhoff and Yang in [12], the logics L ¬ p are closed under a more general substitutionprinciple, that is, classical substitutions : A classical substitution maps every atomic propositionwith a ∨ -free formula. In general, it is expected that for a fixed χ the logics L χ are closed undermore general substitution principles.We can show that χ -logics form a bounded distributive lattice under the set-theoretic con-tainment, as it was the case for intermediate logics. In particular, the meet operation is givenby set-theoretic intersection and the join operation by the closure under modus ponens of theunion—in complete analogy with the case of intermediate logics. Lemma 11.
Given χ a univariate formula and L, M two intermediate logics we have: L χ ∧ M χ = L χ ∩ M χ = ( L ∧ M ) χ L χ ∨ M χ = MP( L χ ∪ M χ ) = ( L ∨ M ) χ Proof.
We consider only the second set of identities, as the proof can be easily adapted for thefirst set. Firstly, notice that L χ , M χ ⊆ ( L ∨ M ) χ . Moreover, since L ⊆ L χ and M ⊆ M χ , forevery χ -logic Λ such that L χ , M χ ⊆ Λ it holds L ∪ M ⊆ Λ; and since χ -logics are closed undermodus ponens it holds L ∨ M = MP( L ∪ M ) ⊆ Λ. So in particular ( L ∨ M ) χ ⊆ Λ. This impliesthat ( L ∨ M ) χ is the least upper bound of L χ and M χ , that is, L χ ∨ M χ = ( L ∨ M ) χ .Secondly, notice that MP( L χ ∪ M χ ) is the χ -logic generated by the set of formulas L χ ∪ M χ .So in particular, since L χ , M χ ⊆ MP( L χ ∪ M χ ), we also have L χ ∨ M χ ⊆ MP( L χ ∪ M χ ).Moreover, since ( L ∨ M ) χ is closed under modus ponens and L χ ∪ M χ ⊆ ( L ∨ M ) χ , it follows7 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro MP( L χ ∪ M χ ) ⊆ ( L ∨ M ) χ = L χ ∨ M χ . From this we conclude that L χ ∨ M χ = MP( L χ ∪ M χ ),as wanted.We will indicate with IL χ the lattice of χ -logics. Notice that the previous proof shows alsothat the mapping L L χ is a lattice morphism.In the next sections we will study the structure of this lattice employing some tools fromalgebraic semantics. But before moving to that, we will tackle one last question in this Section:how many lattices are we dealing with?As noted in Lemma 6, there are only a finite amount of fixpoints, from which the followingresult follows readily. Theorem 12.
There are only 6 Ruitenburg-fixpoints of univariate intuitionistic formulas: ⊥ , p , ¬ p , ¬¬ p , p ∨ ¬ p and ⊤ . Notice also that IL ¬ p = IL ¬¬ p are the same lattice: this follows from Lemma 9, since L ¬ p = { φ ( q ) | φ [ ¬ q / q ] } = { φ ( q ) | φ [ ¬¬ q / q ] } = L ¬¬ p for every intermediate logic L . So in total we are working with only 5 lattices, associated tothe 6 fix-points above. In Section 6 we will see that these are indeed distinct lattices. IL ⊥ IL p = IL IL ¬ p = IL ¬¬ p IL p ∨¬ p IL ⊤ In this section we shall provide a semantic interpretation of χ -logics, by generalizing the alge-braic semantics for inquisitive logic presented in [2] and further developed in [16]. The key togeneralise the algebraic semantics to this context lies in an algebraic interpretation of Ruiten-burg’s Theorem. In this section we will fix a univariate formula χ with index n . H χ [ H ] · · · χ n [ H ]As noted in Section 2, given a Heyting algebra H we can define a polynomial corresponding to χ : χ : H → Ha J χ ( p ) K [ p a ] Ruitenburg’s Theorem tells us that the sequence H, χ [ H ] , χ [ H ] := χ [ χ [ H ]] , . . . is definitely con-stant; and that χ restricted to the set χ n [ H ] isan involution . Henceforth we will call the set H χ := χ n [ H ] the χ -core (or simply core when χ is clear from the context) of H . Notice that the χ -core consists exactly of the fixpoints of χ : Lemma 13. H χ is the set of fixpoints of χ .Proof. By Theorem 5, we have that χ n ≡ χ n +2 . Consider now an element in a ∈ H χ , that is,an element of the form a = b n for some b ∈ H . It follows that χ ( a ) = χ ( χ n ( b )) = χ n +2 ( b ) = χ n ( b ) = a attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro showing that a is a fixpoint of χ . Conversely, let a be a fixpoint for χ . Then it follows that a = χ ( a ) = χ ( χ ( a )) = χ ( a ) = . . . = χ n ( a ) ∈ H χ . For instance, when χ ( p ) = ¬ p the core H ¬ p of H consists of the regular elements of the algebra H , that is, fixpoints of the operator ¬¬ .To obtain an adequate semantics for a χ -logic, it is sufficient to restrict the valuations ofatomic propositions to the core H χ . Let AP be an arbitrary set of atomic propositions. We saythat a valuation σ : AP → H is a χ -valuation if σ [AP] ⊆ H χ . A χ -valuation over H thus sendsevery atomic proposition to some element of the χ -core of H . Algebraic models of χ -logics arethen defined as follows. Definition 14 ( χ -Model) . A χ -model is a pair M = ( H, σ ) such that H is a Heyting algebraand σ is a χ -valuation.The interpretation of a formula φ ∈ L in M = ( H, σ ), in symbols J φ K Hσ , can then be easilydefined recursively, as in the standard algebraic semantics over Heyting algebras: