Proof-Theoretical Studies on Diagrammatic Proofs and Focusing Proofs

Transcription

1 Proof-Theoretical Studies on Diagrammatic Proofs and Focusing Proofs Thesis Submitted to Faculty of Letters Keio University In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Ryo Takemura

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3 cknowledgments First of all, I would like to express my sincere gratitude to my supervisor, Professor Mitsuhiro Okada, for a number of suggestions and constant encouragements; without his advice and help, this thesis would never have been completed. In writing Part II, the attentive guidance of Dr. Masahiro Hamano was invaluable, and I am profoundly grateful to him for his help. I am grateful to Dr. Paul-ndré Melliès for his kind advices and discussions at many occasions on the preparation stage of this thesis. I would like to thank Dr. Olivier Laurent who kindly read a preliminary draft of Part II and gave me fruitful comments to improve the part. I am also grateful to Professor Masahiko Sato for his helpful comments and advices. Part I resulted from the stimulating discussions that I had with Mr. Koji Mineshima on many occasions. I am grateful to him for his fruitful suggestions and advices. 2

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5 Contents Introduction 8 I Diagrammatic proofs with Euler circles 14 1 Introduction to Part I 16 2 diagrammatic representation system (EUL) for Euler circles and its set-theoretical semantics ackground Diagrammatic syntax of EUL Set-theoretical semantics of EUL Diagrammatic inference system GDS Introduction to unification Generalized diagrammatic syllogistic inference system GDS Soundness and completeness of GDS Some consequences of completeness of GDS Unification of any (two) diagrams Decomposition set of an EUL-diagram On normal diagrammatic proofs Structure of canonical diagrammatic proofs EUL-structure 62 5 relationship between EUL-diagrams and Venn diagrams Syntax and semantics of Venn diagrams Transformation of EUL-diagrams to Venn diagrams

6 6 Normal form of diagrammatic proofs of GDS and syllogisms Syllogistic diagrams Syllogistic normal diagrammatic proofs and chains of ristotelian categorical syllogisms Some extensions and future work for Part I 82 II Focusing proofs in polarized linear logic 86 8 Introduction to Part II 88 9 Preliminary 1: Syntax of linear logic and polarized linear logic Second order linear logic LL Second order polarized linear logic Second order polarized linear logic LL pol Second order multiplicative additive polarized linear logic MLLP Focalization and polarization Focalization of linear logic LL into focalized sequent calculus LL foc Polarization of linear logic LL into polarized linear logic with shiftings LL pol LL foc and LL pol are almost isomorphic Preliminary 2: Categories for phase semantics of linear logic autonomous category with products and phase space for MLL Monad and comonad Seely category and enriched phase space for LL Polarized -autonomous category phase semantics for polarized linear logic Polarized phase space for MLLP Second order polarized phase model for MLLP2 and completeness Enriched polarized phase space for LL pol Second order polarized phase model for LL pol 2 and completeness135 5

7 11.5 n application of polarized phase semantics: First order conservativity Second order conservativity of linear logic over its polarized fragment Second order conservativity LL2 is not conservative over LL pol Second order η-expanded system LL η Main proposition for LL η 2: Polarization dditives Weakening and contraction LL η 2 is conservative over LL η pol Some syntactical properties derived from Theorem Second order definability of restricted additives in polarized linear logic Future work for Part II 154 ibliography for Part I 156 ibliography for Part II 162 6

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9 Introduction The purpose of this thesis is to conduct a proof-theoretical investigation into constructions of diagrammatic proofs and focusing proofs. Part I of this thesis is concerned with Euler diagrammatic reasoning. Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by many logicians. Euler diagrams were introduced in the 18th century by Leonhard Euler [1768]. ut it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint, and there are still only few proof-theoretical investigations that have been done. ccordingly, in order to fill this gap, we formalize an Euler diagrammatic inference system and prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We further investigate structure and manners of constructing diagrammatic proofs from a proof-theoretical viewpoint. We also introduce a notion of normal diagrammatic proof, which characterizes basic logical reasoning such as ristotelian categorical syllogisms. In Part II of this thesis, we consider the construction of focusing proofs within the framework of linear logical proof-theory. Linear logical structure is often considered a refined basic logical structure for the traditional logics such as classical logic and intuitionistic logic. In the study of linear logical proof-theory, Jean-Marc ndreoli [1992] introduced focusing proofs to make proof-search or proof-construction effective. Focusing proofs define a complete subclass of the usual normal proofs of linear logic, where many irrelevant choices in searching normal proofs are eliminated. In the study on focusing proofs, the notion of polarity plays a central role. Since Jean-Yves Girard s introduction of the notion of polarity in [Girard 1991], and since Olivier Laurent s formalization of polarized linear logic in [Laurent 1999], polarity has emerged as an important parameter which controls linear logi- 8

