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«Tzu-Keng Fu Dissertation zur Erlangung des Grades eines Doktors der Ingenieurwissenschaften –Dr.-Ing.– Vorgelegt im Fachbereich 3 (Mathematik und ...»

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Universal Logic and the Geography of Thought –

Reflections on logical pluralism in the light of culture

Tzu-Keng Fu

Dissertation

zur Erlangung des Grades eines Doktors der

Ingenieurwissenschaften

–Dr.-Ing.–

Vorgelegt im Fachbereich 3 (Mathematik und Informatik)

der Universit¨t Bremen

a

Bremen, Dezember, 2012

PhD Committee

Supervisor : Dr. Oliver Kutz (Spatial Cognition Re-

search Center (SFB/TR 8), University

of Bremen, Germany)

External Referee : Prof. Dr. Jean-Yves B´ziau (Depart- e ment of Philosophy, University of Rio de Janeiro, Brazil) Committee Member : Prof. Dr. Till Mossakowski (German re- search center for artificial intelligence (DFKI), Germany) Committee Member : Prof. Dr. Frieder Nake (Fachbereich 3, Faculty of Mathematics and Informatics, the University of Bremen, Germany) Committee Member : Dr. Mehul Bhatt (Cognitive Systems (CoSy) group, Faculty of Mathematics and Informatics, University of Bremen, Germany) Committee Member : Minqian Huang (Cognitive Systems (CoSy) group, Faculty of Mathematics and Informatics, University of Bremen, Germany) Sponsoring Institution The Chiang Ching-kuo Foundation for International Scholarly Exchange, Taipei, Taiwan.

To my Parents.

獻給我的父母。 Acknowledgment I owe many thanks to many people for their help and encouragement, without which I may never have finished this dissertation, and their constructive criticism, without which I would certainly have finished too soon. I would like to thank Dr. Oliver Kutz, Prof. Dr. Till Mossakowski, and Prof. Dr. Jean- Yves B´ziau for assisting me with this dissertation. Warmest and sincere e thanks are also due to Prof. Dale Jacquette of the Philosophy Department, the University of Bern, Switzerland, and to Prof. Dr. Stephen Bloom of the Computer Science Department, Stevens Institute of Technology, Hoboken, New Jersey, USA, who provided me with valuable original typescripts on

Abstract

logic.

For their patience, I thank my family, and especially my parents, Mrs.

Mei-Kuei Tseng and Mr. Yu-Lan Fu. I thank my wife Mrs. Jung-Ying Fang for her love and inspirational conversations on cross-cultural psychology. Fi- nally, I thank the Chiang Ching-kuo Foundation for International Scholarly Exchange for having financed my PhD research between 2008 and 2010.

–  –  –

The aim of this dissertation is to provide an analysis for those involved and interested in the interdisciplinary study of logic, particularly Universal Logic.

While continuing to remain aware of the importance of the central issues of logic, we hope that the factor of “culture” is also given serious consideration.

Universal Logic provides a general theory of logic to study the most general and abstract properties of the various possible logics. As well as elucidating the basic knowledge and necessary definitions, we would especially like to address the problems of motivation concerning logical investigations in different cultures.

First of all, I begin by considering Universal Logic as understood by JeanYves B´ziau, and examine the basic ideas underlying the Universal Logic e project. The basic approach, as originally employed by Universal Logicians, is introduced, after which the relationship between algebras and logics at an abstract level is discussed, i.e., Universal Algebra and Universal Logic. Secondly, I focus on a discussion of the “translation paradox”, which will enable readers to become more familiar with the new subject of logical translation, and subsequently comprehensively summarize its development in the literature. Besides helping readers to become more acquainted with the concept of logical translation, the discussion here will also attempt to formulate a new direction in support of logical pluralism as identified by Ruldof Carnap (1934), JC Beall and Greg Restall (2005), respectively. Thirdly, I provide a discussion of logical pluralism. Logical pluralism can be traced back to the principle of tolerance raised by Ruldof Carnap (1934), and readers will gain a comprehensive understanding of this concept from the discussion. Moreover, an attempt will be made to clarify the real and important issues in the contemporary debate between pluralism and monism within the field of logic in general. Fourthly, I study the phenomena of cultural-difference as related to the geography of thought. Two general systems in the geography of thought are distinguished, which we here call thought-analytic and thought-holistic. They are proposed to analyze and challenge the universality assumption regarding cognitive processes. People from different cultures and backgrounds have many differences in diverse areas, and these differences, if taken for granted, have proven particularly problematic in understanding logical thinking across cultures. Interestingly, the universality of cognitive processes has been challenged, especially by Richard Nisbett’s research in cultural psychology. With respect to these concepts, C-UniLog (appendix A) can also be considered in relation to empirical evidence obtained by Richard Nisbett et al. In the final stage of this dissertation, I will propose an interpretation of the concept of logical translation, i.e., translations between formal logical mode (as cognitive processes in the case of westerners) and dialectical logical mode (as cognitive processes in the case of Asians). From this, I will formulate a new interpretation of the principle of tolerance, as well as of logical pluralism.





Chapter 1 Introduction

–  –  –

Tarski’s account of the concept of logical consequence has gained great importance in modern mathematics, and moreover has founded the role of logic in scientific studies. Universal Logic as a general theory of logics is the view that has followed the most rudimentary stage of Tarski’s notion for logical consequence. The conceptual analysis of logic is what the Universal Logic project is about.

There are many questions that one might ask about Tarski-oriented contributions. One – perhaps the most obvious – is why one might suppose the concepts of logical consequence to be mathematical. Another question one might ask is why one might suppose Universal Logic, as a general theory of logics, to be mathematical as well. Yet another question is that of how the logical structures that Universal Logic discusses are mathematical. The short answers might be that a Tarskian contribution is important not for its practical applications, but because a vast quantity of mathematical work assumes that it is true. The aim of this dissertation is to provide not only a theoretical analysis but also application-oriented perspectives for those involved and interested in the interdisciplinary study of logic.