J p K Hσ = σ ( p ) J ⊤ K Hσ = 1 H J ⊥ K Hσ = 0 H J φ ∧ ψ K Hσ = J φ K Hσ ∧ H J ψ K Hσ J φ ∨ ψ K Hσ = J φ K Hσ ∨ H J ψ K Hσ J φ → ψ K Hσ = J φ K Hσ → H J ψ K Hσ where 1 H , H , ∧ H , ∨ H , → H indicate the constants and operations of the algebra H .The key point is that for atomic propositions J p K Hσ = σ ( p ) ∈ H χ , which means that theinterpretation of every atomic proposition is a fixpoint for χ . We say that φ is true in H under σ and we write ( H, σ ) (cid:15) χ φ if J φ K Hσ = 1. We say that φ is valid in H and we write H (cid:15) χ φ if itis true in H under any χ -valuation σ .The algebraic semantics we have introduced differs from the standard semantics of inter-mediate logics in the fact that we consider only a particular class of valuations for the atomicpropositions, namely χ -valuations. The relation between validity at a Heyting algebra and χ -validity is further clarified by the following results. We define, for every valuation V : AP → H ,its χ -variant V χ as the χ -valuation V χ : AP → H χ such that V χ ( p ) = χ n ( V ( p )). With a simpleinduction we can show the following connection between V and V χ : J φ K HV χ = J φ [ χ n ( p ) / p ] K HV .Notice that, since H χ is the image of χ n , every χ -valuation is the χ -variant of some valuation.In fact, given σ a χ -valuation, σ ( p ) ∈ H χ = χ n [ H ]. So, for any valuation V such that V ( p ) ∈ ( χ n ) − ( σ ( p )), we have V χ ( p ) = χ n ( V ( p )) = σ ( p ). This allows us to prove the following Lemmaconnecting validity in the standard algebraic sense and in the context of χ -logics: Proposition 15.
For any Heyting algebra H , H (cid:15) χ φ if and only if H (cid:15) φ [ χ n ( p ) / p ] .Proof. Assume H φ [ χ n ( p ) / p ], that is, there exists a valuation V such that J φ [ χ n ( p ) / p ] K HV = 1 H .Considering the χ -valuation V χ , we then have J φ K HV χ = J φ [ χ n ( p ) / p ] K HV = 1 H , and thus H χ φ .Conversely, assume H χ φ , that is, there exists a χ -valuation σ such that J φ K Hσ = 1 H .As noted above, for some valuation V we have σ = V χ ; from this we obtain J φ [ χ n ( p ) / p ] K HV = J φ K HV χ = 1 H , and thus H φ [ χ n ( p ) / p ].Combining Lemma 9 with the previous Proposition, we obtain the following Corollary. Corollary 16.
Let H be a Heyting algebra and L an intermediate logic, if H (cid:15) L then H (cid:15) χ L χ . attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro The converse of the previous Corollary does not hold in general, as a formula might be true ina Heyting algebra under all χ -valuations but not under all valuations. The next proposition isa weaker version of this converse: Let h H χ i be the subalgebra of H generated by the χ -core H χ ; we say that H is core generated if H = h H χ i . Lemma 17.
Let H be a Heyting algebra, H (cid:15) χ φ if and only if h H χ i (cid:15) χ φ .Proof. Since h H χ i χ = H χ and by the fact that h H χ i is a subalgebra of H , it follows that J ψ K h H χ i σ = J ψ K Hσ , for any formula ψ , from which the result follows. Proposition 18.
Let H be a Heyting algebra and L an intermediate logic. Then we have that H (cid:15) χ L χ entails h H χ i (cid:15) L .Proof. Consider any Heyting algebra H , and suppose that h H χ i L , then there is some formula φ ∈ L and some valuation V such that ( h H χ i , V ) φ . Now, since h H χ i is the subalgebragenerated by H χ , we can express every element x ∈ h H χ i as a polynomial δ x of elements of H χ . We thus have x = δ x ( y ), where for each y i we have that y i ∈ H χ . By writing p = p , ..., p n for the variables contained in φ and δ ( y ) for the sequence of polynomials corresponding to theelements x = V ( p ) , ..., x n = V ( p n ), we get that J φ ( p ) K h H χ i V = φ ( δ ( y )). Since all the elements y in the polynomials δ x are fixed points of χ , we can define a χ -valuation σ : AP → H χ suchthat σ : q i y i for all i ≤ n . Then it follows immediately that J φ [ δ ( q ) / p ] K h H χ i σ = φ ( δ ( y )).But then, since we also had J φ ( p ) K h H χ i V = φ ( δ ( y )), it follows that J φ [ δ ( q ) / p ] K h H χ i σ = J φ ( p ) K h H χ i V .So since ( h H χ i , V ) φ , we also get that ( h H χ i , σ ) χ φ [ δ ( y ) / p ]. So it then follows by Lemma17 that H χ φ [ δ ( q ) / p ]. Now, since L is an intermediate logic, it admits free substitution andso, since φ ∈ L , we also get that φ [ δ ( q ) / p ] ∈ L ⊆ L χ . Finally, this means that H χ L χ , thusproving our claim.Finally, the former results motivate the introduction of suitable χ -varieties, which we willshow being the correct semantic counterpart to χ -logics. Let V be an arbitrary variety ofHeyting algebras, then its χ -closure is the class: V χ = { K ∈ HA | ∃ H. H ∈ V and H χ = K χ } . We say that a Heyting algebra K is a core superalgebra of H if H χ = K χ and H (cid:22) K . Wesay that X is χ -variety if X = V χ for some variety V of Heyting algebras. We then prove thefollowing result which characterises χ -varieties. Theorem 19.
