Study on non-antisymmetric mereology

Study on non-antisymmetric mereology Lidia Obojska1 December 2017

1 Siedlce University of Natural Science and Humanities, Dept. of Mathematics and Physics, ul. 3 Maja 54, Siedlce, POLAND; e-mail: [email protected]

Introduction The presented monograph is a modified and shortened English version of [97]. The field of research comes from foundations of mathematics to history, philosophy and logic. The work presents new formal results in the study of collective sets began by Stanisław Leśniewski in 1916 and continued by Boleslaw Sobociński (Leśniewski’s disciple) and his group at the University of Notre Dame. One can ask a question whether it is still worth to spent time on Leśniewski’s sets if so much has already been said on this subject? We can quote Jan Łukasiewicz, an outstanding logician and a friend of Leśniewski: Outcome [. . . ] should be constantly controlled with data taken from intuition and experience and results of other sciences, especially natural sciences. In case need arises, the system shall be improved with newly formed axioms and assumed new primitive notions. We shall take persistent care to be in touch with the reality in order not to create mythological beings of the type of Plato’s theory of Ideas (Forms) or things-in-itself by Kant, but rather to recognize the essence and construction of the real world in which we live and act and which we want to transform into a better and more perfect place 1 . Recent several dozens of years constitute a period of intense develop-

1

In [p. 56][141], [145].

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ment of natural sciences. Some quanta phenomena, e.g. quantum entanglement, seem to suggest the need of modification of certain formal systems describing reality. Some kind of a theory of relations might be useful here since it seems that there is a specific relationship between entangled particles. In such a state a pair of particles interacts in a very specific manner. Due to quantum mechanics laws, their shared state remains undefined until it is measured. At the moment when we measure this state and define the value of one of the particle in the pair – e.g. its spin, then the other particle from the ”entangled” pair assumes a value exactly correlated with the first one. This correlation is independent of the distance separating the particles, the fact that was very intriguing to Einstein [24], even if the distance is large, and even if we change the characteristics of measurement devices. In the physics environment we have witnessed and we still witness debates on the interpretation of this communication mechanism [107]. Modern physics searches for new solutions [30]. Of course we do not aspire to provide complete answers, but we would like to suggest a certain theory; as Alfred Tarski said: ”some new experiments of a very basic nature may make us change certain axioms of the logic itself”, and as Murawski comments [94, p. 139]: ”New results of the quantum mechanics, according to Tarski, seem to provide such possibility”. Perhaps a modified theory of parts might be applied; we have a certain whole composed by two parts, with perfect symmetry, where these two parts are different. On the abstract level of thought, let us think of two integer numbers: 2 and -2 and of the relation of divisibility: 2 can be divided by -2 and -2 by 2, but they are different. In mathematics, the antisymmetry property which is considered to be natural and very intuitive, will lead us to the conclusion that 2 and -2 are the same number, but they are not. Perhaps we should neglect this ”natural” property and investigate what happens in such a case. These formal investigations on such a theory, we called non-antisymmetric mereology, since mereology is a part-whole theory and we reject the antisymmetry property. The mereology 2 (the theory of collective sets) was founded by Stanisław Leśniewski and presented for the first time in 1916. Its complete presenta-

2 The term mereology comes from Greek word: µερoς – part. We have different formal systems describing mereology – Leśniewski’s mereology (1927-1931), the calculus on indi-

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tion was done in the period of 1927–1931 [63] [64], [66], [69], [71] in Przegląd Filozoficzny 3 . The theory was founded on the primitive concept of part relation, called by Leśniewski ingredient, which partially orders the universum of objects4 . Mereology was investigated by a lot of authors either in Poland or abroad [115], [120], [122], [55], [98], [27], [57], [18], [111], [138], and it was shown that it corresponds to a model of Boolean algebra without a zero element. This theory is also called an extensional mereology because an extensional principle is conserved. Extensional mereology is exposed to different critics. Some people consider it as a non-realistic theory and state that it does not deal with such concepts as ’parts of time’, hence it is difficult to expand it. Others do not accept the mereological axiom of identity of objects stating that objects having the same parts have to be identical. As Peter Simons states „...human beings, have different parts at different times: they are mereologically variable or in flux. An object with different parts at different times cannot be identical with the sum of its parts at any time, for then it would be different from itself” 5 ? We also doubt in extensional property of mereology: two objects can be different but can have the same proper parts. Judith Thomson gave an example of a bouquet of flowers which completely depends on the composition of singular flowers [132]. Recently, Aaron Cotnoir [21] published works concerning non-wellfounded mereology which is a fully non-extensional theory, therefore, according to Leśniewski, it is devoid

viduals of Woodger (1937), Leonard and Goodman (1940), Goodman and Quine (1947), etc. [141, p. 150], [145]. However, Stanisław Leśniewski was the first who gave the formal description of this theory. For the purpose of this work we will investigate mereology as a non-elementary theory, hence we will use the language of logic of the 2nd order. In the case we would like to refer to other models of mereology, we will explicitly write it. 3 English translation one can find in [123]. 4 The term ’part’ will be understood in a universal way, i.e. since Mazovia is part of Poland, and Poland – part of Europe, hence Mazovia is part of Europe. For the purpose of this work, the term ’part’ will always denote an improper part, and not a proper part how it was intended by Leśniewski. This is because of aesthetic reasons; it is easier to speak about a theory of parts than about a theory of ingredients, but the presented work will in fact deal with a theory of ingredients. 5 [111, p. 1] Simons using the term ’part’ has in mind a proper part, of course. Since in the presented book the term ’part’ which will mean always improper part is a primitive notion, hence we will omit the adjective ’improper’ and simply write: part.

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of sense. In the study we propose, it turns out, that if the primitive notion in a theory is understood as the relation of a part (i.e. improper part) which is only quasi-ordering, that is we reject the antisymmetry condition, then the extensionality principle does not have to be conserved [96], hence nonantisymmetry is not interchangeable with non-extensionality. Moreover, if we introduce some basic algebraic concepts, a notion of a collective set seems to be more fundamental to the classical concept of a set defined by George Cantor. In other words, this may suggest that a relational (i.e. collective) structure is more fundamental than distributive. It is worth to mention that Leśniewski alone was an extensionalist that is why he did not take into account non extensional models [93, p. 105], [94]. However, S. Leśniewski, J. Łukasiewicz, A. Tarski and many others belonged to the great tradition called the Warsaw School of Logic characterised with the fact that scientific research was not subjected to any philosophical opinion [93, p. 154], [94]. Consequently, irrespective of opinions, we may try to allow for certain mereology variants on only one condition — that new postulates are not contradictory to the whole theory. Until now, there is no systematic development of non-antisymmetric mereology making the topic worth our attention. Peter Simons [111] describes a small excerpt of non-extensional mereology by analysing the symmetric relation of coincidence and taking into consideration continuous objects, which have the same parts in the same time. Aaron Cotnoir [20] analyses the problem of indiscernibility of objects, taking into consideration various definitions of a proper part. He provides an outline description of a model of non-wellfounded mereology [21], starting with the notion of proper part and imposing it on the condition of reflexivity and transitivity. It may be observed, that recently various theories concerning non-classical models of mereology enjoy quite close attention. Hence, the idea to develop a non-antisymmetric mereology systematically, in details, was born. In the presented work Chapter 1 offers an outline of historical background with Stanisław Leśniewski and his works. It will provide more infor-

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mation about the Warsaw School of Logic and reasons for which mereology was developed. Chapter 2 consists of two main parts. The first two sections present the main features of the relation of being an element and of composing a whole, either in classical or in mereological sense. The second part is a very concise, formal summary of three various manners of modelling mereology (following Stanisław Leśniewski), but with modern approach, realised by the author of this monograph. We will not use the original formalism by Leśniewski, since it might be not understood, but rather the classical notation, applying principles of deduction and proofs. Hence, some knowledge in the proof theory and logic is required from a reader. We will define the theory founded on the relation of part, proper part and disjointedness. We will provide proof for equivalence for three theories and for various mereology notions applied by other authors, for example, the idea of fusion [27], Goodman’s fusion, sum [26] and various definitions of proper part [139]. At the end we will discuss the extensionality principle and its consequences. Chapter 3 will introduce basic terms of non-antisymmetric mereology. We will analyse the fulfilment of supplementation principles, the extensionality principle depending on various definitions of the proper part. At the end we will introduce basic algebraic notions and we will indicate relations of collective sets with distributive sets with the use of those operators.

Chapter 1

Stanisław Leśniewski and the Lvov-Warsaw School of Philosophy 1.1

Lvov-Warsaw School (1895-1939)

Kazimierz Twardowski (1866–1938)1 is recognized as the founder of the Philosophical Lvov-Warsaw School launched when he became the professor at the Jan Kazimierz University in Lvov in 1895. After finishing his studies of philosophy in Vienna (in 1889), and obtaining habilitation (in 1894), Twardowski came to Lvov and took a position at the Department of Philosophy. Since Poland had been partitioned for almost hundred of years, and any scientific tradition was forbidden in that period, Twardowski wanted to found a real Polish philosophical school. The main goal of this school was to teach clear and critical manner of thinking [43, p. 327], [84, p. 607], [2, p. 400–401]. Despite the fact that Twardowski was a student of Brentano, he didn’t impose the master’s opinions on his students. His school was characterised rather by the world-view liberalism. Even though

1 This chapter in its general outline was developed based on the monograph [141], [114, p. 399-461] and [94].

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CHAPTER 1. STANISŁAW LEŚNIEWSKI

there was no single philosophical doctrine, we can discuss ideological unity of this school composed, as Woleński writes: of ”unusually serious approach to philosophical deliberations and teaching and apprehending philosophy as intellectual and moral mission.” [141, p. 13]. This philosophical minimalism was a deeply thought component of creating philosophy in Poland. This intelectual formation known as Lvov-Warsaw School of Philosophy is delineated by the following factors: • Genetic – the activity of Kazimierz Twardowski and his disciples. We consider the representatives of this school either philosophers, the closest disciples of Twardowski, as well as those scholars who were not his direct students, for ex. Hugo Steinhaus considered himself his disciple; he was the founder of the Lvov Mathematical School and attended Twardowski’s lectures. • Geographic – Lvov and Warsaw were geographic centres, but the representatives of the school were also in Krakow, Poznań and Wilno. As a phenomenon, the school had lasted until World War II, however the activity of its members continued and is still continuing. • Temporal – the school was founded at the end of XIX sec. and had been active utill World War II. 15th Nov. 1895 is considered as the exact date of its foundation; the date when Kazimierz Twardowski came to Lvov and became the head of the Department of Philosophy at the University of Jan Kazimierz. • Essential – the stock of common ideas. The common understanding and not the common ideas was the intellectual bond of the members of the school.

