A. v. K z sov On Superintuitionistic Logics as AW SKY Fragments of Proof Logic

Extensions

Abstract. Coming from I and Ci, i.e. from intuitionistie and classical propositional calculi with the substitution rule postulated, and using the sign o to add a new con- nective there have been considered here: Grzegorczyk's logic Grz, the proof logic G and the proof-intuitionistie logic I ~ set up correspondingly by the calculi:

G r z = C l o ~ + ( [ 3 p D p) +(E~(p D q) ~ ([2p ~ ) ) +([]p ~ 71E]p) +(Z] (~(p

= []P) ~ 2 ) ~ p ) + ( A / [ ] A ) , G = C l o A + ( A ( p ~ q) = (AP =Aq) )+ (A(AP

= p) = A p ) + ( A / A A ) , I A = I o A +(p = AP)+((AP = P) = P)§ = q) =/0)

= (Aq ~ p)).

For any calculus /4 we denote by 5//4 the set of aU formulae of the calculus /4 and by 5r the lattice of all logics that are ~he extensions of the logic of the calculus /4, i.e. sets of 5//4 formulae containing the axioms of /4 and closed with respect to its rules of inference. In the logic I e s the sign [] is decoded as follows: [2A = (A &AA). The result of placing [] in the formula A before each of its subformula is denoted by T r A . The maps are defined:

l~ = G r z + { I ' r A I A ~ 1}, l z -~ G + { T r A ] A e 1},

IA = {A e s/IAIT'rA e l } , l V ~ l(~5/I, l,u = {A e s / G ~ l A ~ }

(in the definitions o f , and 2 the decoding of [] is meant), by virtue of which the din. gram is constructed:

2 I ~ , 2 0

In this diagram the maps a, ~ and A are isomorphisms, therefore ~-1 = ~; and the maps ~7 and /~ are the semila~iee epimorphisms that are not commutative with lattice operation 9-. Besides, the given diagram is commutative, and the next equalities take place: a-1 =/~-1A~ 7 and a = V-~g#. The latter implies in parti- cular that any superintuitionistie logic is a superintuitiouistie fragment of some proof logic extension.

If o n l y p ropos i t i ona l logics a re dea l t wi th , a n d this is the case in t h e

sequel , t h e n t h e n o t i o n of a s u p e r i n t u i t i o n i s t i c logic is a l m o s t iden t ica l -to t he n o t i o n of a n i n t e r m e d i a t e one. The f i r s t differs f r o m the l a t t e r

b y a g r e a t b r e v i t y of r e l a t e d f o r m u l a t i o n s a n d b y t h e f ac t t h a t an abso-

l a t e ly i ncons i s t en t logic is be ing super in tu i t ion i s t i c , as well. The d i s t inc t ion b e t w e e n these no t ions increases p r inc ipa l ly , w h e n p r e d i c a t e logics a re

dea l t with~ since a m o n g t h e m even superc lass ica l logics, i.e. i n t e r m e d i a t e

78 A, V. K~,znetsov, A. :Yq~. ,~luravitsky

between classical and absolutely inconsistent ones create an infinite se t [14] and ignoring t h e m may only obscure the general picture. The term ~superintuitionistic logic" goes back to the te rm "superconstructive. calculus" (see [4, 33]), proposed by A. A. 3farkov in 1961~ since by t h a t t ime the intuitionistic logic was~ in our country, more often called the constructive one in connection with the constructive t rend in mathemat ics , and prefix "super" hints a t extension.

In t roducing a logic l by some ca lculus /4 (the Russian "I"), we consider the set of all formulae in the calculus /4 to be the language ) tH of *he calculus (J / is the l~ussian letter "Ya ' ) , and identify the logic I with the set of all formula% deducible in /4. An (normal) extension of a logic of the calculus H is said to be any set of formulae from ~ /4 containing ali the calculus H axioms and closed with respect to rules of inference, postu- lated in H. Particula, rly~ g superintuitionistie (l~ropositional) logic is defined (see [5~ 34~ 10~ 15, 54], cf. [71, 48, 50]) as any extension of the logic of t he intuitionistio propositional eaZoulus I with the subst i tut ion rule postu , laced [51], the formulae of S/I being formed out of propositional variables: /), q , r , . . . (may be with subscripts) by using the connectives of a sig- na ture :

(1) &, v~ ~ , -7,

in the usual way. Examples of superintuit ionistie logics are: the intnit io- nistie logic I, the classical logic Cl = I + ( "7 -O ~ P), i. e. *he extension of *he logic I, obtained by adding a new axiom ( ~ ~p ~ p), the absolu- tely inconsistent logic I+ /o , Dummet t ' s logic [41,54] l + ( ( p ~ q)v v (q ~ p)), and a cont inuum of other logics [34~ 54]. For each of these logics their extensions may be considered as well, the former rules of inference being mean* (i.e. extensions of extension may be considered). ~or example, the classical and absolutely inconsistent logics are extensions of Dummet t ' s logic. I n the meant ime, the difficult question on how many superintuitionistic logics there exist~ each having an enumerable set of extensions, remains open.

Logics with a greater signature than (1) are sometimes introduced for the s*udy of superintuitionistic logics, being the case of the " immersion" of the lat ter into the former and considering the lat ter to be the '~fragments" of the former. Without introducing the general notions of the immersion and fragment , we shall only regard some particular cases of t hem below~ each t ime making their meaning, whenever possible, more precise.

Historically, the immersion of the intuitionistic logic into the modgl logic S4 goes back to the work [44] of tK. G5del, made more precise by" A. Tarski [58] and P. S. Novikov [25], and considered in details below; i* seems to be the first example of this kind. The logic $4 may be defined (cf. [44, 56~ 25]) as the logic of the calculus:

On sqzpe,rintuit,ionislic logics... 79 ,

where the circlet o denotes the extension of the initial calculus language signatm'e, and the sign + means the addition of new axioms and rules of inference (here three axioms and one rule are added permitt ing to pass from any formula A to the formula []A). The immersion is meant~ here as following (see [25, 13], cf. [58]):

(2) I ~ A .~ ~ 4 F T r A

for every A e / / I , where Tr denotes the (immersing) operation of placing the sign [] in front of each subformula in the given formula.

