Strong Equivalence on an Intuitionidtic Kripke Model and Assertorically Equivolumetric Logics

STRONG EQUIVALENCE ON AN INTUITIONISTIC KRIPKE MODEL

AND ASSERTORICALLY EQUIVOLUMETRIC LOGICS

A. Yu. Muravitskii UDC 517.11

INTRODUCTION

The purpose of the present work is to analyze the notion of a ß-morphism of intuition--

istic l~ipke models and, consequently, to apply selected results of this analysis to the so- lution of special problems. Replacing the notion of a ~-morphism by the notion of a strong

equivalence which, in fact, is its kernel, we study the collection of such kernels on an arbitrary intuitionistic Kripke model.

By an intuitionistic Kripke model [14] (or simply a model) we mean a pair (~,~), whereß

is some partially ordered set (p.o.s.) i~,~) and ~ is an intuitionistic valuation on it,

i.e., a relation ~,x)~ A between elements ~E~ and formulas constructed in a usual way

from the propositional variables ~. ~, ~ .... (perhaps, with indices) in the signature

& V m n (i)

and denoted, henceforth, by letters A,~,. ,, such that the following conditions are fulfilled:

O.i) ((~,~)~ ~, ~~~) ~ {~,]#~~ for an arbitrary propositional variable ~;

0.2) (pc)f= (A ~#) ,<->(f,~)~ A ~d(f,x]~ 8;

Note that the property 0.i) easily extends to all formulas

B~,~) ~ A, ~ ~ y ) => (f,~)~ A. (2)

A formula A is, by definition, valid at the point ZEE under the valuation Z if (~,~)~

A. By the logic of a model (/J,f) is meant the set ~(~,~# of formulas strongly verifiable

[13] in (~,~), i.e., valid together with the results of all substitutions at each point of

the given model, and by the logic of a partially ordered set/~is meant the set ~(/Y) of for-

mulas valid at each point of each model (~,f). The relation A &LC~) will sometimes be writ-

ten as /~~A.

we introduce notation necessary in what follows:

~7-~ {æ~EI~,X)~ ~} is the domain of validity of the formula A and /7x Z is the collec-

tion of variables ~ such that (~,X)~~,

By a superintuitionistic logic [2] is meant a set of formulas in the signature (i) containing all axioms of the intuitionistic calculus ~ (see [5]) and closed with respect to its

Translated from Algebra i Logika, Vol. 20, No. 2, pp. 165-182, March-April, 1981. Orig- inal article submitted March ii, 1980.

i12 0002-5232/81/2002-0112507.50 © 1982 Plenum Publishing Corporation

inference rules: substitution and modus ponens. A superintuitionistic logic is called modelable if it is the logic of some p.o.s.

By a ~-morphism of a model ~,?) onto a model (~~ß#) is meant a map i of the set ~onto

~# such that for all I.~6 ~ these conditions are fulfilled:

ù~} ~/='T&.

Conditions i.i) and 1.2) define a strongly isotone map [16]. For the first time, p-

morphisms of models were studied, probably, in the Kripke model theory of modal logics (e.g., see [12, 18]) and they appear implicitly in [17]. Basic properties of ~-morphisms are ex- pressed by the following

Proposition i [12, 16-18]. If / is a Æ-morphism of a model (~,~) onto a model (~',f#),

then for each formula ~ and element ~e~holds

and

{«,«)~ A ~ {f',P(«})~ A (3)

L c/J) c_ D cy ' ) . (4) P r o p e r t l e s ( 3 ) , (4) s h o u l d be t a k e n i n t o a c c o u n t i f we want to p a s s f rom a p . o . s . ~ to

B ' , w i t h o u t c h a n g i n g t h e l o g i c o f t h e f o r m e r b e c a u s e , u n d e r t h e h y p o t h e s i s o f P r o p o s i t i o n i ,

Dc~,f~ = L ~ / , f ' ) . Here, the p . o . s . æ ' may happen to be s impler than p in the sense , say,

o f h a v i n g a s m a l l e r c a r d i n a l i t y .