10 cal proof-theory. In particular, it has been established that polarity provides a natural framework to construct focusing proofs for the first order linear logic. However, it has been an open question whether polarity provides focusing proofs for the second order linear logic. In this thesis we attempt to provide an answer to this open question. Euler diagrams were introduced by Leonhard Euler [1768] to represent logical relations among the terms of a syllogism by topological relations, inclusion and exclusion relations, among circles. For example, a categorical statement ll are is represented by the inclusion relation between two circles named and. Given two Euler diagrams which represent the premises of a syllogism, the syllogistic inference can be naturally replaced by the task of manipulating the diagrams, in particular of unifying the diagrams and extracting information from them. nother well-known diagrammatic representation system for syllogistic reasoning is Venn diagrams, which were introduced by John Venn [1881] and modified by Charles Peirce [1897] originally to overcome expressive limitations of Euler diagrams. In Venn diagrams a novel syntactic device, namely shading, to represent emptiness plays a central role in place of the topological relations of Euler diagrams. ny claim in Venn diagrams is expressed in terms of shading, i.e., logical negation. For example, ll are is expressed by a Venn diagram through a translation to the statement There is nothing which is but not. (See Fig.1.2 of Chapter 1.) ecause of its expressive power and its uniformity in formalizing inference rules, Venn diagrams have been studied thoroughly; formal semantics and inference systems are given, and basic logical properties such as soundness, completeness, and decidability are shown. For a historical review, see [Hammer-Shin 1998], and for recent surveys, see [Stapleton 2005, Howse 2008]. However, the development of systems of Venn diagrams is obtained at the cost of clarity of the representations of Euler diagrams: In Venn diagrams, logical relations among terms are represented not simply by topological relations, but by making use of shadings, which makes the translations of categorical sentences, particularly those with several negations, uncomfortably complex. In contrast to the studies in the tradition of Venn diagrams, we introduce a diagrammatic representation system EUL for Euler circles from the following standpoint: Our diagrammatic syntax and semantics are defined in terms of topological relations between two diagrammatic objects (without using 9

11 shadings as in Venn diagrams). This formalization makes the translations of categorical sentences natural and intuitive. In order to characterize the most basic diagrammatic system, we avoid introducing auxiliary syntactic devices such as disjunctive linkings (between points and between diagrams) which might require arbitrary conventions. lthough our basic system EUL is weaker in its expressive power than usual Venn diagrammatic systems (e.g. Venn-II system of [Shin 1994] which is equivalent to the monadic first order logic in its expressive power), EUL is expressive enough to characterize basic logical reasoning such as syllogistic reasoning. ased on the representation system EUL, we formalize a diagrammatic inference system GDS, which consists of two kinds of inference rules: unification and deletion. In order to keep our diagrams free from disjunctive ambiguity and from representation of conflicting graphical information in a single diagram, we impose certain constraints on unification rules. We define the notion of diagrammatic proof, which is considered as a chain of unification and deletion steps. The inference system GDS is shown to be sound (Theorem 3.3.1) and complete (Theorem ) with respect to our formal set-theoretical semantics. Our proof of completeness of GDS not only implies provability of any valid diagram, but also provides a canonical way of constructing a diagrammatic proof for the diagram. Roughly speaking, the canonical diagrammatic proof consists of the following two steps: Given premise diagrams, 1. first decompose the premise diagrams into minimal diagrams, which constitute general diagrams; 2. then construct the conclusion diagram from the minimal diagrams, while avoiding disjunctive ambiguities. n examination of the canonical construction of diagrammatic proofs in terms of Gentzen s natural deduction system suggests that the structure of each canonical diagrammatic proof essentially corresponds to the structure of a normal (linguistic) natural deduction proof. We also introduce a notion of normal diagrammatic proof, where a unification and a deletion appear alternately. It is shown that for any (linguistic) chain of valid patterns of ristotelian categorical syllogisms there is a corresponding normal diagrammatic proof in GDS (Proposition 6.2.3). Conversely, it is also shown that, in syllogistic fragment of GDS, if there is a diagrammatic proof (not necessarily in normal form) of a diagram which 10

12 corresponds to a categorical sentence, then there is a normal diagrammatic proof for the diagram (Proposition 6.2.4). Hence, the syllogistic fragment of GDS completely characterizes the chains of ristotelian categorical syllogisms. In Gentzen s sequent calculus formulation ([Gentzen 1935]) of logical proofs, the most important normal form is the cut-free normal proof, which provides, for example, a basis for the study of proof-construction or proofsearch. Jean-Marc ndreoli [1992] introduced another important normal form, called focusing proof in the study of linear logical proof-theory. Linear logical structure is considered a refined basic logical structure for the traditional logics such as classical logic and intuitionistic logic. The focusing proofs form a subclass of cut-free normal proofs, where many irrelevant choices in searching a cut-free proof are eliminated. ndreoli classified linear logical connectives into two groups: synchronous, or reversible, connectives are those whose (right) introduction rule is reversible; and synchronous, or focalized, connectives are those whose (right) introduction rule is not reversible. Thus, in proof-search, the reversible formulas are decomposed immediately when they appear in a sequent. lthough the dual focalized connectives are not reversible, ndreoli observed that they can be treated as a cluster in proof-search: Once a focalized formula is selected, i.e., focused, it is successively decomposed up to reversible subformulas. ndreoli formalized the idea, focalization, in his Triadic sequent system [ndreoli 1992]. His system is shown to be equivalent to the usual linear logic, and the focusing proofs defined for his system form a complete subset of proofs of linear logic. It has been pointed out that the dual proof-theoretical properties reversibility/focalization of linear logical connectives can be captured by the notion of polarity. Polarity was invented by [Girard 1991] in his work on LC (Logique Classique), which is a refinement of Gentzen s standard formulation of classical logic LK. In LC, disjunction and existential quantifier are divided in terms of positive/negative polarities, and, for hereditary positive formulas, intuitionistic disjunctive and existential properties hold in the classical logic framework. There is a link between polarity and focalization studied by [Danos-Joinet-Schellinx 1997] and clarified by [Laurent 1999, 2002]. Laurent s formalization of polarized linear logic, in effect, provides a framework to construct focusing proofs in terms of the focalized sequent property. The focalized sequent property of a linear logical system, which is not necessarily a polarized system, means that if a sequent is provable with only polarized formulas, especially in polarized linear logic, it contains at 11