Subsequently, it has been proved that the logic $4 in expression (2). can be replaced by some other logics~ including some of its extensions. I t turned out that there exists a continuum of such extensions [26]~ among them - - t h e logic:

Grz = S 4 + ( O ( O ( p ~ O p ) ~ p ) ~ p)

(see [47~ 67, 63, 18, 13]), formerly denoted sometimes by G. The logic G r z was found out [31,327 36] to be the greatest of thelogic $4 extensions~, which may replace $4 in (2).

I t is relevant to note tha t for each calculus /4 considered here, the, set N'/4 of all the extensions of the calculus /4 logic is a lattice with respect to c with intersection n and union -}-, closed with respect to rules of inference~ as lattice operations in the lattice ~f/4. As a continuation of Dummett~s and Lemmon's research [42] of some $4 extensions I~. L. Maksimovu and V. V. Rybakov [18] investigated the lattice . ~ $ 4 and its connections with .~I~ taking into account the knowledge, accumulated on the lattice ~r of superintuitionistic logics. They have found out t h a t there exists a homomorphism Q of the lattice LfS4 onto the lattice .~eI~ setting to each modal logic 1 e .~$4 a corresponding superintuitionistie logic l~ defined as the set of all formulae A e 5 / / such that T r A e 1 and called [30] the superintui t ionis t ic fragment of this modal logic. The complete inverse image l~ -~ for each l e Lfl was found [18] to have the least element:

l~ = $4 + {TrA IA e l}

and the greatest element la, thereat the map z being the isomorphic embedding of S I into 2zS4. I t is clear from the preceding t ha t I~ = $ 4 and l a ---- G r z . W e need a more general correlation:

(3) la = Grz+l~

for any l e LfI, resulting from [31] (see also [32, 36]), which will help us to prove the following.

:P~oPosImION 1. The map a is a lattice isomorphism of .s onto .SfGrz~ while the restriction of the map ~ o~to S G r z i~ a -~.

80 A. V. Kuznetsov, A. Yu. Muravitsky

l~eally, i t is clear from [18] tha t la e ---- l for every l e ~el. Conversely, for every I e ~ G r z , according to [18] and (3), we have:

1 ~_ lea = Grz~- le~ ~ l -bl = l,

i.e. lea = I. Knowing tha t e is a homomorphism, we conclude tha t a -1, is an isomorphism.

I t is t ime to re turn to work [43] of GSdel and recollect tha t GSdel's intent ion was not merely the immersion of I into $4, but an interpretat ion of intuitionistie logic by means of using the modali ty "provable" and its similarity with the modali ty "necessary". This interpretat ion reflects substantially the intuitionistie idea tha t the t ru th of a mathematical s ta tement consists in its proof. GSdel's interpretat ion [44], interlinks, by our mind, with Kolmogorov's interpretat ion of intuitionistic logic [52] which appeared a year before as the logic of problems. G5del mentio- ned this interpretat ion in [43] as an interpretat ion different from his own and subsequently these two were made more precise in different ways. However, their kinship may be noticed if with each s ta tement (and each of its "substatement") the problem of its proof is being related and with each p r o b l e m - the s ta tement about its decidability. I n addit ion to this, there exists a similarity between the presentat ion of problem solution as algorithms and the process of proof formalization.

GSdel noticed however [43] tha t the a t t empt at deepening his inter- pretat ion of the logic I, the endeavour to immerse it fur ther by means of $4 into the Peano formal ari thmetic P A , specifying the modali ty "provable" as formal provabili ty (deducibility) i~ P A , is hindered by the incompleteness of P A . This hindrance is caused by the fact, tha t the intuit ive meaning of the arithmetical s ta tement B e w (9I), expressing, according to GSdel, the formal provabili ty of the s ta tement 9 / i n P A is being deformed. I t happens this way when B e w (2) gets into the context of formulae and formal proofs regarded as inverse images of their GSdel numbers to the formally described natural range (this "formal" natural range should not necessarily be completely ordered, and therefore the statemen~ - lBew(2&-]9I) is not necessarily true). For surmount ing this difficulty we distinguish [11, 13] two modalities: [] (provability) a n d / k (G5delized provabflity [13]), m e a n i n g / ~ 2 by the formula Bew (9/) ~t the immersion into PA, and the s ta tement about the t ruthfulness of Be w (!~I) in the intuit ively plot ted ("standard") natura l range by the formula ~9/, i.e. connecting the initial intuit ive sense of B e w with [:].

0onstruct ing the proof logic [11~ 13]*, we come from the language

* The collection "The actual problems of logic an4 methodology of science", which comprises the paper [13], contains the selected reports at the VII/kll-Union Symposium on logic and methodology of science, held in October, 1976 in Kiev,

On Supe,'intuitionistiv logics... 81

differing from ~/I by having the signature:

(5) &, v , = , - t , A

(and not (1)). Any formula of this language is said to be GSdel valid, [13], if for each substitution of arithmetic statements into it instead of all variables and decoding • as Bew an arithmetic s tatement provable in P A is obtained from it. The formalization of the notion of the formula GSdel valid is accomplished by us [11, 13] in the form of the following calculus 2( (the :Russian "de"):

2( = C l o A + (A(/~ = q) = (Ap ~ Aq))+ (Ap ~ A A p ) +

§ (A (Ap = P) = Ap) 4- (A/AA) § (AA/A).

Independent ly of our work, a similar calculus was published by R.. Solovay [68] (but without the last rule and with other notations -- [] instead of A) and it proved its complet`eness with respect to the interpretation regarded here, i.e. it`s Gbdel validity. Afterwards, it was found out (see [13]), that discarding from the calculus 2( postulates the rule A A / A and the axiom ( ~ p D ~ p ) we get the calculus equivoluminous t`o it., i.e. with the same set of deducible formulae. We denote this calculus in honour of GSdel (and following [68, 38]) by the lett`er G:

G = C l o A § (A(p = q) = (Ap = Aq)) § = p) = Ap) e ( A f A A )

(cf. with 2(- from [13], and Kd.W from [66, 65]). Let us not`e again that G t- (Ap ~ AAT)) (see [38] as well).

