Rudimentary Beth Models and Conditionally …kosta/Dosen radovi/[P][27] Rudimentary Beth...Rudimentary Beth Models and Conditionally Rudimentary Kripke Models for the Heyting Propositional

Rudimentary Beth Models andConditionally Rudimentary Kripke Modelsfor the Heyting Propositional Calculus

KOSTA DO§EN, Matematicki Institut, Knez Mihailova 35, 11001Beograd, p.f. 367, Yugoslavia

AbstractThis paper continues the investigation of rudimentary Kripke models, i.e. non-quasi-ordered Kripke-style models for the Heyting propositional calculus H. Three topics, whichmake three companion pieces of [3], are pursued. The first topic is Beth-style models forH called rudimentary Beth models. Rudimentary Kripke models may be conceived as aparticular type of these models, which are analogous to rudimentary Kripke models in notassuming quasi-ordering for the underlying frames. The second topic is a first step into thecorrespondence theory for rudimentary Kripke models. It is shown what conditions onframes are now defined by the characteristic schemata of Dummett's logic, the logic ofweak excluded middle and classical propositional logic. The third topic is a generalizationof rudimentary Kripke models that yields models called conditionally rudimentary Kripkemodels. It is shown that, if we don't want to change the usual semantic clauses for theconnectives, conditionally rudimentary Kripke models make the largest class of Kripke-style models with respect to which we can demonstrate the ordinary soundness and compl-eteness of H.

Keywords: Intuitionistic propositional logic, Kripke models, Beth models.

Introduction

The standard conception of Kripke model for Heyting's logic is based on thefollowing assumptions:

(1) valuations must be defined inductively, starting from atomic formulae;(2) the accessibility relation R must be at least a quasi-ordering relation,

i.e. it must be reflexive and transitive;(3) heredity must be satisfied, in the sense that if a formula A holds at a

point * of a model, which we write x V A, then Vy(xRy ^> y tA).

In [3] we began an investigation of Kripke-style models for Heyting's logicwhere these assumptions are changed in various ways.

First, if we reject (1), we may reject (2) too, and get models where R isonly serial (i.e., Vx 3y(xRy)), provided we have assumed in addition to (3)the following converse heredity condition:

J. Logic Computal., Vol. 1 No. 5, pp. 613-634, 1991 © Oxford University Press

613

614/ Rudimentary Beth Models

This condition is trivially satisfied if R is reflexive, but our R need not bereflexive any more. We have called the resulting models rudimentary Kripkemodels.

It was also shown in [3] that, even if we keep (1), but work with bothheredity and converse heredity, (2) is not necessary. Instead of a quasi-ordering, we will have a more general, and somewhat more involved, type ofordering. We have called the resulting models inductive Kripke models. Inall that, the conditions for valuations, and in particular the semantic clausesfor the connectives, are as in the standard conception. We have onlyintroduced converse heredity, which is anyway satisfied in standard Kripkemodels.

This paper is a sequel to [3], and a full understanding of what we want tosay here presupposes an acquaintance with this earlier paper. However, weshall try to make the present paper a little bit more self-contained bysummarizing briefly in a preliminary first section the essential parts of [3].Then we consider three additional topics related to rudimentary Kripkemodels, which make three, rather separate, companion pieces of [3]. As inthis previous paper, we concentrate on propositional logic and leave aside apossible extension of our approach to predicate logic.

In the second section, we introduce a very general notion of Beth modelfor the Hey ting propositional calculus, such that rudimentary Kripke modelsmay be conceived as a particular type of these models. These models, calledrudimentary Beth models, are interesting because for them we make anotherassumption analogous to the converse heredity of rudimentary Kripkemodels. We first consider rudimentary Beth models where both (1) and (2)are rejected, but then we also consider rudimentary Beth models where (1)is kept and (2) is nevertheless rejected.

In the third section, we make some initial steps in the correspondencetheory for rudimentary Kripke models. We show what conditions on framesof various types of rudimentary Kripke model correspond to the characteris-tic schemata of Dummett's logic, the logic of weak excluded middle andclassical propositional logic.

Finally, in the fourth section, we consider a generalization of rudimentaryKripke models that consists in restricting the conditions for rudimentaryKripke models only to those points of our frames accessible from some point.We call the resulting models conditionally rudimentary Kripke models. It isshown that if we don't want to change the usual semantic clauses for theconnectives, conditionally rudimentary Kripke models make the largest classof Kripke-style models with respect to which we can demonstrate theordinary soundness and completeness of the Heyting propositional calculus.

The various types of Kripke-style models introduced in [3] and here aremeant to be primarily an instrument for the analysis of the inner mechanism

Rudimentary Beth Models / 615

of Kripke models. The Beth-style models introduced in this paper shouldserve a similar purpose for Beth models. We don't propose rudimentaryKripke models and similar models as something to replace standard modelsfor the technical investigation of Heyting's logic. For the time being, wewant these new models to be only an instrument that will help us tounderstand better the standard models.

It is not clear whether besides this purely instrumental value our newmodels may also have an intrinsic value for the understanding of intuition-ism. Our semantic conceptions cannot remain the same if valuations are notdenned inductively or if the accessibility relation is not reflexive andtransitive. However, our aim is not to introduce a particular new semanticconception to replace the old ones. It is rather an attempt to delineate a fieldwithin which we can look for new models without ever leaving standardKripke models too far behind.

Kripke models for intuitionistic logic are often taken as a paradigm whenwe try to model other nonclassical logics, such as relevant and linear logic.So, it might be worth knowing all the possibilities inherent in this paradigm,lest we should be stranded by a too narrow imitation of features thatare perhaps accidental. In [2] we have considered models analogous toKripke models for a family of propositional logics weaker than Heyting'sprepositional logic. In this family we find logics based on the Lambekcalculus, logics related to the nonmodal fragment of linear logic, andvariants of relevant and BCK logic. For the models of [2] we makeassumptions analogous to (l)-(3), but in [4] we consider models for logicsbased on the weak implications of our family in which the analoguesof (l)-(3) are changed as they are changed for rudimentary Kripkemodels.

I am indebted to the editor for looking eagerly for connexions withcomputer science, which resulted in the following suggestion. Since Kripkemodels are so important in the study of intuitionistic logic, and since thislogic plays such a central role in computation, as witnessed by the lambdacalculus and domain theory, an advance in the understanding of Kripkemodels may be expected to have repercussions on the understanding ofcomputation. Moreover, some developments in logic programming andartificial intelligence are based on intuitionistic ideas. For example, DovGabbay tried to give an intuitionistic base to nonmonotonic reasoning.