In the school we can distinguish five periods of its development: 1. The initial period which lasted ca. 7 years, and was ended by discussions of first doctoral dissertations written under the supervision of Twardowski. In that period a group of its associates became to form. Jan Łukasiewicz, Witold Witwicki, Władysław Tatarkiewicz, etc. came from that time.

1.1. LVOV-WARSAW SCHOOL (1895-1939)

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2. The period of the formation of opinions and interests of students of Twardowski (1916-1918). 3. The period of taking shape and full crystallization of the school (19181930) – the Lvov School became the Lvov-Warsaw School; The Warsaw School of Logic was born. 4. The flowering – the 30s – in this time, the most important scientific results were obtained, the school had been internationally recognised. 5. If we consider that the activity after World War II was an integral component of the history of the school, this period is characterised by a great geographical dispersion of its members.

The school educated outstanding scientists active in numerous domains from philosophy, logic, mathematics, to psychology and sociology. Only few names will be mentioned here because the list of scientists amounts to over 80 people, e.g.: J. Łukasiewicz, A. Tarski, K. Ajdukiewicz, T. Kotarbiński, Z. Zawirski, S. Leśniewski, W. Tatarkiewicz, A. Mostowski, S. Jaśkowski, W. Sierpiński, K. Kuratowski, W. Witwicki, J. Słupecki, J.F. Drewnowski, J. Hosiassion, M. and S. Ossowscy, E. Poznański, I. Dąbska, M. Kokoszyńska, H. Mehlberg, S. Swieżawski, etc. In general, two main periods may be distinguished in the Lvov-Warsaw School: the Lvov period (1895–1918) and the Lvov-Warsaw period (1918– 1939). The Warsaw School of Logic played an important role in the second period of the school. It comprised two groups of scientists: one was following J. Łukasiewicz and S. Leśniewski and focused on formal logic ([60], [58], [59], [62], [79], [87], [83], [84], [85]), and the second was following T. Kotarbiński and focused on semantics and methodology of sciences ([37], [39], [40], [41], [42]). Apart from the Warsaw School, there was also a centre in Lvov operating under the supervision of K. Twardowski and K. Ajdukiewicz ([5], [6], [1], [2], [3], [2], [4], [134]) Moreover, in 1936, O. J. Bocheński, J.F. Drewnowski, priest J. Sałamucha and B. Sobociński founded a centre in Krakow with the goal to apply logic in humanist sciences, in theology.

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1.2

Warsaw School of Logic (1918–1939)

The reactivation of the University of Warsaw in 1915 was an important event for the Polish intellectual thought. Zygmunt Janiszewski2 , who studied in Paris and did his PhD with H. Poincare, became a head of the Department of Mathematics, and Jan Łukasiewicz – the disciple of Twardowski – that of Philosophy. Together with Łukasiewicz, the posts of professors at the University of Warsaw (Department of Philosophy) took Władysław Tatarkiewicz. In 1916 Tadeusz Kotarbiński came from Lvov, and in 1918 – Stanisław Leśniewski from Moscow. Jan Łukasiewicz and Stanisław Leśniewski are considered the founders of the Warsaw School of Logic. However, the very group of logicians started to gather around Tatarkiewicz and Łukasiewicz much earlier, around 1911 in Lvov. It is linked to publishing Łukasiewicz’s monograph “On the Principle of Contradiction in Aristotle’s Work” (Polish: “O zasadzie sprzeczności u Arystotelesa”) in 1910 [88], becoming the launching point for a discussion in the Przegląd Filozoficzny periodical between Łukasiewicz, Leśniewski and Kotarbiński.

1.2.1

The anticipation: Lvov period

In 1910 Stanisław Leśniewski came to Lvov to do his Phd under the supervision of Twardowski. He supplied his mathematical education attending the lectures of Puzyna and Sierpiński. In that period he read the work of Łukasiewicz ”On the Principle of Contradiction in Aristotle’s Work” (”O

2

Janiszewski launched a programme of renaissance of polish mathematics. In a paper ”On need of mathematics in Poland”, published in 1917, he proposed to focus the efforts of polish mathematicians on one field of investigation: set theory and topology, and to found a new scientific journal dedicated to that discipline. Hence, in 1920, Fundamenta Mathematicae was founded. It turns out that didactic programs of Twardowski and Janiszewski are congruent in many points. Both wanted to make their disciplines modern, they focused on scientific news, international collaboration, proper, national journals, (Przegląd Filozoficzny, Ruch Filozoficzny); both wanted to educate new scholars. Hence, Łukasiewicz, and later Leśniewski (in 1918) found themselves in Warsaw as they were in Lvov.

1.2. WARSAW SCHOOL OF LOGIC (1918–1939)

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zasadzie sprzeczności u Arystotelesa” [88] – he mentioned about it in ”On the foundations of mathematics” [63] (”O podstawach matematyki”) and for the first time he learnt about the symbolic logic. It is interesting that Leśniewski, who studied in Germany, did not hear anything about the famous work of Whitehead and Russell “Principia Mathematica”: In 1911, I came upon a book by Mr Łukasiewicz ”On the Principle of Contradiction in Aristotle’s Work”. The book had considerable impact on the intellectual development of a group of «philosophers» and «the philosophing» scientists of my generation and for me it was a great book from numerous perspectives. I learnt from it for the first time about the existence of «symbolic logic» by Mr Bertrand Russell and about his «antinomy» concerning the «class of classes which are not their own elements» [63, p. 169] in [141, p. 79]. The work of Łukasiewicz evoked Leśniewski’s resistance. As a reply he wrote “Próba dowodu ontologicznej zasady sprzeczności” (1911) – “An attempt to prove the ontological principle of contradiction”, which was published in 1912 in Przeglądzie Filozoficznym (PF). He proved the value of the principle of contradiction of Aristotle, beginning from the Łukasiewicz’s definition of an object, ie. from the definition that no object can have and not have the same feature. As a result of these discussions, in 1913 Tadeusz Kotarbiński published a paper “Zagadnienie istnienia przyszłości” (“The issues of the existence of future”), in which he discussed various definitions of the logical principle of the excluded middle and he proclaimed the possibility of stating truth on the name of different human creativity. Leśniewski joined this discussion, of course. His certainty concerning bivalence of logic and the Aristotle’s principle of contradiction led him to deep conclusions concerning truth and false. Still in 1912 r. during the meeting of the Philosophical Circle at the Lvov University, he gave a lecture “On the principles of the excluded middle” (published in 1913 in San Remo under the title: “Krytyka logicznej zasady wyłączonego środka” [60] – “The critic of the logical principle of the excluded middle”). As a result of the given proof, he realised that Kotarbiński was wrong thinking that there were sentences, which were not true

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nor false. He showed that such propositions do not exist [60]. Additionally, he argued for Twardowski’s position regarding the absolute value of truth (“Czy prawda jest wieczna, czy tez wieczna i odwieczna” [58] – ”Is the truth eternal, or eternal and everlasting”). In this paper Leśniewski analysed temporal phrases [59], [93, p. 107–108] and as a result of the discussion with Kotarbiński (dealing with the problem of determinism), Kotarbiński suggested the existence of the third solution. And this paper contributed, among others, to Łukasiewicz’s conclusions and to his construction of trivalent logic [81], [142, p. 216]. In the middle of 1913 Leśniewski moved to Warsaw and wrote a paper “Czy klasa klas nie podporządkowanych sobie jest podporządkowana sobie?” [61] (“Is the class of classes not subordinated to itself, subordinated to itself?”) being the prodrome of the work on the foundations of the general set theory finished in 1916 and published under the title: “Podstawy ogólnej teorii mnogości” (“Foundations of the General Set Theory”) [62].

1.2.2

Warsaw period

The Warsaw School of Logic is said to have two periods of development: the ‘20s with focus on two and multi-valued logics and their qualities [86], [75], and Leśniewski’s research in the foundations of set theory treated in a separate manner [60], [58], [59], [61], [62], [63]–[71], [65] ([126, 410–605]), [74] ([126, 649–710]); and the ‘30s with numerous publications 3 [37], [40], [41], [79], [87], [83], [84], [85], [86], [131], [128], [129]. In a dozen of years Warsaw became a very important centre of logic research [141, p. 83-85]. Logic in the Warsaw School had broad meaning: it was understood as a formal logic, semiotics and methodology of sciences [40]. Regular seminars, lectures and courses concerning logic commence in 1920. The School was not related to philosophy only at the beginning of its existence; it was always a close cooperation. According to Woleński, students of Łukasiewicz

3

More in e.g. [22], [108], [47].


