~ A ~ Bo~Id A~D

Kos~A Do~E~

Models for Normal Intuitionistic Modal Logics

Abstract. Kripke-style models with two accessibility relations, one intuitio- nistio and the other modal, are given for analogues of the modal system K based on Heyting's propositional logic. I t is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistio analogues of systems stronger than K. A brief survey is given of the existing literature on intuitionistic modal logic.

w 1. Introduction. The aim of our work is to investigate intuitionistie analogues of normal propositional modal logics. We shall present Kripke- -style models for these propositional logics and prove soundness and comple- teness with respect to these models. We shall also investigate modalities in these logics~ their strict implicational fragments, and the embeddings of these logics in modal logics with a classical non-modal basis and two (or more) independent modal constants.

The systems which can be modelled in the class of all the Kripke-style models we shall introduce will be analogues of the system K based on classical logic (we shall call these systems ILK[] and HK~), and systems which are extensions of these analogues and are closed under substitution for propositional variables will be called normal, by analogy with the extensions of K. (We assume that the extensions of a system S are closed under the primitive rules of S.)

In the present paper we treat of intuitionistic analogues of K. In [10], which will be a sequel to this paper, we shall t reat of intuitionistie analogues of modal logics stronger than K~ in particular analogues of $4 and $5. In both papers we shall first deal with systems with [] primitive~ and next with systems with 0 primitive. Unlike what we have in modal logic with a classical basis, these two kinds of systems are not reducible to each other; this means that with the usual semantical definitions of the holding of []A and 5A we cannot use the same models for both kinds of systems. However, in this paper we shall consider a system with both [:] and ~, and models appropriate for it. This system will differ from the others in lacking the disjunction property. Throughout our work our metalogic will be classical.

In a further series of papers, the results of which will be based on this~ paper and on [1017 we shall investigate with our models some intuitionistic

218 M. .Bo.~i6, K. Dow

modal operators with a natura l interpretat ion. In [11] and [3] we shall show how intuit ionist ic double negation can be considered as D, the result ing logic being an extension of HK[:]. I n [12] and [13] we shall investigate with models similar to those int roduced here modal operators in intuitionistic logic which correspond to impossibility and non-neces- sity, and we shall show how intuitionistic negation can be unders tood as

special case of the impossibility operator. On the basis of this paper and of [10] we envisage also to t reat in the

fu ture questions related to the decidability and interpolat ion proper ty of t he logics we have introduced.

The l i terature on intuitionistic modal logic does no t seem to be very extensive. We have thought it useful to give here a brief survey of those works w e came to know about.

In [17] there is a formulat ion of an extension of intuitionistie logic with principles of the modal logic T, [9] (Oh. 8, cf. [8], Ch. 5) envisages exten- sions of intuitionistic logic of the $4 type, [26] (pp. 38-39) formulates a n extension of intuitionistie logic with $5 principles, and [25] (Ch. 6) investigates natural deduct ion formulations of intuitionistie $4 and $5. I n [4] and [5] one can find an algebraic t r ea tment of various intuitionistic modal logics, and [6] presents Kripke-style models for Prior 's intuitionistie $5 ([26]) based on Kripke models for intuitionistie predicate logic. These models are not like those we shall t reat of. I n [24] (pp. 108-109) there are some remarks on Gentzen formulations of Prior 's intuitionistic $5. I n [14] an a t t empt is made to formulate a criterion for f inding an intuitio- nistic analogue of ~ modal system with a classical basis (see also [15]).

Kripke-style models for intuitionistic modal logics using two (or more) accessibility relations between worlds, one of which is intuitionistic and the other (or others) modal, are ment ioned in [30] ([34] refers to a thesis of the author of [30], submit ted in 1979, which probably treats of the same models). I t is this type of models tha t we shall be concerned with. Other works referring to such models are [32] and [21], and they t rea t of intuitionistic analogues of the modal system G and related systems {see [2]); [20] (Ch. 7) treats of an intuitionistic analogue of 614, and [31], [33], [34], [23] and [16] t reat of various problems in intui- tionistic modal logic. The origin of this type of models should probably be t raced to Kripke models for modal logics with two, or more, inde- penden t modal constants, i.e., models which have more t han one accessi- bility relation. These models have already been ment ioned in [22] (p. 91, fn. 1) (cf. also [19], [27], [29], as well as e.g. [28] containing an application of the idea to tense logic).

Our work will presuppose none of the references above. We in tend to present a novel and self-contained t rea tment of models for intuitionistic modal logic, bu t we shall assume a certain familiarity with Kripke models

Models for r intugtgo~istie... 219

for intui t ionist ic proposi t ional logic and Kr ipke models for normal modal logics (for the former the reader m a y consul t [18] or [1], Ch. 9, and for the la t te r [7]).

The system ItK[]

w 2. The syntax of l IKE]. Let /~E] be the language of proposi t ional modal logic wi th denumerab ly 1 m a n y proposi t ional variables~ for which we use the schemata p , q, r , P l , . . . ; the connect ives of Z [] are -% A, v, -7 and the necess i ty opera tor []. We shall use A , B , . . . , A t, . . . as schemata for formulae of this language and other proposit ional languages we shall consider. Capital Greek let ters will be used for sets of formulae. As usual, A++B is def ined b y (A->B) A(B->A) . We shall omit parentheses follow- ing usual convent ions - - i n par t icular we assume tha t A and v b ind more s trongly than -~ and .~. The symbols ~/, 5, :~, ~ and, or, iff, not, and various set- theoret ical symbols will be used in the meta language with the usual meaning t hey have in classical logic. We shall disregard quota t ion marks in the metalanguage.

Now we in t roduce the proposi t ional calculus t IK[] in Z[] . The non- -modal pa,r% of t I K ~ will coincide with t h e t{eyt ing proposi t ional calculus (in L[]) . The axiom-schemata of t IKrn are:

J~l. A -+(B ->A) I t2 . (A-~(B-~C)) ->((A->B)- -+(A~C)) Jz3. (C+A)+((C+B)-+(C+AAB)) H4. A AB-+ A :H5. A AB--> B ~6. A ---> A v B H7. B - + A v B ~ 8 . (A +C)-+((B+C)-+(A vB+C)) H9. (A ---> -]B) ~ (B-+ ~ A ) /-I10. - M ~ ( A ~ B ) D1. DA/, DB-+ [] (A AB) D2. [] (A ->A)

and the rules of HKc~ are

A A - > B 3{P.

B A ~ B

c~A ~ ~ B .

a Although dcnumerability of the set of proposi~ional variables is in principle net essential for the results we shall present, for various reasons it is convenient to take a/~ropositional language with this requirement; for example, we might wish to extend our systems with a series of axioms which require an infinite number of distinct prepositional variables.

2 - Studia Logica 3/84

220 M. B o ~ , K. Dokn

I f we omit the mod~l axiom-schemata D1 and D2 and the modal rule 1~ [D, the system obtained is an axiomatizat ion of the t t e y t i n g propositionul e~leulus in L [3. These modal axiom-schemuta and this modal rule cun b6 replaced by the schema

and the rule [] (A -+B) -> ( [~A --> DB)

A

DA

in order to yield a system with the same theorems as H K D . This schema and this rule a.re characterist ic of the modal proposit ional calculus K,

based on classical propositional logic. The system HK[] is a proper sub- system of an appropriate formula t ion of K.

By a s tandard induct ion on complexi ty it can be shown t h a t any exten- sion of HKE] is closed under the Rule of Replacement

A ~ B C ~ C '

where C' results f rom C by replacing zero or more occurrences of A in C by B.

Next we give the following definition.

DEFINITION 1. I f ~ is a set of formulae in a language L and S a sys tem in L, then r ksA i f f there is a sequence of formulae B1, . . . , B~ (~ >~ 0) such t h a t every formulu in the sequence B1, . . . , Ba, A is either a theorem of S, or belongs to ~, or is obtained by MP f rom formulae preceding i t in the sequence.

Note t h a t MP is the only rule ment ioned in this definition.

We shall write FsA ins tead of 0 FsA, and we shall omit S f rom Fs i~ contexts where it is clear what sys tem S we have in mind.

I t is easy to prove t h a t the Deduction Theorem holds with respect t e ks, where S is a n y extension of the Hey t ing proposition~l calculus in LD, i.e.,

~ u { A } FsB =~ q5 FsA-->B.

(It is essential for this Deduct ion Theorem t h a t N P is the only rule men- t ioned in Defini t ion 1.) ~

2 It is also possible to prove the lollowing kind el Deduction Theorem where S is an extension of HKD and S~-A is the extension of S with the formula A (hence, both S and S + A are closed under the primitive rules of HK~):

q5 FS+A B iff lot some n q~ ~sAA ~ A A ... ^ [] ... EJA~B.