Going one step further from this suggestion of the editor, one can findthat nonmonotonic logic should be closely related to relevant logic, withwhich it shares the rejection of Gentzen's unrestricted structural rule ofThinning (also called Weakening, and also, more recently, referred to byMonotohicity). Since, as we have already surmised, our Kripke-style modelsmay be inspiring for modelling relevant and linear logic, it is perhaps


possible to envisage for them an application in the study of nonmonotonicreasoning. In general, it is as an inspiration for modelling various nonclassi-cal logics interesting for logic programming or artificial intelligence that theideas presented here may have a chance to be applied outside pure logic.

1. Rudimentary Kripke models

Our propositional language has infinitely many propositional variables, thepropositional constant 1 and the binary connectives —*•, A and v. Forpropositional variables, we use the schematic letters p, q, r, . . ., plt . . ., forformulae, the schematic letters A, B, C, . . ., Au . .., and for sets of for-mulae, the schematic letters F, A, 0 , . . ., F , , . . . As usual, A++B isdenned as (A—> B) A (fl—>A), and ->A as A—» ±. By L, we denote the set ofall formulae, and by L+, the set of all formulae in which _L does not occur.In the metalanguage, we use =>, O , &, or, not, V, 3 and set-theoreticalsymbols with the usual meaning they have in classical logic.

The Heyting propositional calculus H in L is axiomatized by the followingusual axiom-schemata:

(AAB)^A, (AAB)^B,

A^(AvB), B-+(AvB), (A v B)^«A^ C)^((fl-»C)^C)),

and the rule modus ponens. It is well-known that this axiomatization isseparative, which means that an axiomatization of a fragment of H involvingsome connectives among which we must have —> is obtained by assumingmodus ponens and all those axiom-schemata from the list above in which theconnectives of the fragment in question occur. So, the positive Heytingpropositional calculus H+ and L+ is axiomatized by rejecting 1—>A fromour list of axiom-schemata.

A frame is (W, R) where"W is a nonempty set and R is a binary relationon W. We use x, y, z, . . ., xu . . . for points in W, and X, Y, Z, . . ., Xx, . . .for subsets of W. For a frame (W, R), an X c W is called hereditary iff, forevery x,

and it is called conversely hereditary iff, for every x,

For a frame (W, R) and X, KcW, we have the binary operationdenned by:

X^R Y={x:Vy(xRy =>(yeX^>ye Y))}.


A pseudo-valuation v on a frame (W, R) is a function from L into PW,i.e. the power set of W, that satisfies the following conditions for everyA, BeL:

(u A) V(A AB) = V{A) n v(B)

(uv) v(A v B) = v{A) U u(fl).

A valuation v on a frame (W, R) is a pseudo-valuation that satisfies:

(A-Heredity) for every formula J4, the set v{A) is hereditary;{Converse A-Heredity) for every formula .4, the set v{A) is

conversely hereditary.

A pseudo-Kripke model is (W, R, v) where (W, R) is a frame and v apseudo-valuation on this frame, and a rudimentary Kripke model is apseudo-Kripke model (W, R, v) where u is a valuation on (W, R). Aformula A holds in (W,R,v) iff v{A) = W, and A holds in {W, R) iff, forevery valuation v on (W, /?), we have that A holds in (W, R, v).

With our usual experience with Kripke models, instead of .A-Heredity andConverse .A-Heredity, we would expect only the following conditions:

(p-Heredity) for every propositional variable p, the setv(p) is hereditary;

{Converse p-Heredity) for every propositional variable p, the setv(p) is conversely hereditary.

Valuations would be defined by specifying v for propositional variables andusing {vl), (i/-»), (VA) and (uv) as clauses in an inductive definition. That>1-Heredity and Converse A-Heredity obtain would be derived by inductionon the complexity of A. Rudimentary Kripke models whose frames haveproperties that guarantee that every v defined on them in such an inductiveway satisfies /1-Heredity and Converse /1-Heredity make an importantproper subclass of the class of all rudimentary Kripke models; models in thissubclass are called inductive Kripke models.

From the instance of Converse /1-Heredity where A is -L, it follows thatfor every rudimentary Kripke model (W, R, v) the relation R must beserial, i.e., V* 3y{xRy). Only seriality is absolutely needed for frames ofrudimentary Kripke models for H, whereas for H+ even seriality issuperfluous. (We call a frame (W, R) serial iff R is serial, and similarly withmodels and other properties.)

In a quasi-ordered frame < W, R), the relation R is reflexive andtransitive, and, because of reflexivity, these frames are serial. RudimentaryKripke models based on such frames, which we call quasi-ordered Kripke


models, are the ordinary Kripke models for H. The conditions for valuationswe have given above are necessary and sufficient for valuations in theseordinary Kripke models.

If T h B means as usual that there is a proof of B in H from hypotheses inT, we can establish the following strong soundness and completeness of Hwith respect to rudimentary Kripke models (Proposition 8 of [3]). For everyT and B, we have r I- B iff

(*) for every (W, R, v), H v(C) = W => v(B) = WCer

or, alternatively, iff

(**) for every (W,R,v),C\ u(C)cu(B).Cer

If r is empty, we obtain the ordinary soundness and completeness of H withrespect to rudimentary Kripke models.

A pseudo-valuation v on a frame (W, R) satisfies A-Heredity iff, forevery B and C, we have v(B) c u((C->C)->B) (Proposition 9 of [3]), andit satisfies Converse A-Heredity iff, for every B and C, we have u((C—»C)—»5) c v(B) (Proposition 10 of [3]). From this we can infer that the classof all rudimentary Kripke models is the largest class of pseudo-Kripkemodels with respect to which H is strongly sound and complete in the sensethat, for every r and B, T\-B iff (**) (Proposition 11 of [3]).

Though (*) and (**) are equivalent for rudimentary Kripke models, (*)does not imply (**) for every pseudo-Kripke model. We will see in the lastsection that the class of rudimentary Kripke models is properly included inthe largest class of pseudo-Kripke models with respect to which H is stronglysound and complete in the sense that, for every F and B, T \- B iff (*); and thelatter class is properly included in the largest class of pseudo-Kripke modelswith respect to which we can prove the ordinary soundness and complete-ness of H.

