Classical Logics « Everywhere

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« Everywhere » and « here » Valentin Shehtman

a

a

Institute of Information Transformation Problems , B. Karetny 19, 0144 , Moscow , Russia E-mail: Published online: 30 May 2012.

To cite this article: Valentin Shehtman (1999) « Everywhere » and « here », Journal of Applied Non-Classical Logics, 9:2-3, 369-379, DOI: 10.1080/11663081.1999.10510972 To link to this article: http://dx.doi.org/10.1080/11663081.1999.10510972

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« Everywhere » and « here »

1

Valentin Shehtman

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Institute of Information Transformation Problems, B. Karetny /9, 0144 Moscow, Russia shehtman@ 1pes. math. msu. ru

ABSTRACT. The paper studies propositional logics in a bimodal language, in which the first modality is interpreted as the local truth, and the second as tl1e universal truth. The logic S4UC is introduced, which is finitely axiomatizable, has the f.m.p. and is determined by every connected separable metric space.

This paper studies some modal logics with universal modality. A systematic research of this area of modal logic was started by papers of Bulgarian logicians in the 80s (cf. [1], [2]). Soon it became clear that the universal modality essentially increases the expressive power of a modal language. We consider the universal modality in the context of topological (neighbourhood) semantics. It is well-known [3) that in neighbourhood semantics a necessity operator D is understood as "locally true": a formula A is true at a point x iff A is true within some neighbourhood of x. In other words, a witness at x can say: "A is necessary true here" if all witnesses rather close to him can confirm that A is true. Having this interpretation in mind, it is natural to enrich the language also with the universal modality V interpreted as "everywhere". Then we get a language telling us about properties of topological spaces, which is more expressive than the language with D alone. This idea was suggested in [11); some results of the present paper were also stated there. Let us pass to formalities now. We consider propositional formulas built from a countable set P L {p, q, .. .} of proposition letters, classical connectives (:>, 1..) and monadic modal operators D, V . We will use also derived connectives, classical (t\, V, -,, T, :=) and modal (, 3). We recall that

=

A= -,0-,A, 3A = -,'t/-,A. A (normal ) bimodal logic (or merely, a logic ) is a set of formulas closed

under the rules of Substitution, Modus Ponens, D- introduction (A/DA), V -introduction (A/VA), containing all classical propositional tautologies and the formulas 1 The work on this paper was supported by the Russian Foundation for Basic Research (project No.96-0l-01378)

Journal of Applied Non-Classical Logics. Volume 9- no 2-3/1999, pages 369 to 379

370

Journal of Applied Non-Classical Logics. Volume 9- no 2-3/1999 D(p :::> q) :::> (Dp :::> Dq),

'V(p :::> q) :::> ('Vp :J 'Vq).

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K2 denotes the minimal bimodal logic. The smallest logic containing a given logic L and a set of formulas r is denoted by L + f; for a formula A, L +A is an abbreviation for L + {A}. We say that the logic K 2 + r is axiomatized by the set r. In this paper the following logics will be often used:

S4 * S5

= K2 + {Dp :J p, Dp :J DDp, 'Vp :J p, 'Vp :J Wp, 3'Vp :J p}, S4U = S4 * S5 + 'Vp :J Dp.

As Kripke semantics for modal logics is well-known, we recall corresponding notions and terminology very briefly. A {bimodal) Kripke frame is a triple F (W, R 1 , R2), such that W # 0 ; R1, R2 ~ W x W. Elements of W are called worlds (or points ) . A Kripke model over the frame F is a pair (F, F) iff f is surjective and

finite The be a from

Equivalently, we can say that f is a c-p-morphism iff .f is an interior map from X onto Top( F). LEMMA 15. (P-morphism Lemma). IfF is a finite generated Kripke frame, X is a topological space and f: X ~ > F then L(X) ~ L(F). PROOF. Let M A( F) be the modal algebra of the frame F = (W, R). Consider the map f" : U .-+ J- 1 (U). Since f is surjective, !" is a Boolean embedding 2w ~ 2x [9]. Since f is a c-p-morphism and F is finite, it. follows also that f" preserves the closure operation. In fact, f*(OU) = f*(R- 1 (U)) =

r

1

(R- 1 (U))

= r 1 ( U R- 1 (w)) = wEU

Ur wEU

1

(R- 1 (w)) =

U cr

1

(w) =

cr 1 (U)(since

U is finite)=

c.f*(U).

wEU

Thus the algebra M A( F) is isomorphic to a subalgebra in the interior algebra of X, and we have that L(X) ~ L(MA(F)) = L(F) . • We will use the following notation. If X is a topological space, Y C X, F is a Kripke frame, a E F, we write: f: (X, Y) ~ > (F, a) to indicate that there exists f : X ~ > F such that f(Y) = {a}. The next statement is a slightly generalized version of l'vlcKinsey- Tarski's Lemma on dissectability from [10]. LEMMA 16. Assume that X is a connected dense separable metric space, Y C X is a closed rare subset, F is a finite quasi-tree, a belongs to the least cluster in F. Then f: (X, Y) ~ > (F, a). PROOF. Repeats the standard one (cf. [9]), with a little modification. It is carried out by induction on the height of F. The essential case is when F is of the height 2. Then F is of the following structure:

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Journal of Applied Non-Classical Logics. Volume 9 -·no 2-3/1999