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and Leśniewski attended Kotarbiński’s lectures and vice versa. The committee of the editorial office of the Fundamenta Mathematicae periodical comprised philosophers, logicians and mathematicians. Logic became an autonomous domain and it was recognized as such, despite its close affiliation to mathematics. Moreover, the “logicizing” methodology of sciences was practised there. We have to remember, that majority of scientists at the Polish school shared Twardowski’s views on methodology. Twardowski believed that philosophy did not differ in its methodology from natural sciences and it could be globally treated not as a single domain, but as a set of disciplines comprising also logic, psychology, pedagogy, aesthetics, ethics, language and culture studies [141, p. 17]. In the Warsaw School the analytical nature of logic and mathematic were rejected. Leśniewski saw ontology as a theory of reality; Tarski believed that dividing terms into logical and extra-logical makes no sense. For Mostowski, mathematics had empirical genesis, and Łukasiewicz changed his opinion several times – from recognizing logic as an a priori science, through its empirical genesis, finally (in 1957) he assumed the pragmatic and relativist approach in the philosophy of logic. However, according to Woleński, there was a unity of opinions connecting the very core of formalism, anti-psychologism of logic and the fact that formal systems describe reality [141, p. 184-185]. On the other hand, in terms of the philosophy of the school (understood as not only philosophy of logic and mathematic) as we have mentioned previously, it was characterised by world-view pluralism. It is worth quoting I. Dąbska cited in [114, p. 407]: [...] because there wasn’t any doctrine shared by Lvov philosophers, [the same for the Warsaw, according to Woleński’s comments] or a uniform world-view. The origins of the spiritual community of those people were not in the content of the science, but the manner, the method of philosophy and the shared by them scientific language. That is why this school could produce spiritualists and materialists, nominalists and realists, logicians and psychologists, philosophers of nature and theoreticians of art.

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Consequently, every logician or mathematician could have different philosophical opinions, sometimes even differing from the topic of scientific research as in the case of Tarski, who, according to Woleński, expressed his sympathy for nominalism and the basis for his scientific research were set theory methods where general and abstract terms are used, contradicting the very nominalism [141, p. 180]. In the Warsaw School much attention is paid to full formalisation of logical systems. Łukasiewicz’s notation without brackets is famous all around the world [84]; functors are written before arguments4 . The Warsaw School principle is the simplicity and efficiency of logical systems, that is the principle that the smaller the amount of primary symbols and axioms, the better. Here, Łukasiewicz’s many-valued logic is born. Also first attempts at formalising paraconsistent logic by Jaśkowski [35] with clear rejection of the principle of contradiction are undertaken at the Warsaw School. Of course there are also philosophical interpretations stemming from the cooperation of logicians and philosophers. Due to the fact that it also triggers numerous controversies, Łukasiewicz finally assumes not to assign any special philosophical content to many-valued logic, and to approach it in a formal manner only [114, p. 415]. Leśniewski believes that the principle of contradiction is uncontested, and he does not share the point of view of Łukasiewicz [75]. In the Warsaw School, Tarski, the only doctoral student of Leśniewski, develops semantic theory of truth [131], [130], despite the fact that the very idea of separating language from metalanguage comes exactly from Leśniewski [180, pp. 138–139]. Thanks to Łukasiewicz and later works by Słupecki, the difference between the Aristotle’s logic and the logic from the Stoa school [80], [82] is presented clearly; it shows and treats both logics as historic representations

4 For example the phrase in the form ’p → q’ is written down: Cpq, where C signifies an implication functor [114, p. 413].


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of the modern logic, which becomes a breakthrough point for the history of logic [141, p. 178].

Jan Woleński, while analysing the reasons for this enormous success of the Warsaw School, believes that we should assign it to the fact that logic became separated from philosophy and that logicians, despite the fact that they originated in philosophical environment, were fully accepted by mathematicians and recognized as equals [141, p. 83–86]. Moreover, as Woleński supposes, it was a favourable moment for logic in terms of development of this discipline, due to the fact that talented and outstanding people were able to use their capacities to work towards proper direction and at a convenient moment of history.

The year 1939 is recognized as the end of the school. During war, nearly half of scientists of the Polish school died, and after war majority of those who survived emigrated (A. Tarski, J. Łukasiewicz, H. Hiż, B. Sobociński, Cz. Lejewski...). Circumstances and atmosphere was not favourable for the rebirth of the beautiful tradition of the Polish intellectual school in postwar Poland. The Vienna Circle faced similar fate – after Hitler’s invasion it stopped operating. Despite the fact that the remaining group of outstanding members, students of Kazimierz Twardowski, educated next generations of the Polish science, the school has not seen its revival.

The dissemination of the idea of the Warsaw School of Logic abroad took place first of all thanks to its representatives. Bolesław Sobociński (1906-1980) worked at the Notre Dame University in Indiana (USA), where research into mereology is still ongoing. Alfred Tarski (1901–1983) worked at the California University in Berkeley (USA) giving foundations to the most important school of logic after war having characteristic features of the Warsaw School, especially when it comes to logic. Jan Łukasiewicz (1878-1956) worked at the Royal Academy of Science in Dublin, Czesław Lejewski (1913-2001) in Manchester; there are also centres in other parts of the world.

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1.3

Stanisław Leśniewski (1886–1939) and his works

Stanisław Leśniewski was born in Russia, in Sierpuchów (near IwanowoWozniesiensko, 250 km North-East from Moscow), 18 March 1886. He was a son of a railway engineer. He graduated high school in Irkutsk, in Siberia. After the matriculation examination in 1904, he went to study philosophy in Germany. He studied in Lipschitz, probably in Heidelberg and Zurich. In 1910 he went to Lvov, to study at the University of Jan Kazimierz under the supervision of Kazimierz Twardowski. There, in 1912 he received his doctoral degree on the basis of the work “Przyczynek do analizy zdań egzystencjalnych” (“A Contribution to the Analysis of Existential Propositions”) published in Przegląd Filozoficzny. In 1913 he settled in Warsaw and there he got to know Tadeusz Kotarbiński. In 1915 went to Moscow and worked on the foundations of general set theory. In 1918 he returned to Warsaw and since 1919 he had delivered lectures at the University of Warsaw, taking over the faculty of philosophy and mathematics. In 1936 he became a professor. He died in Warsaw on 13th May, 1939. [143, p. 39–44]5 . When Leśniewski joined the group of Twardowski’s students (in 1910) in Lvov, he attended lectures by Wacław Sierpiński [143] to fill in the gaps in his mathematical education, especially in the set theory. He had precise philosophical opinions, as one of very few representatives of the Warsaw School, but the year 1916, when he published his first work on the foundations of general set theory [62], was crucial for him. In the letter written to Twardowski in March 1919, he wrote about the breakthrough that took place during the work on foundations. This turn concerned mainly the methodology he applied, not beginning from logic, passing through set theory, in order to reach ontology, as he did earlier, but vice versa: beginning from ontology, passing through set theory, to reach logic: For my whole life I believed that logic is a discipline where all the respectable sciences have their foundations, that the set theory also has logical foundations and I attempted to find proper

5

More information concerning Leśniewski’s biography, go to e.g. [34], [141]

1.3. STANISŁAW LEŚNIEWSKI (1886–1939) AND HIS WORKS

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position for performing a safe leap to the ontological side. However, one evening, completely suddenly, I found myself on the ontological side and everything I previously did changed shape and size under the influence of the view from the new ontological observatory. I believed that the direction of travels should be changed and I should take transportation from the ontologist’s station through the set theory to the logician’s station, not the other way round, as I used to think. I believed that logic may be in fact constructed on this itinerary, if anyone would find it pleasurable; however, the very science together with mathematics, does not need for their justification not a bit of all this what logic is6 .

Leśniewski was only interested in these logical issues which grow up from his foundational concepts. He was not involved in school researches, but followed his own path of investigations7 . He considered himself a nominalist and a supporter of reism, expressing it in his logical theories. Due to him, a language is a collection of inscriptions and propositional expressions being a finite sequence of signs. There is so much how much was written expressions, hence we cannot speak about potentially existing expressions8 . Additionally, a given system has as many claims as have been written; is composed of a finite number of claims. Finally, for Leśniewski general objects, in particular qualities (features), do not exist. He is under strong influence of Twardowski’s views on the theory

6

This text comes from Kazimierz Twardowski’s archives, the Institute of Philosophy and Sociology of the Polish Academy of Sciences (Polish abbreviation: IFiS PAN), Warsaw, quoted in [34, p. 106], (AKT. K.9 – 178). 7 Except the researches on the equivalent propositional calculus. Łukasiewicz during the work on the axiomatization of the propositional calculus, noticed in “Principia Mathematica” two equivalent thesis, but as Woleński quoted, it was Leśniewski who, as the first, presented the axiomatization of the propositional calculus (in 1929) and gave a very simple syntactical criterion according to which “x is a thesis of this calculus if it contains an odd number of occurrence of a functor E and an even number of occurrence of each variable” [94, p. 114-118]. 8 We can notice here the difficulties Leśniewski’s mereology can cause, e.g. it is impossible to define infinite sets within this theory.

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of objects by Edmund Husserl [141, p. 36]. Twardowski represents the view that there are no empty names. Leśniewski takes it over from him in his Mereology. Additionally, since for Leśniewski equivalent systems, e.g. propositional calculus based on negation and alternative or on negation and implication are different, they are never complete and we cannot apply standard methods for their investigations. Moreover, because standard methods define infinite systems, there is a need for a proper logic [94, p. 114-118]. Leśniewski’s nominalism, as he defined it: constructive nominalism, one can combine with intuitive formalism since the language of logic has always to say something about something. As a result, Leśniewski’s formal systems are to give some information about the world; they are a way to express what is intuitively true. This position might seem not to be in agreement with his views since it refers to intuitive knowledge, hence something that was the source of contradictions in Frege and what Leśniewski wanted to eliminate constructing the precision and formal systems. However, such a position is not contradictory for Leśniewski, which is clearly written in [74]. For Leśniewski the logic is a description of the most general features of being (Kotarbiński claimed the same, being under the influence of Leśniewski) and acts as the general theory of objects. This view is consistent with the view of the school which rejected analytical concept of logic9 . Logic and mathematics refer to the formal aspects of reality. Also Sobociński writes that Leśniewski sought through deduction give the most general laws by which reality is constructed. [...] Leśniewski was also a philosopher by education and he also moved away from philosophy [. . . ]. However, contrary to Łukasiewicz, he believed that the «true» system of logic and mathematics can be found. His systematisation of the basics of mathematics was not exclusively postulatory; he wanted to

9 i.e. thesis that logic and mathematics are the set of tautologies speaking nothing about the world.