For a frame (W, R) and k ^ 0 , we define Rk by the following:

xRk+1y&3z(xRkz & zRy).

For an arbitrary X c.W, let:

Cone X = {y: (3x e X)(3k > 0)xRky}

Cone" X = {y : not (3x e X)(3k > 0)yRkx}

and, for m ^ 0, let:

Conem X = {y: (3JC e X)(3k > m)xRky}

Cone" X = { v: not (3x e X)(3k > m)yRkx).


The set Cone X is the least hereditary superset of X (Proposition 13 of [3]),and Cone" X is the greatest hereditary set disjoint from X.

For a frame (W, R), a nonempty subset X of W will be called an co-chainfrom x iff there is a mapping / from the ordinal to onto X such that /(0) = xand (V/i 6 to)f{n)Rf(n + 1). Let co(x) = {X c.W :X is an co-chain from x).For an arbitrary X c W, let:

CLX = {y:(Vy e co(y))YnX±0}.

The set C\w X is the least conversely hereditary superset of X (Proposition16 of [3]), and, for every hereditary X ^W, the set CX^X is hereditary(Proposition 18 of [3]). A frame {W, R) is reflexive iff, for every X c W, wehave Cl^ X = X (Proposition 17 of [3]).

In a frame (W, R), the relation R isprototransitive iff

Vx, z(xR2z => (VZ e o>(z)) 3f(*f?/ & t e C\a Cone{z} & t $ Cl Cone" Z))

and it is protoreflexive iff

VJC(VA'1, X2 € <y(x)) 3y(x/?y & y $ C1M Cone" * , & y $ CL Cone" X2).

Prototransitivity follows from transitivity (let t be z), but not the other wayround, and protoreflexivity follows from reflexivity (let y be x), but not theother way round (the intuitive meaning of these conditions is explained inmore detail in [3]). As the seriality of R is equivalent with the condition that0 is conversely hereditary, so the prototransitivity of R is equivalent with thecondition that, for every X, Y c W, the set Clo, Cone X^>R Clm Cone Y ishereditary and the protoreflexivity of R with the condition that, for everyX, YcW, the set CL Cone X U Cl Cone Y is conversely hereditary. In aframe (W, R), the relation R is serial, prototransitive and protoreflexive iff,for every pseudo-valuation v on (W, R), if v satisfies p-Heredity andConverse p-Heredity, then v satisfies A-Heredity and Converse A-Heredity(Proposition 19 of [3]). We call frames (W, R) where R is serial,prototransitive and protoreflexive inductive frames. An inductive Kripkemodel is then denned as a (W, R, v) such that < W, R) is an inductive frameand v, called an inductive valuation, is a pseudo-valuation that satisfiesp-Heredity and Converse p-Heredity. Proposition 19 of [3] guarantees thatinductive valuations on inductive frames are valuations, i.e. that inductiveKripke models are rudimentary Kripke models.

The largest class of frames such that every pseudo-valuation on a frame inthis class that satisfies p -Heredity satisfies also A-Heredity and Converse/1-Heredity is made of all weakly quasi-ordered frames, i.e. frames thatsatisfy weak reflexivity:

Vx(3k > l)xRkx


and weak transitivity:

Vx, z(xR2z => 3t(xRt & t e Cone{z} & z e Cone{t})).

Every quasi-ordered frame is weakly quasi-ordered, but not vice versa, andevery weakly quasi-ordered frame is inductive, but not vice versa.

2. Rudimentary Beth models

Rudimentary Kripke models resemble Beth models up to a point, by theassumption of Converse j4-Heredity. To show how far this resemblance goeswe introduce a very general notion of Beth models.

A Beth frame is (W, R, JI) where {W, R) is a frame and n is a functionthat assigns to every x e W a set of subsets of W. A chain in a frame ( W, R )is an X c W such that for every x,y eX either x e Cone{y) or y e Cone{x}.Of course, every co-chain is a chain. A path through x is a maximal chaincontaining x. Usually, in a Beth frame, (W, R) is at least quasi-ordered(often it is taken to be a tree) and K{X) is the set of all paths through x, buthere we assume something much more general. For the time being, (W, R)is an arbitrary frame, whereas members of n(x) need not be all pathsthrough x, and, furthermore, they neither need to be paths nor do they needto contain x. It is clear that in an arbitrary Beth frame neither K{X) C W(X)nor co(x) c n(x) must be satisfied. This is so even when n(x) is the set of allpaths through x; for example, a path through x with points preceding x neednot be an co-chain from x, and an co-chain from x that is not maximal is not apath through x. However, it is also clear that with our very general definitionof Beth frames, for some Beth frames, n{x) and (o(x) may even coincide forevery x.

For a Beth frame < W, R, n) and X c W, let:

CL, X = {y:(Vy e a(y))YC\X*0}.

The set Cl,, X contains all the points y such that every Y e jz(y) intersects X.If in the definition of Cl we replace '^ ' by 'to', we obtain the definition ofCla,, but it is clear that CI,, and CI need not coincide, though they may. Ofcourse, if, for every x e W, we have JT(X) = co(x), then, for every I c l V , w ehave GnX = ClmX, but the converse need not hold. A simple coun-terexample (suggested by Predrag Tanovic) is given by (W, R) where R isreflexive and where there is an R' c W2, properly included in R, that isreflexive too. Let n(x) be the set of all co-chains from x defined via R'; then,by Proposition 17 of [3], for every X c W, we have Cl* X = ClaX = X, but,for some x, the sets n(x) and CO(JC) may differ. It is not clear whether there issuch a counterexample if ;r(jt) is the set of all paths through x.

A rudimentary Beth model {W, R, n, v) on a Beth frame (W, R, n) isobtained by specifying a Beth valuation u:L->PW, which satisfies (v±),


(v—>), (VA), A-Heredity, Converse A-Heredity, and the following two newconditions for every A, B e L:

(Ujrv) v(A vB) = v(A) Un v(B) = ax(y{A) U v(B))

(Beth A-Heredity) v(A) = Cl, v(A).As before, A holds in <W, R, K, V) iff v(A) = W.

