partially ordered connectives

2. Moth. Logrk Crundlogen Moth. 38 (19921, 361-372 Q.1992 Johonn Ambrorrur Borth

PARTIALLY ORDERED CONNECTIVES by GABRIEL SANDU and JOUKO V A ~ N A N E N in Helsinki, Finland’)

Abstract We show that a coherent theory of partially ordered connectives can be devel- oped along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results. MSC: 03C80. Key words: Partially ordered connectives, partially ordered quantification, Ehrenfeucht game, undefinability.

1. Introduction

L. HENKIN introduced in [4] the following ordered quantifier, which has since been called the Henkin quantifier

Here, f and g range over unary functions of the universe. T h e references [3] and [9] provides basic knowledge about the Henkin quantifier and its variants. It was proved in [7] tha t in terms of implicit definability the Henkin quantifier is as strong as the full second-order logic.

Our start ing point is the observation tha t the original idea of H E N K I N applies also in the case when the existential quantifiers 3y and 3u of (1) are replaced by disjunctions. We arrive at the following “partially ordered connective”, which we denote by D1,l:

Here f and g denote functions from the universe to the set {O,l}. More general partially ordered connectives immediately suggest themselves and are discussed later. We show tha t the quantifier Dl,l is not first-order bu t strictly weaker than the Henkin quantifier. Moreover we show that the intermediate quantifier D, 1),1

is strictly stronger than D1,l and implicitly even as strong as the Henkin quantifier.

”We are indebted to LAURI HELLA, JAAKKO HINTIKKA, MICHAL KRYNICKI and KERKKO LUOSTO for useful discussions concerning the material of this paper.

362 G . SANDU AND J . V A A N ~ N E N

The structure of this paper is the following: Section 2 contains Insic examples of the expressive power of partially ordered connectives. Section 3 describes Ehren- feucht games for partially ordered connectives and gives an undefinability result . a.5

an application. Section 4 gives a reduction of partially ordered connectives to the infinitary logic L,,, .

2. Basic examples

We demonstrate that Dl,l is indeed not first-order and tiiat D(lJ.1 is genuinely stronger than Dl,l. Moreover, we give an example of the use of a more general variant D?,2 of Dl, l .

The quantifier Dl,l is not firsi-order definable. I n f a d . / h e standard model (N, +, ., 0,1) of anthmei ic can be characterced u p t o isoniorphrsm b y a sentence of Dl, l .

P r o o f . Let us start by considering the following formula p in a language with one unary function symbol S and one constant synibol 0:

P r o p o s i t i o n 1.

where

poo(+, Y, 2) is -(z = y), p o ~ ( t , y, :) is -(r = 01,

CPIO(Z, y, z ) is -(y = =), cpll(z,y. :) is - ( x = y)&-(y = S(r))

Let $J be the conjunction of the sentences

Vr- (S( r ) = O), VzVy(S(z) = S(y) - x = y), Vr(- (r = 0) - 3y(S(y) = r ) ) .

Suppose (p&JI) is true i n a model ,M via the Skolem functions f and 9 for p. I t follows froni the definition of the foriiiulas vij that

f(0)) = 9(=) = 1, g(0) = f(:) = 0,

f(s(o)) = f(sz(o)) = f(s3(0)) = . . . = 1.

f(2) = f(s- '(z)) = f ( P ( z ) ) = . . .= 0.

g(2) = g(s-I(:)) = g ( s - ? ( z ) ) = . . . = 1.

g(s(0)) = g(S'(0)) = g(S3(0)) = . . . = 0,

So in M we have L # S" (0) for any n. Thus : h a s infinitely many predecessors and i r not a standard number. Now we can use (;p&<) to characterize the standard niocl~ of arithmetic by defining S ( r ) = z + 1. 0

The following corollary follows from general results true of any generalized quantifier capable of defining the standard model of arithmetic (see[2]):

PARTIALLY ORDERED CONNECTIVES 363

C o r o l l a r y . The logic wrih the quantifier D1,l i s a non-axtoma2r:able. non- compact e z i ens ion of elementary logic uihich does not satrsJy ihe Craig interpolation theorem n o r the Beth definabthty theorem.