A class of Heyting algebras C is a χ -variety if and only if it is closed undersubalgebras, homomorphic images, products and core superalgebras.Proof. ( ⇐ ) Suppose C is closed under subalgebras, homomorphic images, products and coresuperalgebras. Obviously C is a variety and for any Heyting algebra H such that there issome K ∈ C with H χ = K χ and K (cid:22) H , it follows by closure under core superalgebra that H ∈ C . Therefore, it follows that C = C χ , hence C is a χ -variety. ( ⇒ ) Suppose C is a χ -variety,i.e. C = V χ for some variety V . We show that C is closed under subalgebras, as the othercases follow by an analogous reasoning. Suppose H ∈ C and K (cid:22) H . Since C = V χ there issome H ′ ∈ V such that ( H ′ ) χ = H χ and H ′ (cid:22) H . Then consider K ′ = K ∩ H ′ . Since theintersection of two subalgebras is still a subalgebra and since K ′ ⊆ H ′ , it follows that K ′ (cid:22) H ′ and therefore K ′ ∈ V . Moreover, by a similar reasoning we have that K ′ (cid:22) K . Finally, since( K ′ ) χ = K χ ∩ ( H ′ ) χ and ( H ′ ) χ = H χ , we have ( K ′ ) χ = K χ ∩ H χ = K χ . Therefore, by thefact that K ′ (cid:22) K , ( K ′ ) χ = K χ and K ∈ V , we obtain that K ∈ V χ = C .10 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro It is then easy to show that χ -varieties form a bounded lattice with operations X ∧X := X ∩X and X ∨ X := X ( X ∪ X ), where X ( C ) denotes the smallest χ -variety containing C . We shalldenote the lattice of χ -varieties by HA χ . One can show that the map V 7→ V χ is a latticehomomorphism. Together with the results of the previous sections, we have thus obtained alattice IL χ of χ -variants of intermediate logics, and a lattice HA χ of χ -varieties. In the nextsection we shall see how to relate these two structures in order to prove the completeness of thealgebraic semantics we introduced. In this section we shall show that the lattice of χ -logics IL χ and the lattice of χ -varieties HA χ are dual to each other. We prove this result by relying on the standard dual isomorphismbetween the lattice of intermediate logics and the lattice of varieties of Heyting algebras. Wederive as corollaries of such isomorphism a completeness theorem for χ -logics.Let Γ be a set of formulas and C a class of Heyting algebras, then we define the two maps Var χ and Log χ such that: Var χ : Γ
7→ { H ∈ HA | H (cid:15) χ Γ } ; Log χ : C 7→ { φ ∈ L | C (cid:15) χ φ } . A class of Heyting algebras C is χ -definable if there is a set Γ of formulas such that C = Var χ (Γ).We say that a χ -logic Λ is algebraically complete with respect to a class of Heyting algebras C if Λ = Log χ ( C ). We will next show that Var χ (Γ) is always a χ -variety and Log χ ( C ) is always a χ -logic. This will later allow us to consider Log χ and Var χ as maps between the lattices IL χ and HA χ . Proposition 20. χ -validity is preserved by taking subalgebras, products, homomorphic imagesand core superalgebras.Proof. (Subalgebras) Suppose by contraposition K (cid:22) H and ( K, σ ) χ φ for some χ -valuation σ , then obviously ( H, σ ) χ φ . (Products) Let f : H ։ K be a surjective morphism. If K χ φ , then by Proposition 15 it follows that K φ [ χ n ( p ) / p ]. Since validity is preserved byhomomorphic images, we have that H φ [ χ n ( p ) / p ] and therefore, by Proposition 15, H χ φ . (Homomorphic image) For products we need to show that if Q i ∈ I A i (cid:15) χ φ , then A i (cid:15) χ φ for all i ∈ I . This claim follows immediately by noticing (cid:0)Q i ∈ I A i (cid:1) χ = Q i ∈ I ( A i ) χ , and so χ -valuations over Q i ∈ I A i are all and only the function-products of χ -valuations over the A i . (Core superalgebra) Let K χ = H χ and H (cid:22) K . By reductio suppose that K χ φ . Then forsome valuation σ we have ( K, σ ) χ φ . Since H χ = K χ and H (cid:22) K , σ is a valuation over H and J φ K Hσ = J φ K Kσ = 1. Corollary 21.
For every set of formulas Γ , the class of Heyting algebras Var χ (Γ) is a χ -variety.Proof. It follows from Theorem 19 and Proposition 20.It is a straightforward consequence of Corollary 21 that every χ -definable class of Heytingalgebras is also a χ -variety. The next proposition shows that for every class C of Heytingalgebras its set of validities Log χ ( C ) is a χ -logic. Proposition 22.
For every set of algebras C , the class of formulas Log χ ( C ) is a χ -logic. More-over, Log χ ( C ) is the χ -variant of Log ( C ) 11 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro Proof.
We have: φ / ∈ Log χ ( C ) ⇐⇒ ∃ H ∈ C such that H χ φ ⇐⇒ ∃ H ∈ C such that H φ [ χ n ( p ) / p ] (by Proposition 15) ⇐⇒ φ [ χ n ( p ) / p ] / ∈ Log ( C ) ⇐⇒ φ / ∈ ( Log ( C )) χ . Hence
Log χ ( C ) is the χ -variant of Log ( C ).These two results establish that the maps Log χ : HA χ → IL χ and Var χ : IL χ → HA χ are well-defined. Using the homomorphisms V 7→ V χ and L L χ , and the isomorphism HA ∼ = op IL between intermediate logics and varieties of Heyting algebras given by the maps Log and
Var ,we obtain the following commuting diagrams.