As, from the instance of Converse A-Heredity where A is ±, it followsthat in every rudimentary Beth model R must be serial, which is equivalentwith the condition that for every x eW we have et)(x)#0, so, from BethA-Heredity where A is ±, it follows that, for every x eW, we must haven(x) =£0. The assumption of Converse A-Heredity, which may be written asv(A) = C\w v(A), is analogous to Beth A-Heredity, but, since Cl is ingeneral distinct from Cl^, and, since for rudimentary Beth models we have(i^v), these models are essentially different from our rudimentary Kripkemodels. However, rudimentary Kripke models may be conceived as aparticular type of rudimentary Beth models, as the following propositionshows:

PROPOSITION 1

For every rudimentary Kripke model (W, R, v), there is a rudimentaryBeth model (W, R, K, W) with the same frame (W, R) such that, for everyA, we have v(A) = w(A).

PROOF. On a given rudimentary Kripke model (W, R, v), for every x, letn(x) = (o(x) and, for every A, let w(A) = v(A). That w is a Beth valuationfollows from the fact that v(A) U v(B) is conversely hereditary in(W, R, v), whereas Beth A-Heredity is an immediate consequence ofConverse A-Heredity for v in (W, R, v).

Note that the seriality of R in the rudimentary Kripke model {W, R, v)entails, for every x, that a)(x)3z0. An alternative procedure for provingProposition 1 is to take in (W, R, v) that x(x) - {{x}} and w(A) = v(A);the structure (W, R, K, W) is again a rudimentary Beth model.

We can now verify that H is sound and complete with respect torudimentary Beth models:

PROPOSITION 2

A formula B is provable in H iff B holds in every rudimentary Beth model.

PROOF. From left to right, we have only to verify that every instance of thethree axiom-schemata for H involving v holds in every rudimentary Bethmodel; the rest is verified as in rudimentary Kripke models.

To verify x e v(Br^> (B, v B2)), suppose xRy and y e u(5,). Then, by BethA-Heredity, y e C\n u(B,), and it follows that y e Cl^viB^ U v(B2)). Weproceed analogously for B2-^(BX v B2).

622 / Rudimentary Beth Models

To verify x e v((Bx v B2)-*((BX-^B3)^>((B2^B3)-* B3))), supposexRy, yev(Brv B2) and y * v((Bx-^ B3)^((B2-^ B3)-^B3)). By Beth A-Heredity, there is a Yen(y) such that ynu( (Bi^B 3 )^ ( (B 2 ^B 3 ) -*B3))=0. Since yev(BxvB2), we have either Ynv(B1)*0 or mv(B2)¥=0. Suppose z 6 y and z eu(B,) (we proceed similarly with B2). Itfollows that, for some t and w, we have zRt, tRu, f e t/(5,-»53),u e v(B2-^B3) and M £ v(B3). But, since by >1-Heredity u e v(Bx), we obtaina contradiction.

The other direction, i.e. completeness, is a trivial consequence ofProposition 1 and completeness with respect to rudimentary Kripke models.

In the background of the soundess of H with respect to rudimentary Bethmodels, we find the following fact about frames. Let us say that, in a Bethframe (W, R, n), an I c W is triply hereditary iff X is hereditary,conversely hereditary and C\n X = X. In every Beth frame (W, R, Jt), everyset of triply hereditary subsets of W that contains 0 and is closed under theoperations ->R, fl and U* is a Heyting algebra. (For 0 to be converselyhereditary, we must have the seriality of R, and for Cl* 0 = 0 to besatisfied, we must have, for every x, that n(x) =£0.)

We can easily show that H is strongly sound and complete with respect torudimentary Beth models in the sense that, for every .Tand B, we have thatF\-B iff, for every rudimentary Beth model (W, R, n, v), PlC€rU(C)ev(B). The soundness direction is a simple corollary of the soundness part ofProposition 2, whereas, for the completeness direction, we apply Proposition1 (cf. the proof of Proposition 8 of [3]). This easily entails strong soundnessand completeness in the sense that F\-B iff B holds in every rudimentaryBeth model in which all the members of F hold. So, rudimentary Bethmodels behave as rudimentary Kripke models for the strong soundness andcompleteness of H. (Note that we can prove the equivalence of theanalogues of (*) and (**) for rudimentary Beth models via the respectivestrong soundness and completeness theorems; the notion of generatedsubmodel, used to prove the equivalence of (*) and (**) in [3], is not clearfor rudimentary Beth models and is of no help here.)

Usually, the frame (W, R) underlying a rudimentary Beth model(W, R, n, v) is assumed to be at least quasi-ordered, so that Converse/4-Heredity for v is automatically satisfied and does not figure as an extraassumption, whereas, instead of A -Heredity and Beth A -Heredity, weassume for v only p -Heredity and the following condition for everypropositional variable p:

(Beth p-Heredity) v(p) = Cl,, v(p).

Full ^-Heredity and Beth >l-Heredity should then be derived by inductionon the complexity of A, and we must ensure that our Beth frame (W, R, n)


satisfies sufficient conditions for this induction to go through. We shall nowpresent a type of Beth frame that satisfies such sufficient conditions, butneed not be quasi-ordered.

A structured Beth frame is (W, R, n) where (W, R) is a frame with Rprototransitive and n: W—>PPW satisfies the following conditions for everyxeW:

(1°) (VX € n(x))(3y e X)xRy(2°) (VX 6 w(x))(Vy, z 6 X)X n Condi)'} n Cone{z} # 0(3°) Vy(xRy => (VY e n(y))(3X e n(x)) X is a shadow of Y)(4°) (VX e Ji(x))(yy eX)(3Y e n(y)) Y is a shadow of X(5°) n(x)±0.

Note that, in (2°), we have once Cone, and once Cone. In (3°) and (4°), lX isa shadow of Y' means, as in [3], that X D Cone" Y — 0 , i.e., for every z e X,there is a teY such that (3 k > 0)zRkt. The seriality of structured Bethframes follows from (1°) and (5°). The conditions (l°)-(5°) are analogous,but not identical, to conditions which in [5] are assumed for x overquasi-ordered frames (for easier comparison, we have numbered the firstfour conditions as the corresponding conditions in [5], chapter 3.2). Theconditions (l°)-(5°) are satisfied in serial frames both by paths through x andby co-chains from x. However, these conditions cover also things that areneither paths nor co-chains.

It is not difficult to verify that in a structured Beth frame (W, R, n), forhereditary sets X, Y c W, the operation CU has the following properties(quite analogous to properties mentioned for Clw in [3]):

X <= Ck X

0*0 = 0oa(x n Y) = a, x n ci, Y.