Figure 2 In this case the Lemma means that there exists a splitting r

X=

UG; U UBj

i=l

j=O

such that all G; are open, and for any i, j

•

cG; - G;

= cBj = UBj, j=O

andY~

Bo. Without the latter requirement this is exactly the splitting property. To prove our statement, take the inductive construction from [9]. This construction yields Bj as U Bjn. The only change we make is the following: for the

= =

n>O

=

=

case j n 0 we put Boo Y instead of B 00 0 . The whole argument still works because the set Y is rare . • Now we make the final step in our neighbourhood completeness proof for S4UC. LEMMA 17. Assume that X is a connected dense separable metric space, Y C X is a closed rare subset, F is a finite quasi-park, a belongs to a minimal cluster in F. Then f: (X, Y)--+ > (F, a). PROOF. By induction. If F is a quasi-tree, the statement follows from Lemma 16. Otherwise, let F = (W, R). By definition of a quasi-park, a can be included into a sequence u1, ... , Un ( n 2: 2) of representatives of distinct minimal clusters in F, such that for any i,j E {l, ... ,n}, R(ui)nR(uj)#0 {::}

=

li-jl:::;l.

=

We may assume that a u;, i < n (if i n, put. the same sequence backwards). Let F1, F2, F3 be the restrictions ofF respectively to H/1 = R({u1, ... ,u;}), W2 R({tti+l····,un}), W3 W1 n W2. Then F 1 ,F2 are

=

=

«Everywhere>> and «Here>>

377

quasi-parks, F3 is a quasi-tree. Let b = u;+l, and choose some d such that = R(d). Since every metric space is normal, there exist open subsets X 1, X 2 in X such that Xt 2 Y, X2 # 0, Xt n X 2 = 0 and Xt, X2 are "regular", 1.e. icX1 = X1, icX2 = X 2 . Then let

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w3

By the induction hypothesis and McKinsey - Tarski's Lemma, there exist. c-pmorphisms ft : (eX 1, Y U Yt) -+ > (F1 , a),

h : (cX2, Y2)-+ > (F2, b),

h: x3 -+>

F3.

Figure 3

=

Then the combined map f ft U h U fs is a c-p-morphism. In fact, one can easily see that f is open, so it suffices to show that f- 1(R(e)) is open for any e. To show this, consider three cases. (i) a,b rf: R(e). Then /- 1 (R(e)) /1 1 (R(e)) U /2 1 (R(e)) U /3 1 (R(e)), 1 which is open because /;- (R(e)) is open in eX; (due to the continuity of/;}, f;- 1 (R(e)) ~X; and X; is open. (ii) R(e) = R(a). Then

=

r

1

(R(e))

=r

1

(R(a))

= /1 1 (R(a)) U f:2 1 (W3) U .\'3.

This is again an open set. In fact, /:; 1 (W3 ) is open since it. is open in c.\' 2 and is included in X 2 . It remains to show that f1 1(R(a)) ~ if- 1(R(a)), and for

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Journal of Applied Non-Classical Logics. Volume 9 - no 2-3/1999

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the latter it is sufficient to show Y1 ~ if- 1 (R(a)). So take an arbitrary x E Y1 and show that some open U ~ f- 1 (R(a)) contains x. But this follows from the continuity of fi and h, because x has open neighbourhoods U1 , U3 such that

and so we can take U = U1 n U3- cX2. (iii) R(e)=R(b). Then the argument is the same as in (ii). • THEOREM 18. The logic S4UC is determined by any connected dense separable metric space. PROOF. Let X be a space of this kind. Then S4UC ~ L(X) by Lemma 1 and Lemma 8. For the converse, assume A t/: S4UC; then F it= A for some quasi-park F (by Theorem 14). Then by Lemma 17 and Lemma 15 we obtain that A t/: L(X) . • References [1] GARGOV G., PASSY S., TINCHEV T. Modal environment for Boolean speculations. In: "Mathematical logic and its applications (Druzhba, 1986)", 253-263 .. Plenum Press,l987. [2] GORANKO V., PASSY S. Using the universal modality: gains and questions. Journal of Logic and Computation v.2 (1992), 5-30. [3] SCOTT D. Advice on modal logic. In: "Philosophical Problems in Logic. Some Recent Developments", ed. K. Lambert. Reidel, 1970. pp. 14:3-172. [4] CHAGROV A.V., ZAKHARYASCHEV M.V. Modal logic. Oxford University Press, 1996. [5] BOURBAKI N. General topology, parts I, II. Addison-Wesley, 1966. [6] GERSON N. An extension ofS4 complete for the neighbourhood semantics but incomplete for the relational semantics. Studia Logica, v. 34 (1975), 333-342. [7] SHEHTMAN V.B. Topological models of propositional logics. Semiotika i informatika, No.15 (1980), 74-98 (in Russian). [8] VAN BENTHEM J. The logic of time. Reidel, 1983. [9] RASIOWA H., SIKORSKI R. The mathematics of metamathematics. Warsaw, 1963. [10] MCKINSEY J., TARSKI A. The algebra of topology. Annals of Mathematics, v. 45 (1944), 141-191.

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[11] SHEHTMAN V.B. Modal logic with connectives "Everywhere'' and "Here". In: Modal and Intensional Logics. Abstracts of the 8thA!l-Union Conference in Logic and Methodology of Science, Vilnius,l982., p.124126.(In Russian)