However, we shall have no use for this kind of Deduction Theorem.

Models for normal intuitionistic... 221

A set of formulae r is consistent iff not VA. ~ b A. I t can easily be shown that r is inconsis tent iff 3A. q~ k s -](A->A), where S is a n y extension of the t t ey t i ng proposit ional calculus. We can show tha t the theorems of lIKE] make a conservative, and hence consistent, extension of the t I ey t ing proposit ional calculus in J5 [] wi thout E] by a mapping d which deletes f rom formulae all occurrences of [] -- for if km~[]A, then d(A) is provable in the Hey t ing proposit ional calculus.

A set of formulae r has the disjunction property iff

V A , B (A v B e q5 ~ A e O or B e q~).

I n order to prove tha t the set of theorems of HK[] has the disjunction p rope r ty we in t roduce the following var ian t of Kleene 's slash (cf. [20], 10p. 30 ff):

IP ~ k p IAAB ~>a~ IA and IB IA v B ~d~ lk A or Ik B IA-->~ ~d f Ik A ~ IB tTA "~af not IF A 1 ~ A ~ o l IF A ,

where IFA is short for F A and IA.

] J E ~ 1. The set of theorems of t I K [] has the disjunction property.

1)~ooF. We show by induct ion on the length of proof of A t h a t k A :~IA, where F stands for k~K ~.

I n the basis we shall consider only the modal cases. :For D1 we h~ve

F DAA E]B and iFA and IFB ~ FAAB and IA and IB

f rom which we obtain I E ] A A [ ] B ~ ( A A B ) , whereas for E]2 we have

kA-->A and (lkA :~ ]A)

f rom which we obtain l [] (A -+A). I n the induct ion step we shall also consider only the case with R [].

Suppose FA-->B and IA-->B. I t follows tha t IkA ~ lB. Next , if we suppose ikA, t hen IB and kB. Hence, IFA ~ IkB, i.e., I E]A ~ I[2B. A fortiori, tk DA ~ ] []B, which implies I []A-> E]B. This concludes the induction.

~ o w suppose k A v B . As we have jus t shown, it follows t h a t IAvB, which means t h a t IFA or IkB. This implies t ha t FA or FB. q.e.d.

w 3. HE] models. ~ode l s with respect to which we shall show t h a t HK[] is sound and complete will have a set of "worlds" X and two b ina ry relations ~z a n d / ~ I def ined over X such tha t ( X , / ~ i ) is a Kripke f rame for the ]~eyting proposit ional calculus and k ~ is a modal "accessibility

222 M. Bo~i$, K. DoYen

relat ion". We now proceed t o define these models.

DEFINITION 2. ~ r = <X, RI, RM> is an t I [] frame iff: (i) X is a n o n e m p t y set

(if) Rz -~ X 2 and R z is reflexive and t ransi t ive (iii) R M _ X 2

(iv) / ~ R ~ _~ Rl~/~.

We shall use x ~ y ~ z ~ t, u ~ v, Xl ~ . . . as variables ranging over X. Expressions of the fo rm R1R 2 (abbreviat ing /~oR~) s tand for (<x, y}I3z(xRlz and zt~y)}, and those of the form R -~ s tand for the inverse relat ion o f / &

DEFINITION 3. J4 = <X, Rx~ R ~ V> is an H [] model iff: ( i ) <X~ Rz~/~M> is an HE:] f rame

(if) V~ called a valuation, is a mapping f rom the se~ of proposi t ional variables of L [] to the power set of X such t ha t for every io

W , y e X ( x ~ y ~ (x ~ V(p) * y ~ V(~))).

Note tha t R ~ in F r and ~ / c a n also be empty . W h e n it is empty~ an H [ ] model ~ / is a Kr ipke model for I t ey t ing ' s proposi t ional logie~ modulo inessential ad jus tments .

DEFI~ITION 4. I f ~g is ~n H [] model <X, Rz, RM~ V>~ x e X and A is ~ formula of L ~ , then the relat ion <Jl~ x> ~ A (i.e., A holds in x in Jg; <Jr x> ~ A is abbrev ia ted b y x ~ A where there can be no confusion) is defined b y :

(i) x V p ~ox x e V(p) (if) X ~ B A C ~a~ x ~ B and x ~ C

(iii) x ~ B v C <=>e,t x ~ B or x ~ C (iv) x ~ B->C .~,~ Vy(xR~y ~ (y ~ B ~ y ~ C)) (v) x ~ ~ B r Vy(xR~y :~ not y ~ B)

(vi) x ~ ~ B ~ V y ( x R ~ y ~ y ~ B) .

For the following definitions Jg and A are as in Defini t ion 4.

DEFINITION 5. //{ ~ A (i.e., A holds in ///) i f f Vx. <d[, x> ~ A.

/Vr ~ A (i.e., A holds in Fr) i f f V J / (the f rame of Jr DEFINITION 6.

is F r ~ / / / ~ A).

DEFIh*ITION 7.

•r, t"r ~ A. ~,[] A (i .e, A is H [] valid) i f f for every H [] f rame

Nex t we show the following lemma.

LE]~lT~i 2 (INTUITIONISTIC HEREDITY). In every t l [] mode~ <X, Rz,

Models for normal intuitionistic... 223

RM, V), for every x , y e X and for every A of :L []

x R i y => (x ~ A :~ y ~ A) .

P~ooF. B y induct ion on the complexi ty of A. We shall only consider the case where A is of the form � 9 Suppose xRiy and x ~ []B. I t follows tha t Vt(xRMt :~ t ~ B). Next , suppose tha t y•MZ, With xR1y, we obta in xR~RM z. B u t since in every H � 9 model R~t~ M c_ .~MRI~ we have xRl~R• , i.e. for some u~ x R ~ u and vRIz. Since Vt(xRz~t ~ t ~ B), we have u ~ B~ which w i th u l ~ z and the induct ion hypothes is gives z ~ B. ttence~ Vz (yRMZ => z ~ B), which implies y ~ DB. q.e.d.

Lemma 2 shows tha t every H[~ model is a model for the t t ey t ing pro- posit ional c~lculus in the ex tended l a n g u a g e / , D. The proof of Lemma 2 shows in a cer ta in sense the sufficiency of the condition Rz/~ M c / ~ R z for tha t to be the case. Bu t this condit ion is also necessary -- n~mely~ if we wan t H[ ] models to be models for the I t ey t ing proposi tonal calculus in Z ~ we mus t s t ipulate tha t R I ~ M ~_ RM.R I, This is proved b y the following lemma.

L E n A 3. Zet <X, R ~ RM> satisfy conditions (i)-(iii) of Defini t ion 2 and ~et condition (iv)~ i.e.~ l~iR ~ c ~MRI~ be unsatisfied. Then there is a formula A of L[] a n d a valuation V such that in <X, Rz~ t~M, V} for some x~ y e X

x ~ y and x ~ A and not y ~ A .

1>2~oo~. Since not R I R ~ ~_ R~R~, there are some x and z such ~h~. xR~R~z and not xRz~Rzz. Hence, for some y

(1) x l ~ y and yRMZ and not xRMRzz.

Define V as follows:

l { u [ u e X } if q is n o t p V(q) = [{ulx~MR~u } ii q is p .

~Ye mus t show tha t V is ~ valuat ion. So~ suppose x~R~x~ a.nd x~ e V(q). I f q is different f rom p, x~ will tr ivially belong to V(q). If q is identical wi th p~ then xRMR~x~ which with x~Rzx~ and the t rans i t iv i ty of Rz gives xR,uR~x~. I t follows tha~ x~ e V(q).

Since R~, is reflexive, XRMU ~ X.~I~IRIU. Hence, V u ( x R ~ n ~ u ~io), which implies

(2) x ~ []p.

We also h~ve:

yRMz and not x t t ~ R z z ~ yRMZ and z ~ V(p) ] t ( y R ~ t and not t ~ p) y not ~ ~p .

22~ M. Boz2~, K. Do~en

This, together with (1) and (2), gives us

xl~zy and x ~ ~p and ~ot y ~ []p. q.e.d.

We have shown thut the models with the condition / ~ R ~ _~/r form the largest class of models of the type we have intro- duced with respect to which we can expect to show tha t t I K ~ is sound and complete, ttowever~ as we shall see in w 4, H K ~ can be shown sound and complete also with respect to models which ceteris paribus satisfy the condition R z B ~ ~-tr instead of R~R~ ~_ t t~B~, i.e., with respect ~o a proper subclass of HE] models~ since /~zB~ E / ~ implies R~R~

RMRr, but not conversely. An indication of why this is the case can be found in the following lemmata.