For the accomplishment of the immersion of the logic I into PA, which iS an algorithmic immersion according to the ideas of problem logic, it is necessary, first` of M1, t`o present an Mgorithm admitting to decode []!~ as an arithmetic stat`ement for each arithmetic st`atement !~. ~Not pretend- ing to solve finally this difficult formalization problem, we assume the JBasic Working Hypothesis [13] about the possibility of expressing Dp by a formula of t`he language :/2( so, that the formulae:

(5) ( ~ = p), ( E]p = Ap), [] (p = p)

are deducible in 2( (i.e. these formalae, expressing natural properties of the modMity [] would become GSdel vMid). u t`he deducibili- ty of the formulae (5) is demanded, then the formula (p&~p) expresses ~p . On l~he other hand, for the logic 2( the following v~riunt of the equi-

vMent replacement` law [13] is true:

(6) 2( ~ ((A ~ ) = (A(A ~ ~ ) = (C[A] ~ C[B]))),

where (X ~ 3:) means ((X = :Y) &(:Y = X)), and X [ ] : ] is t~he result of substitution of the formula :Y for the variable p in the formul~ X; taking formulae p, (p ~ p) and s for the variables A, ~ and C, after sIight~

@ ~ S t u d i a Logica 1]86

82 A. V. Kuznetsov. A. X u. Muravitsk.y

transformations using again law (6) and deducibility of formulae (5)~ we conclude (see [13]):

(7) ;I ( o p

Therefore, wherever necessary below the formula [:]A is decoded as (A &hA) . This concludes the construction of the proof logic as the logic of the calculus ~I or calculus G, where [] stands for a derivative connectiv% expressibl~ b y & and fk.

Though the logic S4 and its extensions language use the signa:tm, c:

(8) &, v , ~ , "7, []

(and not (4)), the decoding of O by (7) permits to consider the modal fragment l# of any logic 1 e &~ which is defined as the set of all formulae A e ?$4, lying in 1 after above decoding. The logic l;~e is said to be the superintuitionistic fragment of the logic 1 e 2~G; it is otherwise the super- intuitionistic fragment of the modal fragment of the logic L We have proved [11, 13], that the superintuitionistic fragment of the logic G is the logic I , Le. I = GSe. Thus, it is proved tha t the logic I is really im- mersible into P A by mediation of the proof logic. The main goal of ~h ich [13] was seeking the logic, corresponding most adequately to the aims of the foundations of mathematics, taking into account the intuitionistie and construetivistic critics (and is not so intricate a n d inconvenient for mathematical applications aS, for example, the recursive realizability logic of Kleene [51] -- Rose [62], which claims for this role) and it permit ted us to affirm tha t everything stated there is a good argument for considering the logic I regarded in the context 9 f proof logic to be the most acceptable solution of that problem. Thereat, a n important detail of tha t context turned out to be the logic Grz, since it was shown [12, 13], (cf. [45~ 39])~ tha t the equality G/t = Grz is true. On investigating this context another logic was constructed, lying on the way of immersion of the logic I into the proof logic -- the proof-intuitionistic logic [7, 55] (see [20, 21], as well).

The proof-intuitionistic calculus I ~ is based on signature ( 4 ) a n d is obtained from the calculus I o A by adding the following three axioms:

(9) (io = ~ p ) , 00) ((Ap = p) = p),

(11) (((p = q)= = (Aq = p)).

This form of 1[~ was fh'st published in [7]. I{owever, initially, when i t appeared in the spring of 1977 in oral reports and the subsequent article~ [20, 21], the calculus F was formulated otherwise: instead of axiom (11)~ the following one:

(12) (hp = (qv(q ~ p)))

was postulated.

0.~ superlnt~dtionistiv logics... 83

These two variants of the calculus I z are equivalent (the calculi t/1 and //2 in the same language are said to be equivalent if, for any formulae A and B of this language, /Ii -kA ~ B is equivalent to/42 + A ~ B). Indeed~

x0• ((11) [(q- (q (12t), xoA ~ ((12)[q, p] = (11)),

where the regarded axioms are denoted by theh" number, and X [ Z , Z] denotes the result of subst i tut ion of the formulae :Y and Z into X instead of the variables p and q, respectively.

I t is interest ing to notice the deducibility in I~ of the formulae:

(13) ((p = q) = (Ap = Aq)),

(14) ((p ~ q ) = (Ap ~ Aq)).

(14) is easily obtained from (13) for which:

The deducibili ty of (14) permits extending on I ~ the principle of an equi- valent replacement :

~ ~ ((A ~ B) = (C[A] ~ C[B]))

for any A , B ,: C e ~/I ~, which is valid in I, bur non-valid in $4, G r z and G. I t was mentioned: earlier tha t I = G/x@, i.e. :

(15) I F A ~ G F T r A

for each A e ArI on decoding [] by (7). The logic I a practically appeared when the generalization of correlation (15) to the l~nguage of signature (4) became possible. :Precisely, it was found out [7], tha t :

(16) I ~ F A - ~ G b T r A

for each A e ~ /F at t h e same decoding of [] (see also [24], Lemma 3). As for the proof of correlation (16) from left to r ight (to the converse

proof we will r e tu rn later) it is enough to tes t it for axioms (9), (10) and (12), since the rest is easily implied by (15) and the known laws of the logic $4 c_ G r z -=- Gtz. I t is convenient to replace the immersing transla- t ion Tr by the simplified ~r~nslation Tr' , differing from Tr by [] not being placed in f ront of all subformulae: it is not placed in front of the formula itself, of conjunction, disjunction and just beside /%. By vir tue of the logic $4 known laws, law (6), and easily deducible in G formulae ( []ZSp ~-~/kp), (A []p ~-~AT) for any A e 511 z, the following takes place:

G k T r A ~ G F T r ' A .