By an ~-equivalence (strong equivalence) relation on a model (~,~) is meant an equiva-

lence ~ on ~ satisfying the following conditions:

~.~> CW, f/~E)(~py =--> m f : n ~ ); 2.2} ( Vx,i, i'«E )(-~~'~£)I(«p~i,y~g') ~ (x~~',x',%i'J);

2. ~> { V«, i, «" i'~£ )((=.y', /pf , f.< «', «'n=) ~ «21). I f 2.1) is ignored, one can speak of an #-equtvalence on a p.o.s. ~ .

Obviously, the relat ion ~--~ is the least ~.-equivalence.

A model (/XP~IP) is called the quotient model of the model ~~,~) relat ive to the ~- P ~ - P ~- , , e~uivalence p i~ E ~ E/p «/p~ y/p~ ml >~ ~)lyyp=~/p) and ~2< «/A ~ « ~ C~«~ ~ «

This notion is well defined. Indeed, from 2.2) follows that the relation ~P is well defined and transitive. By 2.3), this relation is antisymmetric; its reflexivity is obvious. That the introduced valuation is well defined follows from 2.1).

The canonical map #; I--+ I/p is obviously ap-morphism of the model (y,f) onto (/tip, f p).

Moreover, the following holds:

Proposition 2 (/~'Morphism). The kernel equivalence p of each p-morphism of a model

(~,~)onto I~I,~ ') is an ~-equivalence and the canonical map of (tWP~f p) onto ~',~#) is an

isomorphism.

113

ùPr0of. Let I be the original~-morphism of (B1.~) onto ~~',~'). If /(~)=/~), then

Furthermore, if /(x)--/(~) and «,y', then, aceording to 1.i) and 1.2), ~(X)~/i(yO

and there exists l>~ such that /(~#)=p(~1).

Finally, from («~~', /[~')=/(y), ~~Œ', /(.T.T')--/{~JJ follows /(=J.<'Pq'j:fq)-<'f(~.';--pcxJ, i . e . , /CX)-=~~I). This shows that the kernel equivalence of the map p i s an 4-equivalence.

We will show that the map ~: ~/~ ~ pcs) is an isomorphism.

If ~/~~<2~/~, then there exists y'~>~ such that /[y')=/(~)o Using i.i), we conclude

that p (m) ~' f,,'~). If, however, /(Œ)~' liq), then for some ~'>~X, by 1.2), we have /(fr')= /(~), i.e.,

It remains to note that z/7= z )"

Remark i. It is clear that the kernel equivalence of any strongly isotone map of a

p.o.s, i~ onto /~' is an ~-equivalence and/~fis isomorphic to /#.

LEMMA i.i.

fined by

i. Lattice of ~-Equivalences

The collection of all &-equivalences on a model ~,~) is a p.o.s, relative to G.

If the definition of an ~-equivalence is restricted to conditions 2.1) and 2.2), one obtains the definition of a ~--equivalence (weak equivalence) on a model (~,~).

If~ is a ~f-equivalence on a model (~~£), then the relation~~on ~ de,

«" y ~ C3,«,/~ E)(«.V,, y?» y.< «, «m~J is an ~-equivalence on ~,~),

Proof. First, we note that

x2/~ «/y. (5) The reflexivity of the relation ~~ follows from (5) and its symmetry from its defini-

tion.

If Xp~~ and ~p~Z, then, using the definition of p~ (twice), 2.2), and then (5), we

obtain ~/Z. Thus, ~~ is an equivalence.

I f ~«Y#' i ß ' "] , f * X#, X~~~ , then there exis t ~w],£ ~ such that « ~ ] " , ~ ' ~ ] , ] *

If x ~ ~ and y*~', then, according to 2.2) , there exists ~ ' ~ ~ such that ~Dk~ I. Using

(5), we conclude that ~~~'.

Finally, using the definition of ~~, 0.i), and 2.1), we verify the property 2.1) for

B: Remark 2. From Lemma i.i and (5) it follows that each maximal UT--equivalence on some

model is an ~-equivalence on the same model; also, each maximal &-equivalence is a maximal ~f -equivalence.

114

THEOREM 1. If X is a nonempty set of 6-equivalences on a model (/.L,~) and a relationp on E is defined by

~M ~ (3«, .. ,«~~EXmp,, ..,p«+, ~x )c«,o,~,p,....~«p«+,/.

then jO*=M.4.pX. Therefore, the colleotion of all ~-equivalences on (/4,Z) forms a complete lattice.