P r o p o s i t i o n 2 . The quantifier D1,l i s definable by the Henkin quaniifier. P r o o f . It

( :; sufices to observe that

v 3 U 3 U ( l ( U = u ) $2

where O(u , u, 2 , y, 2, w ) is the formula ( z = u & w = u & ( p o o ( z , y ) ) v ( z = u & w = t ' & j m ( z , y ) )

( z = u k w = u&p1o(z ,y) ) v ( z = u s ; w = Lv&pll(Z,y)). 0

v

Recall the definition of D(l),l in (3). The proof of Proposition 2 can easily modified to yield

P r o p o s i t i o n 3. Dl,l is definable by D,,,J and D(1),1 is definiable by the Henkin puantrfier

T h u s D(1),1 is also not first-order, non-axiomatizable, non-compact, etc. We shall now show that D(l) , l is not first-order in a much stronger sense than D1,l. This will be demonstrated by means of two other quantifiers.

The following "function quantifier" is studied in [8]:

F P Y V , ' 4 2 , y, u , =) - ~ P V y V d z , Y, f ( + ) , f ( Y ) ) . The idea of this quantifier is that it is like the Henkin quantifier, except that the two Skoleni funktions f and g of (1) are assumed to be the same. The motivation behind studing Fa is that most uses of the Henkin quantifier seeins to be, in fact, applications

We do not know whether F2 is definable by D(1),1, but we define a stronger quan-

of F2.

tifier which can express Fz. Let D ( l ) , ~ be the quantifier

- 3f3fV~VvVwPo, ( , ,w) (~ , f(Z)# 11, w ) ) .

P r o p o s i t i o n 4. The quantifier F:! is definiable by f h e quantifier D(1),1 P r o o f . We shall prove that

364 G. SANDU AND J . VAANANEN

where

cpo(z, Y, v , w ) is (z = v - -.(Y = w ) ) ,

'Pl(Z, Y, v , w ) is 4 x 1 u, Y, w ) .

Suppose F~zuywcp(z, u , y, w ) is true in a niodel M via the Skolem function f. Let g ( u , u ~ ) = 1 if f(u) = w , and g ( u , w ) = 0 otherwise. In order to prove that

v ~ v ' 1 ) v ~ c p g ( u , w ) ( ~ ~ f(E), u , w )

holds in M let x , u, and w be given. If g(ulw) = 1 , then f(u) = w , and we have cpl(z, f(z), u, w ) . If g(ul w ) = 0, then f(u) # w , and we have ~pg(z, f(z), u, w ) . For the converse, suppose that functions f and g are given in such a way that

V z V v V w c p g ( u , w ) ( t , f(t), v , w )

holds in M. Suppose now 2 and u are given. We know that 'p~(~,,(~))(z, f(t), i r , f ( v ) ) holds. If g(u , f ( u ) ) = 0 , then f(u) # f(u). Thus ~ ( t , f(z), v , f ( u ) ) , as claimed. 0

If Proposition 4 is combined with the result of [8] to the effect that the decision problems of F 2 and second-order logic are recursively isomorphic, we obtain the following result:

C o r o 11 a r y . The decision problem of D ( l 1.2 is as difficult as that of second-order logic.

Proposition 4 can be used to define a lot of notions in the logic with D(1),2. This all follows from the strength of F 2 .

P r o p o s i t i o n 5 . In models which have a definable pairing funct ion the quantifier D ( 1 ) , 2 is definable by D(l),l.

P r o o f . Suppose p(z,y) , r ( z ) and q ( t ) are definable functions such that for all E ,

y we have r(p(t ,y)) = t and q ( p ( z , y ) ) = y, and for all L we have p ( t ( r ) , q ( : ) ) = :. Then

By combining Propositions 4 and 5 we can define F 2 by D(l),l in a variety of

C o r o l l a r y . The decision problem of D(1,,1 is as difficuli a s that of second-ordfr

As a final example of the use of partially ordered connectives we consider the

situations. In particular,

logic.

quantifier D 2 , 2 defined by

- 3f39VtVYVuV~Jcp/(,,,),(",",(z, Y, u, v ) .

