a note on modal formulae and relational properties

A Note on Modal Formulae and Relational PropertiesAuthor(s): J. F. A. K. van BenthemSource: The Journal of Symbolic Logic, Vol. 40, No. 1 (Mar., 1975), pp. 55-58Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272270 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 40, Number 1, March 1975

A NOTE ON MODAL FORMULAE AND RELATIONAL PROPERTIES

J. F. A. K. VAN BENTHEM

Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators L- and 0. The semantic structures are frames, i.e., pairs < W, R> with R C W2. Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W. Then the relation

w k a(in <F, V>), we W,

may be defined, for arbitrary formulae ar, following the Kripke truth-definition. From this relation we may further define

F k 4fw] (VV)(w l a (in <F, V))),

F k a (Vw)wew (F a[w]).

Now, to every modal formula a there corresponds some property Pa, of R. A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first- order predicate. For these particular Pa we have

FIka[w] <?FIkPjw]

for all F and w E W. These formulae P. are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when P,, may be taken to be some first-order formula. For example, it can be seen that

F 1= (Dp --O n Ljp)[w] -_ F k (Vy)(Rxy -- (Vz)(Ryz -- Rxz))[w]

for all F and w E W. It is customary to talk about a related correspondence, namely when for all F we have

F k F k P,.

Note that this correspondence holds whenever the first one above holds. The main purpose of this note is to prove THEOREM 1. There is no first-order formula b such that

Flk Fk ELYp-F O Ep for all F.

Received November 20, 1973. 55

? 1975, Association for Symbolic Logic



56 J. F. A. K. VAN BENTHEM

PROOF. Suppose such a formula b does exist: we shall deduce a contradiction. Consider the frame F = < W, R> where

W = {q} U {q. I n E co} U {qn,i I n e co, i ce 2 = {O, 1}} u {rf I ft 20},

R = {<q, qn> I n c co} U {<q, rf> I f E 2w} U {<qn, qn,i> | n E co, i E 2} U<qni qni> I n e w, i e 2} u U {<rf, qn,f(n)> |n c co}.

LEMMA A. F Fop L --> 1p. PROOF. It is easy to see that F h (Flopl - p)[w] for all w e W - {q}; this

hinges on the fact that for all n E cw, i E 2:

qn,i p - qn, i l= OP qn,j 1 Fijp.

Now, suppose that q k lop (in <F, V>) for some V. Then there is an fe 2w such that qn,f(n) k p for all n E co. But then rf k Up and so q k OEp. Q.E.D.

It follows immediately from Lemma A that F I 0. Hence, by the Ldwenheim- Skolem theorem there is a countable elementary substructure F' = < W', R'> of F such that q E W' and qn, qn,O, qn,1 E W' for all n E cu.

LEMMA B. F' V (FIOp --. ' Fjp) [q]. PROOF. Since W is uncountable and W' is countable, we can pick an element rg

of W - W'. Define

V(p) = {qn,g(n) I n o 4}

First, we claim that q k Elop (in <F', V>). It is easy to see that qn k Op. In order to show that rf I Op, proceed as follows. For any f e 20, define -fan) = 1 - f(n) for all n E w. Then if rf E W' it follows that raf e W'. (This may be seen by ex- hibiting a first-order formula which forces it to be true (in F and so in F'). For example, let A1(x) express: Rqx and x has exactly two R-successors; A2(x) express: Rqx and not A1(x). Then take

(Vx)(A2(x) -(y)(A2(y) & (Vz)(A1(z) -? (Vu)((Rzu & Rxu) - Ryu)))).)

Hence, if rf E W' then f =A - g because -j-g = g and rg 0 W'. Therefore,f(k) =

g(k) for some k and so rf k Op because Rrfqkf(k). This completes the proof of our first claim.

Secondly, we claim that q V FLP (in <F', V>). For, Rqnqn, g(n) and so qn = A for all n E a). Also, if rf E W' then f =A g and so f(k) =# g(k) for some k Ec w. Since Rrfqk,f(k) we deduce that rf k )- p. This completes the proof of the second claim and hence the lemma. Q.E.D.