23

present the most general laws of reality construction in deductive form. Due to the fact he benefited little from mathematical and logical theories, which, even if they had been non-contradictory – he would not accept them as compliant to fundamental and structural laws of reality. It was for this reason that he focused his research also on a specific system which he himself constructed with its internal problems. He was convinced that only this system is veritable [120].

Leśniewski was an extensionalist and he rejected any intesional contexts. He supported the bivalence of logic, rejecting strongly many-valued logics, even though in the beginning he also worked on many-valued logics. He even developed two hours long lecture lasting whole year concerning many-valued logics [93, p. 104–105]. The rejection of trivalent logic is connected among others to the fact of lack of interpretation linking the third value to reality. Those systems could be interesting only from the formal point of view10 : Faced with the absence from the world of any satisfactory from the intuitive and formal point of view system of «intensional logic», the speaker does not know any efficient method of reasonable interpretation and logical «mastering» of the mentioned «intensional functions» apart from their «de-intensionalisation» consisting of assigning them subordinate expressions with the same sense of expressions which are already compliant to consistently «extensionalist» principles and we can without any complications deliberate about them on the grounds of normal «extensionalist» and «bivalued» logic [75, p. 236], in [93, s. 106].

Woleński supposes that the theory of de-intesionalisation was rather already developed by Leśniewski, as he had worked on the very intensionality problem for many years. Consequently, his Ontology is a fully extensional

10 R. Murawski quotes also recent research by J. Jadacki concerning the Leśniewski’s attitude to the intensionality and many-valued logics [75], [144, p. 68].

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theory, and Leśniewski tried to “extensionalize” any intensional form. According to Sobociński’s words addressed to O. Bocheński “he is even capable of providing particular examples of such forms and methods of their elimination”[122], [144, p. 72]. He also mentions that if the language is used in a conscious manner, then any intensional form has meaning which can be expressed in an extensional form. However, Woleński suspects that it differed from the Carnap’s method due to the fact that it consisted of two methods, not only one – one on the metalanguage level, and the other not. Unfortunately we do not know anything more on this issue. What was the attitude of the scientific community towards Leśniewski? Wytwicki, in a letter written to Twardowski on 3 December 1920, wrote that due to Leśniewski philosophy was something that did not exist as well as logic, psychology, and epistemology... metaphysics remained only as a theory of objects. In the same spirit were written, among others, articles of Kotarbiński “Sprawa istnienia przedmiotów idealnych” (“A matter of existence of ideal objects”) [37], or “O potrzebie zaniechania wyrazów «filozofia», «filozof», «filozoficzny»” (”On the need of abandonment of philosophical words «philosophy», «philosopher»„ «philosophical»”)[38], which were inspired by Leśniewski’s nominalism. This deviation (in the Warsaw School) from philosophical problems cultivated in Lvov in the spirit of semantic and psychological analysis of Twardowski, resulted in a lively discussion in scientific community and the feedback of Twardowski in a work “Symbolomania i pragmatofobia” [133]. However the critic of Twardowski was directed more towards Łukasiewicz than Leśniewski, the latter (as one of the few) bounded his logical system with reality and his clear philosophical position. As we can see, Leśniewski’s impact on the development of logic in the previous century was impressive indeed. Despite the fact that Leśniewski’s formalisation was seen as complicated, the very author is a pioneer and the driving force in this domain. According to Woleński’s comments, we have to be aware that the article by Łukasiewicz “Bivalued Logic”[80], not yet perfect from the formal point of view, was written in 1920, and the impeccably developed article by Leśniewski was written already in 1922. Woleński again: [...] therefore we may assume that the famous all around the


25

world formalisms of logicians from the Warsaw School were stimulated mainly by Leśniewski. If this assumption is right, then we may recognize Leśniewski as the main scientific «ideologist» of the Warsaw School. [141, p. 152–153], [33].

Mereology as an answer for the Russell’s paradox The set theory paradox discovered by B. Russell in the basis of mathematics11 initiated the work on founding Mereology. Apparently casual operations on certain notions, e.g. the notion “set” or “class” may lead to contradictions. However, the history of paradoxes dates back to Antiquity with the famous liar paradox by Eubulides as an example. On the other hand, in the theory of sets, in 1895 G. Cantor (and in 1897 C. Burali-Forti) noticed the so-called ordinal number paradox [29, p. 254]: let W be the set of all ordinal numbers12 . It is a well-ordered set13 ; therefore a certain ordinal number β corresponds to it. Consequently, for any α ∈ W : β > α. As a result we receive that β ∈ / W and β ∈ W , resulting in contradiction. In 1899, Cantor presented the so-called antinomy of the set of all sets. Let’s assume that A is the set of all sets. Let B be the set of all subsets of the set A. Then the cardinality of the set B (according to Cantor the size of the set B) should be smaller than the cardinality of the set A, but the cardinality of the exponential set amounts to 2card A , consequently, the cardinality of B is bigger than the cardinality of the set A, resulting in contradiction [29, p. 254]. We reach the B. Russell’s paradox, the so-called antimony of nonreflexive classes in the formulation by Kotarbiński [44, p. 160–161].

11

See the letter by B. Russell to G. Frege of 1920 in [137, p. 124–125]. Ordinal numbers are a certain generalisation of natural numbers. Intuitively, numbers corresponding to sets in linear order (ordinal types of well-ordered sets) having certain defined properties. For more information see: [104, p. 140–148]. 13 It means that the relation of order has been defined, in addition, it is a relation of linear order and for every subset of the set W the first element exists. See: [104, p. 140]. 12

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Among classes we may imagine those which are elements of themselves and those which are not. Consequently, we have a class of classes out of which each is an element of itself. However, let’s take e.g. the class of people. A class of people is not a person; therefore it is not an element of itself. Let’s assume K as the class of classes out of which none is an element of itself. “. . . let’s ask ourselves the question if the class of classes out of which each is not an element of itself is one of the classes of the mentioned first type, or one of classes of the second mentioned type. Is it one of the classes out of which each is an element of itself or one of classes out of which none is an element of itself ? In both cases we receive a contradiction. If the class of classes out of which each is not an element of itself (in short: “nonreflexive”) is one of the classes out of which each is an element of itself (in short: “reflexive”), then if it is an element of itself, then it itself is one of such classes, therefore it is not an element of itself. And vice versa. . . Those paradoxes constituted a serious challenge to the basis of mathematics. Various solutions were looked for in order to eliminate them. In 1950, Ernest Zermelo, a German mathematician, commenced his works on axiomatisation of the set theory that is on the formal and precise presentation of its main postulates. In 1908, he presented a setup of axioms, which, having undergone independent modification by A. Frankel and T. Skolem [137], was published with the name of Zermelo-Fraenkel axioms; nowadays it is the most widely applied system of the set theory recognized by majority of mathematicians. Postulates approved of in this manner enabled to reconstruct George Cantor’s results and at the same time avoid antinomies. On the other hand, Russell and Whitehead, proposed their theory of logical types [140], as an antidote to antinomies. According to this theory, we may distinguish individuals, classes, classes of classes, etc. And every class can have only such elements which belong to the logical type directly lower to it. [44, p. 163]. A rapid development of research concerning the foundations of mathematics takes place in the second half of the 19th century and the beginning of the 20th century. Numerous scientists, such as K. Weierstrass, R.


27

Dedekind, and first of all G. Cantor undertake attempts at formalisation of natural and real numbers and they look for the answer to the question of what numbers are. According to Cantor, the differentiation between the relationship of succession and the size (that is cardinality) for sets plays an important role here. According to Gray, before Cantor nobody was interested in what natural numbers are, but situation changed when Cantor said that his infinite sets have qualities of numbers14 and provided proof for it (1874) in the diagonal form, that the line segment [0, 1] has the same amount of elements as the whole straight line of real numbers, in other words, that they are equinumerous, that is, that there exists a one-to-one correspondence between the line segment and the straight line. Gray continues: Cantor not only provided arguments that both sets are equinumerous, if we may find this one-to-one correspondence between elements of the one and of the other set, but also that the cardinality number of the set constitutes a set of such units which have one-to-one correspondence15 . According to Gray, the suggestion, that every quality of objects in the set may be separated from them made G. Frege decide to define the notion of a number. For him it was inconceivable to treat qualities as every other object16 . Frege wanted to indicate those qualities of numbers which make them become numbers and to provide the proof for their logical nature. Therefore for him, numbers constituted qualities of notions, not objects. Having two numbers defined with the use of two notions, Frege defined qualities of equivalence of those notions with the use of extensions and the notion of ’equivalence’17 . He proposed a thesis that every propositional function defining a given notion, that is the extension of a given notion, designates a set of objects of a quality defined by this notion. It is the famous fifth postulate by Frege which led him to antinomy [25, & 45]. According to Gray,

14

”..until Cantor started to suggest that his new infinite sets had numberlike properties.” in [28, p. 129–141]. 15 „... the cardinality of a set is the set of pure units with which it is in a one-to-one correspondence.”[28, p. 160]. 16 [28, p. 160]. 17 We have to add that Leśniewski highly valued Frege’s precision, much more than Russell and Whitehead’s [142, p. 217].