We can also show that, if X c W is hereditary, then Cl* X is hereditary andconversely hereditary.

A structured Beth model (W, R, n, v) on a structured Beth frame{W, R, n) is obtained by specifying a structured Beth valuation u:L-»PW,which satisfies iyl), (v->), (UA), (U^V), p-Heredity and Beth p-Heredity.We can then verify that a structured Beth model (W, R, n, v) is arudimentary Beth model, i.e., by induction on the complexity of A, we canprove that, for every A, the set v(A) is triply hereditary (this proof relies onthe properties of CljT above, and is related to a proof which may be found in[5], chapter 3.2).


In a structured Beth frame (W, R, n) the set s£ of all subsets X of W thatare hereditary, and such that Cl X = X, contains 0 and is closed under theoperations —*R, D and U^. Moreover, every X e si is conversely hereditary,and the structure (si,-^R, D, U,,, 0 ) is a Heyting algebra.

The soundness and completeness of H with respect to structured Bethmodels follows from Proposition 2 and the following proposition analogousto Proposition 1:

PROPOSITION 3

For every prototransitive rudimentary Kripke model (W, R, v), there is astructured Beth model (W, R, n, w) with the same frame (W, R) such that,for every A, we have v(A) — w(A).

PROOF. AS for Proposition 1, in (W, R, v), for every x eW, let JI(X) = w{x)and, for every A, let v(A) = w(A). It is easy to verify that (W, R, JI) is astructured Beth frame, and w a structured Beth valuation.

Note that, for proving Proposition 3, we cannot use the alternativeprocedure for proving Proposition 1, which consisted in taking 7t(x) = {{*}},because (1°) and (2°) may fail. As a corollary of Proposition 3, we havethat, for every inductive Kripke model, there is an equivalent structuredBeth model.

We said that prototransitivity and conditions (l°)-(5°) for structured Bethframes are sufficient to infer, for every A, the triple hereditariness of v(A)from p-Heredity and Beth p-Heredity, modulo (wl), (u->), (DA) and(i^v). However, we have no reason to think that exactly these conditionsare also necessary, and we leave open the question how to formulateconditions on (W, R, n) that would be both necessary and sufficient. Theanalogous conditions on quasi-ordered frames in [5] (chapter 3.2) aresufficient to infer A-Heredity and Beth .4-Heredity from p-Heredity andBeth p-Heredity, but it is not clear how necessary and sufficient conditionsought to look in this context. It would also be interesting to find necessaryand sufficient conditions on {W, R, n) for inferring, for every A, the triplehereditariness of v(A) from the assumption of the triple hereditariness ofu(p) for every propositional variable p. These last conditions would enableus to define inductive Beth models, analogous to inductive Kripke models.Of course, these conditions can trivially be stated in the form: the set of alltriply hereditary subsets of W contains 0 and is closed under —*R, f~l and U^,but we would like to have something more specific involving R and x, as inthe conditions for structured Beth frames.

3. Correspondence for rudimentary Kripke modelsIn the first section of [3], we mentioned rudimentary Kripke models forintermediate propositional logics in connexion with completeness. We shall


now consider these models also in connexion with correspondence, namelyquestions of definability of conditions on frames by schemata for formulae ofL. So we shall make a small step into the field of [6].

First, we introduce the following notation:

&zR2y).

Then we can prove the following:

PROPOSITION 4

In an inductive frame (W, R), we have:

Vy, z{yR-'Rz ^>(ye Cl,,, Cone{z} or z e CL Cone{y}))

iff every instance of (fl-> C) v (C-> B) holds in (W,R).PROOF. (=» Suppose (£-> C) v {C^>B) does not hold in (W, R), i.e.,there is a valuation v such that, for some x, we have x £ u((fl—» C) v (C—»5)). It follows that there are points y and z such that xRy, xRz, y e v(B),y $ v(C), z 6 v(C) and z £ v(B). By Converse ^-Heredity, there is aY € co(y) such that Y n u(C) = 0 and a Z € co(z) such that 2 n u(fl) = 0.But, because of ,4-Heredity, this contradicts either y e C\a Cone{z} orzeCI^Conejy}.

(<=) Suppose, for some x, y and z, we have:

xRy&xRz &y$Cla Cone{z} & z $ CL Cone{.y}

Then, by Propositions 13, 16 and 18 of [5], there is an inductive valuationsuch that v{q) = Cl^ Cone{>>} and v(r) = Cla, Cone{z}. We easily infer that

We have proved this proposition in some detail, as an example. In the restof this section, we will omit such straightforward proofs.

In the sense of Proposition 4, the schema (5—»C) v (C-*fi) defines, onweakly quasi-ordered frames, the condition:

Vy, z(yR-*Rz => (y e Cone{z} or z e Cone{y}))

and, on quasi-ordered frames, the condition:

Vy, z(yR~lRz => (zRy or yRz)), i.e. R~lR c R~l U R.

The schema (5—»C) v (C^^5) when added to H gives Dummett's logic.An alternative way to axiomatize Dummett's logic is to add (B —* C) v

((/?—>C)^>C) to H. As an instance of this last schema, we have weakexcluded middle, i.e. ->B v -i-ifi, for which we can prove that, on inductiveframes, it defines the condition:

Vy, z(yR-*Rz ^> 3t(yRt & t e CL Cone{z}))


on weakly quasi-ordered frames, the condition:

Vy, z(yR~lRz 4> 3t(yRt & t e Cone{z}))


Vy, z(yR~lRz => 3t(yRt & zRt)), i.e. /?"'/? c RR~\

Finally, the schema Bv(B->C) , which when added to H gives theclassical propositional calculus, defines, on inductive frames, the conditions:

Vx, yixRy =>x e Cl Cone{y})

on weakly quasi-ordered frames, the condition:

Vx, yixRy^x e Cone{y})


Vx, y(xRy 4> yRx), i.e. R c R~l.

On prototransitive and protoreflexive frames, i.e. on the inductive framesfor H+, the schema B v (B—»• C) defines the condition:

Vx, y((xRy & to(y)# 0) =>x e CL Cone{y})

whereas, on such frames, the schema (B—>C) v (C^*B) defines again thecondition of Proposition 4. Holding in these frames is defined with respect topositive valuations, for which (u±) does not apply.