Proposition 23.
For every intermediate logic L , Var χ ( L χ ) = Var ( L ) χ . IL IL χ HA HA χ Var ( − ) χ Var χ ( − ) χ Proof. ( ⊆ ) Consider any Heyting algebra H ∈ Var χ ( L χ ). Then we have H (cid:15) χ L χ and byProposition 18 it follows h H χ i (cid:15) L . So we clearly have that h H χ i ∈ Var ( L ) and since h H χ i χ = H χ and h H χ i (cid:22) H also H ∈ Var ( L ) χ . ( ⊇ ) Consider any Heyting algebra H ∈ Var ( L ) χ , thenthere is some K ∈ Var ( L ) such that K (cid:22) H and H χ = K χ . Then we have that K (cid:15) L , so byCorollary 16 above K (cid:15) χ L χ which entails K ∈ Var χ ( L χ ). Finally, since χ -varieties are closedunder core superalgebra, it follows that H ∈ Var χ ( L χ ). Proposition 24.
For every variety V of Heyting algebras Log χ ( V χ ) = Log ( V ) χ . IL IL χ HA HA χ ( − ) χ ( − ) χ Log Log χ Proof.
We prove both directions by contraposition. ( ⊆ ) Suppose φ / ∈ Log ( V ) χ , then φ [ χ n ( p ) / p ] / ∈ Log ( V ) and hence there is some H ∈ V such that H φ [ χ n ( p ) / p ]. By Proposition 15 H χ φ ,hence φ / ∈ Log χ ( V χ ). ( ⊇ ) Suppose φ / ∈ Log χ ( V χ ). It follows that there is some H ∈ V χ such that H χ φ , hence by Lemma 17 h H χ i χ φ . It thus follows by Proposition 15 that h H χ i φ [ χ n ( p ) / p ]. Now, since H ∈ V χ , we have for some K ∈ V that K (cid:22) H and K χ = H χ .Thus it follows that h H χ i (cid:22) K and therefore h H χ i ∈ V . Finally, since h H χ i φ [ χ n ( p ) / p ] we getthat φ [ χ n ( p ) / p ] / ∈ Log ( V ) and hence φ / ∈ Log ( V ) χ .Since the diagrams above commute, it is easy to prove a definability theorem and a completenesstheorem for χ -logics and χ -varieties. Theorem 25 (Definability Theorem) . χ -varieties are defined by their χ -validities: H ∈ X ifand only if H (cid:15) χ Log χ ( X ) . attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro Proof.
For any χ -variety X such that X = V χ we have: Var χ ( Log χ ( V χ )) = Var χ ( Log ( V ) χ ) (by Proposition 24)= Var ( Log ( V )) χ (by Proposition 23)= V χ (by standard duality)Hence Var χ ◦ Log χ = HA χ , which proves our claim. Theorem 26 (Algebraic Completeness) . χ -logics are complete with respect to their correspond-ing χ -variety: φ ∈ Λ if and only if Var χ (Λ) (cid:15) χ φ .Proof. For any χ -logic Λ such that Λ = L χ we have: Log χ ( Var χ ( L χ )) = Log χ ( Var ( L ) χ ) (by Proposition 23)= Log ( Var ( L )) χ (by Proposition 24)= L χ (by standard duality)Hence Log χ ◦ Var χ = IL χ , which proves our claim.The former completeness theorem shows that the algebraic semantics that we have introducedin the previous section is indeed a suitable framework to study χ -variants of intermediate logicsfrom a semantics point of view. Similarly, the definability theorem for χ -varieties allows usto give a first external characterisation of χ -varieties: they are exactly the classes of Heytingalgebras which are χ -definable. Finally, since Var χ and Log χ are lattice homomorphisms, weobtain the following dual isomorphism result. Theorem 27 (Duality) . The lattice of χ -logics is dually isomorphic to the lattice of χ -varietiesof Heyting algebras, i.e. IL χ ∼ = op HA χ . Before turning to the study of specific χ -logics in the next section, we shall also provide analternative characterisation of χ -varieties. First notice that, since χ -varieties are closed undersubalgebras, homomorphic images and products, they are also (standard) varieties and thusBirkhoff Theorem tells us that they are generated by their collection of subdirectly irreducibleelements. It is possible to show more and give an internal characterisation of χ -varieties:they are exactly the classes of Heyting algebras generated (also under the core superalgebraoperation) by their collection of core generated, subdirectly irreducible elements. Recall that,given C a class of Heyting algebras, we indicate with X ( C ) the least χ -variety containing C andwith V ( C ) the least variety containing C . We first adapt Tarski’s HSP-Theorem to the currentsetting. Theorem 28.
Let C be a class of Heyting algebras, then X ( C ) = ( HSP( C ) ) χ .Proof. By definition we have X ( C ) = V ( C ) χ and by Tarski’s HSP-Theorem V ( C ) = HSP( C ).It immediately follow X ( C ) = ( HSP( C ) ) χ .From the former theorem it is easy to prove the following useful result. Proposition 29.