28


Cantor himself had doubts concerning both the extension of the notion to objects and the equivalence of notions (Hume’s Principle), as that operation was completely not determined and only sometimes it might result in size that is in a number. According to him, the size and cardinality should be defined separately. He realised such definition by introducing ordinal numbers and cardinal numbers [28, p. 163]. However, it turned out that Grundgesetze der Arithmetik by Frege contained certain flaws. Frege himself, in one of his footnotes writes: ”I assume, that it is known what the «extension of a notion» means.” [28, p. 164], but unfortunately the system of postulates adopted by Frege was selfcontradictory. In 1902, Frege received a letter from B. Russell (1902)18 , where he writes that Frege’s reasoning leads to a paradox [92, p. 221], [135, p. 70], [91, p. 208]. Russell discovered the set theory paradox resulting from the analysis of the diagonal proof by Georg Cantor. It happened around 1901 and most probably Ernest Zermelo independently discovered the same paradox [99, p. 123]. Russell’s paradox inspired Leśniewski to create his own notion of a set. In 1958, Kotarbiński writes about it in the following manner in [141, p. 148]: And here it happened: Leśniewski (and it was close to the outbreak of World War I) undertook the reading on antinomy by Russell [. . . ]. When he was preparing this reading, the lecturer all of the sudden noticed that the developed by him critique of the discussed antinomy contains an error; it «has fallen into pieces» as he used to say in such situations. Sheer despair. In several hours he was supposed to deliver the reading, listeners would convene and the whole situation may end in a shame. He decided to concentrate as much as possible, helping himself with crunching a bar of chocolate. The result was that, according to his own diagnosis, mereology was born out of chocolate. Because, isn’t it obvious that despite the fact that something is M

18

In [137, p. 124].


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it is therefore the element of the class of M s, but it is completely not true that something which is the element of the class of M s, must be itself the M as well, due to the very same reasoning. In the context of collective theory of sets19 , as Leśniewski used to call it, he formulates the antinomy by Russell in the following manner with the use of the term ’class’ and with the use of the expression of the type ’P is subordinated to the class K’20 , which makes sense if at certain meaning of the word ’a’, K is a class of objects ’a’ an P is ’a’. This expression clearly harmonizes with the set theory notion of being an element [63, p. 185–187]: Most often the class is not subordinated to itself, because as a set of elements it usually has other features than every element separately. A set of people is not a person; a set of triangles is not a triangle etc. In some cases the situation looks different. For example, let’s take the notion of a «complete class», that is such a class where some individuals are members at all. Not all classes are complete, some are empty, e.g. «truly golden mountain», [. . . ]; they are empty because there are no individuals which would be their members. Consequently, we may distinguish them from classes where there are some individuals, and create the notion of a «complete class». [. . . ] The set of all such classes constitutes a new class that is a «class of complete classes». Such a class of complete classes is also a complete class; consequently it is subordinated to itself. Because some classes are subordinated to themselves and others are not, then to differentiate the ones from the others we may create a notion of a «class which is not subordinated to itself». [. . . ] A set of all those classes constitutes a «class of classes which are not subordinated». Let’s call it in short class K. A question arises: Is the class K subordinated to itself or not? If we assume that the class K is subordinated to itself, than,

19

That is mereology. The term ’subordinated class’ shall be understood in such a manner that it is an element of itself. 20

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because each class subordinated to class K is not subordinated to itself, we draw the conclusion that class K is not subordinated to itself. Consequently, contradiction arises, [. . . ]. If we want to avoid this contradiction, we have to assume that class K is not subordinated to itself. And if it is not subordinated to itself, then it is a member of the class K; therefore it is subordinated to itself. And here again we obtain contradiction as a consequence [...]. In this manner, Leśniewski challenged the existence of such classes which are not their own elements. He considers the formulation such as ’class of classes not being an element of itself’ as false, consequently such a class is empty, therefore it does not exist. According to Leśniewski, in the antinomy proof there is a faulty “bridge” applied in “. . . a place where the discussed class is an element of a class of classes out of which each is an element of itself constitutes grounds for concluding that it is its own element. Here we reason according to the formula: if x is an element of the class of M s, then the x is an M . . . ” [44, p. 161], [63, p. 182 and next pages]. It stems from the fact that for Leśniewski, a ’class’ is always a mereological class21 .

Other works by Leśniewski When Stanisław Leśniewski studied “Principia Mathematica”, he thoroughly analysed the symbolics applied by Russell and Whitehead and he drew the conclusion that some expressions are not written in a rigorous manner. He even provides 17 possible interpretations of the proposition ’q. → .p ∨ r’ [63, 169–181], [136, 86–90] always on the grounds of metalogic by Russell and Whitehead. His several years long deliberations were written down only in 1927 (after around 10 years); however, as it is believed, those deliberations gave rise to differentiating between language and metalanguage22 and were

21

The analysis of the solution for the Russell’s paradox in the Leśniewski version may be also found in [7], [136, p. 180]. 22 Some examples by Leśniewski are quoted by Urbaniak in [136, 90–91].


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productive in many previously published works23 . After completing his work on those notions with the use of the language of symbolics, in 1916, Leśniewski wrote an article O podstawach matematyki (“On the Foundations of Mathematics”) (1916) [62] constituting an outline of Mereology, that is the general set theory, but its complete presentation took place only in works published during 1927 and 193124 . Twardowski, in his Diary, on the date of 1 July 1919, wrote about the meeting in Restaurant at the Nowy Świat street in Warsaw, during which Leśniewski, in the presence of Czeżowski, Kotarbiński and Borowski, presented the principles of his new axiology. In 1922, having studied the theory of Russell’s types, Leśniewski formulated the theory of semantic categories being a simplification of Russell’s theory. The theory had three sources: the theory of types, theory of Aristotle categories and Husserl’s theory of meaning categories [141, p. 140]. This is a theory concerning expressions treated as complete sequences of inscriptions which are always concrete but not ’finished’, they can always be extended [141, p. 142]. Leśniewski constructed three logical systems25 : (1) Protothetics which can be understood as expanded propositional calculus, (2) Ontology – the calculus of names and (3) Mereology – the theory of collective sets (the general set theory). Mereology was the first to be developed in 1916, despite the fact that it is a theory constructed over Ontology and Protothetics. Protothetics and Ontology26 were developed from the very beginning without

23

As Woleński suggests, we may say that the inspiration for the theory of metalanguage for Tarski was Leśniewski’s intuition. 24 See [141, p. 136]. 25 In fact he constructed two systems, as Mereology is not considered to be a logical system but rather a theory of collective sets. Apart from ’logical’ texts by Leśniewski [70], [71], Luschei monograph [89] is the first work consisting critical analysis of those systems. 26 For the first time Leśniewski presented the basis of Ontology during lectures at the University in Warsaw during 1919 – 1920 [136, p. 105]. Protothetics was first mentioned during this period, but theoretical deliberations concerning logic in the foundations of Mereology date back to as late as 1930 [136, p. 71].

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reference made to any logic. Leśniewski himself introduced well formalised logic27 . In Leśniewski’s systems there are no free variables, but all of them are quantified. The very comprehension of quantification is disputable. It rather does not get assigned the substitution meaning28 , or classical due to Ontology which is also true for empty domains [141, p. 143]. We also mention another interpretation [141, p. 143], [45], [46]. The first system (not chronologically) – Protothetics is a generalised propositional calculus, defined only with the help of equivalence. According to Woleński, Tarski’s result was important here [127], saying that negation and conjunction may be defined with the use of equivalence and general quantifier [141, s. 144]. Leśniewski, together with Wajsberg and Sobociński, showed that the calculus defined in this manner, with the assumption of the extensionality principle (see Chapter 2.4, p. 68), give rise to all principles for bivalued propositional calculus. Consequently, Leśniewski was convinced that logic is bivalent [142, p. 220]. In 1953, Jerzy Słupecki provided proof for the completeness of this calculus [141, p. 145], [115]. Ontology, the second system by Leśniewski, was founded on Prototethics29 , enriching it with the logical connective from name arguments [141, p. 145]. The only primary term in Ontology is the functor ’is’ (). The adopted names are members of only one category; the division into empty names, unitary names and general names is secondary. Such inscription determines the truthfulness of the proposition ’a is B’ depending on what ’a’ and ’B’ is substituted with. Consequently, the main axiom of Ontology is a developed

27 Leśniewski followed intuition and he applied radical formalism which for him was the means, not the goal [141, p. 138]. According to Bolesław Sobociński, Leśniewski was a metaphysician in logic, because he believed that logical theory describes the real world and, as Woleński adds, from Tarski’s statements we may infer that semantic ideas were formalised by Leśniewski [141, p. 141], [7]. 28 That is the expression ∀x F x is read as follows: “for every substitution for x in F x, F x” [141, p. 143]. 29 We also talk about Protothetics as the sub-theory of Ontology that is what is true for Protothetics is also true for Ontology [136, p. 108].


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axiom30 [69], [72, pp. 364–369], [71], [32, p. 168], [142, p. 221]. Ontology is a theory concerning individuals, not general names. Leśniewski defined two important notions: ’exists’ and ’object’. Informally, we may write it down as follows31 : • ’A exists’ means ’for certain x, x is A’, • ’A is an object’ means ’for certain x, A is x’.

And, as he continues [141, p. 148], those theses do not give grounds for the reasoning that something exists or doesn’t exist, therefore it is possible to introduce abstract objects32 . The power and universality of Ontology stems from the fact that it can be used for the set theory interpretation of the calculus of names. At the end we reach Mereology. Mereology is the last system, it is a superstructure over the two previous ones (therefore we needed the introduction concerning Prototethics and Ontology). It is not a logical system; it is a theory of sets in the collective sense. The primary notion are the relations of ’being a part’ called ingrediens by Leśniewski, but this system will be discussed in detail in (see. p. 54), that mereology by Leśniewski, after adding the zero element and defining the algebra operators forms the Boolean algebra, corresponds to classical bivalued logic. Słupecki [116], tried to generalise Mereology, but he was often criticised among others for the fact that his theory is not extensional while Leśniewski’s theory is [136]. Moreover, Leśniewski wrote two works concerning function and theory of groups (Abelian groups) which were published in the periodical Fundamenta Mathematicae in 1929 [68], [67], [126, pp. 383–409], constituting the

30

∀a,b (ab ≡ (∃c (ca) ∧ ∀c,d ((ca ∧ da) → cd) ∧ ∀c (ca → cb))) ∀a (ex(a) ≡ ∃b ba) [70], [136, p. 107]. Moreover, Leśniewski informally uses capital letters for name variables, if the accuracy of the formula requires that only one object is assigned to name variable. 32 Urbaniak [136] uses the general English term being. 31

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continuation of research over Ontology and the theory of deduction [70], [72] ([126, pp. 606–648]) and works constituting the continuation of his previous works on the foundations of mathematics, prepared for the periodical Collectanea Logica (1938) [73], [74], they were never published but their copies are stored in Harvard College Library [136], [126, pp. 649–710]. After war there were attempts at restoring part of Leśniewski’s conclusions – the author did not write down much. Unfortunately some manuscripts were lost, some were restored from notes of his students and they were published in English [124], [125], [123], and the ’Leśniewski current’ was continued in various centres all over the world33 ,... and during the recent years it enjoys a new interest [109], [110], [111], [112], [113], [8], [78], [7], [138], [139], [20], [21].

