canonical modal logics and ultrafilter extensions

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Canonical Modal Logics and Ultrafilter Extensions Author(s): J. F. A. K. Van Benthem Source: The Journal of Symbolic Logic, Vol. 44, No. 1 (Mar., 1979), pp. 1-8 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2273696 . Accessed: 16/06/2014 14:03 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org This content downloaded from 195.34.79.49 on Mon, 16 Jun 2014 14:03:13 PM All use subject to JSTOR Terms and Conditions

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Canonical Modal Logics and Ultrafilter ExtensionsAuthor(s): J. F. A. K. Van BenthemSource: The Journal of Symbolic Logic, Vol. 44, No. 1 (Mar., 1979), pp. 1-8Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273696 .

Accessed: 16/06/2014 14:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

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THE JOURNAL OF SYMBOLIC LoGic Volume 44, Number 1, March 1979

CANONICAL MODAL LOGICS AND ULTRAFILTER EXTENSIONS

J. F. A. K. VAN BENTHEM

?1. Introduction. In this paper the canonical modal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of an ultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable T7-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a 27-elementary class of frames is canonical.

The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in ?2. The concept of a canonical modal logic is introduced and motivated in ?3,. which also contains the above-mentioned the- orems. In ?4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant opera- tions on frames) is discussed.

?2. Background material. The modal language to be considered here has an infinite supply of proposition letters (p, q, r, ...), a propositional constant I (the so-called falsum, standing for a fixed contradiction), the usual Boolean operators

(not), V (or), A (and), -+ (if ... then ...), and +-+ (if and only if)-with and V regarded as primitives-and the two unary modal operators > (possibly) and El (necessarily)- < being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals. The semantic structures are frames, i.e. ordered couples < W, R> of a nonempty domain W with a binary relation R on W. Frames are denoted by F (= < W, R>), F1 (= < W1, R,>), etc. For a modal formula q6, a frame F with w E W and a valuation V on F taking proposition letters to subsets of W, <F, V> I= q[w] is defined accord- ing to the well-known Kripke truth definition. F l= q[w] is then defined by, for all valuations V on F, <F, V> ,= q[w]. Next, F t= 0 is defined by, for all w E W, F l= q[w]. It is obvious how these notions may be extended to the case of a set of modal formulas.

On the algebraic side, there are modal algebras M consisting of a Boolean algebra with an additional unary operation m satisfying the two laws m(O) = 0 and VxVy m(x + y) = m(x) + m(y). When written in primitive notation, modal formulas may be regarded as polynomials which can be evaluated in these algebras in an obvious way.

Received March 30, 1977.

1

?) 1979, Association for Symbolic Logic

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2 J. F. A. K. VAN BENTHEM

Any frame F (= KW, R>) corresponds to a modal algebra F+ consisting of the power set of W with the operations of set-theoretic complement and union as well as m defined by m(X) = {w E W I 3v E W(Rwv & v E X)}. Conversely, the well- known Stone representation connects modal algebras M with frames M* (= < WM, RM>) defined as follows. WM is the set of all ultrafilters on M and RM holds between two ultrafilters U and V if, for all x E V, m(x) E U. It does not hold for all M that M*+ is isomorphic to M, however, and, therefore, a new concept is needed to ob- tain a complete duality:

2.1 DEFINITION. A general frame is an ordered couple <F, YV> of a frame F (= < W, R>) and a nonempty set #' of subsets of W which is closed under comple- ments, unions and the set-theoretic operation m defined above.

The truth definition is easily extended to this case: <F, WI'> 1= 5 if, for all valua- tions V on F taking values in #' and for all w E W, <F, V> I= O[w]. Now, for any general frame <F, YV>, <F, XY>+ is the obvious corresponding modal algebra. Conversely, for any modal algebra M, the Stone representation yields a correspond- ing general frame SR(M) = <M*, #'M>-where Y'M = {{ U E WM I x E U} I X EM} -such that SR(M)+ is isomorphic to M.