For all these intermediate propositional logics: H plus weak excludedmiddle, Dummett's logic and the classical propositional calculus, we canobtain soundness and completeness results with respect to classes of framesthat satisfy the corresponding conditions (but also with respect to morerestricted classes).

We have said nothing up to now about correspondence for serial frames ofrudimentary Kripke models because in this context correspondence is of arather different type. Namely, conditions defined by our schemata, thoughthey will not mention valuations explicitly, will not amount to much morethan semantic clauses for the holding of the schemata, obtained by a ratherdirect application of the conditions for valuations. For a serial frame(W, R), let si be a set of hereditary and conversely hereditary subsets of Wthat contains 0 and is closed under the operations —»R, D and U. Ourconditions below will mention sets si, and, since these sets are ranges of ourvaluations, our conditions will implicitly mention valuations. Of course, theconditions on frames above also implicitly mention valuations, but in a lessdirect way.

Let us say that, in a frame (W, R), a subset Y of W is R~*-hereditary iff,for every y,

y e Y^>Vx(yR-lx^>x e Y).


Then we can easily prove the following:

PROPOSITION 5

In a serial frame (W, R), for every si, every Y e si is R~^hereditary iffevery instance of B v (B-> C) holds in (W, R).

Let us say that, in a frame (W, R), the subsets Y and Z or W areR~lR-comparable iff, for every y and z,

yR~lRz =>(0> ey=> y e Z) or (z e Z=>z eY)).Then we have:

PROPOSITION 6

In a serial frame (W, R), for every ^ , every Y,Zesd are /?~xi?-comparable iff every instance of (5—» C) v (C—> B) holds in (W, fl).

Let us say that in a frame (W, R) a subset Z of W is R~xR-quasi-hereditary iff, for every z,

z e Z 4 > V X 2 « - ^ 4 > 3 / ( ^ & t e Z)).Then we have:

PROPOSITION 7

In a serial frame (W, R), for every si, every Z e si is i?-1i?-quasi-hereditaryiff every instance of ->B v-i->B holds in (W, R).

For an arbitrary frame (W, R), let J^+ be a set of hereditary andconversely hereditary subsets of W closed under —»R, D and U. So, M+ neednot contain 0 . Then we can easily prove the following analogue ofProposition 5:

PROPOSITION 8

In an arbitrary frame (W, R), for every si+, every Y, Z e si+ and every y,

(yeY&y$ Z)=>Vx(xRyd>x e Y)iff every instance of B v (B —» C) holds in (W, R ) with respect to positivevaluations.

We can omit 'serial' from Proposition 6 and obtain an exactlyanalogous proposition for si+ and positive valuations, without introducingany change in i?"1/?-comparability.

Propositions 5-8 mention implicitly conditions on valuations that in ourpresentation are more or less similar to A-Heredity and Converse A-Heredity. As we said, these conditions do not amount to much more thanthe semantic clauses for the holding of the corresponding schemata. In fact,yl-Heredity and Converse >l-Heredity themselves do not amount to muchmore than such semantic clauses, as we will show below.

In a frame (W, R), a subset X of W will be called conditionally hereditaryiff, for every x,

3z(zRx) ^(xeX^ Vy(xRy ^y


and X will be called conditionally conversely hereditary iff, for every x,

3z(zRx) 4> (Vy(xRy ^>yeX)^>xeX).

Then we can easily prove the following:

PROPOSITION 9

A pseudo-valuation v on a frame (W, R) satisfies that, for every A, the setv(A) is conditionally hereditary iff, for every B and C, we have v(B-+

PROPOSITION 10

A pseudo-valuation v on a frame (W, R) satisfies that, for every A, the setv(A) is conditionally conversely hereditary iff, for every B and C, we havev(((C -+ C) -* B) -* B) = W.

The conditions of A-Heredity and Converse A-Heredity would presumablybe defined by rules rather than schemata, as is indicated by Propositions 9and 10 of [3]. However, we shall leave unexplored here the subject ofdefining conditions on frames by rules rather than schemata.

Correspondence for rudimentary Kripke models is a different matter fromcorrespondence for pseudo-Kripke models, because holding in frames withrespect to valuations of rudimentary Kripke models is essentially differentfrom holding in frames with respect to pseudo-valuations. For example, withrespect to pseudo-valuations, the schema B v {B—>C) defines the conditionR^R°, the schema (fl-> C) v (C-»fl) the condition R~lR c/?°, and theschema n f l v - n f l the condition R-1R^R. With respect to pseudo-valuations, we can also obtain correspondence for theorems of H. Forexample, the schema B—»((C—>C)^5) defines the condition:

VJC, y(3z(zRx) 4> (xRy^>x = y))

i.e. a conditional form of R c R°, whereas the converse schema ((C-> C)—>B)—*B defines the condition:

Vx, y(3z(zRx) ^(x= y^>xRy))i.e. conditional reflexivity.

4. Conditionally rudimentary Kripke modelsThe conditions on pseudo-valuations of Propositions 9 and 10 will define awider class of pseudo-Kripke models for H than rudimentary Kripke models.We will now consider this wider class.

A pseudo-valuation v on a frame (W, R) that satisfies:

(Conditional A-Heredity) for every formula A, the set v(A)is conditionally hereditary;

(Conditional Converse A-Heredity) for every formula A, the set v(A)is conditionally converselyhereditary


will be called a conditional valuation. A conditionally rudimentary Kripkemodel is (W, R, v) where (W, R) is a frame and v a conditional valuation.Conditionally rudimentary Kripke models need not be serial, but they mustbe conditionally serial, which means that they must satisfy:

i.e., every point that is accessible is not a dead end. This is an immediateconsequence of v(l) = 0 . A conditionally serial frame is a disjoint union ofserial frames and isolated inaccessible dead ends. It is clear that, on everysuch frame, we can define a conditional valuation; it is enough that withinthe serial part of the frame, on all accessible points in this serial part, wesecure ordinary ^-Heredity and Converse j4-Heredity.

