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 ...
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 connective 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) ~ 2 )
~p)+(A/[]A),
= p) = A p ) + ( A / A A ) ,
G = CloA+(A(p
+([]p ~ 71E]p) + ( Z ] ( ~ ( p ~ q) = (AP = A q ) ) + ( A ( A P
I A = I o A +(p = A P ) + ( ( A P = 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}, lz -~ 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: 2I ~
,20
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 particular that any superintuitionistie logic is a superintuitiouistie fragment of some proof logic extension.
If o n l y p r o p o s i t i o n a l logics a r e d e a l t w i t h , a n d this is t h e 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 i d e n t i c a l -to t h e n o t i o n of a n i n t e r m e d i a t e one. T h e f i r s t differs f r o m t h e 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 a c t t h a t a n a b s o l a t e l y i n c o n s i s t e n t logic is b e i n g s u p e r i n t u i t i o n i s t i c , as well. T h e d i s t i n c t i o n b e t w e e n t h e s e n o t i o n s i n c r e a s e s p r i n c i p a l l y , w h e n p r e d i c a t e logics a r e d e a l t with~ since a m o n g t h e m e v e n s u p e r c l a s s i c a 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 set [14] and ignoring t h e m m a y only obscure the general picture. The term ~superintuitionistic logic" goes back to the t e r m "superconstructive. calculus" (see [4, 33]), proposed by A. A. 3farkov in 1961~ since by t h a t time t h e intuitionistic logic was~ in our country, more often called the constructive one in connection with t h e constructive t r e n d in m a t h e m a t i c s , a n d prefix "super" hints at extension. Introducing a logic l by some c a l c u l u s / 4 (the Russian "I"), we consider t h e set of all formulae in t h e calculus /4 to be the language )tH of *he calculus (J/is t h e l~ussian letter " Y a ' ) , and identify t h e logic I with t h e set of all formula% deducible in /4. An (normal) extension of a logic of t h e 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, postulated 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 t h e logic of t h e intuitionistio propositional eaZoulus I with the substitution rule postu, laced [51], t h e formulae of S/I being formed out of propositional variables: /), q , r , ... (may be with subscripts) by using the connectives of a signature: (1)
&, v~ ~ , -7,
in the usual way. Examples of superintuitionistie logics are: t h e intnitionistie 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), t h e absolutely inconsistent logic I + / o , D u m m e t t ' s logic [41,54] l + ( ( p ~ q)v v (q ~ p)), and a c o n t i n u u m of other logics [34~ 54]. For each of these logics their extensions m a y be considered as well, t h e former rules of inference being mean* (i.e. extensions of extension m a y be considered). ~or example, t h e classical and absolutely inconsistent logics are extensions of D u m m e t t ' s logic. I n the meantime, t h e difficult question on how m a n y superintuitionistic logics there exist~ each having an enumerable set of extensions, remains open. Logics with a greater signature t h a n (1) are sometimes introduced for the s*udy of superintuitionistic logics, being the case of the "immersion" of the latter into t h e former a n d considering t h e latter 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 h e m below~ each time making their meaning, whenever possible, more precise. Historically, the immersion of t h e intuitionistic logic into the modgl logic S4 goes back to the work [44] of tK. G5del, made more precise by" A. Tarski [58] a n d P. S. Novikov [25], a n d 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 t h e logic of t h e calculus:
79,
On sqzpe,rintuit,ionislic logics...
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 permitting 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 T r 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 t u r n e d out t h a t there exists a continuum of such extensions [26]~ among t h e m - - t h e logic: Grz = S4+(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 m a y replace $4 in (2). I t is relevant to note t h a 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 / / s u c h t h a t T r A e 1 a n d called [30] the superintuitionisticfragment of this modal logic. The complete inverse image l~ -~ for each l e Lfl was found [18] to have the least element: l~ = $4 + { T r A 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 h a 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 while the restriction of the map ~ o~to S G r z i~ a -~.
