Characterization of Boolean Topological Logics Short Form: Boolean Topological Logics Anthony R. Fressola Denison University Granville, OH 43023 University of Illinois Urbana-Champaign, IL USA 61801-61802 afresso2.uiuc.edu Joan E. Krone Department of Computer Science Denison University Granville, OH USA 43023 krone@denison.edu 1-740-587-6484 fax: 1-740-587-5749 Corresponding Author Stoyan Paunov Denison University Granville, OH 43023 Vanderbilt University Nashville, TN USA 37235 Michael D. Westmoreland Department of Mathematics Denison University Granville, OH USA 43023 Westmoreland@denison.edu ABSTRACT We investigate logics that are derived from topological spaces; such logics were first described by Westmoreland and Schumacher in 1993. Here, we explore the relationship between the logic systems and the topologies used to construct them. Special attention is given to Boolean topological logics. We show a characterization of these 1
logics by proving that a topological logic is Boolean if and only if its underlying topology is a partition topology. Keywords: topological logics, Boolean logics, phase space logics, partition topology. I. INTRODUCTION In 1993, Westmoreland and Schumacher developed two algebras of propositions for classical mechanics: a closed phase space logic and an open phase space logic [1]. Both of these logics are based on the simple classical measurement theory that allows the determination of any continuous phase space function to any finite precision. It is surprising that both of these logics are non-boolean; the closed logic fails the law of contradiction, and the open logic fails tertium non-datur. Measurement is a finite physical process. In reality, when we measure something we must tolerate some error. If we want to measure the location of a particular point on a line, our measurement would indicate some open interval around the point, rather than just the point itself. The size of the open interval is dependent of our level of tolerance. That is, no measurement of finite precision can be used to determine whether or not a point x belongs to a set A. This implies that any measurement corresponds to a nonempty open set in the phase space. This suggests that the structure of propositions used to analyze a physical system should reflect the feature of limited precision. Topological phase space logics were developed to explore this possibility. Since the definitions of the logical operators rely only upon interiors and closures [1] any topology can be used as the structure for a logic system under these definitions. We will refer to a logic that is derived from a topological space (X, T) as the derived logic of X associated with the topology T. One can contemplate having different derived 2
logics on a particular phase space X, each associated with a different topology on X. For example, in [2] it is noted that the logic associated with the discrete topology on any nonempty space gives Boolean open and closed logics. Thus, we are lead to ask if any other topologies yield Boolean open or closed logics. II. DEFINITIONS We will adopt the definitions in [1]: Definition 1: In a closed topological logic two sets are said to be equivalent if and only if their closures are the same. Definition 2: In an open topological logic two sets are said to be equivalent if and only if their interiors are the same. Unless stated otherwise, a capital letter will be used to denote a set, while brackets around a capital letter will indicate the equivalence class containing that set. As this paper deals primarily with topological logics we, will drop the modifier topological hereafter. We will also use A and int A to denote the closure and the interior of a set A, respectively If we are in the closed logic, [A] can be thought of as [ A ], the canonical representative of a given equivalence class in the closed logic. Similarly, in the open logic, [A] can be thought of as [int A]. We will frequently use the canonical representative when discussing a particular equivalence class. Furthermore, in both logics, we use 1 to represent the equivalence class containing the universal set X, and 0 to represent the equivalence class containing the empty set. In terms of notation, we have [X] = 1 and [ ] = 0. 3
We also present the definitions for logical operators. In the following definitions, the symbols,, and represent negation, disjunction, and conjunction, respectively. Definition 3: In a closed logic [A] := [( A) c ] [A] [B] := [ A B] [A] [B] := [ A B]. Definition 4 : In an open logic, [A] := [(int A) c ] [A] [B] := [(int A) (int B)] [A] [B] := [(int A) (int B)]. We may now use any topology as the underlying mathematical structure for our logics. In [1] it is noted that the standard metric topology in R 2 yields both closed and open logics that are non-boolean. Recall that a lattice of propositions is Boolean if and only if it satisfies the following axioms [4]: 1. A 0 = A 2. A 1 = A 3. Commutativity: A B= B A A B= B A 4. Distributivity: A (B C] = (A B) (A C] 4