Proof. By definition,~is an equivalence relation [6], and, using 2.1), 2.2), and

Lemma i.i, we conclude that ~~ is an &-equivalence on (~,~).

From the definition of ~ and (5) it follows that for any ~-equivalence @ eX holds ~ ~~+.

Finally, if ~ is an arbitrary upper bound for the set X and ~~~~ for some ~,~~~, then

there exist ~~~t£~ such that ~~~~ ~~~, ~~4~ ~, and~>~. Using the definition of ~, we

conclude that ~~~~ i'~~ , ~~~', and ~~~~. By 2.3), we obtain ~~~.

The last statement of Theorem 1 is later modified in Theorem 2.

2. Description of the Lattice of A-Equivalences

We associate with each model (~,~) two pseudo-Boolean algebras [g]. One of them,~(~),

is the set of all subsets ~ of the set ~closed in the sense of the order ~, so that for all ~,~e~ holds

l~«X,«~f) >yeX with the set-theoretic operations n and U and also operations ~ and ~ , which are defined as fo!lows:

The second algebra

by the elements X~ for all propositional variables.

For any complete [8] pseudo-Boolean algebras ~ and Cwe write:

B ~ C ~ J~ isi~morp~c ~ a compl~esuba~ebraof~ealgebra ~ ,

B~C~ J~ isa com~etcsubalge~aof~ealgc~a C.

LEMMA 2.1.* If there exists a p-morphism of a model (~,~) onto

gebras ~(/u,~) and ~(~',~') are isomorphic.

Proof. It follows from (3)

The following example shows

Example.

X=Y ~ {tee ](V~/>~z)(L/«X~yeY),

J~ (F,~) is the pseudo-Boolean subalgebra of the algebra ~(~), generated

(~, ~l), then the dl-

/

that the map ~:X~--~X~ is the required isomorphism.

that the converse to Lemma 2.1 does not hold.

Consider the model (~,Z) , where

s-~ u { « , < . « , 4 , ~ , « , ~ , » } who,o ~«o «o~ ~,,

uo <_ ~,, ~, -a,~, x, :~~~ zoo« %~«~ y, « , , y,<«~ ~-<a~

*This property was communicated to the author by R. Sh. Grigoliya.

115

Let

are isomorphic though there is no /o-morphism of (/xo,{)onto Bflo,{). nor of (/ffo,~) onto