33 Robert Clay focused in detail on Leśniewski’s Mereology [11], [12], [13], [14], [15], [16], [17], [18], [19], and F. Rickey worked on Protothetics, in particular on the explanation of Leśniewski’s terminology [105], [106], [126]. Moreover, the researches was continued by Sobociński [117], [118], [119], [120], [121], [122], [125], Lejewski [49], [50], [51], [52], [53], [54], [56]), Słupecki [115], [116].

Chapter 2

Leśniewski’s Mereology

2.1

The relation of being an element

Classical set theory by Cantor was founded on two notions: ’set’ and ’relation of being an element’, identified also with the relation of element’s membership in a set – ∈. The very relation of membership is a binary relation with a quality that no object can be an element of itself1 : x ∈ / x. An example of it is the famous antinomy by Russell of a set not being an element of itself (see Chapter 1.3, p. 25). The Relation ∈ is neither transitive, that is, if one object constitutes an element of the second object and the second is an element of the third object, it does not have to be true that the first object constitutes an element of the third object2 . If the ear of a teacup is an element of the teacup and the teacup is an element of a glassware set, then it is not true that the ear of the teacup is an element of the glassware set.

1

We will say that the relation r is irreflexive if ∀x ¬ xrx. Consequently, the relation ∈ is irreflexive. 2 We will say that the relation r is transitive if ∀x,y,z (xry ∧ yrz → xrz). Apart from reflexive and transitive relation we may also distinguish the symmetric relation: ∀x,y (xry → yrx), and antisymmetric relation: ∀x,y (xry ∧ yrx → x = y) and asymmetric relation: ∀x,y (xry → ¬ yrx) [104, p. 65–66]. This is a general classification of relations regarding their features.

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CHAPTER 2. LEŚNIEWSKI’S MEREOLOGY

However, mereology may be founded on the ingredient relation3 . If we mark the relation of a part with – v, then it is assumed that the object p constitutes a proper part of the object x, if p is part of x and p is not x, that is p v x ∧ p 6= x4 . Being a part will be considered synonymous to being an element compliant to Leśniewski’s idea: When, on the grounds of my «general set theory», I introduced propositions of the following type: - «P is the ingredient of the object Q», «P is an element of the object Q» and «P is the subset of the object Q», I introduced three types of propositions adequately equivalent one to another and I unnecessarily complicated in this manner my terminology. Any out of the three presented types of propositions would be sufficient to demonstrate the mutual equivalence of those propositions for all my theoretical goals. [64, p. 278]. According to this definition of a part, any object constitutes an element of itself, because it is its part: x v x. Consequently, the relation of a part is reflexive: I am a part of myself. Similarly, if one object constitutes part of another object and the second object constitutes part of the third object, then the first object constitutes part of the third object. If Warsaw is part of Poland and Poland is part of Europe, than Warsaw is part of Europe. Consequently the relation of part is transitive. If we found Mereology on the notion of proper part which is irreflexive5 and transitive, it will not change the quality of relation of being an element, which remains reflexive and transitive. 3 Chronologically, Leśniewski founded Mereology first on the notion of ’ingredient’ [63], and later on the term ’external’ [71]. He also provided proof that we may construct Mereology on the ’proper part’ notion (for Leśniewski it is always synonymous with a ’part’) and that all systems are equivalent [71]. In this work the term ’part’ will always be synonymous to the notion ’ingredient’ signifying proper part or a part of a given object [63]. In case we discuss a proper part, the term will be explicitly clarified. 4 In the second part of this monograph we will present various definitions of a proper part. 5 See footnote 1.

2.1. THE RELATION OF BEING AN ELEMENT

37

Borkowski6 explains very well crucial differences between the theory of sets and Mereology. First, he explains the meaning of terms ’set’ and ’element’ in mereological sense, and later in the set theory sense: Terms «set», «element of a set» are used in double meaning. One meaning is that the term «set» signifies objects made of parts, collectives, that is conglomerates of various types. Elements of such a set are understood as its any parts, and the term «part» is understood in its common meaning, e.g. a leg of a table constitutes part of the table. A pile of stones in this meaning is a set of those stones. Particular stones and various parts of those stones, that is molecules or atoms are equally elements of this set. According to this meaning the set of given stones is identical with e.g. a set of all atoms constructing those stones. Elements of a set understood in this manner, e.g. a set of all tables, are not only particular tables, but e.g. legs of tables and other parts of tables (. . . ). The second meaning of the term «set» and «element of a set» is used e.g. when we talk about a set of European countries and we recognize particular European countries such as e.g. Poland, France, Italy etc. to be elements of the set, and we do not use various parts of those countries as elements of this set. In this meaning e.g. the Tatra Mountains or the Małopolska Upland are not elements of the set of European countries, despite the fact that they are parts of certain European countries. In this meaning we often use those terms when we talk e.g. about a set of Polish cities and we recognize particular cities e.g. Wroclaw, Warsaw etc. as elements of this set and we do not recognize particular streets, squares and other parts of those cities (. . . ) as elements of this set. In this meaning we cannot identify the notion of an element of a set with the common notion of part. Consequently, being an element in the set theory signifies membership in a set and an element is always different from the very set and it is not

6

It is a large excerpt of Borkowski’s book quoted in [98, p. 14].

38


synonymous to the notion of a part. On the other hand, being an element in Mereology is identified with the meaning of a part which may be identified with the whole set. In addition, mereological being an element of a set or a class of ms does not signify that the element has to be an m, compliant to the example quoted by Borkowski. Those crucial discrepancies between those theories have important impact on the notion of ’composition’ or a ’sum’ of objects.

2.2

The relation of composing a whole

Mereology is seen as a general science of composition. If we have parts, a certain whole composed of those parts has to exist. Consequently, the binary relation of sum is the relation characteristic for Mereology. Some authors use the notion of ’fusion’ instead of ’sum’ [57]. The choice of notion we use doesn’t have much importance in classical theories defining mereology7 because they are equivalent [98]. However, Leśniewski doesn’t use the term ’fusion’ – he uses ’sum’8 . We will also use the term ’sum’.

We will start with Cantor’s theory of sets, in other words called the theory of distributive sets. In the previous section Borkowski clearly explained that the set theory relation of a sum is a type of collection of objects forming a certain abstract whole. It is assumed that for any A and B sets their sum always exists: A ∪ B. A sum of two sets A, B means a set composed of those and only those elements which are members of at least one of the sets

7

We have to be aware that there are many various mereological theories, e.g. (Classical Mereology) – classical minimal model with the relation of order on a set of objects only, a model which was originally proposed by Leśniewski; or the General Extensional Mereology – a model with postulates concerning complementing, which we will take into consideration in the further part of this work and which we will discuss in detail on page 48. 8 However, in the further part of this monograph we will see that this fact doesn’t take place in non-classical mereological theories.

2.2. THE RELATION OF COMPOSING A WHOLE

39

A and B 9 . The term of ’class’ is often used instead of the term of ’set’10 . If in notation we mark the term ’class’ with the symbol K and objects with certain defined quality with S, then a class may be defined in the following manner:

x := K(S) ≡df Kx ∧ ∀y (y ∈ x ≡ y = S),

(2.1)

x x is a class of objects ’S’ if and only if x is a class and every element of this class is ’S’. Distributive classes (sets) cannot be identified with any time-spatial object. As a description of a class we have to carefully adapt the following phrase: «a common gathering», with certain security measures counteracting misunderstandings. Gathering? Perhaps, but not in a sense of complex concrete objects or complex accumulations. The United States are a very large physical body (with arbitrary width) composed of several dozens of states as parts. At the same time, they (the United States) are a physical body composed of several oblasts as wholes. It is the same concrete object irrespective of the realised abstract division. The whole composed of states and the whole composed of oblasts are the same thing. However, a class of states cannot be identified with a class of oblasts because there are many issues we may ascribe to one class and we cannot ascribe to the other [...]. Such classes have to be recognized as two non-spatial and abstract beings contrary to unitary, concrete wholes specified by their ingredients

9

Formally, the Sum Axiom is noted down in the following manner: A ∪ B = {x ∈ U : x ∈ A ∨ x ∈ B}. 10 The notion of ’class’ has broader meaning than the notion of a ’set’: any set is a class, but not every class is a set. It is not true in case of collective sets. Any class is a set and vice versa (see [98]). Quine identifies sets (Cantor’s sets that is distributive sets) with co-extensive classes [103, p. 79], „A class is a set when it is an element of a certain class”.

40


[102, p. 121]11 , [98, p. 17]. Consequently, the United States constitute a different object than the class of states of the USA. Next, as Pietruszczak following Quine’s reasoning concludes, despite the fact that those distributive classes ’exist’ (Polish ’bytują’)12 , they are abstract objects. The fact that classes are universals, that is abstract objects, sometimes becomes blurred by discussing classes as aggregates or gatherings and identifying in this manner e.g. a class of stones with a pile of stones. A pile of stones is in fact a concrete object, as concrete as stones composing it, but a class of stones constructing a pile is not identical with the mentioned pile. If it was so, then certain other class would have to be identical with the same pile, it would be the class of particles the stones in the pile are made of. But those classes have to be recognized as different, because we want to discuss one of them as having, let’s say, one hundred elements, and the other as having trillion elements. Consequently, classes are abstract objects [...]. Of course, providing that they exist [101, p. 157].