I t remains to show the formulae Tr ' (9), Tr' (10), and Tr' (12) to be deducible in G. This is evident for Tr" (9) having the form ( [2p ~ Ap). The formula

84 A. V. Kuzn~tsov, A. Yu. Muravits~y

Tr' (10), which has the form ([](Ap ~ []p)~ [~p), is equivalent in C1 cA to the formula:

= p) A(Ap = Dp))=

easily implied by the lust axiom of the calculus G. The formula Tr' (12)7 having the form:

(•p = ( [ ]qv D( []q = []p)))

is equivalent in C l o ~ to the formula:

( ~ p = ( D q v A ( []q = []P)))7

deducible f rom (Ap -~ A [:]p). Before the proof of correlation (16) f rom the r ight to the left it should

be noticed tha t many properties of the ealouli regarded here are detected and proved by means of the algebraic interpretat ion of these calculi. To begin with7 the calculus G is not equivalent to the e~leulus ~ and the e~lculus I ~ is not equivalent to the calculus:

n ) = x ~ + ( / \ A / A ) 7

introduced in [7] (and equivoluminons to I ~ by v i r tue of (16)7 G and being equivoluminous). Indeed 7 the calculi ~ § 0 and ID § where 0 means (p&-lP)7 are evidently inconsistent. Still, the calculi G + A 0 and I ~ § are consistent~ as it follows easily from their interpretat ion in the set (0~ 1}7 on defining &7 v 7 ~ and -7 in an o~dinary Boolean way (i.e. classically)7 a n d / k p as the constant 1. ~ § v 7P) being inconsisl t en t is shown in [13] thus~ the classical logic is not the superinCuitionistic f ragment of any extension of the logic ~.

Dummett~s logic is shown7 ibid [13] to be the superintuit ionistic f ragment of the most consistent logic ~ extension (defined by the calculus ~ + ( A ( ~ p = q )vA( []q = p)); cf. K4.3W [64]).

I t is well-known tha t pseudoboolean algebras [61~ 1017 i.e. algebras with operations (1) (i.e. of the form <27; &~ v 7 =7 7>) being lattices with respect to & and v with relative pseudoeomplement D and pseudo- complement "77 are an algebraic interpretat ion of the logic L Topoboolean [13] algebras (ol" topologic boolean [61], or interior [36] ones)7 i.e. algebras with operations (8), being boolean algebras with respect to &~ v 7 ~ and -7 with an operation [] such tha t the $4 axioms are identically equal

to 1 ~nd [~1 = 17 where 1 means (p ~ p) are a n algebraic interpreta t ion of the logic $4. Each of these algebras is partially ordered by the relation x ~ y, meaning x&y = x(or x ~ y = 1), having the least e lement 0 and the greatest element 1. Each pseudoboolean algebra is univocally restored by its relation ~ so defined~ bu t it is impossible in n topoboolean algebra for the operation [] to be restored by ~ (this is already evident on the example of the 4-eiement algebras).

On supvrintuitionistic logics... 85

The latt ice s is known to be dually isomorphic to the lat t ice J/l[ of all varieties [3, 54] of pseudoboolean algebras~ and the lattice ~fS4 is dually isomorphic to the latt ice J /S4 of all varieties [18] of topoboo- lean algebras. ~(A variety is the class of algebras with the given signature satisfying the given identities.) Thereat, the logic l e 5e/ / is being ac- companied by the var iety of all algebras f rom ~g//, on which all the equali- ties like A --- 1~ where A e ~, are valid~ i.e. all the formulae A e l are valid in this sense~ and in the ca.so of inverse correspondence t ransforming B --~ C into the form (B ~ {3) ~ 1 is used. The logic of the a~gebra of the given signature is called a set of all valid on tha t algebra formulae of the signature. I t is known for any pseudoboolean algebra tha t its logic lies in .~I~ and for ~ny topoboolean algebra -- t ha t its logic lies in ~$4. The logic 1 is called correspondingly: tabular~ if l coindices with the logic of some finite algebra; pretabular [6]~ if 1 is no t tabular bu t any of its proper (i.e. not equal to l itself) extensions is tabular ; finitely approximable [4, 10~ 54]~ if any formula A ~ l does not lie in any tabular extension Of the logic 5. I t is known tha t there are exactly three pretabular logics in s [15], and exactly five ones in s [19, 16], and for any non-tabular logic from ~ ! or :~$4 there exists its pretabulax extension. There remains the question open whether for every such a superintuitionistic logic t ha t is not finitely approximable there exists its extension which is finitely preapproximable (i.e. it is no t finitely approximable itself, bu t all of its proper extensions are finitely approximable), as well ~s the question of how many finite preapproximable superintuitionistic logics really exist. We will note t ha t for the extension of the logic:

(see [10, 54]) in the paper [2] the answer for the first question is positive.. and for the second one the answer given is: exactly 1.

Between the varieties lying in JIS4 an impor tan t role is played by the variety of Grzegorezylr algebras, i.e. [12~ 131 32] such topoboolean algebras on which the equali ty:

(17) [ ] ( [ ] ( x = [ ]x) = x} =

is true, (i.e. all the formulae from Grz are true). ~;[aksimova has proved (cf. [32]) that , in particular, any topoboolean algebra which is generated by its open elements (i.e. such elements x, tha t E]x = x), as the sys tem of generators~ is Grzegorczyk~s algebra; such topoboolean algebras are called the t raphare t (or special [17], or open-generated [13]). I n particular~ any finite Grzegorezyk~s algebra is the t raphare t one [17]. I t is known [61] tha t for any topoboolean algebra the set of all its open elements with respect to [] is a pseudoboolean algebra with pseudoboolean opera-

86 A. V. Kuznetsov, A. Yu. ~lu~'avitsky

t ions:

(18) x&y, x v y, ~ (x ~ y), [] Nx.

I t is called [32] the trapharet of this topoboolean algebra. Vice versa [61] (cf. [18~ 32])~ for any pseudoboolean algebra there exists its to2oboolean envdope~ i.e. such a t raphare t algebra, the t raphare t of which coincides with this pseudoboole~n algebra (this envelope may be obtained as follows, for example: at first, the boolean algebra is constructed~ and it is set up by the elements of the given pseudoboolean algebra as generators~ lattice correlations between those elements and boolean equalities, includ- ing the ones tha t ensure 0 and 1 being unchanged; then for any element x of this boolean algebra, after t ransforming it with the help of these equalities and correlations to conjunction of several boolean implications of the generators, []x is defined as the conjunction of the pseudoboolean implications of the same generators).