onto .SfGrz~
80
A. V. Kuznetsov, A. Yu. Muravitsky
l~eally, i t is clear from [18] t h a t la e ---- l for every l e ~el. Conversely, for every I e ~ G r z , according to [18] a n d (3), we have: 1 ~_ lea = Grz~-le~ ~ l-bl = l,
i.e. lea = I. Knowing t h a t e is a homomorphism, we conclude t h a t a -1, is an isomorphism. I t is time to r e t u r n to work [43] of GSdel and recollect t h a t GSdel's intention was not merely the immersion of I into $4, b u t an interpretation of intuitionistie logic by means of using t h e modality "provable" and its similarity with the modality "necessary". This interpretation reflects substantially the intuitionistie idea t h a t the t r u t h of a mathematical s t a t e m e n t consists in its proof. GSdel's interpretation [44], interlinks, by our mind, with Kolmogorov's interpretation of intuitionistic logic [52] which appeared a year before as the logic of problems. G5del mentioned this interpretation in [43] as an interpretation different from his own and subsequently these two were made more precise in different ways. However, their kinship m a y be noticed if with each s t a t e m e n t (and each of its "substatement") t h e problem of its proof is being related and with each p r o b l e m - the s t a t e m e n t about its decidability. I n addition to this, there exists a similarity between the presentation of problem solution as algorithms a n d the process of proof formalization. GSdel noticed however [43] t h a t t h e a t t e m p t at deepening his interpretation of the logic I, t h e endeavour to immerse it further by means of $4 into the Peano formal arithmetic P A , specifying the modality "provable" as formal provability (deducibility) i~ P A , is hindered by the incompleteness of P A . This hindrance is caused by the fact, t h a t t h e intuitive meaning of the arithmetical statement B e w (9I), expressing, according to GSdel, the formal provability of the s t a t e m e n t 9 / i n P A is being deformed. It 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 n o t necessarily be completely ordered, and therefore t h e statemen~ -lBew(2&-]9I) is not necessarily true). For surmounting 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 B e w (9/) ~t the immersion into P A , and the s t a t e m e n t about the truthfulness of B e w (!~I) in the intuitively plotted ("standard") natural range by the formula ~9/, i.e. connecting the initial intuitive sense of B e w with [:]. 0onstructing 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 B e w an arithmetic statement 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) = ( A p ~ A q ) ) + ( A p ~ A A p ) +
§ (A (Ap = P) = Ap) 4- (A/AA) § (AA/A). Independently 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 a n d t h e 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 K d . 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 pretending 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 deducibilit y 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 equivMent 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 L o g i c a 1]86
82
A. V. Kuznetsov. A. X u. Muravitsk.y
transformations using again l a w (6) and deducibility of formulae (5)~ we conclude (see [13]):
(7)
;I
( op
Therefore, wherever necessary below the formula [:]A is decoded as (A & h A ) . 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 superintuitionistic 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 t h a t the logic I is really immersible into P A by mediation of the proof logic. The main goal of ~ h i c h [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 permitted us to affirm t h a t everything stated there is a good argument for considering the logic I regarded in the context 9f proof logic to be the most acceptable solution of that problem. Thereat, a n important detail of t h a t context turned out to be the logic Grz, since it was shown [12, 13], (cf. [45~ 39])~ t h a t the equality G/t = G r z 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) 00) (11)
(io = ~p), ((Ap = p) = p), (((p = q)= = (Aq = p)).
This form of 1[~ was fh'st published in [7]. I{owever, initially, when it 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)
( h p = (qv(q ~ p)))
was postulated.
0.~ superlnt~dtionistiv logics...
83
These two variants of t h e calculus I z are equivalent (the calculi t/1 a n d //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
xoA ~ ((12)[q, p]
(12t), =
(11)),
where the regarded axioms are denoted by theh" number, and X [ Z , Z] denotes the result of substitution of the formulae :Y and Z into X instead of t h e variables p and q, respectively. I t is interesting to notice the deducibility in I~ of t h e formulae: (13)
((p = q) = (Ap = Aq)),
(14)
((p ~ q ) = (Ap ~ Aq)).