For frames, this duality inspires the following operation. 2.2 DEFINITION. The ultrafilter extension ue(F) of a frame F is the frame F+*. It is easy to show that F is isomorphic to a subframe of ue(F). Moreover, for any

modal formula 5b, ue(F) I= 0 only if F I= 0. This follows from the fact that, speak- ing algebraically, the modal theory of F+ equals that of SR(F+)+, which-SR(F+)+ being a modal subalgebra of ue(F)+-contains the modal theory of the latter alge- bra.

The familiar algebraic notions of homomorphic images, subalgebras and direct products correspond to the following concepts involving general frames: generated subframes, p-morphic images and disjoint unions (respectively). Since a part of the exact correspondence will be used in the proof of the main theorem (Theorem 3.5), it is stated here. (For a full account, cf. [5].)

2.3 DEFINITION. A frame F1 is a generated subframe of F2 if (i) F1 is a subframe of F2, (ii) for all w E W1, v E W2, R2 wv only if v E W1. A general frame <F1, Y/l> is a generated subframe of <F2, YV2> if (i) and (ii) hold

as well as (iii) v1 = {x nW1 I XEV2}. If g is a homomorphism from the modal algebra M1 onto M2, then the function

f defined by f(U) = ge1 [U] is an isomorphism from SR(M2) onto a generated subframe of SR(M1).

2.4 DEFINITION. A function f from a frame F1 onto F2 is a p-morphism if (i) for all w, v E W1, R1 wv only if R2 f(w) f(v), (ii) for all w E W1, v E W2, R2 f (w)v only if, for some u E W1, Rwu andf(u) v. f is a p-morphism from the general frame <F1, Yr1> onto <F2, Yr2> if (i) and (ii)

hold as well as (iii) for all X E W2, f-1 [X] E W1. If the modal algebra M1 is a subalgebra of M2, then the function f defined by

f(U) = U n M1 (for all ultrafilters U on M2) is a p-morphism from SR(M2) onto SR(M1).

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CANONICAL MODAL LOGICS AND ULTRAFILTER EXTENSIONS 3

2.5 DEFINITION. For a setfFi I i E I of frames, the disjoint union ?{Fj I i E I} is the frame < W, R> with

(i) W ={< i, w> I i EI, we Wil, (ii) R = {<<i, w>, <i, v>> I i E I, <w, v> E Ri}. For a set {<Fi, Wi> I i E I} of general frames, the disjoint union (?{<Fi, WYf> I

i E I} is <F, W/>, where F = <W, R> asaboveand (iii) Y = { X c W I Vi E I fw E WI <i, w> E X} E Yi}. For any set {<Fi, #"i> I i E I} of general frames, (? {<Fi, I i> I i e 1})+ is iso-

morphic to fli <Fi, Wi>+- General frames also play an important role in the modal completeness theory, as

will appear below. For a set T of modal formulas, define ML(F) (the modal logic axiomatized by T) as the smallest set of modal formulas containing T, all proposi- tional tautologies and all formulas of the form K(qs v b) - (> qs v K sb), which is closed under modus ponens, replacement of equivalents, prefixing of C] ("neces- sitation") and the formation of substitution instances. For a while, it had been hoped that the following equivalence could be proven: qs E ML(F) if and only if T t= q, where 1= is the natural notion of semantic consequence defined by T F= 0 if, for all frames F, F ,= T only if F ,= 0. Clearly, qs E ML(F) only if F 1= 5. But, K. Fine and S. K. Thomason have given examples of F and qs such that F 1= s and s 0 ML(F) (cf. [3] and [8]). Call a set F of modal formulas complete if, for all modal formulas s, T 1= qs only if qs E ML(F). This notion will be used in ?3. The general completeness theorem which does hold is the following, however:

5 E ML(F) if and only if 1 k=g sb,

where k=g is defined by F k=g qs if, for all general frames <F, V>, <F, V> I= F only if <F, V> I= 0. This theorem is proven by the familiar Henkin construction of a general model <F, V> (F = < W, R>), where W is the set of all maximally consistent-with respect to ML(F)-sets of modal formulas, R holds between w and v if, for all modal formulas qs, qs E v only if 0 K E w, and, finally, Y is the set of all sets of the form {w E W 1 qs E w} for a modal formula sb.