The algebraic interpretat ion of the logic G is represented by Magari's [12, 13] (or diagonMizable [57~ 35]) algebras, i.e. the algebras in signar (4)~ tha t are boolean in respect to & ~ v ~ ~ ~ and "7 with such an operatioin A~ tha t the axioms of G are identically equal to 1 and/%1 ~ 1. The algebraic interpretat ion of the logic Jr A is represented by /% --~seudoboolean Mgeb, ras, i.e. [7~ 55~ 13~ 21] algebras in the same signature t ha t are pseudoboolean in respect to the above four operations, with such an operation A~ tha t the axioms o f I A are identically equal to 1. We will note tha t ~agar i ' s algebra cannot be restored by the relation ~i defined as above; bu t A - -pseudoboolean algebra can be restored by i t univ0cally~ because after the restoring of its pseudoboolean operations it only remains to note [55] tha t Ax is the least of such elements y of t h e algebra tha t y D x ~ y. The lat ter is implied by the deducibility in I zx of the formula:

(((p = q) p ) ~ (/%q = p)),

(cf. (11)~ ~aking into consideration t h a t IAF ((/% q = q) = Aq). Similarly to what has been said above about s and rig1, and about ~ S 4 and ~ $ 4 , the lattices ~fG and s A are also duMly isomorphic correspondingly to the lattice ~r of all the varieties of ~agari~s algebras and r t h e lattice ~r A of ~ll the varieties of A-pseudoboolean algebras. Let us note~ however~ tha t the lattice s is arranged more in a more complicated way and it is less subdued to the MgebrMc invest igat ion [13, 22]; the same refexs to .5zID. I t is much clearer tha t the calcului G +A0, G +/%p~ 14 +/%0 and IAd -Ap are equivoluminous to C loA- t -Ap; i t permits us (see above) to prove their consistency with the help of the same two-element Mgebra~ which is ~t the same t ime ~agar i ' s and u A-pseudoboolean one. A bit more complicated than the lat ter is a three-element A-pseudoboolean Mgebra~ in which 0 < A0 <: A A 0 = Z~I = 1. The logic of this three-

On superintuitionislic logics... 87

-element algebra, the investigation of which goes back to Ya. S. Smetanich ([27], the calculus H~), may be given by the calculus I A + ( - ] p v A p ) .

In [35] Bernardi has proved (though rather unconstructively) that *he variety of all the Magari~s algebras is generated by its finite algebras, i.e, t h e logic G is finitely approximuble (cf. [66, 65, 68]). The logic I ~ is also finitely approxilnable [7]; the direct proof of that~ not using correla- *ion (16) is given in [21] (when completing the proof of (16) below, it will permit us to: do without using the topoboolean envelope of the finite pseudoboolean algebra, the correctness of which~ indicated abov% pre- supposes the exclusively unconstructive reasons of the fact that the elements of the algebra's t rapharet do not stick together as a result of this construc- tion). The pseudoboolean algebra (E; &, v , o , -1) is called A-enrichable ~8, 55], if there exists such an operation /k on E, that the algebra (E ; & , v , o , - 1 , A ) is A-pseudoboolean; it is equivalent to the fact that any :of its elements is deltable, i.e. [55] there exists for it the least of such 21 e E, tha t y o x ~ y (after all, it is not difficult to see that the calculus l o A +(19) is equivoluminous to IA). Any finite pseudoboolean algebra is A-enrichable [7]. Indeed, as it is easy to understand that the operation A may be introduced in it as follows: A 1 - - 1 , and when x =~ 1, the supremum of all ~toms of its filter {ylx ~ y} can be taken as ~x . All *hese facts and the finite approximability of the logic I imply the next i[7]: if A e 5 / / a n d I a b A then I ~ A.

We have proved [12, 13] that for nny 5Iagari's algebra (E; &, v , ~ , -1, A ) the algebra (E; &: v , ~ , -], []), where:

,(20) [:]x = x&z~x,

is Orzegorczyk's algebra (besides, the operation z~ may be restored uni- vocally by the lat ter algebra ;[12]). The equality:

,(21) D / ~ x = A x

takes place, by virtue of which the trapharet of so defined Grzegorezyk's algebra is a A-enrichable pseudoboolean algebra (owing to what has been proved above concerning (16), and what has been said then about the connection between the translations Tr and Tr'). The A-pseudoboolean algebra obtained in this way, being the algebra with respect to operations (18) and ~ x , is called by us the A-trapharet of the initial Magari's algebra. We call the latter the trapharet algebra, if it is generated byits open elements; i.e. the elements of its A-trapharet .

Vice versa, any A-pseudoboolean algebra is a A-trapharet of some t rapharet 1Kagari's algebra; the latter is called Magari's envelope of this A-pseudoboolean algebra and is obtained as follows: first we construct the topoboolean envelope of the given A-pseudoboolean algebra (as of t he pseudoboolean algebra), then we introduce the operation A on this

88 A. V. Kuzne$sov, A. Yu. Muravitsky

envelope by the equali ty:

(22) ~ x =/% [:Ix

using the fact t ha t the elements of the form E]x of this topoboolean en- velope compose the initial A-pseudoboolean algebra and owing to this fact the operation A has been defined on it. Let us prove tha t the algebra of signature (4) obtained in such a way is ~agar i ' s algebra and tha t equality (20) is true~ supposing the topoboolean envelope under construction to be the algebra in signature (8)~ the initial A-pseudoboolean algebra t~ have (18) and Ax as its operations and the lat ter operation to ex tend on the whole envelope by vi r tue of (22).

Really~ the equality (21) is implied by the fact t ha t Ax lies in the t raphare t of this topoboolean envelope~ and equality (17) is implied by this envelope being a t raphare t algebra. The laws of the boolean algebra on this envelope are evident. Using (17)~ (22)~ topoboolean algebra laws and the correlations on this t rapharet , correspouding to axioms of I ~ we get ident i ty (20) and the equali ty A1 ---- 1 as ~ollows:

~x ~ x&A ~x -- x & A x = x&A [2x

x)) -~ x & ( ~ x v ~ x ) = x & []x = ~ x ,

1 = D I ~ A ~ I = A 1 < ~ 1 .

I t remains to get the correlations:

A ( x = y) <~ A x = A Y , A ( A x = x) <~ A x

corresponding to those axioms of G which contain A. We proceed as follows:

A ( x = y) = A ~ ( x = y) <~/L D( ~ x = ~ y )

[] ( ~ ~ x = A [SY) ~ A [:Ix = A []Y = A x = z~Y,

~ ( A x = x ) = / ~ ~ ( ~ Sx ~ x) ~< A ~( ~ []x = []x)

= ~ [] ( ~ [ ] x ~ [3x) ~</% ~ x = Ax.