Let X be a χ -variety, then X = X ( C ) iff Log χ ( X ) = Log χ ( C ) . Proof. ( ⇒ ) Since C ⊆ X , the inclusion from right to left is straightforward. Suppose now that X χ φ then there is some H ∈ X such that H χ φ . Then since X = X ( C ), it follow by13 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro Theorem 28 that H ∈ HSP( C ) χ . By Proposition 20, it follows that for some A ∈ C we have A χ φ . Hence φ / ∈ Log χ ( C ).( ⇐ ) Suppose Log χ ( X ) = Log χ ( C ). It follows that Var χ ( Log χ ( X )) = Var χ ( Log χ ( C )), henceby the Duality Theorem 27, we have X = Var χ ( Log χ ( C )). Finally, since Log χ ( C ) = Log χ ( X ( C ))by Proposition 20 and Theorem 28, we have Var χ ( Log χ ( C )) = Var χ ( Log χ ( X ( C )); and by Duality Var χ ( Log χ ( X ( C )) = X ( C ), it follows X = X ( C ).A first characterisation is given by following result, stating that every χ -variety X is generatedby its collection of core generated Heyting algebras. We denote by X CG the subclass of coregenerated Heyting algebras of a χ -variety X . Proposition 30.
Every χ -variety is generated by its collection of core generated elements, i.e. X = X ( X CG ) .Proof. Let X be a χ -variety, then for any non core generated H ∈ X we have h H χ i (cid:22) H and H χ = h H χ i χ . So since h H χ i ∈ X CG it follows H ∈ X ( X CG ).By Birkhoff theorem we know that every χ -variety is generated by its subdirectly irreducibleelements and, by the previous proposition, we know that every χ -variety is generated by itscore generated elements. The following result shows that the intersection of these two classesof generators actually suffices. If X is a χ -variety, let X CGSI its subclass of core generatedsubdirectly irreducible Heyting algebras. We prove a version of Birkhoff Theorem for χ -varietiesshowing that X = X ( X CGSI ). Theorem 31.
Every χ -variety is generated by its collection of core generated subdirectly irre-ducible elements: X = X ( X CGSI ) .Proof. By the dual isomorphism between χ -logics and χ -varieties it suffices to show that Log χ ( X ) = Log χ ( X ( X CGSI )). By Proposition 29 this is equivalent to
Log χ ( X ) = Log χ ( X CGSI ).The direction
Log χ ( X ) ⊆ Log χ ( X CGSI ) follows immediately from the inclusion X CGSI ⊆ X .We next show that
Log χ ( X CGSI ) ⊆ Log χ ( X ).Suppose by contraposition φ / ∈ Log χ ( X ), then for some H ∈ X and some χ -valuation σ , wehave ( H, σ ) χ φ and so by Lemma 17 ( h H χ i , σ ) χ φ . Now, it is a well-known fact, originallyshown by Wronski in [22], that for any Heyting algebra B and x ∈ B such that b = 1 B , thereis a subdirectly irreducible algebra C and a surjective homomorphism h : B ։ C such that f ( b ) = s C , where s C is the second greatest element of C . Then, since x = J φ K h H χ i σ = 1 H there is a subdirectly irreducible algebra C and surjective homomorphism h : h H χ i ։ C with h ( x ) = s C . Consider now the valuation τ = h ◦ σ then, since h a is homomorphism, τ is still a χ -valuation. Let p , . . . , p n be the variables in φ , it follows by the properties of homomorphismsthat: J φ ( p , . . . , p n ) K Cτ = φ C [ τ ( p ) , . . . , τ ( p n )]= φ C [ h ( σ ( p )) , . . . , h ( σ ( p n ))]= h J φ ( p , . . . , p n ) K h H χ i σ = s C . From which it immediately follows that (
C, τ ) φ and so that C φ . Now, since H ∈ X , wehave that h H χ i ∈ X and so since h : h H χ i ։ C also that C ∈ X . Moreover, we have that C is subdirectly irreducible and, since C = h [ h H χ i ], also that C is core generated. Finally, thismeans that C ∈ X CGSI and so that φ / ∈ Log χ ( X CGSI ), which proves our claim.14 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro χ -logics In this Section we will consider examples of χ -logics and look at their specific properties andcharacterisation. Recall that in Lemma 6 we have shown that there are only 6 fix-points ofintuitionistic univariate formulas: ⊥ , p, ¬ p, ¬¬ p, p ∨ ¬ p, ⊤ . Since L ¬ = L ¬¬ for every intermedi-ate logic L , this means that there are at most five lattices of χ -logics. We provide a descriptionof these lattices. p -logics: Firstly, the lattice of p -logics IL p actually coincides with the lattice of intermediatelogics IL , since for every intermediate logic L it is clearly the case that L p = L . From thealgebraic perspective, this means that for any Heyting algebra H its p -core is H p = H , thus weare not imposing any restriction on our valuations. ⊤ -logics and ⊥ -logics: The two “limit” cases IL ⊥ and IL ⊤ are more interesting. Notice that H ⊥ = { H } and H ⊤ = { H } , and so under the algebraic semantics that we have introduced ⊥ -models allow only the constant valuation with image 0 H and, analogously, ⊤ -models allowonly the constant valuation with image 1 H . Interestingly, this means that the notion of coresuperalgebra collapses in both cases to that of superalgebra, as we have h H ⊥ i = h H ⊤ i = { H , H } , which is a subalgebra of every Heyting algebra.Thus there is only one ⊥ -variety and only one ⊤ -variety, in both cases the variety of allHeyting algebras. By the duality result of the previous section, this means there are exactlyone ⊥ -logic ( IPC ⊥ ) and one ⊤ -logic ( IPC ⊤ ), which are respectively the ⊥ -variant and ⊤ -variantof every intermediate logic. These two logics are characterised by the following properties: φ ( p , . . . , p n ) ∈ IPC ⊥ iff φ ( ⊥ , . . . , ⊥ ) ∈ IPC iff φ ( ⊥ , . . . , ⊥ ) ∈ CPC φ ( p , . . . , p n ) ∈ IPC ⊤ iff φ ( ⊤ , . . . , ⊤ ) ∈ IPC iff φ ( ⊤ , . . . , ⊤ ) ∈ CPC