13 most one positive formula, in which case we call the sequent focalized. Since the positive formula is always focused, each proof of polarized linear logic gives a focusing proof in ndreoli s sense. Laurent [2002] shows a first order conservativity theorem of linear logic LL over its polarized fragment LL pol. That is, if a (polarized) focalized sequent is provable in LL, then it is also provable in LL pol. Since all the proofs of LL pol are automatically focusing, it follows that any focalized sequent is provable with a focusing proof in LL. Combined with the focalized sequent property of LL, the conservativity neatly captures a main idea underlying polarity in linear logic: the polarity restriction on formulas leads naturally to focusing proofs. Moreover, seen from a logic programming viewpoint (cf. [Miller 2004]), the conservativity is also important since we have only to work with focusing proofs. In his proof of first order conservativity, Laurent made essential use of the subformula property of LL, which ensures that if a focalized sequent is provable then it is provable with only polarized formulas. When we try to extend conservativity to second order linear logic LL2, we immediately encounter a difficulty with the second order -rule, which results in the loss of the subformula property. For this reason Laurent [2002] has left open the question of whether or not the conservativity result can be extended to second order. In order to answer the above question, we introduce a phase semantics for second order polarized linear logic. The main feature of our polarized phase semantics is its employment of a topological structure, which accommodates the positive/negative polarities as openness/closedness. This interpretation is an algebraic instance of the categorical construction developed in [Hamano-Scott 2007] and is based upon the adjunction between interior and closure operators for the topology. To the best of our knowledge, no formulation of phase semantics to completely characterize provability of polarized linear logic has previously appeared in the literature. We prove strong completeness of polarized linear logic by revising Okada s [1999] method, which implies second order cut-elimination. Then we first show by using a counter model construction that LL2 is not conservative over LL pol 2 (Proposition ), which is a rather unexpected by-product of our polarized phase semantics, and which is a negative answer to Laurent s open question. With this result, it appears that LL2 lacks the central idea of polarity in linear logic mentioned above, and that it offers no bridge between polarity and focalization. In order to remedy this shortcoming, we introduce an η-expanded fragment LL η 2 of LL2, in which atoms are exponential forms (i.e.,!x (resp.?x) for a positive (resp. negative) atom). Such a slight restriction was also adopted by Laurent to show 12

14 a correspondence between polarized linear logic and Girard s LC. Under this slight restriction, the conservativity of LL η 2 over its polarized fragment LL η pol2 (Theorem ) is obtained, which is another positive answer to Laurent s open question. 13

15 Part I Diagrammatic proofs with Euler circles 14

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17 Chapter 1 Introduction to Part I Leonhard Euler [1768] introduced Euler diagrams to represent logical relations among the terms of a syllogism by topological relations, inclusion and exclusion relations, among circles. Given two Euler diagrams which represent the premises of a syllogism, the syllogistic inference can be naturally replaced by the task of manipulating the diagrams, in particular of unifying the diagrams and extracting information from them. For example, the well-known syllogism named arbara, i.e., ll are and ll are C; therefore ll are C, can be represented diagrammatically as in Fig.1.1. C D 1 C C D 2 D v 1 C C Dv 2 C C Fig. 1.1 arbara with Euler diagrams E C E v Fig. 1.2 arbara with Venn diagrams nother well-known diagrammatic representation system for syllogistic reasoning is Venn diagrams. In Venn diagrams a novel syntactic device, namely shading, to represent emptiness plays a central role in place of the 16

18 topological relations of Euler diagrams. ecause of their expressive power and their uniformity in formalizing the manipulation of combining diagrams simply as the superposition of shadings, Venn diagrams have been very well studied. Cf. Venn-I, II systems of Shin [Shin 1994], Spider diagrams SD1 and SD2 of [Howse-Molina-Taylor 2000], [Molina 2001], etc. For a recent survey, see [Stapleton 2005]. However, the development of systems of Venn diagrams is obtained at the cost of clarity of Euler diagrams. s Venn [Venn 1881] himself already pointed out, when more than three circles are involved, Venn diagrams fail in their main purpose of affording intuitive and sensible illustration. (For some discussions on visual disadvantages of Venn diagrams, see [Hammer-Shin 1998, Gil-Howse-Tulchinsky 2002]. See also [Sato-Mineshima-Takemura-Okada 2009] for our cognitive psychological experiments comparing linguistic, Euler diagrammatic, and Venn diagrammatic representations.) Recently, Euler diagrams with shading were introduced to make up for the shortcoming of Venn diagrams: E.g., Euler/Venn diagrams of [Swoboda- llwein 2004, 2005]; Spider diagrams ESD2 of [Molina 2001] and SD3 of [Howse-Stapleton-Taylor 2005]. However, their abstract syntax and semantics are still defined in terms of regions, where shaded regions of Venn diagrams are considered as missing regions. That is, the idea of the regionbased Euler diagrams is essentially along the same line as Venn diagrams. We may point out the following complications of region-based formalization of diagrams: 1. In region-based diagrams, logical relations among circles are represented not simply by topological relations, but by the use of shading or missing regions, which makes the translations of categorical sentences uncomfortably complex. For example, ll are is expressed by a region-based diagram through a translation to the statement There is nothing which is but not as seen in D1 v of Fig The inference rule of unification, which plays a central role in Euler diagrammatic reasoning, is defined by way of the superposition of Venn diagrams. For example, when we unify two region-based Euler diagrams as in D 1 and D 2 of Fig. 1.1, they are first transformed into Venn diagrams D1 v and Dv 2 of Fig. 1.2, respectively; then, by superposing the shaded regions of D1 v and Dv 2, and by deleting the circle, the Venn diagram E v is obtained, which is transformed into the regionbased Euler diagram E. In this way, processes of deriving conclusions are often made complex, and hence less intuitive, in the region-based framework. 17

