1. A motivation for algebraic approaches to logics

Size: px

Start display at page:

Download "1. A motivation for algebraic approaches to logics"

Martin Ray
5 years ago
Views:

1 Andrzej W. Jankowski AN ALGEBRAIC APPROACH TO LOGICS IN RESEARCH WORK OF HELENA RASIOWA AND CECYLIA RAUSZER 1. A motivation for algebraic approaches to logics To realize the importance of the research work of Helena Rasiowa and Cecylia Rauszer in the area of algebraic methods in logics let us recall Gottfried Wilhelm Leibniz. In [3] he wrote Things are like numbers. By this he meant that statements about complex things can be derived from statements about their simpler constituents by a process of combination analogous to arithmetic calculation on numbers. An algebraic calculation itself should be done by a machine. Thus he proposed: It is unworthy of excellent [persons] to lose hours like slaves in the labor of calculation which could be safely relegated to anyone else if machines were used [7]. Having this in mind, he invented a device known as the Leibniz wheel for automatic addition, subtraction, multiplication, and division. Leibniz attached great significance to his vision of reducing human reasoning to algebraic calculation, saying: Nobody else, I believe, noticed this, otherwise... he would drop everything to focus on this subject, because a man cannot achieve anything greater [5]. It turned out that applications of algebraic calculus and machines to problem solving is very difficult. Current approaches are based on applications of Boolean algebra (or their modifications). The strong influence of the Polish School of Logic is worth mentioning here, especially Jan Lukasiewicz, Alfred Tarski, Alfred Lindenbaum, Stanis law Leśniewski, Andrzej Mostowski, Helena Rasiowa, Roman Sikorski and others. Modern computer techniques make us aware of many further possibilities and challenges for the researchers of the fascinating idea of automation of reasoning. 139

2 2. An algebraic approach to logics in the works of Helena Rasiowa and Cecylia Rauszer Let us assume that we represent a knowledge about the surrounding world by a structure built from the following elements: 1. a class of admissible worlds we include real objects, but also imaginary objects which could be used as models that illustrate our knowledge, we assume that each admissible world has to it a set of logical values of the admissible world assigned, 2. a set of expressions used as a language for representation of our thoughts about properties and phenomena in the admissible worlds, 3. a truth function, which for each admissible world A assigns another function from the set of all expressions into the set of logical values of A; this function enables us to estimate the degree of credibility and verifiability of expressions in each admissible world, 4. a deductive system that enables inferences about an admissible world based on credibility and verifiability of knowledge about that world. Such a form of knowledge representation about the surrounding world we shall call a logical structure. The above intuitive approach to the concept of logical structure is based on a formal definition from a Ph.D. thesis prepared under the guidance and supervision of Cecylia Rauszer [1]. It is easy to imagine many examples of logical structures and related research problems. Example 1. Let us consider as expressions of a logical structure a set of sentences i.e. formulas without free variables of a theory T of the classical predicate calculus. The class of admissible worlds could be the class of relational structures for the theory T. Logical values are just TRUE and FALSE, i.e. elements of the universe of the two element Boolean algebra. Of course for multi-valued predicate calculus the sets of logical values have more elements. Example 2. Let us consider as expressions the set of all formulas of the theory T of classical predicate calculus with free variables from an infinite set of variables V. Then the admissible worlds could be once again the relational structures. However, the set of logical values and the truth function must be slightly more complex. Namely, for each relational structure A let 140

3 us assume that S is the set of all evaluations of free variables V into the universe of A (i.e. functions from V into the universe of A). Then the set of logical values of A is the set of elements of the Boolean algebra of all subsets of S. The truth function for A is a function which assigns for each formula p the set of all evaluations v of free variables such that the formula p is true in the structure A by the evaluation v. In other words classical predicate calculus could have a class of admissible worlds with more than just two logical values. In general, as a set of logical values for classical logical structures we can use an arbitrary complete Boolean algebra. This approach to semantics for classical predicate calculus is called Boolean models for T and has been introduced by Helena Rasiowa and Roman Sikorski [HR9], [HR14]. The concept of Boolean models for set theory has been applied by Dana Scott and Richard Solovay as an elegant method for a proof of independence of the Axiom of Choice and the Continuum Hypothesis from the axioms of set theory [6], [2]. The most important research problems in the works of Helena Rasiowa and Cecylia Rauszer have been: 1. Find and describe by algebraic methods the most convenient logical structures to be used in research on constructive (algorithmic) problem solving, in particular in research on semantics and inference rules in the area of constructive information processing by one or several agents. 2. Algebraically construct a canonical world for a consistent set of expressions and apply this technique to the study of equivalence between deductive inference and inference based on the truth relation (the completeness theorem). 3. Seek and analyze alternative styles of deductive systems for a given logic, for example based on alternative approaches invented by: Gerhard Gentzen, David Hilbert, Jacques Herbrand, Stanis law Jaśkowski and Helena Rasiowa together with Roman Sikorski [HR31], [HR27]. 4. Analyze the fundamental model-theoretic properties of the admissible worlds for a given logical structure (e.g. Skölem-Löwenheim theorem, compactness theorem, omitting types theorem, etc.). 5. Investigate geometric properties of the space in which points are admissible worlds for a logical structure and the distance expresses similarity between worlds in terms of satisfaction of similar expressions. 141