L E ~ ~. I n H[] models x ~ ~ A ~ y ( x R ~ R ~ y : ~ y ~ A).

P~oo~. F rom Intui t ionist ie t te redi ty and the reflexivity of R z we get

z ~ A

Then we have: x k []A

.r Vy(zR~y ~ y ~ A) .

V z ( x R ~ z ~ z ~ A) Vz(xl~Mz =~ Vy(zR~y =~ y ~ A))

~*. V y ( x t t ~ R i y =~ y ~ A) . q.e.d.

LE~vIX 5. I n the definition of H [] frames the condition ~ 1 ~ c_ t i e R 1 can be replaced by the condition I~II~MR I ~_ 1~M1~ z yielding the same class of frames.

P~ooP. If R I R M ~_ RMRI , then I ~ R M R I c R~RzRI~ which by the t ransi t ivi ty of R z gives t tZRMR I ~_ ~ M I ~ I .

For the converse suppose that 1 ~ 1 ~ R I ~_ -I~MR I . By the reflexivity of /~z we have / ~ / ~ _~ t ~ R ~ R z ~ which together with our suppos i t ion yields ~IRM ~ R~Rz. q.e.d.

So, roughly speaking, out of HI3 models we can make new models by replacing the R ~ R x relation by a new relation Ro such tha t in these new models x ~ ~ A r =~y ~A) and /~zR~ ~ B~ (every R ~ relation disappears f rom the old models, since by the reflexivity of Rz every R ~ is followed by some Rz). The relation R~ is the B M relation of the new models. These new models validate exactly the same formulae us the old ones.

Since R~R~R~ ~_ R~R~, we can fm'ther "condense" these models by m a k i n g / ~ R z _~ R~ (cf. Definition 10).

w 4. Soundness and completeness of H K D. In this section we shall prove tha t HK[] is sound and complete with respect to H[] models.

Let Cl(~) = ~ { A ] r F A}. I t follows immediately tha t for every r �9 _~ CI(r und Cl(Cl(qs)) = Cl(~). A set of formulae tb is deductively

Models for normal intuitionistio... 225

closed iff Cl(r _ r :Now we can give the following definition.

DE~I~ImlO~ 8. A set of formulae 1" is nice iff:

(i) 1" is consistent (ii) 1" is deduct ively closed

(iii) 1" has the disjunction p roper ty .

I n the following lemma ~ stands for ~s, where S is any extension of the Hey t ing proposit ional calculus in L[:], and "nice" means "nice with respect to ~s".

LE)B~A 6. ]Let qb be a set of formulae and A a formula such that not ~ A. Then there is a nice set 1"such that q~ ~_ F and not 1" F A (i.e, A ~ F).

P ~ o o L Let Z =dr {T]q~ ~_ T and not T ~ A}. Since r s Z , Z is nonempty . I t is also easy to show t h a t Z is closed under unions of nonempty chains. Hence , by Zorn~s L e m m s Z has a maximal e lement 1" with respect to ~ . Since not 1" F A (i) 1" is consistent.

:Next, suppose t h a t 1" ~ B and B 6 1". Hence , 1"u{B} is a proper super- set of 1"~ ~nd s ince r _ 1"u{B}~ we have 1"u{B} ~ A. But t h e n 1" F A~ which is impossible. I f follows tha t (ii) 1" is deduct ively closed.

:Next~ suppose tha t for some B and C, B vC e I" and B 6 1" and C 6 F. Since I~w{B} and 1"w{C} are proper superser of 1", and ~b is a subset of bo th of these sets~ we have 1"w{B} ~ A and 1"w{C} b A. Then by the Deduct ion Theorem we obtain 1" k B-+A and 1" ~ C-->A, which implies 1" k B v C ~ A . As B v C e 1", we get 1"P A, which is impossible. I t follows tha t (iii) / ' has the disjunct ion property~ ~nd we can conclude tha t /~ is nice. q.e.d.

On the set of all nice sets we shall build a canonical model def ined ~s follows.

D~FI~ImlO~ 9. :Let S be any extension of H K [] in / )O , ~nd let :

XC =af {1"[/~ is nice wi th respect to bs), 1"R~A "~af1" ~ A, where 1", A e X ~ / ~ / ~ A <=>~1"o - A, where -To =0~(AI []A e F} and 1", A e X c.

Then <X% R ~ t~cM> is the canonical S frame. Let V ~ be a mapping f rom the set of proposit ional variables of L I:]

to the power set of X * such tha t V c (p) =a~ {PIP e 1"}. Then <X ~ t~z, t t ~ , V ~> is the canonical S model.

I n general it will be clear f rom the context when capital Greek let ters range over members of X~ and we shall not always note specially t ha t the sets in question are nice. I n the following two l emmata S shall s tand for any consistent extension of HK[] in 2~O.

226 M. Bo%i6, K. Do~en

I m ~ 7. The canonical S frame is an H [] frame and the canonical S modal is an I t [] model.

PgooF. :For the first pa r t we have: (i) X'=/=O, since the set of theorems of S is consis tent and hence, b y Lemma 6, it can be ex tended to a nice set. (The set of theorems of ILK[] is a l ready nice, since it is consi- stent , deduct ively closed and, b y I~emma 1, it has the dis junct ion property.)

(if) R~ is a reflexive and transi t ive relation over X ~ since _~ is reflex- i re and t ransi t ive.

(iii) R ~ is obviously a rolation over X ~. (iv) 1"R~ :~ 30(1"c_ 0 and 0 o ~_ A)

P B ~ A 1"R~R~A, b y the ref lexivi ty of R~.

So the canonical S f rame is an H O frame.

For the second pa r t we have:

1"R~A ~ Vp(1O e l " vlo (r v~

p e A ) d

So V c is a valuat ion, and the canonical S model is an H I model, q.e.d.

(Note tha t in (iv) of the proof above we have shown tha t the canonical S frame satisfies also the condition /~IR~ _~ /~M.)

L E ~ A 8. I n the canonical S model, for every 1' e X c and for every A

1" ~ A <*- A e1".

l~nooF. B y induct ion on the complexi ty of A. The basis, when A is 1o, is trivial. In the induct ion step we shall consider only the modal case (for the other cases see [18], p. 69). We have:

1"~ []B ~ VA(1"R*MA =~ A ~ B) , b y Defini t ion 4 VA(1" o ~_ A ~ B e A ) , b y Definition 9 and the in-

duct ion hypothesis .

We shall show tha t o B e F <=> VA (i'm - A ~ B e A). F r o m left to r ight we have :

[~B e1" and I" o ~_ A = B e 1 " m ~ 1 3 e A .

:For the other dix'ection suppose [ B 6 1". Then ~ot 1"o k B. Otherwise I'm k B, which implies tha t {A1, . . . , A~} b 33, for some A t ~ I'm (Fo osnnot be e m p t y since [] (A ~ A ) ~ 1"), and b y the Deduct ion Theorem and theorems of HK[:] we have :

[-A1A . . . AAn-->33 k [ ( A 1 A .. . A A n ) ~ o B 1- [A1A .. . A JAn--> o B

.Models for q~ormal ~ntuitionist~e... 227

from which we obtain F k []B and DB ~/~, since I" is nice. Bu t this is a contradiction. By Lemma 6 there is a lfiee set A such tha t I" D c A and B 6 A. q.e.d.

Now we can prove the soundness and completeness of H K ~ with respect to H D models.

TIZ_EO~E~ !. kAKDA <=~. ~RDA.

P~OOF. The soundness pa r t ( ~ ) is proved by a s t ra ightforward induct ion on the length of proof of A in H K ~ , and we shall omit it.

For the completeness p~r t ( ~ ) suppose ~nDA. Then by Lemma 7, A holds in the canonical H K ~ frame, and consequent ly A holds in t h e c~nonical H H ~ model, i.e., VI" e X c. I" ~ A. By IJemma 8~ VI" E XC.A ~ I". Sinee~ as we have seen in clause (i) of the proof of L e m m a 7, the set of theorems of t I H D is a nice set, it follows tha t k~K~A, q.e.d.

(Note tha t we could prove the ( ~ ) pa r t above wi thout knowing whe- ther the set of theorems of l l K ~ is nice. F r o m not Fm~DA by Lemma 6 i t follows tha t the set of theorems can be ex tended to a nice set not contain- ing A, and f rom this it follows tha t A does not hold in the canonical model. The disjunction p roper ty for the set of theorems of lIKE] could be proved model-theoretically, wi thout circularity, by modifying a pro- cedure f rom [22], w 5.3.)

Nex~ we shs, ll consider narrower classes of models with respect t~ which lIKE] can be proved sound and complete. We alluded to the possi- bility of such classes at the end of w 3.