Notice in particular that, although they correspond to the same variety, the two logics aredistinct. ¬ p -logics: Apart from IL , the lattice IL ¬ p is the only lattice of χ -logics that has already beenstudied in the literature, although under a different name. In fact an example of ¬ p -logic is inquisitive logic InqB , which is the ¬ p -variant of the intermediate logics KP , ND and ML as shownin Theorem 10. As a matter of facts, the algebraic semantics for inquisitive logic restrictingvaluations of atomic formulas to regular elements was already introduced in [2], and was latergeneralised in [16] to consider the entire lattice of DNA -logics and their corresponding varieties.This semantics coincides with the one introduced in this paper, since regular elements areexactly the fixpoints of the ¬¬ operator. This approach has proved to be particularly useful:for instance, [16] shows that the lattice of extensions of InqB is dually isomorphic to ω + 1,and also provide an axiomatisation of all such extensions by a generalisation of the method ofJankov formulas. ¬ p -logics have a particularly interesting feature: as mentioned before, the ¬ p -core of aHeyting algebra is the set of its regular elements, which is a Boolean algebra for the signature { , , ∧ , →} . This easily entails the following corollary: Given an intermediate logic L and a ∨ -free formula φ , φ ∈ L ¬ p iff φ is a classical tautology (Theorem 2.5.2 in [7]). That is, ¬ p -logics areare logics whose { , , ∧ , →} -fragment behaves classically, and which present an intuitionisticbehaviour once formulas containing disjunction are concerned. Such intuitionistic behaviourdisappears once also disjunction is forced to be classical, as the following lemma shows: Proposition 32.
Let L be an intermediate logic. Then L ¬ p = CPC iff L extends the logic ofweek excluded middle WEM := IPC + ( ¬ p ∨ ¬¬ p ) . Also referred to as negative variants in the literature ([14, 12, 6]). attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro The original proof of this result is given in [6] (Proposition 5.2.22). Here we present an alter-native proof using the machinery developed in the previous sections.
Proof.
Firstly, notice that L ¬ p = CPC iff q ∨ ¬ q ∈ L ¬ p . The left-to-right implication is trivial.As for the other implication, q ∨ ¬ q ∈ L ¬ p implies that the set of regular elements of an algebrain the variety X ( L ¬ p ) is itself a subalgebra, and a Boolean algebra at that. By Proposition 30and Theorem 26, it easily follows that L ¬ p = CPC .The main statement now follows easily: By Proposition 9, q ∨ ¬ q ∈ L ¬ q iff ¬ q ∨ ¬¬ q ∈ L ,which in turn is equivalent to WEM ⊆ L .We refer the reader to [16] for more information on ¬ p -logics and ¬ p -varieties.( p ∨ ¬ p ) -logics: Finally, let us consider the lattice IL p ∨¬ p . The next proposition gives acharacterisation of the p ∨ ¬ p -core of any Heyting algebra H . Proposition 33.
Let H be a Heyting algebra and let x ∈ H . The following are equivalent:1. x = y ∨ ¬ y for some y ∈ H ;2. ¬ x = 0 ;3. for every y ∈ H , if x ∧ y = 0 , then y = 0 .Proof. (1 ⇒
2) Suppose x = y ∨ ¬ y for some y ∈ H . Then ¬ x = ¬ ( y ∨ ¬ y ) = ¬ y ∧ ¬¬ y = 0 H .(2 ⇒
3) Suppose ¬ x = 0 and x ∧ y = 0. Then ¬¬ x ∧ ¬¬ y = ¬¬
0, hence 1 ∧ ¬¬ y = ¬¬ y = 0.Since y ≤ ¬¬ y , it follows that y = 0. (3 ⇒
1) Suppose x is as in point 3. Since x ∧ ¬ x = 0, itfollows that ¬ x = 0. Consequently, we also have x = x ∨ ¬ x . 0 a bs Var p ∨¬ p ( LC p ∨¬ p )but not in Var ( LC p ∨¬ p ).The circles indicate themembers of the subalgebragenerated by the dense ele-ments. Notice that this al-gebra is not core-generated.The elements satisfying properties 1,2 and 3 above are referred toas dense elements . Notice that property 1 is exactly the conditiondefining the elements of H p ∨¬ p , thus the previous propositionprovides a characterisation of the ( p ∨ ¬ p )-core of H .Now, it is easy to see that the dense elements of a Heytingalgebra form a filter and that they are closed under the operations ∧ , ∨ , → and 1. As a simple consequence of this, we have that forany Heyting algebra H its core subalgebra is h H p ∨¬ p i = H p ∨¬ p ∪{ } . Therefore, the core generated algebras—which by Theorem31 suffice to generate all the ( p ∨ ¬ p )-varieties—are exactly thealgebras containing only dense elements apart from 0.We obtain an interesting example of ( p ∨ ¬ p )-logic by takingthe ( p ∨ ¬ p )-variant of G¨odel-Dummett logic LC . Recall that LC is the intermediate logic extending IPC with the axiom ( p → q ) ∨ ( q → p ). It can be also characterised as the logic of linear Heytingalgebras (Example 4.15 in [5]). Analogously, the logic LC p ∨¬ p forces a similar linearity condition, but now limited to the denseelements of a Heyting algebras. Notice that by Proposition 22,the variety Var p ∨¬ p ( LC p ∨¬ p ) is still generated by the class of linearHeyting algebras—however, the closure under core-superalgebrasleads to a variety properly extending Var ( LC ), as shown in Figure 1. Moreover, notice thatlinear algebras are core-generated since ¬ x = 0 for every non-zero element x ; thus we found aclass of core-generated algebras which generate the whole ( p ∨ ¬ p )-variety.16 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro Finally, we have seen in Proposition 32 that the intermediate logics whose ¬ p -variant is CPC are exactly the extensions of
WEM . So a natural question to ask is what intermediate logicshave
CPC as their ( p ∨ ¬ p )-variant. The next proposition establishes that CPC itself is the onlyintermediate logic with this property.