34 CHAPTER 1. INTRODUCTION

The very nature of Universal Logic is to provide a general theory of logic to study the most general and abstract properties of the various possible logics.

Currently, it is anticipated that many logical researchers will again take an interest in this project. The situation here might be illuminated by analogy with Universal Algebra. Universal Algebra was originally proposed by an interest in a rather precise but intuitive notion to provide a general theory of algebras. Here the notion was that of a conceptually-oriented approach that is closer to our everyday life than the axiomatic approach.

What this dissertation will discuss is the relation between the Universal Logic project and certain core theoretical computer scientific and philosophical notions. The Universal Logic project follows some well-known notions of abstract logic that have their widespread applications.

The first part of this dissertation focuses on such abstract logical investigations. The second part discusses logic translation investigations that have closely followed the techniques adopted in abstract logic. The third part presents logical pluralism, proposed by Rudolf Carnap in 1934 and J.

C. Beall and Greg Restall in 2000, respectively. The fourth part concerns cross-cultural logic, a topic rarely considered in formal logic studies, which we discuss in relation to psychology, anthropology, and in particular, modern cognitive science.

1.1 Universal Logic The meaning of the term Universal Logic is easily misunderstood as “a logic” to unifiy all logics literally, however, this is disputed by Jean-Yves B´ziau and other pioneer universal logicians, who considered logic on a gene eral and abstract level. On the contrary, according to Universal Logic project, the view that there is only one “universal” logic is not possible, claiming that Universal Logic is a general theory of different logics.

In the 1920s, Tarski proposed his theory of consequence operator as a very general theory of logical consequence (see [207], [208], [209], [210]), in the 1930s, Gentzen’s sequent calculus considered a family of formal systems sharing a certain style of inference and certain formal properties ([105]), and in the 1970s, Roman Suszko (with Stephen Bloom and Donald Brown) proposed a concept of abstract logic A, consisting of an algebra A and a closure system ([52], [53]). These three studies considered logics at a general level. They attempted to see logics from a general point of view to

1.1. UNIVERSAL LOGIC 5 discuss the properties, and relations that different logics should have. None of these logicians said that “abstract logic” is the “universal” logic.

It has been explicitly elucidated that Universal Logic is a general theory to study logical structures. The Universal Logic project we discuss follows the same thinking of the three aforementioned studies by combining a general bivaluation semantics with Gentzen’s sequent calculus in order to promote “logic in general” to an even more abstract level. This means, once we recognize which parts of different logics are universal and common, according to the Universal Logic project, we can take them more or less directly to specific logics by the tool kit it provides. Moreover, we could “build” a specific logic of accounting specific situations and problems. (see [29], [31], [34], [38], [39], [46]).

An example raised in [31], states that “the first proofs of completeness for propositional classical logic give the idea that this theorem is depending very much on classical features.” People mistakenly think the concept of maximally consistent 1 is to depend on classical negation when they study the completeness theorem of classical logic. The fact, however, is “one can present the completeness theorem for classical propositional logic in such a way that the specific part of the proof is trivial.” (Ibid, p. 13). This means showing maximal consistency in the proof is trivial. It does not depend on the property of classical negation. It should belong to the universal parts of different logics. In other words, the completeness should be generalized, that is, it should not depend on any specific feature of any given logic.

The Universal Logic project discusses the distinction between what is universal and what is specific for logics. Moreover, it attempts to build a logic for specific problems, just as a doctor prescribes medicine or treatment for an illness. Such a “utopia” provides us a new paradigm to revolutionize the old-fashioned axiomatic approach in Logic (compare e.g. [32], [39]).

1.1.1 Philosophical Background of Universal Logic (I):

Bourbakism Universal logicians are neither pure logicians nor pure philosophers and can be regarded as “Philogicians”. Universal Logic is not considered as a logic but provides a general theory of logics. On one hand, it can be viewed as 1 A theory is a maximal by a consistent set of sentences, if it is consistent and none of its proper extensions is consistent.

6 CHAPTER 1. INTRODUCTION

philosophy-like in terms of a philosophy of logic. On the other hand, it can be thought of as mathematical-like in relation to studies concerning mathematical logic. This is a hybrid theory that includes both mathematical and philosophical aspects where logical structures are considered as mathematical structures within the spirit of Structuralism. This position is hardly a new one, but the Universal Logic project has a profound proposal on the subject

as follows:

• The idea of Universal Logic says precisely that logic (logical structures) are considered as mathematical structures. Universal Logic is to develop a general theory of logics which is an analogy to the idea of Universal Algebra.

Universal Algebra is a field of mathematics which is seen as a special branch of Model Theory. Generally speaking, it deals with “structures having operations”; specifically speaking, it studies “algebraic structures” themselves instead of studying various specific examples or models of algebraic structures. For example, it takes “the theory of groups”, “the theory of rings”, and “the theory of fields” as the objects of study instead of taking “particular groups”, “particular rings”, and “particular fields” as the objects of study.

In other words, Universal Algebra studies the theory of algebraic structures, and is a general theory of algebra. Similarly, Universal Logic claims to study “the theory of logical structures”, and is a general theory of logics.2 Adopting the position of Bourbakism, the Universal Logic project proposed that a logical structure is considered as type S, ⊢ where ⊢ is a consequence relation on P(S) × S. Moreover, S is not specified and any one of the three Bourbakian mother structures could be included in S. This basic conception of logical structure with regard to Bourbakism plays an important role in the Universal Logic Project.



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