Of course, every rudimentary Kripke model is a conditionally rudimentaryKripke model, but not the other way round. However, if from a condition-ally rudimentary Kripke model we excise all inaccessible points (not onlyinaccessible dead ends), we obtain a rudimentary Kripke model, then(Wr, RT, vT) where:

WT = {x e W: (3z e W)zRx}

(VJC, y e Wr)(xRr&xRy)

vr(A) = v(A) D WT

is a rudimentary Kripke model, called the rudimentary core of (W, R, v). Itis clear that, if A holds in (W, R, v), then A holds in the rudimentary core(Wr, Rr,vT), but not vice versa; provided holding in the conditionallyrudimentary Kripke model (W, R, v) is defined as for all pseudo-Kripkemodels, i.e., A holds in (W, R,v) iff v(A) = W.

We shall now demonstrate that H is sound and complete in the ordinarysense with respect to conditionally rudimentary Kripke models:

PROPOSITION 11A formula B is provable in H iff B holds in every conditionally rudimentaryKripke model.

PROOF. For soundness, we proceed by induction on the complexity of B. IfB is a propositional variable p or 1 , then our implication is vacuouslysatisfied, since neither p nor 1 are provable in H. If B is B, A B2, thenB, A B2 is provable in H iff B, and B2 are provable in H. By the inductionhypothesis, for every conditionally rudimentary Kripke model (W, R, v)and every x e W, we have x e u(B,) and x e v(B2). So, for every x € W, wehave x e v(Bx A B2). We proceed similarly if B is B, v B2, using thedisjunction property of H.

If B is B1—*B2, suppose that BX^*B2 is provable in H and that for aconditionally rudimentary Kripke model (W, R, v) and an x e W we havex $ u(B,—» B2). Then, for some y, we have xRy, y € u(B,) and y $ u(B2). If

630 / Rudimentary Beth Models

(WT, Rr, vT) is the rudimentary core of (W, R,v), then y e Wr, ye vT(Bx)and y $ vr(B2). By Converse y4-Heredity and /1-Heredity in the rudimentarycore, there is a z e Wr such that yRrz, z e vT(Bi) and z $ vT(B2), but this isimpossible, since, by the soundness of H with respect to rudimentary Kripkemodels, y e vr(Bx^>B2).

Completeness follows immediately from the fact that every rudimentaryKripke model is a conditionally rudimentary Kripke model.

We can interpret this proposition as saying that Conditional A-Heredityand Conditional Converse v4-Heredity are sufficient conditions on pseudo-Kripke models for obtaining the soundness and completeness of H in theordinary sense that, for every B, in H we can prove B iff B holds in everymodel in our class. Propositions 9 and 10 show that these two conditionalheredity conditions are also necessary. This yields the following proposition:

PROPOSITION 12

The class of all conditionally rudimentary Kripke models is the largest classof pseudo-Kripke models with respect to which H is sound and complete inthe ordinary sense.

So, with pseudo-Kripke models that are not conditionally rudimentary, weloose our grip on H. Arbitrary pseudo-Kripke models, for which, noheredity conditions whatsoever are assumed, are models for the (-3, A, V,1) fragment of the modal propositional logic K, where -3 is strictimplication. (This fragment of K and the same fragments of several otherprominent normal modal logics are axiomatized in [1].)

Much of [3] could now be rewritten in terms of conditionally rudimentaryKripke models rather than rudimentary Kripke models. For example, ananalogue of inductive Kripke models are conditionally rudimentary Kripkemodels whose rudimentary core is an inductive Kripke model. For condi-tions on frames of these models, which are conditional forms of seriality,prototransitivity and protoreflexivity, we could prove an analogue ofProposition 19 of [3] stating the equivalence of these conditions with theimplication from Conditional p -Heredity and Conditional Converse p-Heredity to Conditional A-Heredity and Conditional Converse A-Heredity.We could also reformulate questions (1), (2) and (3) of [3] in terms ofconditionally serial frames.

However, in some important respects, conditionally rudimentary Kripkemodels are different from rudimentary Kripke models. For every rudimen-tary Kripke model (W, R,v), we can show that if #,—» fl2 is provable in H,then u(Z?,)c v(B2), but this may fail for conditionally rudimentary Kripkemodels. If, for x, there is no z such that zRx, then we can have in aconditionally rudimentary Kripke model x e v(B) and x $ u((C-» C)—»fl),


and conversely, we can have xeu((C—»C)—»fl) and x £ v(B) (see Proposi-tion 9 and 10 of [3]).

This indicates that, for a frame (W, R), a set si of conditionallyhereditary and conditionally conversely hereditary subsets of W that contains0 and is closed under the operations —>R, n and U, need not be a Heytingalgebra. (The set W = X^RX must belong to si; for 0 to be conditionallyconversely hereditary, our frame must be conditionally serial.) For someX, Y, Z 6 si, either direction of

may fail; for example, we are not bound to have X c l V - > s X andW^>R X c.X. However, we will always have X-*R (W—»R X) = W and(W^RX)^RX = W, as Proposition 11 shows.

The general phenomenon we encounter here is the following. We have analgebra {sd, —>, A , V, 1, T) , which is a distributive lattice with zero andunit, and, for every theorem B of H and every homomorphism v: L—» si, wehave v(B) = T, i.e. si may serve as an algebraic model for H in the usualsense. (An algebraic model is nearer to our frames than to our Kripke-typemodels; it is quite close to the general Kripke-type frames, like thoseenvisaged in the third section of [3]. An algebraic model si plus ahomomorphism v:L—>si, which corresponds to a valuation, is somethinglike a Kripke-type model.) But this does not imply that our algebra si is aHeyting algebra. It would be a Heyting algebra iff, for every a, b e si, wehad that a —» b = T implies a^b, where a ^ b is defined as a = a A b; butthis implication may fail. (However, the converse implication holds, becausea = a Ab implies a—» b = (a A b)^>b, which, with (a A b)^>b = T, givesfl->fc = T.)

To take a concrete example, let W = {1, 2} and R = {(1, 2), (2, 2)}; so(W, R) is a serial, and, hence, conditionally serial, frame. If si = PW, then{si,^*R, D, U, 0 , W) is a distributive lattice with zero and unit, but it isnot a Heyting algebra because {1} ^ W—»w {1} = 0 and W—>R {2} =W ^ {2}. Note that {1} is conditionally hereditary and conversely her-editary, whereas {2} is hereditary and conditionally conversely hereditary. Itis easy to verify that, for every homomorphism v:L^>si which is hence aconditional valuation on (W, R), and, for every theorem B of H, we havev(B) = W. So a distributive lattice with zero and unit may be an algebraicmodel for H without being a Heyting algebra.