(14) is easily obtained from (13) for which:
The deducibility of (14) permits extending on I ~ t h e principle of an equivalent replacement:
~ ~ ((A ~ B) = (C[A] ~ C[B])) for a n y 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 t h a t I = G/x@, i.e. : (15)
I F A ~ G F TrA
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], t h a t : (16)
I ~ F A-~ G b TrA
for each A e ~ / F at t h e same decoding of [] (see also [24], L e m m a 3). As for the proof of correlation (16) from left to right (to the converse proof we will r e t u r n later) it is enough to test it for axioms (9), (10) a n d (12), since t h e rest is easily implied by (15) and the k n o w n laws of the logic $ 4 c_ G r z -=- Gtz. I t is convenient to replace the immersing translation T r by the simplified ~r~nslation Tr', differing from T r by [] not being placed in front of all subformulae: it is n o t placed in front of t h e formula itself, of conjunction, disjunction and just beside /%. By virtue of t h e logic $4 known laws, law (6), and easily deducible in G formulae ( []ZSp ~-~/kp), (A []p ~-~AT) for a n y A e 511z, the following takes place: G k TrA ~G F Tr'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 t h e lust axiom of the calculus G. The formula Tr' (12)7 h a v i n g 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 from (Ap -~ A [:]p). Before the proof of correlation (16) from the right to t h e left it should be noticed t h a t m a n y properties of the ealouli regarded here are detected a n d proved by means of t h e algebraic interpretation of these calculi. To begin with7 the calculus G is n o t equivalent to t h e e~leulus ~ and t h e e~lculus I ~ is n o t equivalent to t h e calculus: n)
= x~ +(/\A/A)7
introduced in [7] (and equivoluminons to I ~ by v i r t u e of (16)7 G and being equivoluminous). Indeed 7 the calculi ~ § 0 and ID § where 0 means (p&-lP)7 are evidently inconsistent. Still, t h e calculi G + A 0 and I~ § are consistent~ as it follows easily from their interpretation 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 e n t is shown in [13] thus~ the classical logic is not t h e superinCuitionistic f r a g m e n t of any extension of the logic ~. Dummett~s logic is shown7 ibid [13] to be the superintuitionistic f r a g m e n t of the most consistent logic ~ extension (defined by the calculus ~ + ( A ( ~ p = q ) v A ( []q = p)); cf. K4.3W [64]). I t is well-known t h a t pseudoboolean algebras [61~ 1017 i.e. algebras with operations (1) (i.e. of the form ) being lattices with respect to & and v with relative p s e u d o e o m p l e m e n t D a n d pseudocomplement "77 are an algebraic interpretation 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 ~ a n d -7 with an operation [] such t h a t the $4 axioms are identically equal to 1 ~nd [~1 = 17 where 1 means (p ~ p) are a n algebraic interpretation of the logic $4. Each of these algebras is partially ordered by t h e relation x ~ y, meaning x&y = x(or x ~ y = 1), having t h e least element 0 and the greatest element 1. Each pseudoboolean algebra is univocally restored by its relation ~ so defined~ b u t it is impossible in n topoboolean algebra for t h e operation [] to be restored by ~ (this is already evident on t h e example of the 4-eiement algebras).
On supvrintuitionistic logics...