4 This is a metaphor of Stone s representation theorem for Boolean algebras 1. As it has been noticed by Alfred Lindenbaum and Alfred Tarski, if we glue together all the sentences which represent the same thought in a deductively closed theory in classical logic, we get a Boolean algebra. Intuitively, this is an algebra of thoughts for the theory. Algebraic operators in the algebra of thoughts correspond to logical connectives. In particular, implication defines a partial order of thoughts. Greater value in the order intuitively means more true. Disjunction corresponds to the supremum in the order generated by the implication, and conjunction corresponds to the infimum in the same order. If we treat the existential quantifier of classical logic as infinite disjunction, then its algebraic interpretation is the supremum. Similarly, if we treat the universal quantifier as infinite conjunction, then it corresponds to the infimum. The logical connectives could be generalized to an arbitrary set of any finite operators and constants (truth values), and the intuition of quantifiers could be modified to infinite logical operators. Let Q be a set of generalized algebraic operators (i.e. finite and infinite operators). In the standard way we define the concept of Q-algebra that includes all operators from the set Q [HR49], and concept of Q- homomorphism between two Q-algebras which preserves all Q-operators. In this paper we assume that Q is enumerable and each Q-algebra has a special constant 1 for the logical value of truth. An interesting introduction to logics based on Q-algebras with implication (extensions of implicative algebras) can be found in [HR49]. We say that a logical structure is a Q-logical structure, if: the algebra of thoughts for any theory is a Q-algebra, the logical values for each admissible world form a Q- algebra, the truth function is based on a Q-homomorphism from the set of expressions into a Q-algebra of logical values for each admissible world. The main idea of algebraic approach to predicate calculus in the research papers of Helena Rasiowa and Cecylia Rauszer is based on two concepts: Q-representability and Q-characterizability. They have used the concepts in several variants and have applied then as the main tool for solution of several important problems. 1 One of my most exciting personal intellectual experiences was a lecture given by Helena Rasiowa, where she presented a proof of the Rasiowa - Sikorski Lemma using topological properties of the space of models (Baire property). 142

5 In order to introduce both concepts let us assume that we have two classes M and K of Q-algebras. We say that M is Q-representable by K, if for any algebra from the class M there exists a Q-isomorphism to a Q-algebra in K. For example, if Q includes standard finite Boolean operators, then Stone s representation theorem for Boolean algebras means that the class of Boolean algebras is Q-representable by the class of all Q-fields of sets. The Stone representation theorem has been generalized for infinite Boolean operators by Rasiowa and Sikorski. If Q is a set of standard finite Boolean operators and a countable set of operators corresponding to infinite infima and suprema in such Q-Boolean algebras, then the Rasiowa-Sikorski representation theorem says that the class of all such Q-Boolean algebras is Q-representable by the class of all Q-fields of sets. By the definition it means that for any countable set Q of infima and suprema in a Boolean algebra there exists a Q-isomorphism to a Q-field of sets. We say that M is Q-characterizable by K, if for any algebra A from the class M and for any element x of A such that x is different from 1, there exists a Q-homomorphism to a Q-algebra from the class K such that the value of x after the transformation by the Q-homomorphism is different from 1. Q-representability is a very close concept to Q-characterizability. Usually we use Q-characterizability in order to prove Q-representability. For example, if Q includes standard finite Boolean operators, and K has only one two-element Q-Boolean algebra, then each Q-homomorphism from a Boolean algebra A into the class K could be identified with a Q- prime filter of A, and any element a of A could be identified with the set of all Q-prime filters including a. Thus Boolean algebras are characterizable by two element Boolean algebras and this proves that Boolean algebras are representable by the class of Boolean algebras of sets (Stone representation theorem). Originally the Rasiowa-Sikorski Lemma could be expressed as follows: if Q is a set of standard finite Boolean operators and a countable set of operators corresponding to infinite infima and suprema in such Q-Boolean algebras, then the class of all such Q-Boolean algebras is Q-characterizable by the class which has only one two-element Q-Boolean algebra. The logical meaning of the concept of Q-representability of the class of Q-algebras of thoughts by Q-algebras of logical values for admissible worlds of a logical structure L is that L satisfies the completeness theorem. The 143