Let us note~ tha t except the laws ment ioned above~ including (21)~ th~ following is being used here: the quasi-identity

x ~ y ~ A x ~ ~ y ,

implied by valid formula (13)~ and the law of the distr ibution of Ak~ cor- responding to the deducibility in I ~ of the formula:

(23) (A(p =q ) ~ (Ap ~ Aq)),

On superintuitionistic logics... 89

the formnl~

is deducible in I o A ) . l~eturning to the proof of correlation (16), we will p rove the counter-

-position of wha t remained to be proved. Le t t he formula A be undeducible in I ~. Then owing to the finite approximabi l i ty of I s there exists a f ini te Z~,.pseudoboolean algebra on which the formula A is no t valid. We cons t ruc t the ~aga r i ' s envelope of this algebra. On the A - t r a p h a r e t of this envelope the formula A is no t valid. Then we come to the recording of the operat ions on the A- t r apha re t as (18) and ~ x , and, keeping in mind the above-men- t ioned connection of t ranslat ions Tr and Tr', we conclude, t ha t on this Magari 's envelope, as on an algebra in s ignature (4) the formula TrA is no t valid and, hence, is undedueible in G. Thus correlat ion (16) is proved.

When considering some calculus /4 and some logic l e .~e/4 we call [20] the set l n f l l the assertorieal fragment of the logic l. For example, for a n y of the logics $4, G r z and G the assertoricM f ragmen t is represen ted by the logic C1, and for lo&~ it is represented by L Let us in t roduce the following maps V, ~ and 4. Namely, for a n y 1 e s ~ we define the set l V as an assertorical f r agmen t for the logic l, and :

l~ = G + {TrA I A e 1},

(cf. ~ wi th v and a); and for a n y l e 2gG we define:

l~ = {-4 e ) t la [TrA e l}

(in the definit ions of ~ and 2 the decoding of [] is meant) . F r o m the wha t has been said above, including correlation (16), i t is clear, t ha t I a V = I , l ~ z = G , G2 = I ~ ; the map V is the map of t he lat t ice .LzI ~

into -~I, u is t he map of the same lat t ice into .~eG, and ~ is the map of the la t t ice s into s

The calculi /4~ and /42 are called assertorically equivalent [20] if

/ / ~ + A F B ~ / 4 ~ + A F ~

for a n y formulae A, B e S/L For several years the question whe the r 14 and I are assertoricMly equivalent remained open; in other words, i t is the question [55] whe the r any superintuit ionist ic logic l is A-closed, i.e. [8] if I ~ + l F A implies A e 1 for any A e ) /I (i.e. [20] ( l a + l ) V = 1). The likelihood of the negat ive answer seemed to suggest itself firsr of all because ID and I are no t assertoricMly equivalent (for the calculus ID + ( -] -liP -~ P) is inconsistent [7], because I o ~ F (( "7 -lp ~ p) [A0] ((10) [0] ~ z~O))); or, in other words, no superintuit ionist ic logic is D-closed,, i.e. such logic, t ha t ID + 1 F A implies A e t for a n y A e f / l (even the logic CI is no t D-closed: ( ID+C1)V = Cl+O r C1). I t was noticed [55] t h a t t h e

90 A. V. Kuznetsov, A. Yu. Muravitsl~y

greatest consistent D-closed superintuitionistic logic is Dummet t ' s logic. Different examples of such pseudoboolean algebras tha t are not A-enri- chable have been discovered; for example [8]: the free pseudoboolean algebra of rank 2 -- its free generators are not deltable. Though~ recently there has been proved:

P~oPosITIOE 2 [9]. The calculi I ~ and I I are assertorically equivalent.

Proposition 2 easily implies the following corollaries: any superintu- itionistic logic is A-closed, the map V is one of the lattices ~ I ~ onto the lattice 5eI (i.e. surjection); i.e. any supcrintuitionistic logic is the asscrtorical f ragment of some extension of the logic 14. This connection can be cont inued towards the extension of the proof logic by vir tue of the next recently proved proposition (it is the reformulation of Theorem i from [24] with different, in particular~ denoting of the mapping) :

P~oeosI~Io~ 3. The map ~ is the lattice isomorphism of the ~ I ~ onto AeG~ and ~-~ = 2.

Proposition 3 implies at once tha t the lattice ~r J is also isomorphic to the lattice ~ G . Propositions 2 and 3 imPlY easily the following coro- llary as well: for any superintuitionistic logic 1 there exists such logic l' e LfG tha t ~ =/'A~7(~' = (I ~ § may be taken, for example). Recent ly it has been proved as well ([24]~ Theorem 2), tha t the m~p ~ preserves the tabular i ty of the logic and preserves its nontabular i ty . All t ha t und Proposit ion 3 imply tha t t he maps u a n d 2 preserve the nontabularity~ pretabular i ty and finite approximabi l i ty . . In particular~ knowing from [37] tha t there exists an exactly countable set of the pretabular extensions of the logic G, we can conclude [24] tha t there exists an exactly countable set of :the pretabular extensions of the logic I ~, as well.

Propositions 1, 2 and 3, their consequences and all the remarks:concern- ing the map ~ a-~ ~, V , ~ and ~-z = 2 led us:to the idea of depicting the connections between the lattices .~I, .~q'Grz~ ~fI ~ and LfG represented by the above mappings in the form of ~he below presented arrow diagram {Diagram l). As it is accepted, for example, in the c~tegorical logic [46], the diagram is c~lled commutat ive, if any o f the two ways along the

.~ .T 6" ~ .~ O r z

arrows starting from one ~nd the same object (a lattice in Our case) and also ending in one and the same objeet~ give the same mapping in the

Oq~ su.perintuitionistic logics... 91

composition. I n this case a n y map ~ is unders tood as the set of all pairs like {x, x~}, where x is t aken f rom the domain of the map ~; the compo- sition ~ , where ~ and W are the sets of pairs, is unders tood as the set of all pairs ( x , y } for which there exists such z t ha t (x , z} e ~ and (z , y} e p; and e -1 is unders tood as the resul t of tu rn ing over the pairs in t he set ~. The connections be tween two ways of immers ion of the intuitionistic logic into the proof logic are specified and s t rengthened by:

TttEORE~ :1. Diagram 1 is commutative.