DEFX~ITIO~ 10. An H [::] f rame (model) is condensed iff R I R ~ _~ R~z.. An H ~ f rame (model) is strictly condensed iff /~MR• ~ /~M"

I t is easy to show tha t str ict ly condensed HD frames (models) fo rm a proper subclass of condensed HE:] f rames (models), which form a p rope r subclass of HI2 f rames (models). I t is equally easy to show tha t in con- densed HE] frames ~I~:~M = RM~ whereas in strictly condensed HE] f ra- mes /~zR ~ = t~MR z = / ~ 4 . (All the connections between R~ and /~M in strictly condensed HE] frames follow f rom -~zRMR~--~ N~z-)

We can prove the following theorem.

T~mo~E~ 2. k~gDA ~ for every conde~sed H [] frame .~r~ _~r ~ A .~ for every strictly condensed HE] frame Fr~

~,r ~ A .

P~ooF. F r o m left to r ight we use the ( ~ ) pa r t of Theorem i . F o r the other direction it is enough to show t h a t the canonical HK[] f r a m e is str ict ly condensed -- it will follow tha t it is condensed. We have I'= _~ A and A c_ 0 ~ I'D _c 0, i.e., ~MRZe ~ ~_ R~M. (In the parenthet ical r e m a r k

228 M. Boz2~, K. Do,on

~after the proof of Lemma 7 we have already stated that the canonical HK[] frame is condensed.) q.e.d.

Our models easily provide another proof that HK[] is a conservative extension of the Keyting propositional calculus in Z [] without []: a Kripke model falsifying a non-theorem of this calculus is an H[] model where R ~ is empty, modulo some inessential adjustments.

w 5. Strict implication in H K [] and its extensions. We can introduce in the language 2~ [] a strict implication connective -3 by the usual defi- nition

A - 3 B =dr [ ] ( A ~ B ) .

fin this section we shall make some remarks on the behaviour os this connective in ILK[] and extensions of HK[]. First we give the following lemma.

L]~ i :~ 9. In H[] models, x ~ A -3 B <:~ V y ( x R ~ R x y => (y ~ A ~ y B)).

P]~00F. We have:

x ~ A - 3 B �9 r x ~ ~ ( A - + B )

Vy(xR R y (y a =, u B)). q.e.d.

In Kripke models for modal logics with a classical basis we have

x ~ A -3 B ~ V y ( x R ~ y ~ (y ~ A :~ y ~ B)).

This equivalence can replace the equivalence of I~emma 9 in strictly ,condensed H[] models, since in these models R~/~I = RM. Hence, if a system is sound and complete with respect to a class of strictly con- densed HI2 models, as HK[] is, then the Rz relation of these models will be irrelevant us far as the modelling of its strict implicational fragment is concerned. (The same applies to the strict implication-conjunction-dis- junction fragment.) This means in particular that the strict implieational fragment of ILK[] will coincide with the strict implicational fragment of t he modal system K based on classical propositional logic. The same will hold also for some extensions of HK[] and their classical analogues. So, in a sense, intuitionistie strict implication coincides with classical strict implication.

w 6. Embedding of H K [] in systems in the language J5 [] [] based on classical propositional loglc. The ]~eyting prol~ositional calculus can be embedded in the modal system $4 by a translation which prefixes [] to every (proper) subformul% save (possibly) subformulae with A or v us main connective. We shall show that the system HK[] can be embedded in a similar way in modal systems based on classical propositional logic

Modds for normal intuitio~istio... 229

wi th two squares: one an $4 square which serves for the t ransla t ion of intuitionistic connectives, an4 the o ther ~ K square. Axioms connect ing these two squares will correspond to the conditions connecting the R x a n d Rz~ relations in HO models.

Le t the language ~ [] [] be ~ [] ex tended with a new necessi ty operator oz. For perspicui ty we shall wri te O~ instead of [] for the old necessi ty operator of Z � 9 0. Next we give the following defiuitions.

D~NITION 11. <X, RI, R~> is a birdationa~ frame iff X is a n o n e m p t y set and RI and/~zz are relations defined over X.

<X, RI, R ~ , V> is a birelational model iff <X, Rz, tiM> is a birela- t ional f rame and V is u mapping f rom the set of proposit ional variables of J5 [] [] to the power set of X.

I t follows immedia te ly t ha t an H � 9 f rame is u birelationM frame, and t h a t an H � 9 model is a birelational model (modulo the subst i tut ion of Z O O for LO).

DEFINITION 12. I f d4 is a birelationM model <X,/~x~ RM, V>~ x e X and A is a formula of Z � 9 O~ then the relat ion <d//~ x> ~ A (abbreviated by x k A) is def ined by :

(i)-(iii) a re as in Defini t ion 4 (iv) x ~ B-+C - ~ (x ~ B => x ~ C) (v) x ~ -1B <=>at not x k B

(vi) x ~ 0 I B ~ Vy(xRxy ~ y ~ B) (vii) x ~ OM B ~af Vy (XRMy :z- y ~ B).

Analogues of Definitions 5, 6 and 7 hold when we replace H � 9 frames (models) by birelationM frames (models). Next we prove the following lemma.

L E ~ 10. Zet ~r = <X~RI~RM> be a birelational frame. Then:

(i) ~r ~ OM o l A --> o i []M- A- ~ .t~IRM ~-- -t~MRI (ii) 2~r ~ [:]MA -> Oi OM A r R I R M _ ~M

(iii) 2Fr ~ o~A--> O~ oxA ~ I~MR ~ ~ R ~ .

P~OO~. (i) We have ~ha~ x ~ O~ o~A -> Oz o ~ A is equivalent with

(1) Vy(xt~MRIY :*- y ~ A) ~ Vz(xRzI~MZ ~ z ~ A) .

I t is easy to show tha t I~IRM ~-- RMR ~ implies (1). To show the converse suppose tha t in a birelationM f rame not I~ZRM ~_ ~MI t~ i . e , for some x and z, xR~RM z and not xRMR~z. Then let u ~ ~ "~d~ XRMRx~. ~u have

~ y ( x R ~ R z y =~ y k p) and ]z(x t t zRMz and not z ~p) ;

i . e , (1) does not hold. For (ii) and (iii) we proceed analogously, q.e.d.

We now int roduce three systems in Z [] [] which are obtained by extend- ing an axiomatizat ion of the classical preposi t ional calculus. More pre-

230 M. Bo~6, K. Do~w

eisely, these systems are obtained by extending ttl-]~10 and N P wi~h=

P.

V]I1. []zAA[]zB-~ o~(A AB) Ox2. [~(A ~ A ) 0• E]zA -->A []zd. o~A --> []z [~z A

A-->B A ~ B Oz. t~ OM.

Ol A --> 0IB oMA --> []MB.

OM1. OM 2.

O~:sAAo:uB-> []~(A AB) o ~ ( A ~ A )

The system S4K is obtuined by adding moreover

o S d K . 0 ~ r] IA --> 01 OM A �9

The system S4Kc ("c" stands for "condensed") is obtained by extending SdK with

oS4Kc. []~A-~ Ox []M A

and the system S4Ksc ("sd' stands for "s$rictly condensed") is obtained by extending SdK with

oS4Ksc, o2~A-> []~ ozA.

By using fairly s tandurd methods frmll modal logic and Lemm~ 10 iV can be shown tha t SdK is sound and complete with respect to birela- tional frames in which/~z is reflexive and transitive, and Rzt~ M ~_ RMR x

- - i.e., S4K is sound and complete with respect to H[] fra~mes. To obtain soundness and completeness for S4Ke (SdKsc) these frames will satisfy also R~R i ~_ R~ (RMI~ I ~_ I~) , i.e., they will be condensed (strictly condensed) / / 0 frames. Then it is easy to prove tha t SdK is a proper subsystem of S4Kc, which is a proper subsystem of SdKsc. (:Note t h a t oS4K is superfluous in SdKc.)

:Next we define the following translat ion (mapping) from L [] to Z [] [] :

t(p) =d~ ozp t(AAB) =a~ t(A)At(B) t (AvB) =d~ t(A)vt(B) t(A ~B) =(~ Oz(t(A)~t(B)) t(-lA) =a~ 0I~t(A) t (oA) =d~ O~ut(A).

In the following lemmata (cf. [18], pp. 43-44) ~'H will be an H[] model and d/B a birelational model on the same H[] frame.

LE)fi~A 11. I f for every x and every p, (~r x) ~ p ~ (JIB, x) ~ t(p), then for every x and every A, (-gin x) ~ A .~ (dtm x) ~ t(A).

Models for normal intuitionistiv... 231

We omit the proof of this lemma, which consists of a straightforward induction on the complexity of A.