Let us call a unary function h cycle-free if h" ( z ) # x for all n.


L e m m a . A unary funct ion h i s cycle-free af and only if there as a binary relniion

( i ) R ( z , y) - R(t, h(y)), P r o o f . Suppose first that h is cycle-free. Define

R such that for all r and y

(ii) R ( z , h ( z ) ) , ( i i i ) -R(t,t).

R(r.y) - h " ( t ) = y for some n 2 1.

Then (i)-(iii) hold. Conversely, suppose h has a cycle z , h ( z ) , . . . /I"(+) = z. Then R(r ,h ( r ) ) , R ( r , h ? ( ~ ) ) ~ . . . , R(r ,h" ( z ) ) , i.e. R(z,z), a contradiction. 0

P r o p o s i t i o n 6. The cycle-freeness of a unary functzon can be defined by 01,~.

P roof. Consider the following sentence 9:

where

V O O ( ~ , 9, u , 1) ) is '(y = h ( t ) ) ,

pol(t9y,u,u) is - ( z = u & y = u ) ,

plo(z,y.u,v) is l((~=z~r,r:h(y))Si-(t=u6ry=u),

p~l(r. y, u , 11) is -(y = r).

Suppose Q is true via the Skoleni functions f and g. Then f = g. Let

4 t . y ) - f ( z , y ) = 1.

LVe show that R satisfies the conditions (i)-(iii) above:

( i ) If R ( r , y ) J t - R ( t , h ( y ) ) , then f ( z l y ) = 1 and g(z ,h (y ) ) = 0, contrary to

( i i ) If l R ( z , h ( z ) ) , then f ( z , h ( r ) ) = 0, contrary to poo(z ,h ( t ) , z ,+) .

( i i i ) If R ( z , z), then f ( t , z) = 1, contrary to (p11(z, z, z.t).

Conversely, suppose that R(r , y) satisfies (i)-( i i i) . Define

VlO(Z, y,z, h(Y)).

f ( z , y ) = g(z,y) = 1 - R(z,y). (a ) If ,J~o(E, y, z, h(y)), then R(r , y) & -R(r, h(y)). contrary to ( i ) .

(b) If plo(r. y, z, y), then f ( z , y) # y ( r , y). a contradiction.

(c) If poo(z9h(z), u , D ) , then i R ( r , h ( z ) ) , cont.rary to ( i i ) .

( d ) If gll(r.z.gq ( J ) , then R ( t , z ) , contrary to ( i i i ) .

(e) If + " ~ ( z . y ~ r , y ) , then f ( z , y ) # g ( r , y ) , a contradiction. 0

25 Ztschr. f. math. Logik


3. An Ehrenfeucht game We shall define an Ehrenfeucht game which can be used to prove that structures are equivalent relative to a logic with partially ordered connectives. To avoid complicated notation, we define the game only for the following quantifier 0 2 . 1 :

- 3f39VzvrVucpl(,,,)g(”)(t, 2 , u). We use the Ehrenfeucht game to show that certain notions, like the finiteness of

a unary predicate, are not definable by the quantifier D2,l. The proof is based on a technique used in [8].

The game G , ( M , N ) is played by two players V and 3 on two models M and n/. The game has n < w moves. During the game a sequence of 3n pairs ( z , ~ ’ ) is produced. At each move the players perform the following sequence of actions:

1. Player V chooses a model, say M , and two functions f : M’ - (0, I} and

2. 3 chooses the other model, say JV , and two functions f’ : N’ - {0,1} and

g : M - {0,1}.

g’ : N - {0,1}.

3. V chooses 3 elements a , b , c from 1v and 3 chooses 3 elements u’, b’, c’ from A1 such that f(a’, b’) = f ’ ( a , b) and g ( b ’ ) = g ’ ( b ) .