¢,~o,«). LE~~4A 2.2 (see [16]). For each complete pseudo-Boolean algebra ~änd p.o.s. ~, if

~~~(/), then the relation2oon Edefined by

is an %-equivalence on#and the algebras ~and J~(,(,(~'f') are isomorphic. Conversely, for

each &-equivalenceßon/Zthere exists a complete [8] monomorphism #;J~(/I#)--*'~(/g), defined

by the condltion

B(X) -~ {«e~ 1 ~ / 2 ~ X ] . (7)

Proof. Suppose that ~~~(#). We will show that (6) defines an ~-equivalence on~.

Let Xm~--.~{X£~I~£XJ for ~E~. It is clear that (6) ±mplies

BX ~ (~~,~)' where ~z~-,{#6~ l~~ßJ'. Then the algebras '~(/..Z~.o, «) and J~(/.£,~~ ,~)

Note, furthermore, that

~. We hav. ,2EX.~,~'EX«, X~, c_Xz, andXËc~X.t ,. ==>. First~ note that

Xy). (9)

(VX«~)(xi c_x ~=»XynE«=_X ), (lO)

E x~X~ ~X E./3, and,

(il)

B~X' then, by (I0), we

Thus, (9)

Indeed, if~#cX, th«n X~¢]E~~X.

O E£ C-X, then, according to Chap. i, Sec. 12, (i) in [8], If X~ X~~XymX'Xz6]X~ =X" and X~ C-X'" therefore,

Furthermore» we have

Indeed, had there existed X~~ such that X c X~ and Xr]Ex=X #

would have X~ ~ X. Contradiction.

Next, the faet that U[XE~ IXcA~) eX~ implies, by (ii), that

Ler «'~XffnE=\u{_A~:nXlXeY~, XcXl}. I~ is clear that ~-<Æ and X~,=X~j. is proved.

From (9) follows that ~ is an &-equivalence on/.

It remains to show that the algebras J~ and J~(/l P) are isomorphic.

116

To this end, wc consider the map ~: ~ "'~~(~ß), defined by ~(X)~{£/~ ]«6~}. Ob-

v±ously, this map is well defined. It is also an "onto" map, for if XB6 J~ß) then, putting

X ~U{Xzl«/ff£XP~, we have ~(X)=X :. Ind«ed, if X/:6f(X), then zöX~ for some y such that ~/ffeX2. Hence X~~X~, and,

by (9), there exists ~'~~ such that X~,=~~, i.e., ~'/~=~/~. Therefore, ~/jo6~ ~.

If ~/~6X p, then Xz~~. Hence ~6%, i.e., x/~~~(~).

Finally, we have X ~ Y <~--~>~(X) ~ ~(v).

=~. If Z126¢(X), i.e., I6X, then ZeV, i.e., Z/:eß(Y). ~:==. If X£X, i.e., æ/ff£f(X), then =/:6p(Y). Therefore, for some y such that

X/ff = ~/ff, wehave F6 Y. By definition of ~, we obtain £6 ~.

Suppose now that for some ~-equivalencejo on/~the algebras~ßand~ß(/P) are isomorphic. Just as in Theorem 4.6 in [16], we show that (7) defines a complete monomorphism of the al-

gebra ~(~P) to J~(/) (indeed, as has been noted in the Introduction, the map ~: 2--~~/~

is strongly isotone). Thus, Lemma 2.2 is proved.

A p.o.s. /is said to be dually embeddable into a p.O.So ~' if there exists a map

B:/__+/I such that for all ~,J£E we have

If ~ is an "onto" map, then/~ is said to be dually isomorphic to ~/.

Let ~~c~.~)denote a complete lattice on the set [~/J~~,:)~~ ~ ~~)} with the usual set-

theoretic operation n as gn{, where (~,~) is some fixed model.

THEOREM 2. The lattice of all &-equivalences on a given model (~,~) considered as a

p.o.s, is dually isomorphic to the lattice ~(/,f).

Proof. We define the map ~ of the lattice ~(~,Z) onto the lattice of all ~-equiva-

lences of the model ~,~) according to (6). From J~~,:) ~~ and (6) follows 2.1), i.e., yo

is an~-equivalence on (~,~).

LetjObe an arbitrary &-equivalence on (/u,~). By Le=as 2.i and 2.2, ~(~,~)---~~Cß~

BP) ~«~[/U2) ~.~(/7), where the monomorphism ~ satisfies (7).

Since (/P,~P) is aß-morphic image of (/,~), we have, by (3), that ~(XÄP)=X;: for any

formula A. Therefore, ~ (/J,:) =~ [~~~~~~, i.e., ~ E~ (/.zP~ ~ ~ ~/4~,"

Ler aO'~~~ ~~(/z~P~ ). Then

~2,~ 4=> (Vx ~~(~:))(.~~(x)~~«?(x)) <=>(vx ~~F:,)(~I2~X~=,,~/:~x)~>~/: = ¢12,

i.e., ~'=p. Therefore, ~ is an "onto" map.

Consider algebras ~0.~i 6~

~c :4)- Suppose that ~o ~~I" Since

We will verify that ~a ~4 if and only if ~ (~I)~ ~,Z)"

(6) holds for ~C~ i) (1:0,4), we have ~ CJ:~,)_C~(~o)o

117

Suppose now that ~{~I )~~~~o )" Denote X~ ~~{XE~II~6X } {~:0,«). We will show that

« Xe~0 »X = U Xt. (12)

zEX

, ~X~ ~ Let X~~o. Note that, according to (S), the inclusion ~l%)=~(~o) implies (X~

f The inclusion X ~Xg xx is obvious.

« If - ~~~~£~~vXx' i.e., for some ~6X holds ~~X~/, then [~~X~. By (9), there exists II>x

« -

xt=× ~;, . x ; , ~ x , . such that ] and, thus, = X] Since æ'£X, and therefore ~6X

It remains to note that the inclusion ~0 C_~i follows from (12).

Theorem 2 is proved.

COROLLARY i. The lattice of &-equivalences of a given model is dually embeddable, as a p.o.s., into some complete algebraic [15] lattice. Moreover, if ~ is the embeddingpointed above, then

where ~p and b~ß are operations on the lattice of $-equivalences and the constructed algebraic lattice, respectively.

Proof. Indeed, if the lattice of &-equivalences of a model(/X,~) is considered, then

the lattice of subalgebras of the algebra ~(/) is (see Sec. 6, Theorem 5 in [15]) algebraic and complete. Then the assertion of the corollary follows from Theorem 2.

3. Assertorically Equivolumetric Logics