6 concept of Q-representability of the class of Q-algebras of thoughts by Q- algebras of logical values for admissible worlds of L is used for construction of very special admissible worlds, called canonical models. We say that a Q-logical structure L is a Rasiowa-Sikorski logical structure, if the class of Q-algebras of thoughts is Q-characterizable by the Q- algebras of logical values for admissible worlds of L. There are several examples of Rasiowa-Sikorski logical structures based on logics such as: intuitionistic [HR31], Heyting-Brower [CR12], intermediate [CR6], modal [HR31], Post [HR49], Semi-Post [HR98] and algorithmic [4]. It is possible to apply several schemes of analysis to research of Rasiowa-Sikorski logical structures. An excellent compendium of the schemes can be found in [HR49], [HR31] and [HR98]. The schemes can be used for typical model theoretic applications like completeness theorem, construction of canonical model, etc., and also for not very typical ones, like naturaldeduction systems in the Rasiowa-Sikorski style [HR27] and [HR31], proof of decidability of formulas in prenex form for constructive intuitionistic theories [HR31]. 3. Further development of the algebraic approach to logics: A prognosis Let us try to make a prognosis of further development of the algebraic approach to logics, which we can learn from the research papers of Helena Rasiowa and Cecylia Rauszer. In order to do this let us notice that: 1. Architecture of today s computer is based on algebraic approach to classical propositional calculus based on Boolean algebras with finite operators. Let us observe that it took about one hundred years from the discovery of Boolean algebras to their application to computer architecture. 2. We gradually introduce into computer applications some areas of predicate calculus, for example application of relational calculus to relational databases, logic programming, program specification and verification, artificial intelligence, logic of programs, etc. 3. The phenomenon called software crisis enforces applications of more formal and verifiable procedures in software engineering. 144

7 Having in mind these three observations, it is reasonable to expect that the algebraic approach to predicate logics will be very important in the future computer science, as algebraic approach to propositional logic has been very basic for the traditional computer architecture. Of course, it is impossible to fully motivate and extend this forecast based on our current knowledge. In fact this is science-fiction. Anyway let us try: There are many possible directions of continuation of research done by Helena Rasiowa and Cecylia Rauszer in the area of applications of the algebraic approach to computer science. Probably the most important streams are the following: applications of non-classical logic, more sophisticated algebraic treatment of logical connectives. In the area of applications of non-classical logic it is especially important to continue research of Helena Rasiowa and Cecylia Rauszer in the area of formal frameworks for logics of several agents which use approximate logic. For example in [HR88] an epistemic logic is designed, which formalizes approximating reasoning performed by groups of agents who perceive reality using perception operators or in [CR44], [CR45], [CR47] and [CR48] we have an algebraic approach to problems concerning representation of rough and distributed knowledge for groups of agents. A more sophisticated algebraic treatment of logical connectives should be useful for Leibniz s vision of reducing human reasoning to algebraic calculation. Of course, now we have some theorems which suggest that direct implementation of Leibniz s idea is impossible (the Church theorem on undecidability of predicate calculus or even the Karp theorem on NPcompleteness of the complexity of satisfiability of propositional calculus). But, maybe for some nontrivial classes of formulas of predicate calculus it is possible to get computer support using algebraic methods for logic? References [1] A. Jankowski, Zanurzenia struktur logicznych, Institute of Mathematics Polish Academy of Sciences, preprint No. 10 series B, [2] T. J. Jech, The Axiom of Choice, North-Holland [3] G. W. Leibniz, De Principio Individui, [4] G. Mirkowska and A. Salwicki, Algorithmic logic, Warszawa PWN, Reidel Publ. Comp

8 [5] A. Mostowski, Logika matematyczna, Warszawa-Wroc law, Czytelnik [6] D. Scott and R. M. Solovay, Boolean valued models for set theory, preprint, [7] A. Tucker, A. P. Bernat, W. J. Bradley, R. Cupper and G. W. Scragg, Fundamentals of Computing I, pp. 3, McGraw-Hill, Institute of Mathematics Warsaw University Polish-Japanese Institute of Computer Techniques Warszawa, Poland 146

CHAPTER 11. Introduction to Intuitionistic Logic

CHAPTER 11. Introduction to Intuitionistic Logic CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated