“Everywhere” and “Here”.
“Everywhere” and “Here”.
复制标题
“到处”和“这里”。
DOI：
10.1080/11663081.1999.10510972
复制
发表时间：
1999
影响因子：
5.6
通讯作者：
V. Shehtman
中科院分区：
文献类型：
--
作者：
V. Shehtman
本文用双峰语言研究命题逻辑，其中第一模态被解释为局部真，第二模态被解释为普遍真。引入了可公理化的、具有f. m. p.的、由每个连通可分度量空间确定的逻辑S4 UC。本文研究了具有泛模态的模态逻辑。对模态逻辑这一领域的系统研究始于80年代保加利亚逻辑学家的论文（参见：[1]，[2]）。很快，人们就清楚地认识到，普遍情态本质上增加了情态语言的表达能力。我们认为，在拓扑（邻里）语义的背景下的普遍模态。众所周知[3]，在邻域语义学中，必然性算子D被理解为“局部真”：公式A在点x处为真当且仅当A在x的某个邻域内为真。换句话说，在x点的见证人可以说：“A在这里必然为真”，如果所有与他相当接近的见证人都能证实A为真。考虑到这一解释，自然也要用被解释为“到处”的普遍方式V来丰富这一语言。这样我们就得到了一种告诉我们拓扑空间性质的语言，这种语言比单独使用D的语言更有表现力。这一思想在[11]中提出，本文的一些结果也在[11]中得到了阐述。现在让我们进入正式程序。我们考虑从可数集PL = {p，q，. . }的命题字母，经典连接词（：>，1..）和一元模态算子D，V .我们还将使用派生的连接词，经典的（t\，V，-，，T，：=）和情态的（<>，3）。我们记得<>A= -，0-，A，3A = -，'t/-，A.一个（正常的）双峰逻辑（或仅仅是一个逻辑）是一组公式封闭的规则下的替代，前件，Dintroduction（A/DA），V -介绍（A/VA），包含所有经典的命题重言式和公式1本文的工作是由俄罗斯基础研究基金会（项目编号96 - 01 -01378）应用非经典逻辑杂志。第9卷，第2-3/1999，第369至379页，[ Si m on F ra se r U niverity]在21：12 16 N ovem be r 20 14 370 Journal of Applied Non-Classical Logics。第9卷第2-3期/1999 D（p：：：> q）：：：>（Dp：：：> Dq），'V（p：：：> q）：：：>（' Vp：J 'Vq）。K2表示最小双峰逻辑。包含给定逻辑L和一组公式r的最小逻辑用L + f表示;对于公式A，L +A是L + {A}的缩写。我们说逻辑K2 + r是由集合r公理化的.本文中经常用到的逻辑是：S4 * S5 = K2 + {Dp：Jp，Dp：JDDp，'Vp：Jp，'Vp：JWp，3 'Vp：Jp}，S4 U = S4 * S5 + 'Vp：JDp。由于模态逻辑的Kripke语义是众所周知的，我们非常简要地回顾了相应的概念和术语。（双峰）Kripke框架是三元组F =（W，R1，R2），使得W # 0 ; R1，R2 ~ W × W。W的元素称为世界（或点）。框架F上的Kripke模型是一个对（F，<p），其中<p：P L-+2 w是任何函数（赋值）。拓扑框架（本文中）就是拓扑空间。我们分别用i和c表示内部和闭包运算。拓扑框架X上的拓扑模型是一个对（X，<p），其中<p：P L -+ 2x是一个任意函数（赋值）。在（克里普克或拓扑）模型M中，公式A在点x处的真值记为M，x FA（或简称为x FA）。回想一下拓扑情形下M，xFA的归纳定义：M，xFA> xE <p（A）（对于AEPL）;
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, Dintroduction (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 9no 2-3/1999, pages 369 to 379 D ow nl oa de d by [ Si m on F ra se r U ni ve rs ity ] at 2 1: 12 1 6 N ov em be r 20 14 370 Journal of Applied Non-Classical Logics. Volume 9no 2-3/1999 D(p :::> q) :::> (Dp :::> Dq), 'V(p :::> q) :::> ('Vp :J 'Vq). 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, R1 , 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, <p), in which <p: P L-+ 2w is any function (valuation ) . A topological frame (in this paper) is just a topological space. We denote the interior and the closure operation by i and c respectively. A topological model over a topological frame X is a pair (X, <p ), where <p : P L -+ 2x is an arbitrary function (valuation ) . The truth of a formula A at a point x in a (Kripke or topological) model M is denoted by M, x FA (or briefly, by x FA). Recall the inductive definition of M, x F A for the topological case: M,x FA¢> x E <p(A)( for A E PL);