The duality between general frames and modal algebras was used by R. 1. Gold- blatt and S. K. Thomason to characterize the modally definable classes of frames, i.e. the classes of the form {F I F t= T} for some set F of modal formulas. This general result [6, Theorem 3] uses a rather ad hoc notion, however, viz. that of a "state of affairs frame". A more elegant result is Theorem 8 in the same paper, characterizing the modally definable 2z-elementary classes of frames (which is proven as Corollary 3.11 below). Now, if a class _X" of frames is modally definable at all, it is defined by the complete set {qs I VF e -X: F l= 0}. Moreover, Corollary 3.10 below (which is our version of Theorem 2 of K. Fine [4]) says that any F which is complete and preserved under elementary equivalence (i.e. F defines a 27- elementary class of frames), is canonical in the sense of Definition 3.3. It follows that the main result of this paper (Theorem 3.5), which characterizes the canonically definable classes of frames, may be viewed as an extension of the above-mentioned result from [6] characterizing the canonically definable 7z-elementary classes of frames.

?3. Descriptive frames and canonical sets of formulas. K. Fine characterized

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4 J. F. A. K. VAN BENTHEM

modal Henkin models in [4] by means of three properties: "tightness", "1-satu- ratedness" and "2-saturatedness". In a more convenient form, these notions were used by R. I. Goldblatt in [5]:

3.1 DEFINITION. A descriptive frame is a general frame <F, WY> satisfying (i) Vxe-yWVyEW(VXeYf((yeX--xeX)--x =y),

(ii) Vx E W Vy E W (VXe Y (y E X -x E m(X)) -> Rxy) and (iii) any ultrafilter on 1f has a nonempty intersection. Goldblatt used descriptive frames because they are fixed points of the Stone

representation: 3.2 LEMMA. For any general frame <F, Y >, <F, IYt> SR(<F, WY>+) if and only

if <F, WM> is descriptive. (The connection with Henkin models is not surprising. A Henkin type complete-

ness theorem is nothing more-speaking algebraically, that is-than a combination of a simple Lindenbaum completeness theorem and a representation of the verify- ing algebra as an algebra of sets.)

There exist frames F such that, for no V, <F, Yf> is descriptive. An example is provided by the positive integers with the "greater than" ordering. To see this, note that, for any frame F, there exists a set #min of subsets of Wsuch that, for any general frame <F, )V>, W min C a'- WSmin = {{w e W I F :- q [w] } I q6 is a modal formula in which no proposition letters occur), i.e. it consists of the subsets of W definable by means of closed modal formulas. Clearly, <F, Wmin> is a general frame itself. Now, for the particular case of the positive integers, this means that any general frame on them will contain, for each n> 1, the set Xn = {w E W I F k= o *. "(n times ...K ' I [w]} ( = {k I k ? n + 1 }). {Xn I n > 1} has the finite intersection propertybut its intersection is empty, whence condition (iii) of Definition 2.1 can never be satisfied. On the other hand, for any general frame <F, #,> SR(< F, 1>+) is a descriptive frame with the same modal theory.

The next definition introduces the notion of "canonicity" (inspired by K. Fine [4]), which is called "d-persistence" by Goldblatt.

3.3 DEFINITION. A set T of modal formulas is canonical if, for all descriptive frames <F, MY>, <F, ir> I= T only if F t= P.

Thus, canonicity consists in being preserved in passing from descriptive frames to the underlying frames. (The notion in [4] is not exactly the same, being formu- lated in terms of Henkin frames. Fine calls a set ML(F) "canonical" if it holds on the frame of its Henkin model, regardless of the cardinality of the set of proposition letters. Because Henkin models may be regarded as descriptive frames, canonicity in our sense implies canonicity in Fine's sense. It is not obvious if the converse holds. But, a natural extension of Fine's definition does yield an equivalent to our notion. Call ML(F) generally canonical if it holds on the frame of the Henkin model for any modal logic containing ML(F), regardless of the cardinality of the set of proposition letters. Again, canonicity in our sense implies general canonicity. Moreover, a simple proof establishes the converse implication. If <F, i'> is a descriptive frame on which ML(F) holds, then, for a suitable choice of proposition letters, <F, *r> is isomorphic to the Henkin model for {q6 1 <F, YF> 1= q6}, which is a modal logic containing ML(F).)

