Singular Propositions and their Negations in Diagrams Lopamudra Choudhury1, Mihir Kumar Chakraborty2, 1 School of Cognitive Science, Jadavpur University 2 Visiting Professor, School of Cognitive Science Jadavpur University Jadavpur, Kolkata 700032 India 1 choudhuryl@yahoo.com 2 mihirc4@gmail.com Abstract— This paper deals with the visual representation of S is non-P‘) and ‗Not some S is P‘( equivalently ‗No S is P‘) negation involving particular propositions. The underlying are particular negative and universal negative respectively. consideration depends on the three basic desirable aspects of The negation of a universal proposition is a particular visual representation viz. simplicity, visual clarity and proposition and vice versa and the negation of an affirmative expressiveness [9]. For incorporation of constants in diagrams we proposition is a negative proposition and vice versa. In this discuss Venn-i (2004), Swoboda’s diagrams (2005) and Spider diagrams with constants (2005). We also discuss representation sense we can say that the class of categorical propositions is of negation in these diagrams. To depict negation in Venn-i the closed with respect to negation. concept of absence is brought in from the conceptual schema of Traditionally singular propositions like ‗a is P‘ is Indian philosophy. The advantage of Venn-i over spider diagram considered to have the syllogistic form (b) as ―All a‘s are P‖. is discussed. The notion of absence naturally calls for the concept But sentences of the form (b) admit both contradictories and of an open universe. A brief discussion on open universe is contraries, while sentences of the form (a) have only presented at the end. contradictories. The two sentences (a) and (b) behave differently under negation. So their logical forms are to be Keywords- singular propositions, negation, absence, simplicity, considered different. Less attention has been paid to this fact. open universe For an ongoing debate on the issue we refer to the site: http://www.tandfonline.com/doi/abs/10.1080/0004840538520 I. INTRODUCTION 0151. Negation of sentences like Pa makes the distinction Negation plays a crucial role in all logics [12]. The notion between (a) and (b) more transparent. We are however of inconsistency and contradiction are primarily understood in interested in the representation of sentences of the form (a) terms of negation. Although there is the notion of absolute and their negations rather than the debate. From this inconsistency that does not involve negation and that turns out standpoint it would be clear that (a) should be considered to be equivalent to inconsistency in classical logic, for a differently from (b) if the desirable parameters of diagrams pictorial representation absolute inconsistency is not useful. In viz. simplicity, visual clarity and expressiveness are to be diagram logic inconsistency is attributed to a diagram whereas taken into account. It should be mentioned that the above contradiction is a relation between diagrams. Of course two three qualities of diagrammatic representation is to be diagrams are contradictory if and only if the conjoined understood only informally. diagram is inconsistent. A set of propositions is absolutely inconsistent if any proposition what so ever follows from it. It is interesting to note that although representation of This notion originally proposed by Peirce, is presently called logical propositions through diagrams and reasoning thereby ‗explosiveness‘ and has gained much importance after the is an issue that has engaged pioneering logicians like Euler L advent of paraconsistent logics [6]. In classical logic one deals (1707-1783), Venn J (1834-1923) and Peirce C S (1839-1914) with the following three types of basic propositions: for more than three centuries it is only recently that researchers are paying serious attention to representation of (a) a is P: ( P a) where a represents an individual, propositions of type (a) as well as their negations. Diagram (b) all S is P: ( x Sx Px)), systems, as alternative system of reasoning faces the challenge (c) some S is P: ( x Sx Px)). of incorporating all the basic items that are involved in logical reasoning. The latter two types and their negations constitute Aristotle‘s categorical propositions which were used In section 2 diagrams involving individuals are presented. specifically for syllogistic reasoning. Types (b) and (c) are In section 3 negations of diagrams involving individuals are known as universal affirmative and particular affirmative discussed in detail. Section 4 deals with open universe. propositions respectively. ‗Not all S is P‘ (equivalently ‗some Section 5 consists of some concluding remarks. 