19 In contrast to the studies in the tradition of region-based diagrams, we proposed a novel approach in [Mineshima-Okada-Sato-Takemura 2008] to formalize Euler diagrams in terms of topological relations. Our system has the following features and advantages: 1. Our diagrammatic syntax and semantics are defined in terms of topological relations, inclusion and exclusion relations, between two diagrammatic objects. This formalization makes the translations of categorical sentences natural and intuitive. Furthermore, our formalization makes it possible to represent a diagram by a simple ordered (or graph-theoretical) structure (cf. Chapter 4). 2. Our unification of two diagrams is formalized directly in terms of topological relations without making a detour to Venn diagrams. Thus, it can directly capture the inference process as illustrated in Fig We formalize the unification in the style of Gentzen s natural deduction, a well-known formalization of logical reasoning in symbolic logic, which is intended to be as close as possible to actual reasoning ([Gentzen 1969]). This makes it possible to compare our Euler diagrammatic inference system directly with natural deduction system. Through such comparison, we can apply well-developed prooftheoretical approaches to diagrammatic reasoning. From a perspective of proof-theory, the contrast between the standpoints of the region-based framework and the topological-relation-based framework can be understood as follows: t the level of representation, the contrast is analogous to the one between disjunctive (dually, conjunctive) normal formulas and implicational formulas; at the level of reasoning, the contrast is analogous to the one between resolution calculus style proofs and natural deduction style proofs. The contrast between the two systems is summarized in the following table: Venn diagrams Our Euler diagrams Representation Region with shading Topological relation Disjunctive normal formulas Implicational formulas x( ), x( ) x( ), x( ) Reasoning Superposition of shadings Unification Resolution calculus style Natural deduction style From a perspective of cognitive psychology, our system is designed not just as an alternative of usual linguistic/symbolic representations; we make 18

20 the best use of advantages of diagrammatic representations so that inherent definiteness or specificity of diagrams can be exploited in actual reasoning. See [Sato-Mineshima-Takemura-Okada 2009] for our experimental result, which shows that our Euler diagrams are more effective than Venn diagrams or linguistic representations in syllogism solving tasks. The rest of this Part I is organized as follows. In Chapter 2, we introduce a topological-relation-based Euler diagrammatic representation system EUL. We first roughly review previous work on Euler and Venn diagrammatic systems, and we make clear our underlying conception for our formalization in Section 2.1. We give a definition of an Euler diagrammatic syntax EUL in Section 2.2 and a set-theoretical semantics for it in Section 2.3. In Chapter 3, we formalize a diagrammatic inference system GDS. We introduce two kinds of inference rules: unification and deletion. We define in Section 3.2 the notion of diagrammatic proof (d-proof, in short), which is considered as a (possibly long) chain of unification and deletion steps. The inference system GDS is shown in Section 3.3 to be sound (Theorem 3.3.1) and complete (Theorem ) with respect to our formal set-theoretical semantics. In Section 3.4, we discuss some consequences of completeness of GDS. In particular, a class of ±-normal diagrammatic proofs of GDS is defined, and a normal form theorem (Theorem 3.4.3) of GDS is shown. ased on the completeness and the normal form theorems, we give a proof-theoretical analysis on structure of diagrammatic proofs. In Chapter 4, we give a graph theoretical representation of EUL-diagrams based on a partially ordered structure, called an EUL-structure. Then, using the graphical representation, we discuss validity of our unification rules of Section 3.2. In Chapter 5, we investigate into a relationship between our EUL-diagrams and Venn diagrams. We first review informally syntax and semantics of Venn diagrams. Then we give a translation of each EUL-diagram into a semantically equivalent Venn diagram. In Chapter 6, we give a characterization of chains of ristotelian categorical syllogisms in our diagrammatic inference system GDS. For the characterization, an important subclass of the ±-normal diagrammatic proofs, called syllogistic normal diagrammatic proofs, are introduced. Then it is shown that each chain of ristotelian categorical syllogisms corresponds to a syllogistic normal diagrammatic proof (Proposition 6.2.3). Furthermore, we also show that the syllogistic fragment of GDS completely characterizes the chains of ristotelian categorical syllogisms (Proposition 6.2.4). Finally, in Chapter 7 we discuss some possible extensions of our system 19

21 and outline some future work. 20

22 Chapter 2 diagrammatic representation system (EUL) for Euler circles and its set-theoretical semantics In this chapter, we introduce a diagrammatic representation system, EUL. In order to characterize our system, we begin with a short review of Euler and Venn diagrammatic representation systems in Section 2.1. Then we introduce the syntax of EUL in Section 2.2, and its set-theoretical semantics in Section ackground Euler diagrams were introduced by Leonhard Euler [1768] to illustrate syllogistic reasoning. In Euler diagrams, logical relations among the terms of a syllogism are simply represented by topological relations among circles. 1 With Euler diagrams the universal categorical statements of the forms ll are and No are are simply represented by the inclusion and the exclusion relations between circles, respectively, as follows: ll are No are 1 Throughout this thesis, we mean by a circle a simple closed curve. 21