Proposition 34. L p ∨¬ p = CPC iff L = CPC .Proof.
The left-to-right implication is trivial. As for the other implication, suppose L p ∨¬ p = CPC . In particular, q ∨¬ q ∈ CPC = L p ∨¬ p . By Lemma 9, we have ( q ∨¬ q ) ∨¬ ( q ∨¬ q ) ≡ q ∨¬ q ∈ L ,which means L = CPC .Now that we have described the five lattices of χ -logic more in detail, we are ready to showthey are distinct. However, we need to clarify what we mean by distinct lattices: as we havealready seen, IL ⊤ and IL ⊥ both contain only one logic, so these lattices are isomorphic; but wealready observed that the logics IPC ⊤ and IPC ⊥ are different, so we will consider these latticesdistinct. More generally, we will say that IL χ and IL θ are equal if for every intermediate logic L we have L χ = L θ —as in the case of IL ¬ p and IL ¬¬ p ; otherwise we will say the lattices aredistinct.This suggests to study the relation between these lattices in a more systematic way: definethe pointwise extension relation IL χ (cid:22) IL θ to hold if for every intermediate logic L we have L χ ⊆ L θ . The relation (cid:22) is a partial order between the lattices of χ -logics. In particular, IL χ (cid:22) IL θ (cid:22) IL χ if and only if the two lattices IL χ and IL θ are equal. The following Theoremcharacterises the properties of this relation. Theorem 35.
Let χ and θ be univariate formulas. Then the following are equivalent:1. IL χ (cid:22) IL θ ;2. IPC χ ⊆ IPC θ ;3. ( θ ( p ) ↔ p ) → ( χ ( p ) ↔ p ) ∈ IPC ;4. For every Heyting algebra H , H θ ⊆ H χ .Proof. (1 ⇒
2) It follows from the definition of (cid:22) . (2 ⇒
3) Since
IPC χ ⊆ IPC θ , we have inparticular that χ ( p ) ↔ p ∈ IPC θ . This means that IPC + ( θ ( p ) ↔ p ) (cid:15) χ ( p ) ↔ p ; and so bythe deduction theorem of IPC we have ( θ ( p ) ↔ p ) → ( χ ( p ) ↔ p ) ∈ IPC . (3 ⇒
4) We prove thecontrapositive of the implication: suppose that H θ * H χ for some Heyting algebra H . Consideran element a ∈ H θ \ H χ . By Lemma 13 we have θ ( a ) ↔ a = 1 H and χ ( a ) ↔ a = 1 H . So inparticular H ( θ ( p ) ↔ p ) → ( χ ( p ) ↔ p ), showing that this is not a formula in IPC . (4 ⇒ L and take an arbitrary formula φ / ∈ L θ . By Proposition 22 L θ = Log θ ( Var ( L )), and so there exists an algebra H ∈ Var ( L ) and a θ -valuation σ such that( H, σ ) φ . Since H θ ⊆ H χ by hypothesis, it follows that φ / ∈ Log χ ( Var ( L )) either. Againby Proposition 22, L χ = Log χ ( Var ( L )), and thus φ / ∈ L χ . Since φ was generic, it follows that L χ ⊆ L θ , as wanted. 17 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro IL p = ILIL ¬¬ p IL p ∨¬ p IL ⊥ IL ⊤ Figure 2: The Hasse diagramof the 5 lattices of χ -logics,ordered under the relation (cid:22) .The diagram is computed usingTheorem 35. Corollary 36.
There are exactly 5 lattices of χ -logics, for χ a univariate formula: IL ⊥ , IL p = IL , IL ¬ p = IL ¬¬ p , IL p ∨¬ p , IL ⊤ . Proof.
What remains to be shown is that the lattices aredistinct. By Theorem 35, we can to this by exhibiting aHeyting algebra H for which the χ -cores are all distinct.Indeed, the algebra in Figure 1 is an example of such analgebra: H ⊥ = { } H ⊤ = { } H ¬ p = { , a, b, } H p ∨¬ p = { , s, } H p = H In this article we introduced χ -logics and a sound and complete algebraic semantics for them,based on Ruitenburg’s Theorem. In Section 3 we defined the notion of χ -logics and studiedthem from a syntactical perspective, showing that for a fixed χ they form a bounded lattice, andthat we have only 5 such lattices. In Section 4 we defined an algebraic semantics for χ -logics,by relying on an algebraic interpretation of Ruitenburg’s Theorem originally described in [20]and we introduced χ -varieties as the semantic counterpart of χ -logics. In Section 5 we provedthat χ -logics are indeed complete with respect to their corresponding algebraic semantics andwe proved some of their properties. Finally, in Section 6, we have looked more in detail ateach of the 5 lattices of χ -logics and characterised explicitly the point-wise extension relation (cid:22) between the lattices.The results of this article provide a first approach to generate and study new logics andcorresponding algebraic semantics in a systematic fashion. This work can be extended inseveral directions: Firstly, we limited ourselves to univariate formulas but the approach basedon Ruitenburg’s Theorem can be generalised to the lattice produced by an arbitrary formula—although with a more complex technical machinery. Secondly, even in this more general settingcores are required to be definable, but it is natural to consider more general notions of core (i.e.,fixpoints of some operator) and their corresponding logics. This step would also allow to movefrom the setting of intermediate logics to other families of logics. Another interesting directionof work would be to interpret the results presented in this paper in terms of topological duality,that is, Esakia duality for Heyting algebras ([10]). Giving a topological interpretation to theresults and constructions presented (such as the core-superalgebra operation) would give noveltools to study the structure of the lattices of χ -logics. References [1] G. Bezhanishvili and W. H. Holliday. A semantic hierarchy for intuitionistic logic.