43 II. DIAGRAMS WITH INDIVIDUALS It is already mentioned that sentences of the form Pa or Home School singular propositions were treated as universal propositions. In this approach instead of individuals, the subject term was viewed as a singleton set. We have also stated before that this Jill Jill treatment is somewhat problematic. Representation of individuals was however not dealt with in the diagrams of Euler[11], Venn – Peirce[7,2], Shin[15] and Hammer[3]. Fig.3 Propositions like ‗Socrates is not mortal‘ cannot be expressed in any of these diagrams. As a matter of fact, Fig. 1 and Fig. 3 are similar. Recent incorporation of individuals in diagrams may be II.I REPRESENTATION OF INDIVIDUAL IN VENN-I discussed now from the perspective of Venn-i [8](2004), Swoboda‘s system [13](2005) and Spider diagram[4] (2005). Spider diagrams are extensions of Venn II diagrams [15]. In Venn-i we adopted Venn-Peirce convention of expressing universal and particular propositions. A rectangle In spider diagrams centrally connected clusters of nodes called is used to represent the universe. Besides, Venn-i introduces spiders are used for representing individuals. two additional diagrammatic objects: one is constant symbol Spiders are of two types: existential spiders expressed by representing individual and the other absence of individual. round nodes and constant spiders expressed by square nodes. An individual‘s presence in a region is represented by placing Existential spiders are similar to the sign x introduced by Peirce the difference being in the nature of connectivity. the individual‘s name in that region (Fig.4). Absence of an individual a is represented by placing ā in that region (Fig. Constant spiders are labelled and are similar to constants in 16). Additionally, broken lines are introduced to express first order logic. The habitat of a spider is represented by possibility (intended exclusive disjunction) of an individual‘s placing nodes at different regions and joining them in pairs by presence in different regions. lines all originating from one of the nodes. Distinct spiders Before entering into the discussion on representation of represent distinct elements. Following is an example: Constant spider: ‗Web is either a dog and not a cat or Web negation we shall make some general observations. An individual is assumed to occur along with some properties or is a cat and not a dog‘ (Fig. 1) predications (positively or negatively). In diagrammatic representations of Venn-i, Spider or Swoboda this means that Dog Cat each individual ‗a‘ shall occur within the region of a closed curve or its complementary region in the universe. On the Web other hand a closed curve can be drawn without any individual depicted in the rectangle. The simplest picture with individual hence, would contain one closed curve and one individual. Thus there are two possibilities: Fig.4 or Fig.5 Fig. 1 Thus in spider the singular proposition Pa will be represented by: P P P a a a Fig.2 Fig.4 Fig.5 Swoboda‘s (2005) system is again an extension of Venn II These pictures may be extended with one individual and two system by incorporating constants. predicates giving rise to pictures in the series: Following is an example representing ‗Jill is either in home and not in school or Jill is either in school and not in home‘. Fig.6 44 representations of negative statements of the above form viz. by using ā in M and by using broken lines showing the possible alternative locations of a in the complement MC in the universe. The basic advantage had been simplicity and directness of visual representation. The completeness proof had been carried out on the basis of this equivalence. However, during subsequent developments we have noticed a deeper significance of the use of the symbol ‗ā’ and have Fig.7 preferred to shift from this equivalence. The subtle difference between the two will be discussed in the following section. For a formal presentation of the alphabet and formation rules we refer to [10]. A cue to the depiction of absence by a symbol directly may be traced in the knowledge system of ancient India. The Indian logicians (Nyaya Vaisesika thinkers) admit a distinct ontological category called abhāva (absence) with a view to Fig.8 accounting for negative statements [14]. It is important to note that absence was also considered to be real. The fact that an absence is always an absence of some entity shows that absence as a category presupposes the existence of positive entities. Absence has to be admitted as the object of negative form of cognition. Russell in his ―Philosophy of Logical Atomism‖ maintains similar view when he considers two kinds of atomic facts: Fig.9 positive atomic fact and negative atomic fact. To quote from o Russell ―.. I think you will find it better to take negative facts A further extension with more than one individuals and as ultimate. Otherwise you will find it so difficult to say what more predicates is possible in a natural way. In spider it is that corresponds to a proposition. When, e.g., you have a diagrams the pictures would remain the same except in that false positive proposition, say ‗Socrates is alive‘, it is false they would use square dots for each individual along with because of the non-correspondence between Socrates being their names. The difference will be noticed in depicting their alive and the state of affair. A thing cannot be false except negations which will be treated in section III. But we need to because of a fact, so that you find it extremely difficult to say say a few words about the representation of the absence of an what exactly happens when you make a positive assertion that individual. The connection between absence and negation will it is false, unless you are going to admit negative facts‖.