23 However, things become complicated when existential statements come into the picture. In Euler s original system, any region in a diagram is assumed to represent a non-empty set, and this existential import destroys the simple correspondence between Euler diagrams and categorical statements. For instance, the diagram of Fig.2.1 can be read as the following four categorical statements (1) (4): Fig Some are, 2. Some are, 3. Some are not, 4. Some are not. John Venn [1881] overcame this difficulty by removing the existential import from circles. Venn fixed such a diagram of Fig.2.1 as a so-called primary diagram, which does not convey any specific information about the relation between and. Thus Venn diagrams can represent partial, not fully specified, information between circles. Meaningful relations between circles are then expressed by specifying which regions are empty with the novel syntactic device of shading, which corresponds to logical negation. Observe that ll are is equivalent to There is nothing which is but not, and the statement is expressed as the following Venn diagram by making use of the shading: ll are in Venn diagram (There is nothing which is but not ) In Venn diagrams, existential claims are expressed by using another syntactic device,, which was introduced by Charles Peirce [1897], and which represents non-emptiness of the corresponding region. Existential categorical statements of the forms Some are and Some are not are represented by using the symbol as follows: Some are Some are not 22

24 In order to make Venn diagrams more expressive, Peirce further introduced another syntactic device: a linear symbol which connects symbols to represent disjunctive information. 2 For example, the following diagram expresses that There is something or : ased on Venn s and Peirce s work, Sun-Joo Shin [1994] formalized a diagrammatic reasoning system called the Venn-II system. The following devices were adopted, which with some modifications came to be regarded as the set of standard devices in subsequent studies: 1. Venn s shading (for emptiness); 2. Primary diagrams (for non-specific information); 3. Peirce s (for non-emptiness); 4. Peirce s linking between s (for disjunctive information on objects); 5. Linking between diagrams (for disjunctive information on diagrams). Furthermore, some options can also be considered: 6. Constant symbols (for existence of particular objects); 7. Linking between constants (for disjunctive information on particular objects). See, for example, [Stapleton 2005, Molina 2001] for surveys of various syntactic devices. The Venn-II system has its own formal semantics and inference rules, and some basic logical properties such as soundness and completeness are proved. Furthermore, Venn-II system is shown equivalent to the monadic first order predicate logic in its expressive power. ecause of its expressive power and its uniformity in formalizing the inference rules, Venn diagrams have been very well-studied, and they have been developed into various systems, such as heterogeneous inference system, [Hammer 1994]; Spider diagrams SD1 and SD2, [Howse-Molina-Taylor 2000, Molina 2001]; etc. For recent surveys, see [Stapleton 2005, Howse 2008]. 2 Cf. [Hammer-Shin 1998] for the other device o, which Peirce introduced in place of Venn s shading. 23

25 Recently, Euler diagrams with shading were introduced: E.g., Euler/Venn diagrams of [Swoboda-llwein 2004, Swoboda-llwein 2005]; Spider diagrams ESD2 of [Molina 2001] and SD3 of [Howse-Stapleton-Taylor 2005]. However, their abstract syntax and semantics are still defined in terms of regions, where shaded regions of Venn diagrams are considered as missing regions. That is, the idea of the region-based Euler diagrams is essentially along the same line as Venn diagrams. We may summarize the underlying conception in the literature of Venn diagrammatic systems as follows: Region-based formalization: Logical relations among terms are represented by shading (or erasing) regions. Emphasis on expressive power: In order to make diagrams as expressive as possible, various syntactic devices are introduced. In contrast to the studies in the tradition of Venn diagrams, we introduce our Euler diagrammatic system based on the following conception: Topological-relation-based formalization: Our diagrammatic syntax and semantics are defined in terms of topological relations between two diagrammatic objects (circles and points). Preservation of visual clarity of diagrams: In order to keep the inherent definiteness or specificity of diagrams, we avoid introducing auxiliary syntactic devices such as shading and linking, which may require arbitrary conventions. (See, for example, [Hammer-Shin 1998, llwein-arwise 1996] for some discussion on the nature of diagrams.) ased on the conception, we start our study on the relation-based Euler diagrammatic representation system EUL by concentrating on the following basic syntactic devices: 1. Inclusion and exclusion relations between two diagrammatic objects (circles and points). 2. Crossing relation between circles, which does not represent specific information between circles as it does in Venn diagrams. 3. Named points (constant symbols) to represent the existence of particular objects. 24