Firs t we prove t h a t :

Le t real ly 1 e ~ G . Then /~-IV and ll~a -I belong to ~el; and for a n y A S/I by v i r tue of Propositions 1 and 3 the following takes place:

A e l~ - 1 V ~ A ~ ~ ' i ~ A e l ) . ~ T r A ~ I

~> T r A e I# r A e Itt@ ~ A e l~a -1.

~Vithout in ter rupt ing the proof~ we in t roduce the nex~ three "un i t " maps :

'~1 = {<X, X> lX e Z l } , S~] = {(X, X) [X e Z l n } , e3 -= {(X, X} [$ e .5r

The following details of the eommuta t i v i t y of d iagram 1 are implied by (24) wi th the help of Proposit ions 1 and 3:

It is interesting to add to Theorem 1:

CO]~OI,LAI~Y I. c = ~-1~/t~ a -~ =/t-1~-~.

Indeed~ the second of these equMities may be obtained from the former by turning over its pairs./~et us prove the first equality using Proposition 2. The la t te r implies: ~1 = - ~ - 1 ~7. ]~enee:

C O R O L L ~ 2. A n y extension of the logic G r z is the modal frag- m e n t of some extension of the logic G.

Le t l e . ~ G r z and V = (I a + la -1) ~. Then, l' e 2"G, and with the help of Proposit ions 1, 2 and 3 we get :

~'# --- l'~ -~ V o" -- (I A + h;-~) ~-~ ~

- (I a + 1~-1)~,. V~r = (I '~ + l~ -~) ~ a

92 A. Y. Kuznetsov, ~rr A. Muravitsk~

The direct answer to one of the questions s tated in [13], is given by the following, implied by Corollary 2 and Proposit ion 1

COROLLARY 3. Any superintuitionistic logic is the superintuit io- nistic f ragment of some extension of the logic G.

Unlike the maps a and u, V and /, are not lattice isomorphisms. Speaking more precisely we have:

Tm~o~v.~ 2. The map V is the semilattiee homorphism of the ~attic~ s z~ onto s and the map ff is the semilattice homomorphism of the lattice .~G onto La Grz , neither V nor # being commutative with the lattice opera- tion +.

Let us begin with the proof of V being the semilattice homomorphism~ i.e. being commuta t ive with c~. Let really 1,~' e . ~ I '~. Then:

(1 c~l') V = I c~l' c~ / l = (ln ~/I) n (I'c~ 5//) = IV n l ' V .

Now, we show tha t the map ff is also commuta t ive with n . Let really l, l' e .~G. Then, owing to everything just proved, Theorem i and Propc- sitions :t and 3, we get:

(lnl')/~ = ( lnV) . -I V a = lu -x V a n l ' . -x ~ a = l[~nt'l~.

I n the same way, the eommutat iveness of the m a p / z with the lat t ice operation + , Theorem 1 and Propositions 1 and 3 would imply the com- mutat iveness of V with + if I, V e ~ I z~ then :

(I + ~') V = (l + ~') ~Fa - i = luffa - i + l' ~Fa -1 = l ~ + l ' v .

Ig remains to show with t he help of what has been proved above that, the map V is not commuta t ive with the lattice operation + . I t is enough to consider the formulae:

(25) ( ~ 0 = ( ~ p v -1 7p)) ,

(26) (p Ip = 7q))), (27) (pv = q)) to prove the following correlations:

(28) (In + (25)) V + (IA + (26)) V = I + ( 2 6 ) ,

(29) I + ( 2 6 ) c I + ( 2 7 ) ,

= i + ( 9 , 7 ) .

Despite the locations of these correlations we shall begin with t he proof of (30). The lat ter is implied by Proposition 2 from the equality :

z" + (25) + (26) = 1" + (27),

<)~ 8u~crintuitionistic logics... 93

which follows f rom:

zo z~ ~ ((27)[ ~p , p] = (2~)), I ~ ((27) = (26)).

I t is clear f rom all the above ment ioned tha t :

1,1,(26) _~ I + ( 2 7 ) .

Before we start to prove the strictness of the inclusion (29), it is worth recollecting tha t formula (27) has already been basically ment ioned in Umezawa's paper [70] (in the form of the sequence R, ; cf. [48]), and also compare it with Smetanich's formula:

(31) ((p ~ q)v(q = r )v ( r = s))

f rom [33]~ and to notice tha t :

I ~ ((~7) [q, r] = (31)),

I ~ (((I = p)v(p = q)~(q = 0)) = (27)).

I n connect ion with tha t it is known that, the logic 1,1,(27) coinciding with the logic 1,1,1,(31), is the logic of a three-element pseudoboolean ,algebra, (let us note Chat:

I A +(27) = I A .1.( -3pvAp) ;

see above). We compare formula (26) with Yankov% formula, [33]:

with its equivMent from [~5]:

and with Nagata 's formula [59], us well:

(see [48, 50], formula P2). I t is easy to see tha t :

(35) x+(26) - - x + ( 3 2 ) = I + ( 3 3 ) = x + ( 3 ~ )

(it is wor th not ing tha t I zx +(26)----I a + / \ A 0 ) . We know [15, 54] tha t ~he logic ! .1. (26) coinciding with the logics fi'om (35) is one of the pretabu- la.r superintuit ionistic logics. ~'or the well-known part i t ion of Che lattice ..~~ in slices by t{osoi [48] the logic 1.1. (26) is the least logic of the second slice (see [49]), and the logic 1-I-(27) is the greatest one of this slice. (The

94 A. V. Ku~netsov, A. Yu. Muravltsky

logics l, t ' e ~ ! belong to one and the same slice if and only if:

I § ~ q)v(q ~ p ) ) § - ~ I § q)v(q = p ) ) §

I n the same way as ~[aksimova [17] has transferred this part i t ion into slices f rom ~ I onto ~ $ 4 , it would be interest ing to transfer it onto ~ I ~ considering the logic l e Lfl ~ to lie in ~he n- th slice if and only if l~7 lies in the q~-th t tosoi slice. I t may be noted thpt the least logic of the n-th slice of the lattice .5flA~ when n is finite~ i s t h e logic I A ~-A~0, where A ~ means n t imes A.)