Canonicity implies completeness. For, if T is canonical and q6 ? ML(F), then q6 is falsified in a descriptive frame (obtained through the Henkin construction) on

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CANONICAL MODAL LOGICS AND ULTRAFILTER EXTENSIONS 5

which T holds, whence 0 is falsified on the underlying frame on which T holds as well. Examples of canonical sets are provided in abundance by Fine's result, for- mulated below as Corollary 3.10, that any complete T which is preserved under elementary equivalence is canonical. But not every canonical T is preserved under elementary equivalence: Fine's counterexample is {<>L0 p - O(Clp A (Clq V L] - q))}.

The following preservation property is needed for the proof of the main theorem. 3.4 LEMMA. Any canonical set of modal formulas is preserved under ultrafilter

extensions. PROOF. Let P be canonical and let F l= T. It follows from the Stone construction

that SR (F+) (= <ue (F), WI"F+>) I= T. By Lemma 3.2, SR (F+) is a descriptive frame, whence ue(F) I= F. QED

Lemma 3.4 implies that not every complete set is canonical. For, consider L1(ELp -U p) -O l p. This modal formula holds on the positive integers with the "greater than" ordering, but it does not hold on the ultrafilter extension of this frame. To see this, note that, for any free ultrafilter U on this frame, RF+

UU. This implies that our modal formula cannot hold on ue(F); for, any frame on which it holds is irreflexive. So, {E](ELp -- p) -- Elp} is not preserved under ultrafilter extensions, and, therefore, it is not canonical. It is complete, however, as was shown in K. Segerberg [7].-

The main theorem of this paper characterizes the canonically definable classes of frames:

3.5 THEOREM. A class X- of frames is of the form {F I F l= F} for a canonical set T of modal formulas if and only if it is closed under generated subframes, p-morphic images and disjoint unions, while both )C and its complement are closed under ultrafilter extensions.

PROOF. If a class of frames is modally definable, then it is closed under generated subframes, p-morphic images and disjoint unions. These are well-known facts. It follows from the remark following Definition 2.2 that the complement of a modally definable class is closed under ultrafilter extensions. If the class is, in addition, canonically definable, then it will be closed under ultrafilter extensions itself, by Lemma 3.4 This proves the implication from left to right.

Next, let .' and its complement be closed under the operations described above. First, we show that X' is modally definable by applying Birkhoff's theorem (char- acterizing equational classes of algebras) and the duality results of ?2. Consider .+ = {F+ I Fe r}. By Birkhoff's theorem, the smallest equational class con- taining )+ is HOM SUB PROD (,+), where HOM, SUB and PROD are the ob- vious operations. Let T+ be the modal theory of this equational class (expressed in algebraic notation) and T the corresponding set of modal formulas. We will show that, for any frame F, if F+ E HOM SUB PROD (X'+), then F E A3. From this it follows immediately that )C is {F I F # F}.

Let F+ E HOM SUB PROD (,'+), i.e. F+ is a homomorphic image of a subal- gebra of a product of algebras in ,+. Now this product is isomorphic to an algebra F1j in ,+, by the result following Definition 2.5 and )h's being closed under dis- joint unions. It follows that F? is a homomorphic image of some subalgebra of F{. This subalgebra is of the form <F1, IV>+, for some *1 such that <F1, 1> is a general frame. Letf be a homomorphism from <F1, *1,>+ onto F+. We have:

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6 J. F. A. K. VAN BENTHEM

F,+ <F,, *'i>+ f F+

This diagram may be expanded, using the duality results of ?2, to

SR(F+,) 9 SR(<Fl, l l>+) < h SR(F+)

F,+ <F,, 'W j>+ f ,F+

where g is a p-morphism from ue(F1) onto <F1, -#r>+* and h is an isomorphism from ue(F) onto a generated subframe of <F,, *1>+*. Now F1 E Xf, so ue(F1) E Y'

(," is closed under ultrafilter extensions) and, therefore, <F1, 4,X>+* E )C (-X" is closed under p-morphic images), whence ue(F) E X, being isomorphic to a gen- erated subframe of <F1, #l>+* (.,C is closed under generated subframes, and isomorphisms are p-morphisms). But, then, F e a", because the complement of )r is closed under ultrafilter extensions.