The players switch to the next move. We say that during this move the pairs ( u , r i ‘ ) .

(b , b’) and (c , c’) are produced. 3 wins the game if after n moves the pairs produced in the game form a partial isomorphism between M and n/.

We set M En n/ iff M and JV satisfy the same sentences of L(D2,1) of quantifier rank 5 n. We forget the quantifiers 3 and V since they are definable by D L ) , ~ .

P r o p o s i t i o n 7 . I f p l a y e r 3 h a s a w i n n i n g s i r a t e g y i n C n ( M , N ) , t h e n ~ M EnN. P r o o f . Easy. 0

P r o p o s i t ion 8. The not ions “u is infiniie” and “U is f inite and euen” are not definable in the logic L(DZ,l).

P r o o f . We construct for every n two models M = (hi, U ‘ ) and A/ = (N, ””), such that IU’l = 2 ( n - 1)+9, lU”l = JlJ’l+ 1 and M E,, N. This shows that evenness of 1111 cannot be defined by D2,1. The same idea can be used for finiteness and a variety of other similar notions.

We define for every model M = (M, U, a l , . . . , a t ) the following codings t : M - { 0, . . . , k + 1 } and i : M --L { 0 , 1 , 2 , 3 , 4 } :

0 if a E U - { a l , ..., Q k } *

k + l if a $ U U { a 1 , . . . , a k } , if a = a,,

PARTIALLY ORDERED CONNECTIVES 3 6 i

and

0 if a = b = c , 1 if a = b # c ,

3 if a = c # b , 4 if a # b # c # a .

2 if (I # b = c,

If M = ( M , U , a l ,

T ( M , f , g ) =

If 0 1 , . . . ak and bl , . .

we define: . . , a t ) is a model, f : M 2 - (0, I } and g : .\I - (0, I } . then

_ . , bt are sequences of elements of models M and A*, respectively, and if the mapping a , H b , , i = 1, . . . , k, is a partial isomorphism from M to ,t*, then we write ( a l , . . . , a t ) Y ( b l , . . ., b k ) .

The proof of Proposition 8 is based on the following two lemmas (similar to lemnias 6.2 and 6.3 in [a]):

L e m m a 1. Lei M = ( M , U’, 0 1 , . . . , a t ) , ,%r = ( N , ( : ‘ I , b l , . . . , b k ) be models such fhaf N _> M , IN1 = IMI = u, Ilr’l 2 k + 9, 1/1”1 = IU‘I + 1, k < n , a n d ( a l , . . . , a t ) 2 ( b l , . . ., b t ) . T h e n for a n y f : M’ - {0,1} a n d g : M - {0,1} fherc are f’ : N 2 -. {0,1} a n d g’ : N -.. { O , 1 } such f h a f T ( J M , f , g ) 2 T ( N , f ’ , g ’ ) .

L e m m a 2 . Lei Jvi = ( M , U’, a l , . . . , a t ) , hf = (N , U“, b l , . . . , b k ) be mode l s such that M _> N , IN1 = IMI = w , ILI”I 2 k + 9 , IU‘I = IU”I + 1, k < n, a n d (a l , . . . , a t ) 2 ( b l , . . . , 6 k ) . T h e n for a n y f : M ? - { 0 , 1 ) and g : M - { 0 , 1 ) there are f’ : N’ - { O , ] } a n d g ’ : N -+ {0,1} such fhaf T(,U,f,g) _> T ( N ? f ’ , g ’ ) .

Proof of L e m m a 1. We may assume, that N = A1 u { p } and li” = U‘U { p } . We define the following equivalence relation on V = U’ - { a l , . . . , a k } :

368 c. SANDU AND J . VAANANEN


whence a E T ( M , f, 9 ) . If a = t, 6 # t, then a = (0, t ( b ) , 0 , f(z, b),g(t) , 4 ) . If 6 # E‘ ,

then the situation is similar to the previous one. If 6 = E ‘ , then we have

= ( t ( E ) , t ( t ’ ) , t ( z ” ) , f(z, t’), d z ” ) , 41,

a = ( t ( a ) , t ( b ) , 0 , f(a, b ) , g ( ~ ) , 4 ) = (t(a), t ( b ) , t ( t ) , f ( a , 6 ) , d x ) t 4).

whence a E T ( M , f, 9 ) . 0

Lemma 2 is proved similarly.