Pussing to t h e proof of correlation (28) we note tha t it follows~ by vir tue of Proposi t ion 2, f rom the equali ty:

(36) (I ~ + (25)) W = I

(el. with what has been said in [7 ,55] 9bout the formula

((( 7p): A0) A0) equivalent to (25) in I~). ~or the proof of equality (36), we have to speak ~bout the relational in te rpre ta t ion of some logics considered here, tha t we often prefer to consider close to the algebraic interpretat ion of these logics and with theh ~ topological interpretat ion, which cannot be left wi thout a t tent ion, either.

The topological interpretat ion of the intuitionistic logic has been known for a long t ime from the papers by Tprski ([69], for example} and some other investigations (see the books [25, 61]). Briefly, it consists in the following: for an arbitrary topologic91 space the psendobodean algebra of all its open sets with respect to ~ is considered; the formula is said to be valid on this space if it is v91id on this algebra. The set of 911 formulae tha t are valid on the given space (or all the given spaces) is called the logic of this space (correspondingly~ these spaces). I t is known tha t the logic of 911 topological spaces (or, ut least, finite ones) coincides with the logic I. The relational interpretat ion of the logic I (pnd its ex- tensions) may be t reated in two ways: at first, us the part icular case of the topological interpretat ion, and secondly, as the interpretat ion by the Kripke's models [53]. We begin with the first var iant of the relational interpretat ion. I t consists (cf. [50, 60]) in considering ~ particula~r case of the topological space, namely, an arbi trary partially ordered set oa which there is given its r ight topology, i.e. [40] such one tha t its open sets are represented by the subsets closed in respect to the increase (in the sense of the partial order). One may puss f rom any finite topological space without changing its logic to ~ finite space of this kind, the same transit ion may occur as well f rom any finite pseudoboolean algebra.

There still remains open the following question s tated in [54] which is impor tan t as one about the role and the adequacy of the topological interpretat ion for the logics from ~ I , namely, whether any superintui-

0.~ suTeri~.tuitionislic logics.,. 95

tionistic logic is topologizable, i.e. if it is a logic of some topological space. The open question by V. B. Shehtman whether any superintuitionistie logic is the logic of some full pseudoboolean algebra comes close to it. However [28], not any superintuitionistic logic is modelable (or Kripke complete), i .e. a logic of some partially ordered set as a space with its right topology; moreover, V. :B. Shehtman [29] has shown an example of such superintuitionistic logic that is topologizable but not modelable, which gives an answer to another question from [54] (the answer for one more question from [54] adjacent to them is given in [23]: there exists in s such a modelable logic that is not approximable by the pseudoboolean algebras with the Descending Chain Condition, i.e it cannot be represented as the intersection of the logics of such algebras; cf. [20], Theorem 5.

The second variant of the relational interpretation is based on Kripke's models. A Kripke 's model can be understood as a partially ordered set with the valuation, i.e. with putt ing in correspondence to any propositional variable and any element of this set a meaning "true" or "false", so that the domain of truthfulness of any variable can be an open set in the right topology. This valuation may be transferred from the varia,bles to other formulae of the language 5/I with the help of induction by the construction of formula so that the connectives in the formula would correspond to the pseudoboolean operations on the domains of truth- fulness of the subformulae in the algebra of open sets. In particular (see [43]), the formula (A ~ B) is said to be t rue in the element (point) x, if for any point y that x ~ y the truthfulness of A in y implies the truth- fulness of B in y. The formula is said to be valid on the give~ model if it is t rue in any point of it. The logic of the given partially ordered set (with its right topology) coincides with the set of all formulae that are valid on it as on Kripke's model by all the valuations.

If we limit ourselves to the second variant of the t rea tment and use the finite approximability of the logic I a, we may lay the relational interpretation of the logic I a (and its extensions) on the finite Kripke's models, extending the valuation of the formulae of the language f/Ia: it is enough for that to agree [7, 21] to consider the ~ruthfulness of the formula GA in the point x as the truthfulness of the formula A in all such points y that x < y (such an extension of the valuation was in reality given by A. V. Bessonov in [1], though without any connection with modalities and by using another denotation -- ~ instead of A ; and the calculus I0 ~ constructed there is not complete with respect to this inter- pretation).

Returning to the proof of equality (36) we note the evidence of the following:

x (I + v .

96 A. V. Kuznetsov, A. Zu. Muravitsl~y

I t remains to prove the inverse inclusion. Le t the formula A e ~ I be deducible in I ~ § (25). Now, we show t h a t A is valid on all f inite Kripke 's models and, hence, is deducible in I (owing to its finite approximabili ty). Le t us consider an arb i t ra ry finite model, i.e. a finite par t ia l ly ordered set 9Yt wi th u valuat ion ~. Ze t a~, :..~ a~ be the list of all maximal points of the set !)~. We can extend 2)2 by adding some new points b~ , . . . , bn to it, so t h a t correlation a i < b i t~kes place if and only if i = j . I t is e~sy to unders tand t h a t on this new part ial ly ordered set ~0~ + with any valua- t ion the formula /k/k0 is t rue in the points al , . . . , a~, b~, . . . , b ~ and only there and in all these points (and perhaps some others) the formula ( - ] p v -1-]p) is t rue. Hence, formula (25) is valid on 9~ +. Tha t is why the formula deducible in I ~ § is also valid on !fit +. I t is eusy to see t h a t the pseudoboolean algebra of the open subsets of the se t !~ as a space is isomorphic to some subalgebra of open subsets of the set !~ + us a space (namely~ to the subalgebra of all such open sets t h a t either a~ and b~ belong to a set, or nei ther of t hem belongs to it). F r o m tha t we conclude t h a t t h e formul~ A is valid on9~ by any valuat ion, and in par t icular by ~.

References*

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On superintuitionistio logics... 97

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~):EPT. OF .-~xLG]EBRA AND ~%[kTII]~ATICAL LOGIC

�9 -~][ATH3~ATICAL INSTITUT]~ OF CO~PUT]~I~ C]~NT]~I~

O]~" ~OLDAVIAN SS]~, KISHINI]~V USSR

Received December 10, 1984.

Studio Logica XLV, 1