Considering pr0positional logics with different signatures and the same set of propositional variables, it is reasonable to ask whether the sets of formulas constructed using only their common signature part (assuming it is not empty) and valid in these respective logics coincide; also, if the notion of inference is defined for the given logics, whether their deductive possibilities for the same sets of formulas are equivalent.

Making the above discussion more precise, we consider, along with signature (i), also the signature

V -D q A (13)

where the last sign ~ means a unary propositional connective.

Henceforth, we will mean by ~ some superintuitionistic logic and by ~; a logic in the

signature (13), i.e., a set of formulas in the signature (13) closed relative to the rules

of replacement and modus ponens, such that L ~~L a. In particular, by D ~

will be meant

the logic in the signature (13) whose axioms are formulas valid in L and also the following:

3. ~) Cp = ~p),

3. 2) ([~p =p) np),

3 3) (~p=(9 v c~=p))).

Formulas 3.1)-3.3) have been found by Kuznetsov and are modifications of formulas (2)-

. [~ (4) in [i] The calculus obtained from the above is equivolumetric (as has been noted by Kuznetsov) to the calculus of the same name in [3] (see p. 224 in [5]).

By the assertoric fragment A(~;)of a logic /~4 is meant the set of all formulas in

the signature (i) contained in /~L 4. It is clear that ~ ~ A (/~;),

118

Logi«s ~ and /~~ are said to be assertorically equivolumetric if ~ = A(/~~ )o

For instance, as has been noted in [3], the logics I and Z a are assertorically equiv01- umetric.

In connection with study of the logic /%//, yet another question arises: whether ~- formulas (or ~-axioms, such as 3.1)-3.3)) define a new operation. Here, the term "new operation" is sometimes interpreted differently by different authors (see [i, 3, 9]).

We say that a weakly new operation is defined in /~~ if for each formula A[p] with a

single variable p holds (~~]~~~} ~/~~, and a strongly new operation is defined if for

each formula ~~3 with a single variable ~ the logics ~ and ~~~ I~~]~ ~p) are not asser-

torically equivolumetric.*

K 4) and strongly new operation is defined in , then this LEMMA 3.i. If ~= ~(~& a

operation is weakly new in '~L"

The proof is obvious.

LEMMA 3.2. If a weakly new operation is defined in ~a, then this operation is strongly

new in L~.

Proof. Suppose that for some formula A~J the logics ~ and ~~+(A~]~~p) are asser-

torically equivolumetric. This implies that the formula (~ ~3 D (~v(9 bp}))6 ~, and that the

following sequence of formulas is easily completed to the validity in ~4

cA [pl = (~p v p))

Similarly, one can show that (A B -~ A ~.]) 6 ~ ~ p) Dp) belong to ~. A contradiction.

(A~] ~ ap).

because the formulas (pDA[p~) and «A[p~D

Lemmas 3.1 and 3.2 show that in the case ~=A I~ ~) one can simply speak of a new oper- tion. Is it possible to remove the last restriction? In this respect, of interest is the following

THEOREM 3. If ~ =- A [~~), then formulas 3.1)-3.3) define a new operation in L ~ if and

only if ~ is neither an absolutely contradictory nor a classical logic.

Proof. First, we note that if Ce is a classical logic, then the following sequence of

formulas is easily completed to the validity in C~~:

(~p v(Ap~p)). (,~p vp), ~p.

*In [i, 9] a strongly new operation is called new and in [3] a weakly new operation is called new (by Novikov). Both definitions have been introduced in connection with Novikov's hypothesis on the impossibility to add a new and unique operation to I in such a way that the calculus describing it would be noncontradictory (by Novikov) and complete (by Novikov).

119

Therefore, ((p DB) ,~/lB) 6 ~~ A

On the other hand, we will start with proving two statements in which the symbol ~3 de-

notes the three-element pseudo-Boolean algebra [4] and the symbol p~ (~ being a natural num-

ber) denotes the n-th power of the formula p: o~ (~& 7~),p~ ~ ~p, pZ__~~,p~~ ( ~+I /)~) and p~+~-~ (~~+'v /y 2n+2 ),

The first:

Indeed, if Æ~n < 6,

can be easily seen.

On the other hand,

Brmma ip ~p)~p) isvalidon ~j<==~ ~ ~e. (14) (

then the formula [(p~Dp)D/~) is not valid on the algebra ~j, as

for ~ ~ 6 we have Z~-(C/TaDÆ)m~). This is readily seen by assign-

ing to powers of the formula p elements of the free cyclic pseudo-Boolean algebra.

The seeond:

~>« --> ~rmula (pc= I~v{g»p))) is not valid on Z3 ' (15)

Taking into account that for ~~ 6 we have ~F- lqpD,Æn), it remains to refute the for-

mula Cp~m Ig v (~ D~))) on ~j, evaluating the variable p with zero of the algebra Z3 and f with

the intermediate element of this algebra.

Suppose now that Z= ~(L ~) and ~ is neither an absolutely contradictory nor a classi-

cal logic. If for some formula ~~], ~A~]~~p)6~ 4 had taken place, then the formulas