([1] be discussed in the following section. p.214). That the absence of Socrates belongs to the collection of living humans may be considered as a negative fact. In So long in the diagram literature there was no symbol to Venn-i this fact is directly depicted. represent the absence of an individual. Placing individual in the complement would indicate its absence from the class as III. NEGATION OF DIAGRAMS WITH INDIVIDUALS free ride. In the context of closed universe i.e when the discourse is limited within a fixed universe represented by the We now consider negation of diagrams involving individuals. rectangle in diagrams, mark of absence goes along with In Spider diagrams Pa is represented as Fig.10 simplicity of representation. Say for example, when the teacher marks the attendance of the students and a student is not found in the class, puts absence mark against the student meaning there by that the student is not present in the class — the teacher is thus depicting the absence of the particular student. a with an upper bar (ā) placed within M represents literally that absence of a belongs to M that is, not that a belongs to M. Fig.10 Thus absence of an individual is used for negating some ~P(a) is represented as Fig.11 predication about the individual. In the system Venn-i this semiotic device should be considered as an additional means to represent negation. It is to be noted that when the paper was published in 2004, the diagram system Venn-i considered only classical negation. The negative statement ~(a M) has been taken as equivalent with ā M. There have been two diagrammatic Fig.11 45 which is equivalent to Fig.12 And ~ (P(a)&Q(a)) would be the following Fig.18 Fig.12 Extension of this with two predicates would be Fig. 18 P(a)&Q(a) (vide Fig.13) and which is equivalent to Fig.19 Fig.13 Fig.19 ~(P(a)&Q(a)) as Fig.14 One may wonder what might be the advantage of representing negation of singular proposition by using ā-like diagrammatic entities. Let us consider the following cases: Case1. Negation of the content a S ∩P∩ M in Spider would be S P Fig.14 which is equivalent to Fig. 15 M P Q Fig. 20 which is equivalent to: Fig. 21 S P Fig. 15 In Venn-i P(a) is represented as Fig.4 and ~P(a) as Fig. 16 M Fig. 21 The first diagram does not convey information to our cognition immediately. While the second one is quite Fig.16 complex, the complexity will increase with the increase in the which is equivalent to Fig.5 number of predicates and individuals. Whereas in Venn-i its representation of the same information is Fig.22 which says With two predicates P(a)&Q(a) it would be Fig.17 directly that ‗a‘ is absent in S ∩P∩ M. Fig.17 46 S a P S P ā b Fig.26 M S P Fig. 22 a b In Venn-i there is an equivalent representation with dotted lines. M S P Fig.27 a a a Case 3. a a a In order to express both a and ā in the same region the a representation in Venn-i (Fig.28) and Spider(Fig.29) can be M compared Fig.23 This diagram is visually more elegant than the spider (Fig. 21) S P since one has to intersect lesser number of bordering curves. One may argue that the visual complexity of the figure in our aā diagram will increase if the location of absence in our diagram M will increase i.e if the absence of ‗a‘ is to be depicted in many zones. But since we have at our disposal the sign to represent Fig.28 presence too, it is possible to have a trade-off and decide S P which diagram to take. a a M Fig.29 Fig. 24 Apart from visual simplicity of Fig.28 over Fig. 29, the direct Thus Fig.24 may be replaced by Fig 25 since a is absent in cognition of contradiction in the first diagram may also be more zones than its possible presence. taken into account. S P a a IV. MODIFICATION OF THE INTERPRETATION OF ABSENCE IN VENN-I M Fig. 25 So long we have interpreted ā P equivalently with a PC. But we have mentioned in the introduction that such a Case 2. representation has a deeper significance. With increase in the number of individuals the picture loses its In the modified version we assume that absence of a in the set visual clarity in spider diagrams. For example, one can M does not necessarily imply that a belongs to the compare the following diagrams Fig.26 of spider and Fig. 27 complement of M with respect to the universe although a is in of Venn-i. M implies that absence of a viz. ā is in the complement of M. Thus from Fig.4, in Venn-i modified follows Fig.30. M ā Fig.30 but not conversely. 47 Also from Fig. 5 follows Fig.16 not conversely. complementation. In such a situation, negation of ā P turns The main idea behind this unidirectionality lies in the into the presence of the absence of a in P. Absence in Venn-i following consideration. From the fact of a M it is outright (closed universe) draws complement as free ride. But Venn-io inferred in classical set theory and logic that a belongs to the (the system for open universe [10]) does not allow the free complement of M irrespective of the fact whether a is ride. What is given is exactly an individual not appearing in a locatable in the complement M or not. Actual locatability is set. The open universe and absence would change the our concern. So if we do not see a in M, we do not infer that a ontology of the Euler Venn diagram. is in the complement of M until or unless a becomes locatable in the complement of M. For a more detailed discussion we Incorporation of absence leads towards the admission of the refer to [8, 10]. third possibility viz. the ‗know not‘ situation. Introduction of At this point the notion of absence gains significance and open universe along with the absence of a particular will now there are two kinds of negation viz. classical negation render the diagrammatic system more natural language and absence. Let us state our