JJElV~h 12. ~ot d4~ ~ A =~ 3///B not . ~ ~ t(A).

P~ooF. Let d/B =at d4~. We have:

<J4B, x> ~ t(p) "~ <~B, X> ~ []Zp ~:~ Vy(xRzy :*- <J-~B, Y> ~ P)

Then use Lemma 11. q.e.d.

Z~3ka,~A 13. Not ~II n ~ t(A) =~ 3~4~ not ~ /~ ~ A.

P~oo~. Define on the frame of ~/~ a H [] model ~//~ such that

<~H, x> ~ p ~at (~B, x> ~ t(p).

Then use Lemma 11. q.e.d.

Using these lemmata and the soundness and completeness theorems for HKc2, S4K, S4Ke and S4Ksc, we can show the following theorem.

T ~ o 1 r 3. Zet S be S4K, S4Kc or S4Ksc. Then Fnw7 A r Fst(A),

Since in all the systems S of Theorem 3 we can prove;

[~zAAnzB ~ Uz([2zAA~zB) 5zA v []zB ~ E]z( ~zA v c3zB) ~ []xA ~ E1z []z~ []z x

(compare the last theorem with I~emma 5), we could use the translation t 1 from J5 [] to L [] [] which prefixes ~z to every subformula, in order to embed H K ~ in S in the sense of Theorem 3. And since for all these sys- tems S

[]zA ~ t-A

the translation Q which differs from t~ by prefixing []z only to prol)er subformulae could serve the same purpose.

We shall conclude this section with some remarks on the embedding of K (i.e. HK[] extended with ((A-~B)-+A)-~A) in tlKr7 with the help of translations based on double negation. I t is easily shown that Gli- venko's Theorem does not hold for K and H K ~ : using a suitable falsifying model, we can show that "1-7[](77A-+A) is not a theorem of t IKD. In ot]aer words, it is not the case that for any A the following holds: ~ A ~ ~HK~-7-tA. On the other hand, using the translation g which prefixes a double negation to every subformula we can show that ~ K A -~ F~a-~g(A). From left to right this is proved by a straightforward

232 M. ~o~i~, K. Do~en

induct ion on the length of proof of A in g . Since in the ~ey t i ng proposi- tional calculns we can prove -]'~(-]-]A-->~-]B) ~--~(-~TA--->'-]'~B), "~'-]('-I-]AA-]-]B) *--~ "-'I~AAT-]B and -]-]-]A ~ -]A, the same embed- cling could be obtained when using the translat ion gl which differs f rom g in not prefixing double negation to subformulae which are implications, conjunctions or negations.

The system HK 0

w 7. The syntax of H K O. With this section we start our t rea tment of systems with (> primitive. Sections 7, 8, 9 and 10 will be parallel to sec- tions 2, 3, 4 and 6, respectively, and the results presented will be in a cer- tain sense dual.

Let Z(> be the language which differs from Z [] only in having the possi- bility operator (> instead of []. We introduce the propositional calculus ILK(> in Z(>. As for IlK[]~ the non-modal axioms and rules of ILK(> are given by H1-H10 and MP. The modal axiom-schemata and the nmdal rule of HK(> are

(>1. (> (A vB) -+(>A v(>B (>2. qoo-7(A---.'.A)

A .-> B (>a--,(>B.

I t can easily be seen tha t they correspond to []1, ~2 and ~ [2. As well as the extensions of HK[], the extensions of HK(> are closed

under the l~ule of Replacement of equivalent formulae. Also, the De duction Theorem holds with respect to ~s, where S is any extension of the Heyt ing propositional calculus in Z(>. I t is also easy to show tha t ILK(> is a conservative, and hence ccnsistent~ extension of the Heyt ing propositional calculus in Z(> without (> by a procedure analogous r the procedure ment ioned for IIK[] in w 2.

I n order to prove tha t the set of theorems of ILK(> has the disjunction proper ty we extend the slash defined in w 2 with the clause

] (> A "*> at IPA. We shall also need the following lemma.

LF_~A 14. ~o formq, La of She form (>A is a theorem of I tK (>.

P~ool~. Define the following mapping:

f(lo) =~f p f (AaB) -~d~ f(A)af(B), where a is A~ v~ or -> f('-]A) =af ~ f ( A ) f ( (> A ) ~- ,u "-](A-+A).

I t is easy to show by induction on the length of proof of A tha t F~KoA implies tha t f (A) is ~ theorem of the Heyt ing propositional calculus. ~ r o m this the Lemmu follows immediately, q.e.d. ~ o w we can prove the following lemmu.

Models for normal iutuitionistic... 233:

L v , ~ 15. The set of theorems of I IK 0 has the disjunction ~roperty.

:P~oo~. As for Lemma 1~ we show by induction on the length of proot~ ' of A tha t FA ~ [A, where F stands for ~ o "

In the basis we consider only the modal cases. For 01 we proceed as follows. I n vir tue of Lemma 14 k 0 (A vB) is false, and hence

F 0 ( A v B ) and [0 (AvB) ~ [OAvOB

from which we obtain [<~ (A vB) ->0A v0B. ~or 02 we have

not (t-O ~(A--->A) and ~- ~ ( A ~ A ) and [-l(A-~A))

f rom which we obtain [ '~O~(A~A) .

In the induct ion s~ep we proceed for 1~0 as we did for 1~ [] in the p roof of Zemma 1~ and by following this proof fur ther we obtain the Zemma~ q.e.d.

w 8. H 0 models. Models with respect to which we shall show tha t tIKO is sound and complete differ f rom HKD models only in the condi- t ion connecting the /~x and R~z relations.

DEFI~ITIO~ 13. (X~ R I , / ~ ) is an H O frame iff (i)-(iii) are as in Definition 2 and

(iv) R-ilRM ~_ tt•R71.

The definition of tIO models is obtained f rom Definition 3 by subst i tut ing everywhere 0 for ~. I n Definition 4 we again make this subst i tut ion and we replace clause (vi) by

(vii) x ~ 0B ~af 3 y ( x ~ y and y ~B).

Definitions 5 and 6 remain unehanged~ and in Definition 7 we again subst i tute 0 for 71. Next we can show the following 1emma by induct ion on the complexi ty of A.

L E ~ I ~ 16 (INTuITIONISTIC ]~E~.EDITY). In every l I o model (X~t~z~ t~.1i, V)~ for every x, y e X and for every A of _L 0

xt~ly :~ (x ~ A =~ y ~ A).

This lemm~ proves tha t the condition ~ 7 ~ _ c / t ~ R i ~ is s~fficien$ to make every H 0 model a model for the t Iey t ing propositional calculus in the extended language 150. That this condition is also necessary is shown by the following lemm% whose proof is analogous to the proof of I~emma 3.

LE~I~)~ 17. Zet {X~ R ~ t t ~ satisfy conditions (i)-(iii) of Definition 1~ (see Definition 2) and let condition (iv)~ i.e.~ , ~ ~ ~ ~_ ~ ~ ~-~ ~ be q~nsatisfied,

234 M. Bo~i$, K. Dow162

Then there is a formula A of Z<> and a valuation V such that in ( X , Rx , R ~ ~ V} for some x , y e X

xR~y and x ~ A and not y ~ A .

So, in a certain sense, models with the condition RT~Rz~, R ~ R 7 ~ form the largest class of models with respect to which we can expect to show tha t tlK<> is sound and complete. But we also have the following lemmata which indicate t h a t a proper subclass of H 0 models might be suff ic ient

T,E)X~A 18. I n H 0 models, x ~ OA .~ 3 y ( x l ~ R ' i ~ y and y ~ A).

This lemma is proved using z ~ A .~ 3y(yl~]z and y ~ A), which is obtain- ed from Intui t ionist ic Heredi ty and the reflexivity of /~] .

LE3~vL~ 19. In the definition of H <>frames the condition l~-i~ tgx~ ~_ R~1~7 ~ can be replaced by the condition t~7~RMR-i ~ ~_ RMR7 x yielding the same class of frames.

This lemma is proved by using the reflexivity and t ransi t ivi ty of RI. So, roughly speaking, out of HO models we can m a k e n e w models by

replacing the R~uR7 x relation by a new relation Ro such tha t in these new models x ~ 5 A ~ 3 y ( x R ~ y and y ~ A) and RT~Rr _ R<) (every R ~ relation disappears f rom the old models, since by the reflexivity of /~z every ~ is followed by some Rye). The relation /~o is the /~M relation of the new models. These new models validate exactly the same formulae as the old ones.

Since RMRT R7 we can fur ther "condense" these models by making RoR-~ ~ ~_ R o (cf. Definition 15).

w 9. Soundness and completeness of H K ~ . In this section we shall prove tha t HKO is sound and complete with respect t o HO models.