~~~]Dp)o~) and (A~]DIgv[~mp})) would both belong to Zaand, therefore, to ~. But this

contradicts (14) and (15) because all formulas valid in L ere also valid on the algebra ~3"

In particular, Theorem 3 gives a method of finding logics ~ ~ for which d-axioms 3.1)-

3.3) define a new operation. It obviously suffices that the logics L and ~abe assertoric-

.ally equivolumetric and L be neither absolutely contradictory not classical.

On the other hand, study of the assertorical equivolumetricity of logics is related to

the question of their assertorical equipotentiality. Namely, logics L and ~4 ere seid to be assertorically equipotential if for any formulas A and ~ holds

L+A~- ~<-~~L~+A~ ~, where the sign ~- in both parts is understood as inference using two rules, replacement and

modus ponens (see p. 212 in [5]). It is easily seen that if for each logic Z' such that

(J- ~' the logics ~ i and ~~ ere assertorically equivolumetric then the logics ~ and ~ ~ are assertorically equipotential.*

We will make an attempt to describe a certain class of logics ~ such that L = A (L 4).

A model (~,~) (a p.o.s.~) is ealled nonwide if/ has no infinite set of pairwise-in-

comparable elements. An example of a nonwide p.o.s, is given by a p.o.s, of width th(see [i0]), i.e., having no a+ / pairwise-incomparable elements.

*The question (still open) on the assertorical equipotentiality of the logics I and I a was raised by A. V. Kuznetsov. Also this question is open: whether there exists a superintuitionistic logie L such that L CA(LA)?

120

Following [12], a model C~,«) will be called weak if the set ~~E/]: is finite.

LEMMA 3.3. If (/~~ ~ß) is a quotient model by the greatest &-equivalence on a weak

nonwide model (~.~~, then yP contains no infinite ascending chains.

Proof. We introduce definitions necessary for the proof, relating to any weak non-

wide model (~l ~t). Let ~,__~{~t E~Iæ~~,~I, i ,t is not the beginning of an infinite ascending

chain} and Vsr~-~I/7~~l~~) ~, ~l is the beginning of an infinite ascending chain}. We call

the beginning of an infinite ascending chain ~'eE ~ static [12] if for each ~'6~ ~ such that

~t is the beginning of an infinite ascending chain and ~6~~ l holds ~~, =~~I, and we call

this beginning stationary [12] if for each ~'EE z such that ~t is the beginning of an infinite ascending chain and ~O~X I holds V~i= ~fl •

Note that if I~ is a static (stationary) beginning and ~' the beginning of an arbitrary

infinite ascending chain such £~I~ I, then jt is also a static (stationary) beginning. And

if ~t is a stationary beginning, then ~~l={RT}. We now go to the proof of the lemma.