26 Compared with Shin s Venn-II system, our system lacks shading, linking between points and linking between diagrams; and hence our system is weaker than Venn-II in its expressive power. However, it is shown that EUL is expressive enough to characterize syllogistic reasoning. Cf. Chapter Diagrammatic syntax of EUL We introduce the diagrammatic syntax of EUL. Each EUL-diagram is defined as a set of named simple closed curves and named points in a plane. We further consider some equivalence classes of concrete diagrams in terms of topological binary relations, called EUL-relations, between pairs of diagrammatic objects. Let us start by defining the diagrams of EUL. Definition (EUL-diagram) n EUL-diagram is a plane (R 2 ) with a finite number, at least two, of named simple closed curves 3 (denoted by,, C,... ) and named points (denoted by a, b, c,... ), where each named simple closed curve and named point has exactly one name; any two distinct named simple closed curves and named points have different names. EUL-diagrams are denoted by D, E, D 1, D 2,.... In what follows, a named simple closed curve is sometimes called a named circle. Moreover, named circles and named points are collectively called objects, and denoted by s, t, u,.... We use a rectangle to represent a plane for an EUL-diagram. 4 3 See [lackett 1983, Stapleton-Rodges-Howse-Taylor 2007] for a formal definition of simple closed curve on R 2. 4 Several Euler diagrammatic representation systems impose some additional conditions for well-formed diagrams. E.g., at most two circles meet at a single point, no tangential meetings or concurrency etc. Cf. e.g., [Rodgers-Zhang-Fish 2008, Stapleton-Rodges-Howse-Taylor 2007]. However, for simplicity of the definition, those are all considered to be well-formed in our system EUL. (See also well-formed diagrams of EUL in Fig.2.3 below.) 25

27 The following are typical examples of non well-formed diagrams of EUL: a b b D 1 D 2 D 3 D 4 D 5 D 6 Fig. 2.2 Non well-formed diagrams of EUL D 1, D 2 and D 3 respectively consist of at most one object; a circle has two names in D 4 ; two distinct objects have the same name in D 5 and D 6. mong EUL-diagrams, there are particularly simple diagrams which consist of only two objects: Definition (Minimal diagram) n EUL-diagram consisting of only two objects is called a minimal diagram. Minimal diagrams are denoted by α, β, γ,.... We study mathematical properties of EUL-diagrams in terms of the following topological relations between two diagrammatic objects: Definition (EUL-relation) EUL-relations are the following binary relations between diagrammatic objects which have distinct names: b b a b the interior 5 of is inside of the interior of, the interior of is outside of the interior of, there is at least one crossing point between and, b is inside of the interior of, b is outside of the interior of, a is outside of b (i.e. a is not located at the point of b). Observe that EUL-relations and are symmetric, while is not. Note also that all EUL-relations are irreflexive. Each of the EUL-relations is illustrated in the following EUL-diagrams of Fig.2.3: 5 Here, the interior of a named circle means the region strictly inside of. Cf. [lackett 1983]. 26

28 b b b a b b b a b Fig. 2.3 EUL-relations Proposition Let D be an EUL-diagram. For any distinct objects s and t of D, exactly one of the EUL-relations s t, t s, s t, s t holds. More precisely, 1. for any distinct named simple closed curves and, exactly one of,,, and holds; 2. for any named point b and any named simple closed curve, exactly one of b and b holds; 3. for any distinct named points a and b, a b holds. Observe that, by Proposition 2.2.4, for a given EUL-diagram D, the set of EUL-relations holding on D is uniquely determined. We denote the set by rel(d). For example, consider the EUL-diagram D 1 below, composed of named circles,, C, and a named point a. C a D 1 The set of EUL-relations rel(d 1 ) is {, C, C, a, a, a C}. The following properties, as well as Proposition 2.2.4, characterize EULdiagrams. (See also Chapter 4.) 27

29 Lemma Let D be an EUL-diagram. Then for any objects (named circles or points) s, t, u ob(d), we have the following: 1. (Transitivity) If s t, t u rel(d), then s u rel(d). 2. ( -downward closedness) If s t, u s rel(d), then u t rel(d). 3. (Point determinacy) For any point x of D, exactly one of x s and x s is in rel(d). 4. (Point minimality) For any point x of D, s x rel(d). In order to develop our study on mathematical properties of our diagrammatic system, it is convenient to talk about equivalence classes (or types) of diagrams rather than drawn tokens of diagrams. We first identify objects (named circles or points) which have the same name. For example, if s is a circle named by in one diagram and t is a circle also named by in another diagram, then s and t are identified up to topological isomorphism. Intuitively, the circles (resp. points) s and t are intended to represent the same set (resp. element). For diagrams, we define their equivalence in terms of the EUL-relations: Definition (Equivalence of EUL-diagrams) When any two objects of the same name appear in different diagrams (planes), we identify them up to isomorphism. ny EUL-diagrams D and E such that ob(d) = ob(e) are syntactically equivalent when rel(d) = rel(e), that is, the following condition holds: For any objects s, t ob(d) and any {,,, }, s t holds on D iff s t holds on E. For example, the following diagrams D 1, D 2, and D 3 of Fig.2.4 are equivalent since exactly the same EUL-relations, C, C, a, a, and a C hold on them. (Cf. also Chapter 7 (1-1) for an extension of our representation system EUL, where D 1, D 2, and D 3 are distinguished.) On the other hand, D 1 and D 4 (resp. D 1 and D 5 ) are not equivalent since different EUL-relations hold on them: C holds on D 4 in place of C of D 1 (resp. C and C hold on D 5 in place of C and C of D 1 ). Our equation of diagrams may be explained in terms of a kind of continuous transformation (deformation) of named circles, which does not change 28