It remains to be shown that the set T defining )is canonical. To see this, let <F, YF> be a descriptive frame such that <F, YF> t F. Again, a diagram may be drawn like above:

SR(F+t SR(<F, W>+) < h SR(<F, # Y>+)

F+ - ~<F J,, f <F. f >+

The same argument establishes that <F, Y>+* e S. It follows from Lemma 3.2 that <F, W>+* is isomorphic to F, whence F is in C and F t- F. QED

3.6 COROLLARY. A complete set F is canonical if and only if it is preserved under ultrafilter extensions.

PROOF. It suffices to prove the direction from right to left. If F is preserved under ultrafilter extensions, then {F I F t= F} satisfies the closure conditions of Theorem 3.5 and is, therefore, of the form {F I F t= F} for some canonical F. It follows from the proof of 3.5 that F' may be taken to be {g5 I VF(F --F r= F # ), whence F C F. On the other hand, T' c ML(F ). For, otherwise, for some 0 E F', 0 ? ML(F). By F's being complete, there exists a frame F such that F kl F

and F s 0. But, then, F t F' and F Wq 0, a contradiction. It follows that F is canonical: if <F, *-> is descriptive and <F, YF> I= F, then <F, WI> t ML(F), and so <F, WI'> t FT, whence F - F' and F r- F. QED

The above proof depends essentially on the assumption that F is complete. For the general case it only establishes

3.7 COROLLARY. A set F is preserved under ultrafilter extensions if and only if the class offrames on which it holds is defined by a canonical set of modalformulas.

S. K. Thomason's incomplete modal logic L(cf. [8]) shows that 3.7 cannot be improved upon. L holds on exactly the same frames as the canonical logic S4 (that S4 is canonical follows from its being first-order definable and complete), so it is preserved under ultrafilter extensions. But, L is not canonical, being in- complete. In other words, the converse of Lemma 3.4 fails.

Apart from canonical sets of formulas, there are other interesting examples of preservation under ultrafilter extensions, e.g. all sets of modal formulas defining a 2J-elementary class of frames. This follows from

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CANONICAL MODAL LOGICS AND ULTRAFILTER EXTENSIONS 7

3.8 LEMMA. For any frame F, ue(F) is a p-morphic image of some F' elementarily equivalent to F.

PROOF. Let F = < W, R>. Add, for each X c ' W, a unique unary predicate con- stant cx to the first-order language with identity and one binary predicate constant R. Expand Fto a structure F1 for this language in the obvious way. Using a familiar model-theoretic construction, take an elementary extension M = <F', Cx>xcw of F1 which is saturated with respect to sets of formulas containing at most one para- meter from W'. (Note that F' is elementarily equivalent to F.) The function f defined for w e W' by f(w) = { X c W I w e c[ } is a p-morphism from F' onto ue(F), as will be shown now.

(i) f is well-defined, for f(w) is an ultrafilter on F, in view of the equivalences Vz (-1cXz -+ cw-xz) and Vz ((cxz A cyz) +-? cxnyz), which are true in F1 and, therefore, in M.

(ii) f is onto, since any ultrafilter U on F corresponds to the finitely satisfiable set {cXz I X e U} on M, which is satisfied by some w e W', because M is saturated with respect to such sets.