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A Proof of Lemma 6
Proof of Lemma 6.
As shown in [15, 17], the following is a presentation of all the non-constantunivariate intuitiornistic formulas modulo logical equivalence. β := p β n +1 := α n ∨ β n α := ¬ p α n +1 := α n → β n We will consider the following two properties for a univariate formula φ : 19 attices of Intermediate Theories via Ruitenburg’s Theorem Gianluca Grilletti, Davide Emilio Quadrellaro ¬¬ φ ≡ ⊤ .2. If ψ has property 1, then φ [ ψ / p ] ≡ ⊤ .In particular, if φ has both properties then φ ≡ ⊤ , that is, the fix-point of φ is ⊤ .Firstly notice that α = (( ¬¬ p ) → p ∨ ¬ p ) → ( ¬ p ∨ ¬¬ p ) and β = (( ¬¬ p ) → p ∨ ¬ p ) → ( ¬¬ p → p )have both properties. α has property 1: ¬¬ α ≡ ¬¬ ((( ¬¬ p → p ) → p ∨¬ p ) → ( ¬ p ∨¬¬ p )) ≡ ¬¬ (( ¬¬ p → p ) → p ∨¬ p ) →¬¬ ( ¬ p ∨¬¬ p ) ≡ ¬¬ (( ¬¬ p → p ) → p ∨¬ p ) →⊤≡ ⊤ β has property 1: ¬¬ β ≡ ¬¬ ((( ¬¬ p → p ) → p ∨¬ p ) ∨ ( ¬¬ p → p )) ≡ ¬¬ ( ¬¬ (( ¬¬ p → p ) → p ∨¬ p ) ∨¬¬ ( ¬¬ p → p )) ≡ ¬¬ ( ¬¬ (( ¬¬ p → p ) → p ∨¬ p ) ∨⊤ ) ≡ ¬¬⊤≡ ⊤ α has property 2: for φ with property 1, α ( φ ) ≡ (( ¬¬ φ → φ ) → φ ∨¬ φ ) → ( ¬ φ ∨¬¬ φ ) ≡ (( ¬¬ φ → φ ) → φ ∨¬ φ ) → ( ⊥∨⊤ ) ≡ (( ¬¬ φ → φ ) → φ ∨¬ φ ) →⊤≡ ⊤ β has property 2: for φ with property 1, β ( φ ) ≡ (( ¬¬ φ → φ ) → φ ∨¬ φ ) ∨ ( ¬¬ φ → φ ) ≡ (( ⊤→ φ ) → φ ∨⊥ ) ∨ ( ¬¬ φ → φ ) ≡ ( φ → φ ) ∨ ( ¬¬ φ → φ ) ≡ ⊤∨ ( ¬¬ φ → φ ) ≡ ⊤ Moreover, we can show that, if α n and β n have properties both properties, than this holds for α n +1 and β n +1 too. α n +1 has property 1: ¬¬ α n +1 ≡ ¬¬ ( α n → β n ) ≡ ¬¬ α n →¬¬ β n ≡ ¬¬ α n →⊤≡ ⊤ β n +1 has property 1: ¬¬ β n +1 ≡ ¬¬ ( α n ∨ β n ) ≡ ¬¬ ( ¬¬ α n ∨¬¬ β n ) ≡ ¬¬ ( ⊤∨⊤ ) ≡ ⊤ α n +1 has property 2: for φ with property 1, α n +1 ( φ ) ≡ α n ( φ ) → β n ( φ ) ≡ α n ( φ ) →⊤≡ ⊤ β n +1 has property 2: for φ with property 1, β n +1 ( φ ) ≡ β n ( φ ) ∨ α n ( φ ) ≡ ⊤∨⊤≡ ⊤ So, by induction all the formulas α n , β n with n ≥ ⊤ . Asfor the remaining formulas, one can easily show their fix-points are as follows: β = ( p ) ≡ p = ⇒ ( β ) ≡ ( β ) β = ( p ∨ ¬ p ) ≡ p ∨ ¬ p = ⇒ ( β ) ≡ ( β ) β ≡ ( ¬ p ∨ ¬¬ p ) ≡ ⊤ = ⇒ ( β ) ≡ ( β ) β ≡ ( ¬¬ p ∨ ( ¬¬ p → p )) ≡ ⊤ = ⇒ ( β ) ≡ ( β ) α ≡ ( ¬¬ p → p ) ≡ ¬¬ p → p = ⇒ ( α ) ≡ ( α ) α = ( ¬¬ p ) ≡ ¬¬ p = ⇒ ( α ) ≡ ( α ) α = ( ¬ p ) ≡ ¬ p = ⇒ ( α ) ≡ ( α ) α ≡ (( ¬¬ p → p ) → p ∨ ¬ p ) ≡ ⊤ = ⇒ ( α ) ≡ ( α )4