This phenomenon arises with the classical propositional calculus too. Let,for (W, R) as in the preceding paragraph, si = {W, {1}, 0 } ; then we canverify that (si,^*R, n, U, 0 , W) is a distributive lattice with zero and unitthat is not a Heyting algebra. Note that {1}, besides being conditionallyhereditary and conversely hereditary, is also R~'-hereditary (see Proposition5). It is easy to verify that, for every homomorphism i/:L—»si (which is


hence a pseudo-valuation on (W, R) such that for every A the set v(A) isconditionally hereditary, conversely hereditary and R~'-hereditary), and, forevery theorem B of the classical propositional calculus, v(B) = W. For everyB that is not a theorem of the classical propositional calculus there is a v ofthe type above such that v(B) = 0 (for every A the set v(A) may berestricted to W and 0 ) . So, a distributive lattice with zero and unit may bean algebraic model for the classical propositional calculus without being aHeyting algebra, and hence also without being a Boolean algebra.Moreover, every formula which is not a two-valued tautology may befalsified on this algebraic model.

All this shows that for H we cannot prove strong soundness andcompleteness with respect to conditionally rudimentary Kripke models. Weknow already from Proposition 11 of [3] that strong soundness in the sensethat, for every F and B, if Fl-fi, then (**), must fail for conditionallyrudimentary Kripke models, but we can also not prove, for every F and B,that, if TVB, then (*), i.e.

if T\- B, for every conditionally rudimentary Kripke model (W, R, v),if all the members of F hold in (W, R, v), then B holds in (W,R,v).

As we saw in Proposition 8 of [3], this obtains when we omit 'conditionally',but with 'conditionally', we have the following counterexample. We have{(C->C)->p}hp, but, on our frame where W = {I, 2} and R = {{I, 2),(2, 2)}, we have a conditional valuation v for which v(p) = {2}, whereas

A conditionally rudimentary Kripke model (W, R, v) that satisfies:

(Global Converse A-Heredity) for every formula A,Vx(3z(zRx)d$>x € v(A)) => v(A) = W

will be called a globally rudimentary Kripke model. Global Converse.4-Heredity says that if A holds in the rudimentary core of ( W, R, v), thenA holds in (W, R, v). So, holding in conditionally rudimentary Kripkemodels of this particular type is equivalent with holding in their rudimentarycores. Note that Global .4-Heredity, i.e. the other direction of GlobalConverse v4-Heredity, is trivially satisfied for every pseudo-Kripke model.The class of globally rudimentary Kripke models is properly in between theclass of conditionally rudimentary Kripke models and the class of rudimen-tary Kripke models. In globally rudimentary Kripke models, we need nothave either A-Heredity or Converse .4-Heredity, and their frames must beonly conditionally serial.

This new class of conditionally rudimentary Kripke models is interestingbecause for it we can prove the following analogue of Proposition 11 of [3]:


PROPOSITION 13

The class of all globally rudimentary Kripke models is the largest class ofpseudo-Kripke models with respect to which H is strongly sound andcomplete in the sense that, for every T and B, T\-B iff (*).

PROOF. That T\-B implies (*) for globally rudimentary Kripke modelsfollows immediately from the definitions and Proposition 8 of [3], and theconverse is a trivial consequence of Proposition 8 of [3]. This shows thesufficiency of Global Converse A-Heredity. The necessity of this condition isa consequence of the fact that for H we have {(C-^C)—*B}\-B. So, forevery B, we must have that v{{C^C)—>B) = W implies v(B) = W, fromwhich Global Converse A-Heredity follows.

For every class of models for H of the type (W, R, v) wider thanconditionally rudimentary Kripke models, we would have to change some-thing either in the conditions (v±), (y—»), (DA) and (uv) for pseudo-valuations or in the definition of holding in a model. For example, we couldhave:

v(±) = {x:not 3y(xRy)}

and leave it to Converse A-Heredity to ensure that if x ev(±), then forevery A we have x e v(A). This would exclude the need for seriality inframes for H. Or we could replace (VA) and (uv) by clauses for 'strictconjunction' and 'strict disjunction':

v(A AB) = {X:Vy(xRy => y e v(A) n v(

v(A v B) = {x: Vy(xRy => y e v{A) U v(B))}.

All these new conditions for v are satisfied in ordinary Kripke models for H.Holding of A in a model (W, R, v) could be defined in another manner byrestricting v(A) to a suitable subset of W; for example, we could say that Aholds in a conditionally rudimentary Kripke model {W, R,v) iff vr(A) = Wr,where (Wr, RT, vr) is the rudimentary core of (W, R, v). We have also seenthat we can take rudimentary Kripke models as a particular type ofmodels based on general rudimentary frames (see the third section of [3]) ora particular type of rudimentary Beth models. Now we could try togeneralize conditionally rudimentary Kripke models in the same directions.

However, we will not go here beyond conditionally rudimentary Kripkemodels. Our intention was only to show that there are wider classes ofmodels for H of the type we have called pseudo-Kripke models, sufficientlyclose to ordinary Kripke models for H in their conditions concerningvaluations to deserve the name Kripke models. Rudimentary Kripke models,and to a lesser degree conditionally rudimentary Kripke models, make suchclasses.


Acknowledgments

I would like to thank the Alexander von Humboldt Foundation forsupporting my work on this paper with a research scholarship. I would alsolike to thank the Centre for Philosophy and Theory of Science of theUniversity of Constance for its hospitality.

References

[1] G. Corsi (1987) Weak logics with strict implication, Zeitschrift fiir mathematische Logikund Grundlagen der Mathematik 33, 389-406.

[2] K. Dosen (1988/9) Sequent-systems and groupoids models, I and II, Studia Logica 47,353-85; 48, 41-65.

[3] K. Dosen (1989) Rudimentary Kripke models for the Heyting propositional calculus (toappear in the proceedings of Logic Colloquium '89).

[4] K. DoSen (1990) A brief survey of frames for the Lambek calculus, Zeitschrift furmathematische Logik und Grundlagen der Mathematik (to appear).

[5] A. G. Dragalin (1979) Mathematical Intuitionism: Introduction to Proof Theory (inRussian), Nauka, Moscow (English translation: American Mathematical Society, Provid-ence, 1988).

[6] P. H. Rodenburg (1986) Intuitionistic Correspondence Theory, doctoral dissertation,University of Amsterdam.

Received on 4 December 1990

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