On the set of all nice sets of formulae of Z(> we shM1 build a canonical model defined as follows.

DEFINITION 14. Let S be any extension of H K 0 in Z 0, and let X c = {F]F is a nice set with respect to ks} F t ~ A ~af i ' c A, where F , A e X c FRCM 3 "~a~ A <> ~_ F, where A ~ = a ~ { 0 A I A s A } and / ' , A e X ~.

Then <X c, R~, R ~ is the canonical S frame.

The canonical S modet is obtained from the canonical S frame as in Definit ion 9 when we replace Z[:] by iS(>.

I t is easy to verify tha t Lemma 6 holds also for extensions S of the t t ey t ing propositional calculus in Z0 . In the following three lemmata S will s tand for any consistent extension of HKO in ZO.

3lodels for normal ~ntu~tionistiv... 235

LEPTA. 20. The canonical S frame is an H 0 frame and the canonical S mode~ is an l I 0 model.

P~ooP. For the first pa, r t we have:

(i) X ~ = O, since the set of theorems of S is consistent and hence by Lcmm~ 6 it can be extended to a nice set. (The set of theorems of HKO is already nice, as it is consistent~ deductively closed and, by Lemma 15, it has the disjunction property.)

(ii) / ~ is a reflexive and transitive relation over X c.

(iii) R ~ is obviously a relation over X c.

(iv) ~(Ri) - I I~MA ~ ~ 0 ( 0 ~_ I ~ and Av ~_ O) => A~ c_ F

c c - 1 / ' ~ ( R z ) A, by the reflexivity o f / ~ .

So the canonical S frame is an H0 frame. For the second purr we proceed as for the second part of the proof

of Lemma 7. q.e.d. In the following l e m m a "nice" means "nice with respect to bs".

: L E ) ~ 21. Suppose F is nice and 0A e P. Then there is a nice set A such that A e A and A o _ F .

:PI~oo:F. Let Z :d~(~]A e r and qS~ ~ F and C/(~5) ~ ~}. First we show that C I ( ( A ) ) e Z . For this we only need to show that (CI((A))) ~ c F, the rest being trivial; Since Z is nonempty~ and since it is easy to prove that it is closed under unions of nonempty chains, by Zorn's Lemm~ it has a maximal element A with respect to ~. Then we show that A is nice. q.e.d.

L:E~A 22. I n the canonical S model~ for every I" e X ~ and for every A

I ~ A ~ A e F .

P~ooF. By induction on the complexity of A. As in the proof of Lemma 8~ only the modal case of the induction step will be considered. We have:

F ~ O B ~ 3A(FR~M A and A ~B) .~ 3 A ( A ~ ~ F and B e A)~ by Definition 14 and the induction

hypothesis.

~u shall show thut o B e F ~ ~ A ( A ~ c F and B e A). From left to right we use Lemma 21. The other direction is an immediate consequence of the definition of A% q.e.d.

l~ow we cain prove the soundness and completeness of HK0 with respect to HO models.

3 - Studia L o g i c a 3/84

236 M. Bo~i~, K, Do$en

Tm~ORE~ 4. t-~K~A ~ ~ o A .

P~OOF. The soundness part ( ~ ) is proved by a straightforward induction on the length of proof of A in HKO, and we shall omit it.

For the completeness part ( ~ ) suppose ~n~A. Then by Lemma 20, A holds in the canonical HK~ frame, and consequently A holds in the canonical HKO model, i.e., V/" e X c . / ' ~ A. By Lemma 22, V/~ e X c. A e/~. Since, as we have seen in clause (i) of the proof of Lemma 20, the set of theorems of HK(~ is a nice set, it follows that ~nK~A. q.e.d.

(Note that analogously to what we had for Theorem 1, we could prove the ( ~ ) part of Theorem 4 without knowing whether the set of theorems of HK5 is nice.)

:Next we shall consider narrower classes of models with respect to which HKO can be proved sound and complete. We alluded to the possi- bility of such classes at the end of w 8.

DE~NI~IO~ 15. An H ~) frame (model) is condensed iff R 7 ~ / ~ _ R M.

An HO frame (model) is strictly condensed iff / ~ R i ~ ___ R~. I t is easy to show that strictly condensed HO frames (models) form a pro-

per subclass of condensed H~ frames (models), which form a proper sub- class of H(~ frames (models). I t is also easy to show that in condensed H~ frames 27~R~ =/~M, whereas in strictly condensed HO frames "~il '~M = ~ M ' ~ I 1 = ~ M ' (Al1 the connections between /~i and R M in strictly condensed H0 frames follow frem / ~ / ~ R 2 ~ ___ R~x. )

We can prove the following theorem.

T~EO~E~ 5. ~u~<)A ~ f o r every condensed H ~ frame Fr, Yr ~ A .~ for every strictly condensed H O frame Fr,

~r ~ A.

2~OOF. From left to right we use the ( ~ ) part of Theorem 4. For the oCher direction it is enough to show that the canonical HKO f rame is strictly condensed -- it will fellow that it is condensed. We have

AO ~_ T' and O ~ A => O(~ ~_ A o =~ 0 0 ~ .F~

I ~ ( t ~ ) ~_ tr i.e., ~ ~ -~ ~ (~ote that in (iv) of the proof of I~emma 20 we have already shown that the c~nonical HKO frame is condensed.) q.e.d.

Our models can easily provide another proof that HKO is a conser- vative extension of the Vreyting propositional calculus in Z(~ without ~.

w 10. Embedding of H K ~ in systems in the language ~ [] 0 based on classical propositional logic. In this section we shall show that with the help of translations like those of w 6 the system HK~ can be embedded in modal systems bused on classical propositional logic with a square

.Models for normal intuitio,~istic... 237

and a d iamond: the square, which is of the $4 type, will serve for the t ransla t ion of intuit ionistic connectives, and the d iamond will be a K diamond. Axioms connect ing the square and the d iamond will correspond to the conditions connect ing the R~ a n d / ~ i relations in H<~ models.

Let the language ZOO be LO extended with <~. I n the present section we shall wri te O~ for O, and < ~ for <>, in order to make clearer the role of [] and <). T e x t we ex tend Definit ion 11 and Definit ion 12 by substi- tu t ing .LO<> for Z O O and adding x ~ (>MB <=> 3y(xt~z~y and y ~ B).

We can show the following lemma.

L E p t A 23. .Let _Fr = <X, RI , RM} be a birelational frame. Then:

(i) Fr ~ <>M 0I A ~ OzO~A <:~ RT~R~u ~- RMR-i 1 (ii) _Fr ~ (}MA ~ []I~M A. "r R/1R3I __~ R3f.

P~ooF. (i) We have tha t x ~ (}M o I A -+ Oi #MA is equivalent with

(1) 3y(xRMy and Vz(yt~zz ~ z ~ A)) ~ Vt(xRxt ~ 3u(tRMu and u ~ A)).

:Firs~ we show tha t R-[l t t~ c_ R ~ t t - [ ~ (1). Suppose tha t x R ~ y and V z ( yR i z ~ z ~ A) and xRIt. I t follows f rom /~Y~/~M ~ RMR-[ ~ t ha t there is a v such tha t yRiv and tRMV. F r o m Vz(yR~z ~ z ~ A) we obtain v ~ A; hence, ]u( tRMu and u ~ A).

To prove tha t (1) ~ / ~ i ~ R ~ E R,~R~ 1, suppose tha t in a birelational f rame not lr ~_ RMR~ ~, i.e., for some x, y and t, x l ~ t and x t ~ y and not t R ~ R ~ y . Then let v ~ p ~d~ YRzV. I t follows immedia te ly tha t

(1.1) xR~-~y and Vz(yR~z ~ z k p) and xR•

F r o m not tRMR-~y it follows tha t V u ( t R ~ u ~ not u ~ p). This and (1.1) imply the falsi ty of (1). For (ii) we proceed analogously, q.e.d.

Till now our results concerning HK(> have been all in a certain sense dual to results concerning H K o . So we would expect an addit ional clause in L e m m z 23 sgating what formula A of L O(> is such tha t

(iii) ~ r ~ A ~ / ~ / ~ __ R ~ .

~u leave ~he question open if there is such s formula. (This question looks similar to the question whether there is n formula of modal logic such tha t in an ord inary Kripke f rame /Tr, ~ r ~ A ~_~R-~ _~/~. This condi- t ion concerning /~ is obtained by a k ind of inversion of the condition satisfied by an euclidean /~, viz. /~-~/~ ~_/L)

We now int roduce two systems in .L [] 5 which are obtained by ex tending t t1 -~10 , t ) and MP (i.e., the classical propositional calculus) with Ozl- O~4 and /~Oz (see w 6), and <~M 1, QM2 and R<>:r obtained f rom <>1, 02 and 1 ~ by subst i tut ing <>M for <~. For the sys tem SdK' we add moreover

238 ~ . Bo~i~, K. Do~en

and for the system S4Kc'

(>S4Kc'. o ~ A -~ ~OMA.