Finally, it follows immediately from the second lemma that F' V b. This contradiction proves the theorem. Q.E.D.

In order to place the main theorem above in perspective, we conclude the paper with a positive result which we state without proof. Let ~0p = 00p = p and F]n+1p = D-np, onfl+p = 0onp for all n fl w.

THEOREM 2. For every modal formula b of the form



MODAL FORMULAE AND RELATIONAL PROPERTIES 57

where M1, ** X Mn, are modal operators (i.e., [a or 0), there exists afirst-orderformula +* (in R and =) such that

F k q*[w] Fq F ![w]

for all F and w e W. (Our convention is that if n = 0 then <M1,**, Mn> is empty.) Although we shall

not prove this result, we shall describe how to obtain +* from b. To do this, we need some more notation.

Let Q(O) = 3, Q(E1) = V, C() =&, C(D)=-. If u = <ul,*, u> is an n-tuple of variables, define Q(M1. Mn, u, v) to be (i) empty if <M1,**, Mn> is

empty, (ii) (Q(M1)ul)(RvuC(Ml)) if n = 1 and (iii)

(Q(Ml)ul)(RvuC(M1)) ... (Q(M.) u.)(Ru. _ 1unC(M.))

if n > 1. Also define ROxy = "x = y"; R'xy = Rxy; R2xy = (3z)(Rxz & Rzy), etc.

Now let y = <Y1, Yk>; Z = <zl, X*,XZ>. Let v be x if k = 0 and Akotherwise; let w be x if <M1, , M,> is empty and zn otherwise. Finally, define

= Q(Dk, y, X)Q(((M ... M., z, x)Rlvw) *. )

REMARK. Following a suggestion from the referee we have discovered that a more general result than Theorem 2 is contained in a paper by H. Sahlqvist: Completeness and correspondence in the first and second order semantics for modal logic, which is to appear in the Proceedings of the Third Scandinavian Logic Sym- posium, Uppsala 1973, North-Holland, Amsterdam.

REMARKS (added in proof, July 1974). 1. In the proof of Theorem 1 it is actually sufficient to take any countably

infinite elementary substructure of F. For every such structure will contain q and infinitely many q,'s. A counterexample for 00p -0- ODp can be found as before. So we have shown the Ldwenheim-Skolem theorem fails for modal logic in the following sense: There exists an uncountable frame with no countably infinite elementary subframe satisfying the same modal formulas.

Of course we were using a hybrid formulation since "elementary substructure" was taken in its predicate-logical sense. If we try to define more purely modal notions, however, the situation becomes rather trivial. E.g.,

(F1 = <W1, R1>, F2 = <W2, R2>).

Define F1 'm F2 by (i) W1 , W2; (ii) R1 = R2 n (W1 x W1); (iii) for all modal

0, w E W1, valuations V(V1 = V [ W1): <F1, V1> k f[w] iff <F2, V> k f[w]. It turns out that F1 '-m F2 iff (i) F, ' F2 and (ii) for all w Ec W1, v Ec W2, R2wv: v E W1.

It is obvious now how the Lbwenheim-Skolem property fails with respect to this notion of elementary substructure.

2. Yet another modification of the proof of Theorem 1 enables us to prove that

Dop -- OlDp has no first-order equivalent on countable frames. For this one needs a set S of first-order formulas describing a point like q with R-successors of two kinds. (Those of the first kind have exactly two R-successors, those of the second kind share exactly one R-successor with every point of the first kind; also some



58 J. F. A. K. VAN BENTHEM

additional requirements should be included.) Add formulas requiring the existence of n different points of the first kind for every n. Also add the purported first-order equivalent. It is clear that S is finitely satisfiable, so it should be satisfiable in a countably infinite domain. But a contradiction can be obtained through a counterexample like before.

On the other hand it is easy to see that for all transitive frames F, w E W: F I loYp --* OLp[w] *-4 F k (3y)(Rxy A (Vz)(RyZ -- z = y))[w].

INSTITUUT VOOR GRONDSLAGENONDERZOEK, UNIVERSITEIT VAN AMSTERDAM

AMSTERDAM, THE NETHERLANDS



a note on modal formulae and relational properties

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