(iii) If w, v e W' and R'wv, then RF+ f (W) f (V); for, it suffices to show that, if v e c , then w e cm'(X), and this follows from the truth in F1 (and, therefore, in M) of Vy VZ((cxz A Ryz) -cm(x)y)

(iv) If RF+ f (W)U, then a v e W' may be found such that R'wv and f(v) = U. Consider 2 = {cXz I Xe U} U {Rwz}. This set is finitely satisfiable in M; for, if X1, ..., X, e U, then {cXZ ..., CXz, Rwz} is satisfiable in M. (To see this, note

3that X = .1 n .. n Xk e U. By the definition of RF?, m(X) ef(w), so W e C'(X). Then use the truth in F1 (and M) of Vy(cm(x)y -- 3z(Ryz A CXZ)). Since M is saturated with respect to sets like 2 , a v as described exists. QED

The construction in the preceding proof is due to K. Fine (cf. [4]). 3.9 COROLLARY. If a class of frames is closed under elementary equivalence and

p-morphic images, then it is closed under ultrafilter extensions. The converse of 3.9 fails: Fine's formula mentioned above yields a counter-

example. 3.10 COROLLARY (K. FINE). If r is complete and preserved under elementary

equivalence, then it is canonical. PROOF. Any set F of modal formulas is preserved under p-morphic images, so,

if F is preserved under elementary equivalence, Corollary 3.9 applies and F is preserved under ultrafilter extensions. F's being complete as well makes Corollary 3.6 applicable: F is canonical. QED

3.11 COROLLARY (R. I. GOLDBLATT AND S. K. THOMASON). If a class offrames is closed under elementary equivalence, then it is modally definable if and only if it is closed under generated subframes, p-morphic images and disjoint unions, while its complement is closed under ultrafilter extensions.

PROOF. One direction is obvious, so it suffices to prove the direction from right to left. Let )C be any class with the mentioned closure properties. By Lemma 3.8, it is closed under ultrafilter extensions, so Theorem 3.5 applies, which means that ,) is definable by a (canonical) set of modal formulas. QED

Finally, it should be remarked that any modally definable class which is closed under elementary equivalence is (not only 2J-elementary, but also) A-elementary. This follows from a result in van Benthem [2]:

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8 J. F. A. K. VAN BENTHEM

3.12 LEMMA. If a class offrames is closed under elementary equivalence, disjoint unions and generated subframes, then it is closed under ultraproducts.

?4. Preservation results. This section is devoted to the subject of first-order definable relational properties and ultrafilter extensions. Clearly, not all first-order sentences are preserved under ultrafilter extensions. E.g., Vx -'Rxx is not (cf. the above example involving L (El p -- p) - lip). Which first-order sentences are preserved under ultrafilter extensions? This general question has not been solved yet, but there are a few relevant results in van Benthem [1]:

4.1 DEFINITION. Let u be any individual variable. A u-formula is any formula obtained from atomic formulas of the forms Rux, Rxu, U = x, x = u (x different from u) and atomic formulas in which u does not occur, by applying the Boolean operations and using quantification taking formulas 0(u) to 3y(Ruy A [y/u] ) or 3y(Ryu A [y/uIoq, where y does not occur in 0(u).

4.2 LEMMA. For any frame F, any u-formula 0 = 0(u, x1, ..., Xk) and any w1, wk E Wand ultrafilter Uon F, ue(F) - 0 [U, w4, ..., w*] if and only if {v E WI Ft 0[V, W1, ..., Wk]} E U; where w*(1 < i < k) is the ultrafilter {X c W I wi E X}.

It follows from 4.2 that, for any u-formula 0 = 0 (u, xI, ..., Xk) and any w, W, ..., I E W, F* - 0[w*, w , w*] if and only if F # 0 [w, w1, ..., wk]. So, existential closures of u-formulas are preserved under ultrafilter extensions. This does not exhaust the class of such formulas, however, as is shown by examples like VxRxx or VxVyRxy, which are also preserved under ultrafilter extensions.

For the other three notions used in Theorem 3.5, a general preservation result was proved in the same work:

4.3 THEOREM. A first-order sentence is preserved under generated subframes, p-morphic images and disjoint unions if and only if it is equivalent to a sentence of the form Vxq, where ? = 0(x) is a formula constructed from atomic formulas of the form Rzy and the falsum I using A, V and restricted quantification of the forms 3y(Rzy A and Vy(Rzy -- (with y distinct from z).

REFERENCES

[1] J. F. A. K. VAN BENTHEM, Modal correspondence theory, Dissertation, Amsterdam, 1976. [2] , Modal formulas are either elementary or not IJ-elementary, this JOURNAL, vol. 41

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