Suppose the contrary. Si~ce the p.o.s. ~~ is nonwide, the proof of Theorem 4.2 in

[12] implies that there exists a static and stationary (simultaneously) Neginning ~~ in

Io /~: «0/~< F «,/~ < ~ . . . . (16)

We define a binary relation /0 ~ on ~as follows:

~~) ~ Z~~ or (~~, ~#) are static stationary beginnings;

It is clear that

X]O~ + XjO'], (17)

We will show that the relation p / is an &-equivalence on ~,~).

On one hand, the condition 2.1) is obvious.

Suppose Iß~ and ]~y. If ~pp, then, sincep satisfies the condition 2.2) and (17),

we obta±n the validity of 2.2) for ~ t too. And if not ~7o~, then we consider two cases.

First case: ]~p~~ = %. Then, obviously, there exists ~~~ such that ~~~t. Apply-

ing (17), we obtain ~~t~~ Second case: ~t/~~ ~f/p. Then ]~ß is the beginning of an in-

finite ascending chain in ~P. Using properties stated above, we deduce that ~7~ is a

static and stationary beginning and also that ~~= U~/p = Um~ and /Ty~ =/7~'. Thus ~pt]t,

By Remark 2 and (17), we conclude that ~=~I

On the other hand, it is clear, by (16), that ~o~t~+, but not ~op~1.

In connection with studying formulas in the signature (13), we extend the validity conditions 0.1)-0.5) to formulas in signature (13); we define for an arbitrary formula a in the extended signature:

(cf. [1]).

121

LE~flqA 3.4. For any p. o. s./Z holds :

3) y~ ((~p=p)=p)~~> in,Æ contains no infinite ascending chains.

Proof. The verification of statements i) and 2) is straightforward.

Suppose that for some valuation of 2' ~,~)~{~BD/P)Dp) is false. Then there existsyo~~ ~

such that ~,~o)~ (~BDÆ) and (~~o)~ß is false. Therefore, there exists ~>~o such that

~,~I)~ ~ is false. However, by property (2) extended to formulas in the signature (13),

we have (~,~~)~ (d~=~). Thus, we obtain a required chain.

Conversely, suppose that ~ has an infinite chain ~0<~I< .... We define the valuation

as follows:

(~,~) ~ ~ ~ th=eisno ~~ such that Yi ~ x "

I t i s e a s i l y seen t h a t (~,~a) ~ I ( A p ~ p ) ~ } ? ) i s f a l s e .

A modelable l o g i c i s s a id [5] to be de termined by a g iven c o l l e c t i o n of p . o . s , i f t h i s i o g i c co inc ide s wi th the i n t e r s e c t i o n of l o g i c s (as s e t s of formulas) of each p . o . s , from this collection.

A superintuitionistic logic L is said [2] to be approximated by pseudo-Boolean alge-

bras of class ~if for each ~ not belonging to ~ there exists an algebra in~on which all

formulas from ~ are identically equal to the identity of this algebra and the formula A is not equal to it.

THEOREM 4. If a logic ~ is determined by nonwide p.o.s, or approximated by algebras

in each of which the set of simple filters is nonwide as a p.o.s, by ~, then ~ and ~a are assertorically equivolumetric.

Proof. The first statement of the theorem immediately follows from (4) and Lemmas 4.1 and 4.2,

The second statement follows, by virtue of (3) and Lemmas 3.3 and 3,4, from the fact

that each pseudo-Boolean algebra ~is isomorphic to the algebra ~~,~), where ~ is the set

of simple filters of the algebra B ordered by the relation ~ (see Lemma 3 in [7]).

COROLLARY 2. There exists a continuum of superintuitionistic logics for which ~-axioms 3.1)-3.3) define a new operation.

Proof. It suffices to note [2] that the continuum of logics satisfies the hypothesis of Theorem 4 and to apply Theorems 4 and 3.

COROLLARY 3. If a superintuitionistic logic L is modeled by a p.o.s, of width ~ (n be- A

ing a natural number), then the logics ~ and ~ are assertorically equipotential.