11

Quine comes back to identifying classes with qualities and continues his reasoning: When we finally liberated classes from any misleading trace of tangibility, there are few reasons left for distinguishing them from qualities. It doesn’t’ matter whether we will read «x ∈ y» as «x is an element of the class y» or «x has the y quality». If any difference between classes and qualities exists, it consists of only the following: classes are the same when they have the same elements; in terms of qualities, their identity has not been broadly recognized when they are ascribed to the same objects. [...] However, we are allowed to see classes as qualities if the last notion is described in the following manner: two qualities are identical when they have the same objects. 12 In footnote 25 p. 18 in [98] he writes: „Some philosophers do not identify the content of words ’exist’ and ’be’. They believe that everything which exists – is, but not the other way round (or in other words: all objects are but there are objects which do not exist.). Consequently, the term ’Pegasus’ signifies a (fictitious) object which does not exist. [...] Other philosophers identify the content of words ’exist’ and ’are’. E.g. Quine provides the following response to the question ”What exists?”: everything. [...] he emphasises that «What exists, exists, remains a statement only. As a result, a space for difference of opinions remains concerning particular cases; that is why this issue remains alive since many centuries».” [101, p. 9].


41

If a class of ’something’ took any place in the space, then it would be identified with this ’something’. On the other hand, a mereological object13 always has its place in space-time. Mereology is sort of ’gluing’ objects in a concrete whole. Perhaps this is the reason for using the term ’fusion’ more often than ’sum’ as it represents better the nature of this operation. In general, we may say that we deal with two different perspectives of creating a ’collectivity’. The set theory proposes a ’bottom-up’ approach to creating sets: first we distinguish certain elements either by enumeration or by extension14 and only then we treat them as a certain abstract whole. On the other hand, Mereology offers as if ’top-down’ approach: first we have a certain whole, and only then, as a second step, we distinguish parts of this whole. Therefore, the whole has to be a concrete, physical object. We cannot divide ’nothing’ into parts. An important feature of distributive classes is the fact that if we have two identical objects and a certain object indicates a class of one of the objects, then the same class is indicated by the second object:

∀y (y = S ≡ y = M ) ∧ x = K(S) → x = K(M ),

(2.2)

x = K(S) ∧ x = K(M ) → ∀y (y = S ≡ y = M ).

(2.3)

In order to assure that the terms we deal with – the class of Ss, is not a general name [98, p. 27], it is assumed that the so-called extensionality axiom (or the extensionality principle which states that different classes having the same elements do not exist.

13

For Leśniewski only concrete, physical objects exist. Such a creation as distributive class does not mean anything to him, therefore it doesn’t exist. In addition, it is said that Mereology is a theory where unitary variables represent general names contrary to ontology by Leśniewski, which is considered to be a theory of distributive classes. This differentiation, in a sense, is similar to the one realised on the family of sets and a set being their sum. For more details on this subject see [101], [98], [100]. 14 Defining a given notion by extension comprises providing the scope of its name, that is all objects which are designates of this name.

42


Kx ∧ Ky ∧ ∀z (z ∈ x ≡ z ∈ y) → x = y.

(2.4)

In fact, if x is a class of Ss and y is a class of Ss then pursuant to the Definition 2.1 (p. 39), x and y are classes and ∀z (z ∈ x ≡ z ∈ y). Then, pursuant to (2.4) we receive the equality of classes x = y [98, p. 27]. Leśniewski did not permit the existence of distributive sets or classes. He treated them as empty terms [63, pp. 204–205], [98, p. 34]. I do not know what Mr Whitehead and Mr Russell mean as a class in their comments to their system. The information that «class», compliant to authors, should be the same as «range» does not help me here at all, because I don’t know either what authors understand as the range. Moreover, when they deliberate those issues, I don’t know what exact objects’ existence or lack of existence they deliberate. For him, only collective classes make sense, only this, what is concrete. For him, the notion of a ’class’ is equivalent to the notion of a ’sum’. Mereological definition of a sum by Leśniewski states that the sum of any not empty set of objects15 X (where, in the distributive set theory, ’set of objects a’ corresponds to the term ’set of Ss’) is simply a sum of those parts of it for which elements with X exist which are overlapping with them16 [66]. Definition 2.1. P17 is a sum of objects ’a’ if and only if if the following conditions are satisfied18 :

15

P is a set of objects a if and only if (i) P is an object, (ii) for any Q, if Q is an ingredient of the object P , then a certain ingredient of the object Q is the ingredient of certain a which is the ingredient of the object P [64, p. 270]. 16 Formal definition of a sum is presented on the page 50, similarly to the relation of overlaping (p. 46). 17 We have left the general notation applied by Leśniewski. 18 Leśniewski defines the notions of ’object’ and ’being external’ in the following manner:


43

(I) P is a class of objects ’a’, (II) At any Q and R and if Q is ’a’ and R is ’a’, then Q is the same object as R, or is external in reference to R.

According to Pietruszczak’s suggestion [98, p. 66], if a given name is represented by the symbol ’a’ and we assume that the distributive set {y ∈ M : y is a} [98]19 , is its range, then instead of saying that x is a class we may say that x is a mereological sum of all elements of the set {y ∈ M : y is a}, that is the sum of all ’a’. Next, instead of discussing a mereological sum of all ’a’ we may discuss a sum of all elements of a given subset of the set M . In this manner, the term ’class’ (Def. 2.1) assumes the form: Definition 2.2. ’x’ is a class of elements of Z if and only if: (i) ’x’ is an object, (ii) every element of Z is an ingredient of the object ’x’, (iii) at any ’w’, if ’w’ is the ingredient of the object ’x’, then certain ingredient of the object ’w’ is an ingredient of a certain element Z.

A class may be composed of various groups of its parts. Europe may be perceived as a class of their countries, or as a class composed of eastern part and western part of our continent. We will once more use the example of Lewis depicting how the meaning of the notion ’sum’ in mereology differs from the meaning of the same notion

P is a class of objects ’a’ if and only if (i) P is an object (ii) every ’a’ is an ingredient of the object P, (iii) at any Q, if Q is the ingredient of the object P then certain ingredient of the object Q is the ingredient of certain ’a’ [64, p. 264]. P is external to Q if and only if (i) P is an object, and (ii) no ingredient of the object Q is an ingredient of the object P [64, p. 278]. 19 Here we deal with an extension of a given notion ’a’ to a set of objects having it.

44


in the set theory [77, p. 78]. Let’s assume that we have two words ’master’ and ’stream’ which are made of the same six letters. We have two manners of writing a word composed of the same signs. If we assume that letters are certain arts objects which may be displaced and we are in certain spacetime, then the letter ’m’ in the word ’master’ has other place than the same letter in the word ’stream’ and the object which is a composition (a sum, fusion) of all those six letters written as one object and called a term ’master’ corresponds to other moment in time than the object ’stream’. Consequently, those both objects are composed of the same six letters but they have different parts. We have time difference between those objects. They are two different mereological objects. Now, from the perspective of the set theory context we have again the same six letters used to form a sequence of signs where letters are not repeated. We have objects ’master’ and ’stream’. They are two sets composed of the same elements. The only problem emphasized by Lewis is that it is not a mereological composition, but a sum of objects in distributive sense. This set is not perceived as a composition of its proper parts, but as a group of its elements where none of the elements constitutes a part of this set. In common set theory sum, the order of elements in a set is not important, therefore from this perspective the sets {m, a, s, t, e, r} and {s, t, r, e, a, m} would be the same. On the other hand, for us the order of elements in the set is important. To sum up, Lewis states that the example of letters and words does not constitute a threat to the principle of extensionality of sum, in other words for the principle stating that if we assume that we have two objects which are the same, then what is true for one of them should be also true for the other (more details in Section 2.4), because the type ‘word’ has been composed of simpler universalia: six letter types and the relation of setting. Consequently, the composition is made of not only letters, but also relations, and in case of those two words we have different relations [76]. To sum up: in both theories we realize the same operation of ’collection’, but in case of Mereology, it is always collection of concrete objects, that is

2.3. AXIOMATISATION OF EM

45

objects which ’exist’ in space-time [98, p. 16]. Consequently, the formed collective set is also a concrete object. In case of distributive sets, a complex object has to be always understood as an abstract notion. In addition, it seems that the role of certain relations is also important here, e.g. the relation of order in case of Mereology, this relation is by rule skipped in case of distributive sets.

2.3 2.3.1

Axiomatisation of Extensional Mereology Mereology founded on the relation of part

Stanisław Leśniewski founded mereology on three various notions. In this work we will adopt the model of Mereology grounded on the relation of ingredient – v, that is the improper part, as we would like to develop a model on non-classical mereology proposed in Chapter 3 founded on the rejection of the anti-symmetricity principle for this relation. Let M be our universum of objects, as Leśniewski calls it, and v – the binary relation of being an ingredient. It is assumed that the relation v satisfies three fundamental conditions: reflexivity, antisymmetry and transitivity, in other words it is a relation which partially orders the universum M:

∀x∈M x v x, 20

(M1)

∀x∈M ∀y∈M (x v y ∧ y v x → x = y),

(M2)

∀x∈M ∀y∈M ∀z∈M (x v y ∧ y v z → x v z) .

(M3)

Those three postulates express the natural conviction that every object constitutes part of itself21 and that if we have two objects and one of

20

The sign ’∈’ is used to express the relation of being an element. We would like to remind the reader that for the purpose of this paper the notion of a part is always synonymous with the improper part. 21

46


them constitutes part of the other and the other constitutes part of the first object, then we discuss the same object. In general, transitivity, after numerous discussions, is usually approved of. With the use of the relation v we may define three auxiliary relations: the relation of overlapping – ◦, disjointedness – o and the relation of a proper part – @, defined on the M × M product in the following manner:

∀x∈M ∀y∈M (x ◦ y ≡df ∃z∈M (z v x ∧ z v y)),

(2.5)

∀x∈M ∀y∈M (x o y ≡df ¬ (x ◦ y)),

(2.6)

∀x∈M ∀y∈M (x @ y ≡df x v y ∧ x 6= y).