~ote that if to S4K' and S4Ke' we add ~MA ,~ " ]DITA, i.e., the usual definition of ~ in terms of D, the schemata ~M1, ~M2 and the r u l e / ~ become superfluous, and the systems obtained are K T 4 G (in the termi- nology of [7]; the old name is $4.2) and $5, respectively. I t is well known that these two systems are proper extensions of $4. On the other hand, the $4 character of S4K, S4Kc and S4Ksc would be preserved by adding OMA ~ []IA tO these systems. At this point the duality between our treatments of [] and ~ ceases to hold.

By using fairly standard methods from modal logic and I~emma 23 it can be shown that S4K' is sound and complete with respect to birela- tional frames in which R I is reflexive and transitive a n d / ~ i 1 ~ c t~MRi 1 -- i.e., S4K' is sound and complete with respect to H(> frames. To obtain soundness and completeness for S4Ke' these frames will satisfy R-1R~ c R ~ , i .e , they will be condensed H~ frames. Then it is easy to show that S4K' is a proper subsystem of S4Ke'.

~ e x t we define the translation t from LO to J5[]6 which is obtained from the translation t defined in w 6 by substituting

t (~A) =dr <>~t(A)

for the l~st clause. Using Lemmata 11-13, which hold when we substitute H0 for H ~ and the soundness and completeness theorems for HK~, S4K' and S4Ke', we can show the following theorem.

THEOREiK 6. .Let S be S4K' or S4Ke'. Then F'HKoA .*~ bst(A).

Since in both S4K' and S4Ke we can prove:

E]IAAE]IB ~ ~([:]IAA[3IB) [~zA V [31B ~ []1( [:]z A v ~I.B) OM [~Z A ~-~ [ 3 I ~ [:]~A (cf. Lemma 19)

~ A ~ bA

we could use the translations t~ and t, (see w 6) Crom Z~ to Z ~ 5 , in order to embed HKO in S4K' or S4Ke'.

We shall conclude this section with some remarks on the embedding of K (i.e. HK(> extended with ((A-->B)--->A)--->A)in HK(> with the help of translations based on double negation. As we had for H K ~ at the end of w 6, Glivenko's Theorem does not hold for K and HK~. Since -1-1 (~ -~-]A -+~A) is not a theorem of HK(>, as can be shown with a suitable falsify- ing model, it is not the ease that for any A the following holds: ~KA

k~K~ -]-]A. But now, in contradistinction to what we had with H K ~ , we cannot use the translations g or g~ to embed K in HK~. This follows

Models for normal intuitio~istie... 239

from the fact tha t

nn (nno nn(n-qA vnnB)- nn(nno vnno nnB))

is not a theorem of HKO, as can be shown using a suitable falsifying model. This again indicates a certain lack of duality between [] and 0 in intuitio- nistic modal logic.

The system I l K [DO

w l l . H [] frames and systems in the language ~ [] O. When presented with systems in the language L[20 (i.e., L[] extended with O) based on the Heyting propositional calculus, we could ~ind models for them in trirelational frames of the form <X, R~, R~ , ~ > , where <X, ~ z , / ~ > is an HE] frame, <X, Rz,/tM> is an HO frame, the holding of [2A is defi- ned in terms of/~M and the holding of 0A in terms of R~ . This is, in prin- ciple, the course suggested in [30], where trirelational frames are based on what we call strictly condensed H[2 and HO frames. We could perhaps imagine that these trirelational frames arise out of a birelational frame by replacing Rz~Rz by R a and ~z~R7 ~ by Ro, following the suggestions at the end of w and w

Another possibility, which we shall investigate in this section, is to find birdatio~.al models for intuitionistic modal logics in ZE]<>, i.e., models in which the holding of both [2A and OA is defined in terms of the same relation ~M.

t~irs% we shall introduce a system in L[2<>, related to HHD~ which we shall call HK[20. The axioms and rules of HHD0 are those of HH[] extended with:

[]01. 0A v[]NA D02. ~ ( O A A D - ] A ) .

I t is not surprising to find that in systems in L[20 for which we wa, nt to find models with a s i n g l e / ~ relation there will be axioms connecting [] and 0. Our axiom-schemata [201 and D02 are in a certain sense "in- duced" by the completeness proof which we shall give for I t K [ ] o and which goes via a canonical model where/ 'R~zA is defined as F a ~ .4, i.e. in terms of D.

I t is easy to show that OA ~-. -1[::3-1A is provable in HK[]O, since in the Heyting propositional calculus we can prove:

N(<>AADNA) *-~ (oA--> N [2NA) O A v [2 - IA -+ ( -1E] -3A ---> o A ) .

The converse of the second formula holds classically, but not intuitionistic- ally, and the system S obtained by extending HK[2 with <>A ~ NDNA would be weaker than H K [ ] o . To show that, let g(A) be obtained from A

240 M. B o ~ , K. DoWn

by replacing every occurrence of (> in A by -70-% Then it is easy to show that FsA ~ FuKog(A). But not ~HKog(<)A vo-TA), since by I~emrna 1 the set of theorems of H K o has the disjunction property. Hence o<>1 is not u theorem of S.

We can easily prove that H K o 0 is consistent with the help of a map- ping which deletes from formulae all occurrences of [] und <> (el. w 2); however, in this way we map H I ; o 5 not in the Heyting but in the classic- al propositional calculus. The system H K o ~ , as well as H K o and HK(>, is a subsystem of the modal propositional calculus I;, suitably formulated. As before~ we can show that I I K o 5 is dosed under the Rule of l~epla- cement of equivalent formulae, and that the Deduction Theorem holds with respect to FH~oO . But in contradistinction to / / K o and / /K0, the set of r of I tKo(> does not have the disjunction property, us

consideration of O(>1 shows. A strictly condensed H o frame (model) which is at the same time

an HO frame (model) will be called an tlo<> frame {model); i .e , in HOO frames we have:

"RI BM ~" RM'~I : ~ M 1~7~1~M ~ RMR-i ~.

In Definition ~ of the relation ~ we add clause (vii) for OB from w 8, und in Definition 7 we repluce / /O by HO~.

Intuitionistic Heredity for HO~ models follows immediutely from Lemmata 2 and 16.

Next~ we proceed to prove that HKO~ is sound and complete with respect to H ~ models. The c~nonica l / /KO~ frume (model) is obtained as in Definition 9 when we replace LO b y / ~ O 0 and S i s / / K o ~ . We can show the following lemmata.

L ~ X 24. The canonical H K [] (> frame (model) is both a strictly condensed I t [] frame (model) and an I I <> frame (model).

P~ooF. To show that the canonical H K [] 0 frume (model) is a strictly condensed H[] frame (model) we note only that X c r O, since the set of theorems of HKo<) is consistent and by Lemma 6, which holds in the present context to% it can be extended to a nice set. For the rest we proceed as in the proofs of Lemma 7 ~nd Theorem 2.

To show that this frame (model) is also an H(> frame (model) we need to prove only that c -~ r (RI) R M ~_ R~(R~) -~. Suppose, on the contrury~ that for some F, A e X ~

~ X ( X =_ F and X o ~ zJ) and not 3 I I ( F D ~_ I i and A =_ II) .

From not 311(F o =_ 11 and A ~ 11) it follows that FowA is inconsistent -- otherwis% by Lemma 6, it could be extended to a nice s e t / / . Hence, for some Ai s F o and some Bj e A, {A1, . . . , A,~, B1, . . . , B~} ~- 7 ( C ~ C ) .

MoRels for norma~ in$uifionistie... 241

:From this, by the Deduction Theorem and theorems of H K ~ O , we obtain:

F A ~ A ... AA,~-~ -TB F [3A~A ... A[z]Au--> []TB

where B is BxA ... A B e , and B e A as A is nice. Since A i e F~ ~nd since / ' is nice we have [:]-7B e F. But then it follows from [302 that 0B ~ F. From Z _~ /~ we obtain 0B ~ Z, but from ~01 it follows that [3-7B e Z, since X is nice. ~tence, -~B e Z~, and as Z~ _~ d, we have -~B ~ A. But this gives u contradiction since B e A. q.e.d.

L ] ~ A 25. I n the eanonieal I t K [] 0 modal, for every 1 ~ ~ X ~ and for every A

F ~ A ~:~A e F .