Proof. It suffices to note (see Theorem 1 in [12] or Lemmas 3, 5, 8 in [i0]) that each

extension of the logic L is determined by some class of p.o.s, of width e. Then, by Theo-

rem 4, for each such extension ~I , ~~ we have L = ~(~ ). Therefore, L and L ~ are assertori-

cally equipotential.

THEOREM 5. There exists a modelable superintuitionistic logic which is not determined by any class of nonwide p.o.s.

Proof. Consider the logic of the p.o.s.y from [ii, 13]. As has been shown in [ii]

for the formulas ~and ~ defined there, fi~/-~ is valid and ~ ~ ~ is not valid, and also,

for any p.o.s. ,#~, if ,~'~/2 is valid and ~'~ E is not valid, then DZcontains an infinite

ascending chain. It remains to apply Lemma 4.1.

122

The question of existence (»f a finite]v axiomat[zablt, ]oZi« ;:[th the, pr«»p«r'~/ stated in Theorem 5 is open.

Note final]v that in []O], formulas ~ and < are cor~~tructed such that / is stq~arated

from J bv a nonwide p.o.s, but is not separated from it bv anv u.o.s, of finite widtil. As

folh)ws from [ii] and Lemma 4.1, formulas ~ and [ are suci~ that is separated from ~ bv some p.o.s, hut is not separated from it by anv nonwide p.o.s.

The author is sincerely grateful to A. V. Kuznetsov and L. L. Ésakia for valuable dis- cussions.

LITERATURE CITED

I. A. V. Bessonov, "New operations in the intuitionistic calculus," Mat. Zametki, 22, No. i, 23-28 (1977).

2. A. V. Kuznetsov, "Superintuitionistic logics," Mar. Issled., I0, No. 2, 150-158 (1975). 3. A. V. Kuznetsov, "Provably intuitionistic logic," in: Modal and Intentional Logics

(Theses of the Coord. Conference, Moscow, June 5-7, 1978) [in Russian], Moscow (1978), pp. 75-79.

4. A. V. Kuznetsov and V. Ya. Gerchiu, "On superintuitionistic logics and finite approx- imability," Dokl. Akad. Nauk SSSR, 395, No. 5, 1029-1032 (1970).

5. A. V. Kuznetsov and A. Yu. Muravitskii, "Provability as modality," in: Actual Problems of Logic and Methodology of Science [in Russian], Naukova Dumka, Kiev (1980), pp. 193- 230.

6. A. G. Kurosh, Lectures in General Algebra, Pergamon (1965). 7. L. L. Maksimova, "Pretabular superintuitionistic logics," Algebra Logika, ii, No. 5,

558-570 (1972). 8. E. Rasiowa and R. Sikorski, Mathematics of Metamathematics [Russian translation],

Nauka, Moscow (1972). 9. Ya. S. Smetanich, "On completeness of the propositional calculus with an additional

operation on one variable," Tr. Mosk. Mat. Obo, 9, 357-371 (1960). i0. S. K. Sobolev, "On finite-dimensional superintuitionistic logics," Izv. Akad. Nauk

SSSR, Ser. Mar., 4_~i, No. 5, 963-986 (1977). Ii. V. B. Shekhtman, "On incomplete propositional logics," Dokl. Akad. Nauk SSSR, 235, No.

3, 542-545 (1977). 12. K. Fine, "Logics containing K4o Part I," J. Symb. Logic, 39, No. i, 31-42 (1974). 13. K. Fine, "An ineomplete logic containing $4," Theoria, 40, 23-29 (1974). 14. M. C. Fitting, Intuitionistic Logic, Model Theory and Forcing, Amsterdam (1969). 15. G. Gratzer, Universal Algebra, Princeton (1968). 16. D. H. J. de Jongh and A. So Throelstra, "On the connection of partially ordered sets

with some pseudo-Boolean algebras," Indag. Math., 28, 317-329 (1966). 17. H. Ono, "Kripke models and intermediate logics,"Publ. RIMS, Kyoto Univ., 6, 461-476

(1970). 18. K. Segerberg, Results in Nonclassical Propositional Logic, Lund (1968).

123

Strong Equivalence on an Intuitionidtic Kripke Model and Assertorically Equivolumetric Logics

Documents

Transcript of Strong Equivalence on an Intuitionidtic Kripke Model and Assertorically Equivolumetric Logics