5. Tertium non datur: A A = 1 A (B C) = (A B) (A C) 6. Law of Contradiction: A A = 0. Before going any further, we state and prove two general theorems: Theorem 1: For any topological space X, the associated closed and open logics will be commutative, associative, and distributive. Furthermore, [A] 0 = [A] and [A] 1 = [A] for any subset A X. Proof: The proofs for commutativity, associativity, and distributivity follow directly from Theorems 1 and 4 in [1]. Under the closed logic, [A] 0 = [ A ] = [ A ] = [ A] = [A], and [A] 1 = [ A X] = [ A] = [A]. Under the open logic, [A] 0 = [(int A) (int )] = [(int A) ] = [int A] = [A], and [A] 1 = [(int A) (int X)] = [(int A) X] = [int A] = [A]. Theorem 2: For any topological space X, the associated closed logic l satisfies tertium non-datur. Dually, the open logic will satisfy the law of contradiction. Proof: Let A X. Then, in the closed logic, we have [A] [A] = [A] ( A ) c = c c A ( A) A ( A) = [ X ] Thus, it must be the case that [A] [A] = [X] = 1, so tertium non-datur is satisfied. In the open logic system, we have. 5
[A] [A] = [A] [(int A) c ] = [(int A) int((int A) c )] [(int A) (int A) c )] = [ ]. Thus, it must be the case that [A] [A] = [ ] = 0, so the law of contradiction is satisfied. III. PARTITION TOPOLOGIES We examine several topologies that result in Boolean logic systems. Consider the indiscrete topology on a set X. Recall that the only open sets in the indiscrete topology are the empty set and the entire set itself. If X is nonempty, there are exactly two equivalency classes in both the closed and open logics: [ ] and [X]. Under the closed logic, any nonempty set is in the same equivalency class as X, and under the open logic, any proper subset of X in the same equivalency class as the empty set. By Theorem 1, we know that the logics produced by the indiscrete topology are commutative, associative, and distributive, and that [A] 0 = [A] and [A] 1 = [A]. Theorem 2 implies that we only need to check that the law of contradiction holds in the closed logic and tertium non-datur holds in the open logic. Since every set is either in the equivalency class [ ] or [X], we only have two cases to check in each case. We state and prove a simple but useful lemma: Lemma 3: For any topological space X, [ ] = [X] and [X] = [ ] in both the closed and open logic systems. Proof: In the closed logic system, [ ] = [( ) c ] = [ c ] = [X] and [X] = [( X) c ] = [X c ] = [ ]. In the open logic system, [ ] = [(int ) c ] = [ c ] = [X] and [X] = [(int X) c ] = [X c ] = [ ]. 6
We now return to the indiscrete topology. Using the lemma, we have [ ] [ ] = [ ] = 0 and [X] [X] = [ ] = 0 in the closed logic and [ ] [ ] = [X] = 1 and [X] [X] = [X] = 1 in the open logic. Thus, the closed logic satisfies the law of contradiction and the open logic satisfies tertium non-datur in all cases. Since all of the Boolean axioms are satisfied, we see that the indiscrete topology produces Boolean open and closed logics. We have just shown that the weakest topology on a set X has associated logics that are Boolean. We now consider the logics associated with the strongest topology on as space X.: the discrete topology. Recall that the discrete topology is the finest of all topologies because every subset of X is an open set. Since every set in the topology is open, every set is also closed. Thus, the closure and the interior of any set is the set itself, so every set is in its own equivalency class in both the closed and open logic systems. This implies that in both systems, [A] = [A c ] for every subset A of X. It follows that the closed logic system satisfies the law of contradiction and the open logic system satisfies tertium non-datur. Since the other axioms hold for any topology, we can conclude that the logics associated with the discrete topology are Boolean. Next we look at the odd-even topology. In the odd-even topology, our space is the set of integers, Z, and a set is declared open if it is one of the following sets:, O, E, or Z, where O is the set of odd integers and E is the set of even integers. Immediately we see that any open set is also closed, and vice versa. Under both the closed and open logic 7
systems, there are exactly four equivalency classes: [ ], [O], [E], and [Z]. Since, O, E, and Z are both open and closed, we have that [A] = [A c ] for every subset A of X. Thus, both tertium non dater and the law of contradiction hold, which indicates that the odd-even topology produces Boolean logic systems. The three topologies just discussed all examples of partition topologies. In a partition topology, the subsets formed by a partition of X, along with the empty set, serve as a basis for the topology. A subset of X is then declared open if it is the empty set or a union of the subsets formed by the partition. The trivial partitions yield the indiscrete and the discrete topologies. In the odd-even topology, the integers were partitioned into the odd and even integers. In every partition topology, any open set of a partition topology is also closed [3]. Note that the three topologies examined thus far are all partition topologies and all produced Boolean logics. This is not a coincidence: Theorem 3: A derived logic on a space X associated with topology T is Boolean if and only if T is a partition topology. Proof: By Theorem 1, we know that [A] 0 = [A] and [A] 1 = [A] for all subsets A of X, and that the logic is commutative, associative, and distributive. Furthermore, by Theorem 2, the closed logic system satisfies tertium non-datur and the open logic system satisfies the law of contradiction. Thus, we only need to show that the closed logic system satisfies law of contradiction and the open logic system satisfies the tertium non- 8