30 C C C C a a a a C a D 1 D 2 D 3 D 4 D 5 Fig. 2.4 Equivalence of EUL-diagrams. any of the EUL-relations in a diagram. The named circle C in D 1 of Fig.2.4 can be continuously transformed, without changing the EUL-relations with, with and with a in such a way that C covers the intersection region of and as it does in D 2. Similarly, C in D 1 can be continuously transformed, without changing the EUL-relations with, with and with a in such a way that C is disjoint from the intersection region of and as it is in D 3. In what follows, the diagrams which are syntactically equivalent are identified, and they are referred by a single name. We summarize our notations. Named points: a, b, c,... (x, y, z,... for meta variables) Named circles:,, C,... (X, Y, Z,... for meta variables) Objects: s, t, u,... EUL-diagrams: D, E, F,... Minimal diagrams: α, β, γ,... When D is an EUL-diagram, we denote by pt(d) the set of named points of D; by cr(d) the set of named circles of D; by ob(d) the set of objects of D; by rel(d) the set of EUL-relations holding on D. 2.3 Set-theoretical semantics of EUL In this section, we give a formal semantics for EUL. Here, we adopt the standard set-theoretical semantics. 6 Intuitively, each circle is interpreted as a set of elements of a given domain, and each point is interpreted as an 6 For similar set-theoretical approaches to semantics of Euler diagrams, see [Hammer 1995, Hammer-Shin 1998, Swoboda-llwein 2004, Howse-Stapleton-Taylor 2005] etc. Our semantics is distinct from theirs in that our diagrams are interpreted in terms of binary relations, and not every region in a diagram has a meaning. 29

31 element of the domain. However, observe that each point of EUL can be considered as a special circle which does not contain, nor cross, any other objects. This observation enables us to interpret the EUL-relations and uniformly as the subset relation and the disjointness relation, respectively. Definition (Model) model M is a pair (U, I), where U is a nonempty set (the domain of M), and I is an interpretation function which assigns to each diagrammatic object s a non-empty subset of U such that I(x) is a singleton for any named point x; I(x) I(y) for any points x, y of distinct names. Definition (Truth-condition) Let D be an EUL-diagram. M = (U, I) is a model of D, written as M = D, if the following truth-conditions (1) and (2) hold: For all objects s, t of D, (1) I(s) I(t) if s t holds on D, (2) I(s) I(t) = if s t holds on D. Note that when s is a named point a, for some e U, I(a) = {e}, and the above I(a) I(t) of (1) is equivalent to e I(t). Similarly, I(a) I(t) = of (2) is equivalent to e I(t). Remark (Semantic interpretation of -relation) y Definition 2.3.2, the EUL-relation does not contribute to the truth-condition of EULdiagrams. Informally speaking, s t may be understood as I(s) I(t) = or I(s) I(t), which is true in any model. The well-definedness of the truth-conditions in Definition follows from Proposition 2.2.4, which ensures that the EUL-relations holding on a given diagram D are uniquely determined. Definition (Validity) n EUL-diagram E is a semantically valid consequence of EUL-diagrams D 1,..., D n, written as D 1,..., D n = E, when the following holds: For any model M, if M = D 1 and... and M = D n, then M = E. Let D be an EUL-diagram. Let β be a minimal diagram consisting of two objects s and t which is obtained from D by deleting all objects other than s and t. Then, by definition, we have D = β. (See also Section ) 30

32 Chapter 3 Diagrammatic inference system GDS In this chapter, we introduce Generalized Diagrammatic Syllogistic inference system GDS for the EUL-diagrams defined in Section 2.2. There are two inference rules of GDS: unification and deletion. We first give an informal explanation of our unification in Section 3.1, and we then formalize it in Section 3.2. We give an inductive definition of diagrammatic proofs of GDS as is usual in the study of symbolic logical systems. In Section 3.3 our GDS is shown to be sound and complete with respect to the set-theoretical semantics given in Section 2.3. In Section 3.4, we discuss some consequences of the completeness theorem of GDS. In particular, we define a class of normal diagrammatic proofs of GDS and we show a normal form theorem. 3.1 Introduction to unification efore giving a formal description of our diagrammatic inference system, we motivate our inference rule unification. Let us consider the following question: Given the following diagrams D 1, D 2 and D 3, what diagrammatic information on, and c can be obtained? (In what follows, in order to avoid notational complexity in a diagram, we express each named point, say c, simply by its name c.) c c D 1 D 2 D 3 31

33 c c c c c c D 1 c D 2 D 3 D 1 D 2 D 3 3 c c D 1 D 2 c c D 3 D 1 + D 2 c (D 1 + D 2 ) + D 3 Fig. 3.1 D 1 + D 3 c D 2 + (D 1 + D 3 ) Fig. 3.2 D 2 + D 3 c D 1 + (D 2 + D 3 ) Fig. 3.3 Figs.3.1, 3.2, and 3.3 represent the three ways of solving the question. In Fig.3.1, at the first step, two diagrams D 1 and D 2 are unified to obtain D 1 + D 2, where the point c in D 1 and D 2 are identified, and is added to D 1 so that c is inside of and overlaps with without any implication of a relationship between and. Then, D 1 + D 2 is combined with another diagram D 3 to obtain (D 1 + D 2 ) + D 3. Note that the diagrams D 1 + D 2 and D 3 share two circles and : holds on D 1 + D 2 and holds on D 3. Since the semantic information of on D 3 is more accurate than that of on D 1 + D 2, according to our semantics of EUL (recall that means just true in our semantics), one keeps the relation in the unified diagram (D 1 + D 2 ) + D 3. Observe that the unified diagram represents the information of these diagrams D 1, D 2, and D 3, that is, their conjunction. Figs.3.2 and 3.3, illustrate other procedures to solve the question. t the first step of unifying diagrams D 1 and D 3 in Fig.3.2 (and D 2 and D 3 in Fig.3.3), there are two possible positions of the point c. However, EULdiagrams do not have syntactic devices to represent such disjunctive information about positions of a point. One solution to this problem is, as illustrated in Figs.3.2 and 3.3, to introduce Peirce s linking of points. However, following the conception we explained in Section 2.1, we keep our diagrams free from such disjunctive ambiguity. For that purpose, we impose some constraint on unification, called the constraint for determinacy: ny two diagrams are not permitted to be unified when the relations between each point and all circles of the two diagrams are not determined. Thus D 1 and D 3 of Fig.3.2 (respectively D 2 and D 3 of Fig.3.3) are not permitted to be unified. We impose another constraint on unification called a constraint for consistency, in order to avoid complexity due to conflicting graphical informa- 32