Now, by repeating application of Lemmas 1 and 2, the player 3 can win the play

An interesting consequence of the above Proposition is that t.he quantifier

Gn(M,N). 0

Qozy(t) ,-+ { E : y(z ) is finite}

is not definable by D2,l.

4. A reduction t o infinitary logic

In this section we describe a reduction of partially ordered connectives to the infinitary logic L,,,. This reduction works in countable models only, but it nevertheless dernon- strates the relative weakness of partially ordered connectives. As applications we get many non-definitability results, for example, the non-definitability of well-ordering.

Let us consider the quantifier Dn, ,..., n b ( l l , . . . ,Zk) defined by

where Z1, . . . , Zk are arbitrary but ,finite. The semantics of ( 4 ) is readily defined in view of the definitions of Dl,l and D2,l.

T h e o r e m 9. Let L(Dn,, , , , ,nk) denofe the eztension o f t h e first-order logic b y the partially ordered connectives Dn1, .,,,b(Z1,. . . , Ik) for all finite 11, . . . , Zk.

If9 E L(Dnl,.. . ,nb), then there is a y‘ E L,,,, such that M + (p - 9’) for countable models M .

P r o o f . To simplify the notation, we consider the quantifier D2,2. I t should be clear that the proof does not depend on this restriction. Suppose y is as in (4) and


all pij are in L,,,. In 111 BARWISE observed that partially ordered quantifiers can be defined by game-expressions. Thus let us consider the following game-formula 0:

C C W

where $iEi: . . .ikik 1 2 (z:1,zi2r zil, lowing formulas:

. . . , ztl, zf2, zgl, zf2) is the conjuction of the fol-

C l a i m 1. M + (p - 0 ) for any M . P r o o f . Suppose M is a structure and there are functions f and y

such that M k VzVyVuVv~p,~ , , , )~~~ ,~ , ( z , y, u, v ) . Then the player 3 can win M k 0 by playing i{ = f(zil,z12') and i; = f(zS1,&), 1 5 P < w .

C I a i m 2. M + (a - 9) for countable M. P r o o f . Suppose M is a countable model of 0. Let ( a ~ 1 , a ~ 2 , a ~ 1 , a ~ 2 ) , 1 c LJ, he

a list of all quadruples from M . Let us play M k 0 in such a way that the player V lets t i j = aij for 1 < w , 1 5 i , j 5 2. Let ( i \ , i ! , ) , I < w , list the corresponding moves of the player 3, playing his winnig strategy. Let f(z, y) = i{ for the smallest 1 such that (t, y) = (a{l, and g ( x , y) = i!, for the smallest 1 such that (2, y) = (ail, a!,?). We claim that

I f (6) M + ~,(a:I ,~:~)g(o~l ,a~,)(0:1.0:?.0?1.0?2)

for all I < w . Suppose I is given. Let. I + be the smallest I+ such that = (a::,a{:) and I++ the smallest I++ such that (dl, a&) = (a$+, a ! , , ) . In view of ( i ) we know that M + p.1 .I ( ~ ~ ~ . a ~ ~ , a ~ ~ , a ! , ~ ) . Sin?e a:1 = a:: and a:2 = a::, we have by (ii), f(a:l,a\2) = i:. Similarly, by (iii), g(a2,,aL) = i!,. Thus (6) is proved.

To finish the proof of the theorem we consider an "approsimation" of ( 5 ) . If n < LJ

let 0" be the following formula:

I 2112

. . . I - - , , - - \ I (7) vz' * ' '

12

I& 2 I .

Note that 0" is in L,,,, since n is finite.