datur. Recall that in a partition topology, every open set is also closed. We now consider the associated closed logic. By definition, [A] [A] = [A] ( A ) c. = A ( A) c Since A is closed, we have that ( A ) c is open. But in a partition topology, every open set = = = 0, and the law of c c is also closed, so we have A ( A) A ( A) [ ] contradiction is satisfied. Now consider the associated open logic. By definition, ( ( )) c [A] [A] = [A] [(int A) c ] = ( int A) int ( int A). Since the interior of A is an open set, its complement is closed. But in a partition topology, every open set is also closed, so we have ( ) ( ) c c ( ( )) ( ) ( ) [ ] int A int int A = int A int A = X = 1, and tertium non-datur is satisfied. Therefore, all of the Boolean axioms are satisfied, and the partition topology yields a Boolean logic system for both the closed and open logic systems. We now assume that the logic associated with T is Boolean. Let A be an arbitrary open subset of X. Since tertium non-datur is satisfied in the open logic, we have [A] [A] = 1. Using Definition 4, we see that [A] [(int A) c ] = 1, which implies that[(int A) int((int A) c )] = [X]. Hence int((int A) int((int A) c )) = X by Definition 2. Since the interior of any set is contained in the set itself, it must be that (int A) int((int A) c ) = X. Since A was open, this is equivalent to A int (A c ) = X. It follows that A c int(a c ). However, it must be that int(a c ) A c, so in fact, we have int(a c ) = A c. It follows that A c is open, which implies that A is closed. Since A was 9
arbitrary, it has been shown that every open set is closed, so the underlying topology is a partition topology. Theorem 3 completely characterizes Boolean topological logics. Topological logic systems are Boolean if and only if the underlying topology is a partition topology. A partition topology, in turn, is characterized by whether every open set is also closed. A topology with this quality is sometimes also referred to as almost discrete. This indicates that the structure of a topological logic system strongly depends on connectedness. A topological space X is connected if and only if the only sets that are both open and closed are and X. We conclude with a corollary: Corollary: Every nontrivial connected topological space has non-boolean associated logics. While Theorem 3 settles the issue of when a derived topological logic is Boolean, it is unknown, in general, when a derived logic is Boolean or not. Another class of derived logics is the Birkhoff - von Neumann type quantum logics. These logics are derived from the underlying vector space that models the behavior of a given quantum system. We leave as an exercise for the reader to show that such a logic is Boolean if and only if the underlying vector space is one dimensional. Thus, the structure of Boolean logics derived from topological spaces is a bit richer than the corresponding structures derived from vector spaces: one dimensional quantum systems are trivial but discrete topologies on phase spaces are the standard way of considering classical systems (i.e. one generally pretends that infinitely precise 10
π measurements can be made). Yet, in both systems there is a type of triviality in the theory of measurement. For one dimensional Birkhoff von Neumann logics, measurements on the mechanical system can tell us only about the presence or absence of a system. Theorem 3 tells us that the underlying theory of measurement for a Boolean logic must be, in a sense, trivial: we can measure for the clopen sets in the topology and nothing else. In view of this, it is natural to wonder if Boolean derived logics are always associated with some sort of triviality in measurement. There are many possible mathematical structures that can be used to derive a logic: a group and its lattice of subgroups, a group and its lattice of normal subgroups, a ring and its lattice of two - sided ideals, etc. It may be interesting to explore if and when Boolean logics can be derived from other mathematical structures. REFERENCES [1] Westmoreland, M. and Schumacher, B., "Non-Boolean derived logics for classical systems", Physical Review A, Volume 48, Number 2 (August, 1993), p. 977-985. [2] Westmoreland, M. and Krone, J., Derivation Schemes in Twin Open Set Logic, Collision-Based Computing, A. Adamatzky, ed., Springer-Verlag, New York, 2002, p. 201-229. [3] Steen, L.A. and Seebach, J.A.Jr., Counterexamples in Topology, Springer-Verlag, New York, 1978. [4] Suppes, P, Introduction to Logic, Dover Publications, Mineola, NY, 1957. [5] Munkres, J.R., Topology, Prentice Hall, Upper Saddle River, NJ, 2000. 11
SYMBOLS We have used symbols and notations consistent with those in [4] and [5]. int A A A c [A] interior of the set A closure of the set A complement of set A or and equivalence class which is a proposition union intersection 12