34 tion represented in a single diagram. For example, it is not permitted to unify two diagrams D 1 and D 2 when, as is shown in Fig.3.4, they share two circles C and such that a C and a hold on D 1 and C holds on D 2. Note that these relations a C, a, and C are incompatible in the same diagram. The diagrams D 3 and D 4 in Fig.3.4 are also not permitted to be unified in our system. Recall that each circle is interpreted by non-empty set in our semantics of Definition 2.3.1, and hence D 3 and D 4 are also incompatible. C a C D 1 D 2 D 3 D 4 Fig. 3.4 Inconsistency 3.2 Generalized diagrammatic syllogistic inference system GDS In this section, we introduce unification and deletion of GDS. We formalize our unification of two diagrams by restricting one of them to be a minimal diagram, except for one rule called the Point Insertion-rule. Our completeness (Theorem ) ensures that any diagrams D 1,..., D n may be unified, under the constraints for determinacy and consistency, into one diagram whose semantic information is equivalent to the conjunction of that of D 1,..., D n. (We will return to this issue in Section ) We give a formal description of inference rules in terms of EUL-relations: Given a diagram D and a minimal diagram α, the set of relations rel(d + α) for the unified diagram D + α is defined. It is easily checked that the set rel(d + α) satisfies the properties of Lemma 2.2.5, and hence there is a concrete instance of the set. (See also Chapter 4, where we give a graphical representation of unification.) We also give a schematic diagrammatic representation and a concrete example of each rule. In the schematic representation of diagrams, to indicate the occurrence of some objects in a context on a diagram, we write the indicated objects explicitly and indicate the context by dots as in the diagram to the right below. 1 For example, 1 Note that the dots notation is used only for abbreviation of a given diagram. For a formal treatment of such backgrounds in a diagram, see, for example, [Meyer 2001]. 33

35 when we need to indicate only and c on the left hand diagram, we could write it as shown on the right. D E b F c c Definition (Inference rules of GDS) xiom, unification, and deletion of GDS are defined as follows. xiom: 1: For any circles and, any minimal diagram where holds is an axiom. 2: ny EUL-diagram which consists only of (at least two) points is an axiom. Unification: We denote by D + α the unified diagram of D with a minimal diagram α. D + α is defined when D and α share one or two objects. We distinguish the following two cases: (I) When D and α share one object, they may be unified to D + α by rules U1 U8 according to the shared object and the relation holding on α. Each rule of (I) has a constraint for determinacy. (II) When D and α share two circles, if the relation which holds on α also holds on D, D + α is D itself; otherwise, they may be unified to D + α by rules U9 or U10 according to the relation holding on α. Each rule of (II) has a constraint for consistency. Moreover, there is another unification rule called the Point Insertion-rule (III). (I) The case D and α share one object: U1: If b holds on α and pt(d) = {b}, then D and α may be unified to a diagram D + α such that the set rel(d + α) of relations holding on it is the following: U1 is applied as follows: rel(d) {b } { X X cr(d)} D b b U1 D + α 34 b α C b D 1 U1 C b D 1 + D 2 b D 2

36 U2: If b holds on α and pt(d) = {b}, then D and α may be unified to a diagram D + α such that the set rel(d + α) of relations holding on it is the following: U2 is applied as follows: rel(d) {b } { X X cr(d)} D b b U2 b α D 1 C b b U2 D 2 C b D + α D 1 + D 2 U3: If b holds on α and cr(d), and if X or X holds for all circle X in D, then D and α may be unified to a diagram D + α such that the set of relations rel(d + α) is the following: U3 is applied as follows: rel(d) {b } {b X X rel(d)} {b X X rel(d)} {b x x pt(d)} b C b D U3 b α D 1 U3 D 2 C b D + α D 1 + D 2 U4: If b holds on α and cr(d), and if X holds for all circle X in D, then D and α may be unified to a diagram D + α such that the set of relations rel(d + α) is the following: U4 is applied as follows: rel(d) {b } {b X X rel(d)} {b x x pt(d)} 35

37 b b D b U4 α D 1 U4 b D 2 D + α D 1 + D 2 U5: If holds on α and cr(d), and if x holds for all x pt(d), then D and α may be unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { } { X X or X rel(d)} U5 is applied as follows: { X X rel(d)} { X X rel(d)} {x x pt(d)} F E C D U5 α D 1 F U5 D 2 E C D + α D 1 + D 2 U6: If holds on α and cr(d), and if x holds for all x pt(d), then D and α may be unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { } {X X or X or X rel(d)} U6 is applied as follows: {X X rel(d)} {x x pt(d)} E C D U6 α D 1 U6 D 2 C E D + α 36 D 1 + D 2