85
The lattice s is k n o w n to be dually isomorphic to the lattice J/l[ of all varieties [3, 54] of pseudoboolean algebras~ a n d t h e lattice ~fS4 is dually isomorphic to t h e lattice J / S 4 of all varieties [18] of topoboolean algebras. ~(A variety is t h e class of algebras with the given signature satisfying the given identities.) Thereat, t h e logic l e 5e// is being accompanied by t h e variety of all algebras from ~g//, on which all the equalities like A --- 1~ where A e ~, are valid~ i.e. all the formulae A e l are valid in this sense~ a n d in t h e ca.so of inverse correspondence transforming B --~ C into t h e form (B ~ {3) ~ 1 is used. The logic of the a~gebra of t h e given signature is called a set of all valid on t h a t algebra formulae of the signature. I t is known for a n y pseudoboolean algebra t h a t its logic lies in .~I~ a n d for ~ny topoboolean algebra -- t h a t its logic lies in ~$4. The logic 1 is called correspondingly: tabular~ if l coindices with t h e logic of some finite algebra; pretabular [6]~ if 1 is n o t tabular b u t a n y of its proper (i.e. n o t equal to l itself) extensions is tabular; finitely approximable [4, 10~ 54]~ if a n y formula A ~ l does n o t lie in a n y tabular extension Of the logic 5. I t is known t h a t there are exactly three pretabular logics in s [15], a n d exactly five ones in s [19, 16], a n d for a n y non-tabular logic from ~ ! or :~$4 there exists its pretabulax extension. There remains the question o p e n whether for every such a superintuitionistic logic t h a t is n o t finitely approximable there exists its extension which is finitely preapproximable (i.e. it is n o t finitely approximable itself, b u t all of its proper extensions are finitely approximable), as well ~s t h e question of how m a n y finite preapproximable superintuitionistic logics really exist. We will note t h a t for t h e extension of t h e logic:
(see [10, 54]) in t h e paper [2] t h e answer for t h e first question is positive.. and for t h e second one the answer given is: exactly 1. Between the varieties lying in JIS4 an i m p o r t a n t role is played by the variety of Grzegorezylr algebras, i.e. [12~ 131 32] such topoboolean algebras on which the equality: (17)
[]( [](x = []x) = x} =
is true, (i.e. all t h e formulae from G r z are true). ~;[aksimova has proved (cf. [32]) that, in particular, a n y topoboolean algebra which is generated by its open elements (i.e. such elements x, t h a t E]x = x), as the system of generators~ is Grzegorczyk~s algebra; such topoboolean algebras are called t h e t r a p h a r e t (or special [17], or open-generated [13]). I n particular~ a n y finite Grzegorezyk~s algebra is the t r a p h a r e t one [17]. I t is known [61] t h a t for a n y 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
tions: (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 r a p h a r e t algebra, t h e t r a p h a r e t of which coincides with this pseudoboole~n algebra (this envelope m a y be obtained as follows, for example: at first, t h e boolean algebra is constructed~ and it is set up by t h e elements of the given pseudoboolean algebra as generators~ lattice correlations between those elements a n d boolean equalities, including the ones t h a t ensure 0 and 1 being unchanged; t h e n for any element x of this boolean algebra, after transforming it with t h e 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 interpretation of the logic G is represented by Magari's [12, 13] (or diagonMizable [57~ 35]) algebras, i.e. the algebras in signar (4)~ t h a t are boolean in respect to & ~ v ~ ~ ~ and "7 with such an operatioin A~ t h a t the axioms of G are identically equal to 1 and/%1 ~ 1. The algebraic interpretation of t h e logic JrA is represented b y / % --~seudoboolean Mgeb, ras, i.e. [7~ 55~ 13~ 21] algebras in the same signature t h a t are pseudoboolean in respect to t h e above four operations, with such an operation A~ t h a t the axioms o f I A are identically equal to 1. We will note t h a t ~ a g a r i ' s algebra cannot be restored by t h e relation ~i defined as above; b u t A -pseudoboolean algebra can be restored b y it univ0cally~ because after t h e restoring of its pseudoboolean operations it only remains to note [55] t h a t A x is t h e least of such elements y of t h e algebra t h a t y D x ~ y. The latter is implied by t h e 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 w h a t has been said above about s a n d rig1, and about ~ S 4 and ~ $ 4 , t h e lattices ~fG a n d s A are also duMly isomorphic correspondingly to the lattice ~r of all t h e varieties of ~agari~s algebras and r t h e lattice ~r A of ~ll t h e varieties of A-pseudoboolean algebras. Let us note~ however~ t h a t t h e lattice s is arranged more in a more complicated way a n d it is less subdued to t h e MgebrMc investigation [13, 22]; the same refexs to .5zID. I t is m u c h clearer t h a t t h e calcului G + A 0 , G +/%p~ 14 +/%0 a n d I A d - A p are equivoluminous to C l o A - t - A p ; it permits us (see above) to prove their consistency with t h e help of t h e same two-element Mgebra~ which is ~t t h e same time ~ a g a r i ' s a n d u A-pseudoboolean one. A bit more complicated t h a n the latter is a three-element A-pseudoboolean Mgebra~ in which 0 < A0