conventions regarding friendly in the sense that we will be able to talk about individuals and their absence. First, an individual a cannot fictitious objects, like ghosts or fairies, unidentified objects of occur in more than one regions in a diagram but ā can occur. science fiction like UFO or life in other planets etc. However, in case of closed universe ā cannot occur in all regions because in that case a would be a non-entity. VI. CONCLUSION Secondly, there is no notion like absence of absence, there is In conclusion we present a summary. We have attempted to no diagrammatic object like a with double bar in the system represent negation ( of sentences of type Pa) by the absence of Venn-i. This feature particularly makes absence different from a (ā ) in P. This representation gives more visual clarity. This intuitionistic negation although there is a flavour of representation pertains to the philosophical position of constructivity in the above mentioned aspect of locatability. considering absence as a positive category similar to abhāva of Indian philosophy and Russell‘s negative fact. Diagrams With this standpoint one can see that there emerge two with absence of individuals represent negation in a way term types of negations in Venn-I modified: negation is used in logic with the exception that here negation (absence) is placed with the subject term which is the name of The modified interpretation of ā is compatible with the notion an individual. In the context of open universe, representation of open universe to be discussed in Section V. of negation by absence seems to be an essentiality. A more formal treatment of the notion of absence is presented in [10] V. OPEN UNIVERSE however a formal way of measuring the clutter of a diagram and comparing various systems in terms of these definitions The representation of absence gives way to the notion of are still open issues. open universe [10], where the description of the universe admits to be incomplete. A reading of the information given in ACKNOWLEDGMENT Fig. 16 may be thus: we do not see a in P but know not This research is supported by DIT funded NPPE project and where. In the context of open universe the notion of absence UGC UPEII under the School of Cognitive Science, Jadavpur becomes more significant. Absence of a in P does not University, Kolkata, India. necessarily imply a is in the complement of P since the We are thankful to Mr Arindam Bhattacharya Project Fellow complement is not known because the universe is open. (UPEII) for composing the diagrams for us. . In depicting open universe, diagrammatically [10], there is no rectangle outside the closed curve indicating that the domain REFERENCES of discourse is not fixed. Objects here continuously appear [1]. B. Russell, Philosophy of Logical Atomism, in Logic and and disappear; at one instant it is existent and may be Knowledge, Unwin Hyman,1988. [2]. C.S.Peirce (1933), Collected Papers of C.S.Peirce, Vol. iv, HUP. nonexistent at another instant. Here, a A does not necessarily [3]. E.Hammer (1995), Logic and Visual Information, CSLI Pubs. imply a B for some B since, although ‗a‘ had been an object [4]. G. Stapleton (2005) A survey of Reasoning System Based on of the universe, or because of the universe being in flux, ‗a‘ Euler Diagrams, Electronic Notes in Theoretical Computer may have disappeared altogether. In the classical case B is AC. Science, www.elsevier.nl/locate/entcs [5]. G.Stapletion (2007), Incorporating negation into visual logics: A This means that it becomes meaningful to assert the law of case study using Euler diagrams, visual language and Computing excluded middle a A or a A although the latter does not pp 187-194 entail that a AC. In fact, AC is not at all determined since [6]. J Y Beziau et al ed (2007) Handbook of Paraconsistency, King‘s College Publication, London there is no fixed universe or even if fixed initially, it is subject [7]. J.Venn (1880), On the diagrammatic and mechanical to change. Absence of an individual here in this room does not representation of propositions and reasonings. The London, entail her presence outside, she might nt be locatable or she Edinburg and Dublin Philosophical Magazine and Journal of might have disappeared altogether. It should be realized that Science. [8]. L. Choudhury and MK Chakraborty (2004), On Extending Venn once open universe is accepted, classical negation fails to Diagram by Augmenting Names of Individuals, A. Blackwell et al operate since there exists no notion of absolute (Eds): Diagrams 2004, LNAI 2980 p.142-146. 48 [9]. L. Choudhury and MK Chakraborty (2005), Comparison between [12]. L.R Horn (2001) A Natural History of Negation, CSLI Pub Spider Diagrams and Venn diagrams with Individuals Proceedings [13]. N. Swoboda (2005), Heterogenous Reasoning with Euler/ Venn of the workshop Euler Diagrams 2005, INRIA, Paris. Diagrams Containing Named Constants in FOL, Electronic Notes [10]. L. Choudhury and MK Chakraborty (2012)On representing Open in Theoretical Computer Science Universe, Studies in Logic, Vol 5, No 1, pp 96-112. [14]. S.Dutta (1991), The Ontology of Negation, Jadavpur Studies in [11]. L.Euler(1761), Lettres a une princesse d‘allemagne . Sur divers Philosophy, in collaboration with K.P. Bagchi and Co,. sujets de physique et de philosophie, letters No. 102-108 vol 2 [15]. S.J.Shin (1994), The Logical Status of Diagrams, CUP. Basel, Birkhauser. 49