Pgoor . By induction on the complexity of A. We shall consider only the modal case of the induction step which was not t reated in the proof of Lemma 8. We have

I ~ O B <:~ 3 . A ( P R ~ A and A ~B) <:> 3A(Fr7 c_ A and B ~ A), by Definition 9 and the induction

hypothesis.

We shM1 show that <)B e F ~ 3 A ( I ~ ~ A) and B e A). From left to right we proceed us follows. Suppose 0B e / ' . Then, by D02, D-1B ~/ ' , and it follows that I~ u{B} is consistent. (Otherwise, /'D F 7B, and for s o m e A i e I '~, F AxA . . . AA n->-7B. This implies F DA~A ... ^ []A n -> ~-TB~ and hence []77B e/ ' . ) So by Lemma. 6, I'Dk){B } can be extended to ~ nice set A.

For the other direction we have:

O B ~ / ' ~ [ ] - ] B E / ' , by [301 V A ( r D c_ A => - lB ~ A) VA( I" D ~_ A ~ B f A), since A is nice. q.e.d.

ZNow we can prove the soundness and completeness of H K o 0 with respect to H ~ 0 models.

Tm~0~E~ 7. FAKDoA ~ for every H ~ 0 frame ~r, ~ r ~ A.

P~ooF. ( ~ ) By ~ straightforward induction on the length of proof of A in HK[]O. We shall consider only the cases with [301 and []02. F o r [:]01 we u s e R M R I c_ RM to s h o w t h a t

(1) 3y(xRMl:lzy and y ~ A) ~ 3z(xRMz and z ~ A)

and from this we can easily derive Vx. x ~ 0A v ~ - l A . For [:]02 we use ~M c_ R ~ R ~ to show the converse of (1), from which we can easily derive Vx. x ~ -~(0AA[]TA).

( ~ ) Suppose for every Hi20 frame ~r~ ~ r k A. Then by Lemma 24, A

242 M. BoSi~, K. Doge~

holds in the canonical H K ~ 5 frame and, consequently~ A holds in the canonical H K ~ model~ i.e. VF e X ~. T' ~ A. By I~emma 25, V/ ' e X c. A e/~. Next, suppose not t'HK[]~A. Then, by Lemma 6, the set O of theorems of HK770 can be extended to a nice set A such that 0 _ A and A r A. This contradicts V/ ' e X c. A e/~, a n d hence ~AKD~A. q.e.d.

Our models can easily provide a proof that H K [] 0 is a conservative exten- sion of the Heyting propositional calculus in 3~ [] without []. ~rom results in w 6 and w 10 we easily deduce that the system HK[]O can be embedded by the translation t in the system in L[]0 (whose modal operators are ;7~ and 5M) extended with an additional operator []i, the axioms and rules of which are obtained by taking the union of the axioms and rules of S4Ksc and S4K', and also O Az~ ~ - 3 ~ M T A .

I t follows immediately from Theorems 7 and 4 that H K 5 (in L ~ 0 ) is included in H K ~ O (for / /K[] this is trivial). Of course, this can be shown syntactically as well. I t is of some interest to consider the proof of O1 in H K []5 which we proceed to sketch:

( O A v D - ] A ) A ( O B v [ ] y ] B ) , by ~01

-3([]-3AA~-7B) -+ O A v O B

-] [] ("-]A A -3B ) -+ O A r O B

-3[]-](A vB) -+OA v<)B ~ by []52

0 (A vB) -+0A v0B.

We also have that "7 []-3(A vB) -+ -3 []-]A v-3 [2-]B is a theorem of HK[20, since every %heorem o f / / K 5 in which we replace O by "] [:]-7 is provable i n / / K [ ] O . But the same does not hold f o r / / K [ ] , i.e 4 not every theorem of IIKO in which we replace 0 by -1[]-7 is provable i n / / K ~ , and in par- ticular -7 [] -3 ( A v B ) -+ -] [5 -]A v -3 [] -3B is not provable in / /K[] . This can be shown with a suitable H[] model falsifying this formula. (Dually~ not every theorem of / /K[] in which we replace [5 by -70-7 is provable i n / / K O , and in particular 7~- ]AA-3OTB- + "]O - ' I (A A B ) is not provable in ~~KS. ~ow this is shown with a su i t ab le / /5 model falsifying this for- mula.) So the use of []51 in the proof above is essential.

Another way to formulate ILK[50 is to take I tK[] in L[50 and extend it with:

[]3. "q []-TA v [] --]A [203. 5 A ~ 7[] - ]A .

l%r strictly condensed / / ~ frames /~r it is possible to show that 2~r -3[]TA v [ ] T A .r c_ ~ M R i ~ (the techniques needed to prove

such equivalences will be considered extensively in [10]; cL also the proof of Lemma 3). Then it is clear that with a completeness proof which goes via a canonical model which is a strictly condensed / / ~ model we must admit []3. On the o ther hand~ []3, which is valid with respect to

~odds for qwrmal intuitionistic... 2~3

all H O 0 models, would not be val id if in the definition of these models we replace the condit ion tha t t hey are strictly condensed H � 9 models b y the weaker condit ion tha t t hey are H � 9 models (or the weaker condi- t ion tha t t hey are condensed H � 9 models). This explains the a s y m m e t r y in the definition of H O 0 models, which might have puzzled the reader.

I t is not difficult to modi fy proofs we a l ready have in order to show tha t the sys tem which is as / / K o 0 in this new formulat ion, save t ha t ins tead of 03 it has - 1 o A v o A ~ is sound and complete with respect to strictly condensed H � 9 models which are also condensed t Io models. Since - ] o A v o A can be falsified in an H O 5 model, H K o 5 is a proper subsys tem of this system. Fo r str ict ly condensed H � 9 f rames l~r it is possible to show tha t ~'r ~ - ] o A v o A ~ / ~ i l R M _ /~M"

We shall conclude this paper with some remarks on the dual i ty of [] and O. Suppose we in t roduce in our language the opera tor [] or the ope- ra tor (>z for which we give the following clauses in the definition of ~ in Kr ipke models <X, Rz, V> for t t ey t ing ' s logic:

x ~ o I A "r VY(xRzY :~ Y ~A) m ~ S i A ~af ] y ( x ~ I y and y ~ A).

_Now it can easily be shown tha t we have ~ � 9 ~ A, since f rom Intui- t ionistic H e r e d i t y ~nd the ref lexivi ty of 1~i,x ~ A -,.. Vy(xl~iy => y ~ A). B u t not ~ OzA ~ A, and in order to preserve Intui t ionis t ic He red i ty we mus t require tha t in our models / ~ 7 ~ z -~ i~i/~i 1 (el. L e m m a t a 16 and 17). I t is well known tha t this condit ion for /~I is equivalent to ~ -TA v-7-~A, and -TA v- t -TA is no t a theo rem of Hey t ing ' s logic. So, Oz and O• would not be exact ly dual.

We have seen in this section how a canonical model, w h e r e / ~ is defi- ned in t e rms of [:], can be bo th an H � 9 model and an HO model. The sys tem of which this is the canonical model lacks the dis junct ion proper ty , and is perhaps intuit ionist ically spurious, bu t still it is a conservat ive extension of the t t ey t i ng proposi t ional calculus~ and it is not far f rom H K o and IIKO. I t can be asked whether a canonical model for a sys tem which is intui t ionist ical ly plausible a t least as much as I t K o 0 , in which / ~ i is defined in te rms of ~, can also be bo th an H � 9 model and an H(> model. We shall leave this p rob lem open. t towever , it seems tha t results concerning H K o 0 and H O 0 models cannot be s t ra ightforwardly "duu- l ized ' . This nondual i ty be tween [] and ~ (which also consists in a k ind of p r imacy os [] over O) will occur again at some points in [10], where we sh~ll deal with intuit ionist ic analogues of sys tems stronger than g a.

a We are indebted to a referee for drawing our attention to a paper by H. On�9 "On Some Intuitionistic l~[odal Logics", t)ubl, l~es: Iq~st. 21lath. Sci. (Kyoto) 13 (1977), pp. 687-722. Among other things, this paper covers some of the topics treated in [10] ; in particular, it gives completeness proofs for analogues of $5 with respect to specific classes of HE] models. It also deals with the decision problem for intuitionistic modal logics, including some os the logics in [10].

2 ~ 4 M. Bo~i~, 1L Do~ea

R e f e r e n c e s

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Models for normal intuitionistie... 2~5

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2~[ATEMA~I~Kr INST~TUT KNEz MIrIArLovA 35 BELGI~ADE, YUGOSLAVIA

l~eceived ~ebruary 2, 1983, revised September 28, 1983

~Studia Zog~r X L I I I , 3