C P C

a i m 3 . M b 0 impl i e s JU b 0" for a n y JU and a n y n < w.

o o f . This is trivial, since 0" is clearly easier for the player 3 than 0. 0

a i m 4. I'M t= 0" for all n < w , then M + 0 , for a n y M . P r o o f . Suppose not M b 0. By GALE-STEWART'S theorem the player V has

a winning strat,egy r in M b 0. Let T be the set. of sequences (ii, i:, . . . , i:, i!.) of moves of the player 3 in M /= 0 against 7, such tha t he has not lost yet, i.e.,

\Ve can think of T as a finitely branching tree with no infinite branches. By K ~ N I G ' S lemma, this tree is in fact finite. Let n exceed the cardinality of T. Since M a", the player 3 can play n choices ( i i , i i , . . . , i;, i s ) in M 0 without losing. T h u s T has a branch of length n, a contradiction. This ends the proof of the theorem, since we can let

the formulas +iii i . . .i:i5(zi1, z!?, zil, r2?, 1 . . . , zt1, L':?, r21, k k E??), A. 5 I , are all true.

9' * A an, n < w

which is in L,,,. 0

L(F?) i s n o t a sublogic of L,,,...,,, .

count,able st.ructures. 0

C o r o l l a r y . T h e nofion of wel l -order ing is not definable in L(Dn,,.. . ,n,). Hence

P r o o f . I t is well-known tha t well-ordering is not definable i n L,,,, even on

We may conclude tha t while the intermediate quantifier D(1j.2 is essential of second-order, capable as it is of defining F?, the "proper" partially ordered connectives &, , , , , , , ( I l , . . . , lk) are essentially non-second-order, a t least on countable structures.

Let us finally note tha t the partially ordered connective

where I, J , I<, L are finite, the semant,ics of which is easily adapted from the semantics of Dl,l~ is trivially first-order definable. However, if (8) is added to L,,, and I, J . I<, L are allowed to be countably infinite, a proper extension of L,,, results. This is because the well-roundedness of a binary relation R ( r , y) on (N, +, ., 0 , l ) be- comes definable. To see this, let c, denote the term of which value in (N, +, ., 0, 1) is n. Consider the sentence

where g i j h / is choosen as follows:

cg # cg if k = i and j # 1, 9il1.1 = R ( ~ 1 , c j ) if k = i + 1. { cg = co otherwise.

;This sentence says that R ( r , y) has an infinite descending chain.

372 G. SANDU AND J. VAANANEN

References

(11 BARWISE, J., Some applications of Henkin quantifiers. Israel J. Math. 25 (1976), 47-63. (21 BARWISE, J., and S. FEFERMAN, Model Theoretic Logic. Springer-Verlag, New York,

1985. [3] ENDERTON, H.B., Finite partially-ordered quantifiers. This Zeitschrift 16 (1970), 393-

397. [4] HELLA, L., Definability hierarchies of generalized quantifiers. Ann. Pure Appl. Logic 43

(1989), 000-000. [S] HENKIN, L., Some remarks on infinitely long formulas. In: Infinitistic Methods, Perga-

mon Press, Oxford-London-New York-Paris, 1961, pp. 167-183. (61 KUWCZYK, A., and M. KRYNICKI, Ehrenfeucht games for generalized quantifiers. In:

Set Theory and Hierarchy Theory, Springer Lecture Notes in Mathematics 537 (1976), 145-152.

[7] KRYNICKI, M., and A. LACHLAN, On the semantics of the Henkin quantifier. J. Symbolic Logic 44 (1979), 184-200.

181 KRYNICKI, M., and J. VAANANEN, Henkin and function quantifiers. Ann. Pure Appl. Logic 43 (1989), 273-192.

[9] WALKOE, W., Finite partially order quantification. J. Symbolic Logic 35 (1970), 535-550.

G. Sandu Department of Philosophy University of Helsinki Unioninkatu 40B 00170 Helsinki Finland J. VI in inen Department of Mathematics University of Helsinki Helsinki, Finland

(Eingegangen am 4.Oktoher 1990)

partially ordered connectives

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