(2.7)

The relation of overlapping defines mutual overlapping of objects and states that two objects overlap (or there is an intersection of them) if there is one object which constitutes a part both of the first and the second object. The relation of disjointedness states that two objects x and y are disjoint if there is no other object which constitutes part of x and y. The last relation is a relation of a proper part stating that a given object x is a proper part of the object y if x constitutes part of y and x is not y. Taking into consideration the postulate (M1) together with the Leibniz’s Principle (p. 118) we receive the following thesis:

{(M 1), (LP )} ` x = y → x v y.

(ML)

which, pursuant to the transposition principle (TOLL)22 results in equivalent principle in the following form:

{(M 1), (LP )} ` x 6v y → x 6= y.

22

(MLa)

The list of all logical principles and rules applied in the text is given in Appendix.


47

Consequently, the principle (M2) may be noted down in equivalent form in the following manner:

∀x∈M ∀y∈M (x 6= y → x 6v y ∨ y 6v x),

(M2’)

providing basis for the following definition of proper part and a theorem:

∀x∈M ∀y∈M (x @ y ≡df x v y ∧ y 6v x).

(2.8)

Theorem 2.1. {(M 1), (M 20 )} ` (x v y ∧ x 6= y) ≡ (x v y ∧ y 6v x). Proof: (→) (1) x v y ∧ x 6= y {ass.} (2) x v y ∧ (x 6v y ∨ y 6v x) {(1, (M 20 )), (AC), (CON ), (M P )} (3) (x v y ∧ x 6v y) ∨ (x v y ∧ y v 6 x) {(SEP ) 2} (x v y ∧ y 6v x) {(DA) 3} (←) (1) x v y ∧ y 6v x {ass.} (2) y 6v x → y 6= x {(M La)} (3) x v y {(DC) 1} (4) y 6v x {(DC) 1} (5) y 6= x {(M P ) 2, 4} x v y ∧ x 6= y {(AC) 3, 5}

The second part of the proof is independent of the (M2)’ postulate. Consequently, in non-antisymmetric Mereology we may presuppose that the mentioned two definitions of proper part will not overlap23 .

23

See also [96].

48


Apart from conditions (M1)–(M3), in classical Mereology the so-called Strong Supplementation Principle is also assumed – we will mark it in a symbolic way as (M4). This principle states that if one object is not part of the second object then the first of them has to contain a part which does not intersect with the other. It is a postulate introducing certain separation principles.

∀x∈M ∀y∈M (x 6v y → ∃z∈M (z v x ∧ z o y)).

(M4)

This principle is independent of the partial order characterising Mereology and it cannot be derived from (M1)–(M3) postulates only. Gorzka [27, p. 13] provides the example of concentric circles. Let’s assume that M is a set of such circles in R2 , and the relation v is interpreted as the relation of inclusion ⊆. Postulates (M1)–(M3) are satisfied: each circle is included in itself if one circle is included in the other and vice versa, then we have the same circle. Moreover, the quality of transitivity is quite obvious. However, let’s take two circles x, y, where y * x. Due to the fact that we have concentric circles, then a circle included in y which would not intersect with x does not exist. Moreover, as presented in [27], this condition implies the so-called Weak Supplemetation Principle:

∀x∈M ∀y∈M (x @ y → ∃z∈M (z @ y ∧ z o x)).

(WSP)


49

Theorem 2.2. {(M 1), (M 2), (M 4), (2.7)} ` (W SP ). Proof: (1) (b 6v a → ∃z∈M (z v b ∧ z o a)) ∧ a @ b {ass.} (2) (b 6v a → ∃z∈M (z v b ∧ z o a)) ∧ (a v b ∧ b 6v a) {(LP )(T h.2.1, (2.7)), 1} (3) ∃z∈M (z v b ∧ z o a)) ∧ a v b {(M P ) 2; (COM M )(CON )(DC) 2} (4) ∃z∈M (z v b ∧ z o a)) {(DC) 3} (5) a v b {(DC) 3} (6) c v b ∧ c o a {(D ∃) 4} (7) c v b {(DC) 6} (8) c o a {(DC) 6} (9.1) b 6= c {ad. ass.} (9.2) c v b ∧ b 6= c {(AC) 7, 9.1} (9.3) c @ b {(2.7)} (9.4) c @ b ∧ c o a {(AC) 9.3, 8} (9.5) ∃z∈M z @ b ∧ z o a {(D ∃) 9.4} c @ b → ∃z∈M z @ b ∧ z o a {(AI) 9} Moreover, the Week Supplementation Principle together with axioms (M1), (M3) imposes the antisymmetry condition for the relation of part. Theorem 2.3. {(M 1), (M 3), (2.7), (W SP )} ` (M 2). Proof: (0) a @ b → ∃z∈M (z @ b ∧ z o a) {ass.} (1) ∃x∈M ∃y∈M (x v y ∧ y v x ∧ x 6= y) {N DP } (2) ∃y∈M (a v y ∧ y v a ∧ a 6= y) {(D ∃) 1} (3) a v b ∧ b v a ∧ a 6= b {(D ∃) 2} (4) a @ b ∧ b v a {(2.7), (COM M ), (CON ) 3} (5) a @ b {(DC) 4} (6) b v a {(DC) 4} (7) ∃z∈M (z @ b ∧ z o a) {(M P ) 0, 5} (8) c @ b ∧ c o a {(D ∃) 7} (9) c @ b {(DC) 8} (10) c o a {(DC) 8}

50


(11) c v b ∧ c 6= b {(M P )((DE) (2.7), 9)} (12) c v b {(DC) 11} (13) c 6= b {(DC) 11} (14) c v a {(M 3) 12, 6} contradiction {14, 10}

Therefore in the classical Mereology by Leśniewski, the Strong Supplementation Principle is deductively stronger than the Week Supplementation Principle. In non-classical theories such dependence does not take place at all, rather to the contrary, roles of those both conditions change, as we will see later in the Chapter 3, compliant to the way it was presented in the paper [96]. Characteristic relation in Mereology is the relation of a sum – Sum corresponding to the notion of a class by Leśniewski [64, p. 264] (see Section 2.2)24 .

x Sum X ≡df ∀y∈X y v x ∧ ∀z∈M (z v x → ∃w∈X w ◦ z) .

(2.10)

The X symbol signifies a non-empty distributive set of all those elements which are the range of the name Sum. Consequently, if we think about an x, we may think about all elements of a certain non-empty set X 25 , which is composed of all its parts which overlap with a certain part of

24 This relation can be also expressed with propositional schemata – FUS [31], which is a collection of formulas defined with the language of the first order logic, where φ(x) is a form with free variable x:

F us(φ) := ∃x∈M φ(x) → ∃z∈M ∀y∈M (y ◦ z ≡ ∃w∈M (φ(w) ∧ y ◦ w)).

(2.9)

25 That is why, consequently, the definition of Sum is a formula of the second order. This definition corresponds to the presented previously schemata of fusion – FUS := {F us(φ)}, where φ is a formula of a language of the first order logic. Due to this reason Mereology is defined either as an elementary theory with infinite number of axioms or as a nonelementary theory. Various authors adopt various manners of axiomatisation depending on goals of developments. We will see Mereology as a non-elementary theory.


X. The Sum relation defined in such a way is a monotonic relation: Theorem 2.4. ` ∀x ∈ M ∀y∈M ∀∅6=X,Y ⊆M (X ⊆ Y ∧ x Sum X ∧ y Sum Y → x v y). Proof: (0) ∀x∈M ∀y∈M (x 6v y → ∃z∈M (z v x ∧ z o y)) {ass.} (1) X ⊆ Y ∧ x Sum X ∧ y Sum Y ∧ x 6v y {N DP } (2) X ⊆ Y ∧ (x Sum X ∧ y Sum Y ∧ x 6v y) {(CON ) 1} (3) X ⊆ Y {(DC) 2} (4) x Sum X {(DC), (CON ), (DC) 2} (5) x Sum Y {(DC), (CON ), (DC) 2} (6) x 6v y {(DC), (CON ), (DC) 2} (7) ∃z∈M (z v x ∧ z o y) {(D ∀) 0 − 2x, (M P ) 0, 6} (8) c v x ∧ c o y {(D ∃) 7} (9) c v x {(DC) 8} (10) c o y {(DC) 8} (11) ∀m∈X (m v x → ∃w (w ∈ X → w ◦ m)) {(DE) (2.10), (M P )((2.10), 4)} (12) c v x → ∃w (w ∈ X → w ◦ c) {(D ∀) 11} (13) ∃w (w ∈ X → w ◦ c) {(M P ) 12, 9} (14) b ∈ X → b ◦ c {(D ∃) 13} (15) b ∈ X {X 6= ∅} (16) b ◦ c {(M P ) 14, 15} (17) b ∈ Y {(DE), (D ∀), (M P ) 3, 15} (18) ∀m∈Y (m v y) ∧ ∀v∈M (v v y → ∃w (w ∈ X → w ◦ m)) {(DE) (2.10), (M P ) 5, (2.10)} (19) ∀m∈Y m v y {(DC) 18} (20) b ∈ Y ∧ b v y {(D ∀) 19} (21) b v y {(DC) 20} (22) ∃k∈M (k v b ∧ k v c) {(M P ) (2.5), 16} (23) d v b ∧ d v c {(D ∃) 22}

51

52


(24) d v b {(DC) 23} (25) d v c {(DC) 23} (26) d v b ∧ b v y {(AC) 24, 21} (27) d v y {(DE)(M 3) 26} (28) d v c ∧ d v y {(AC) 25, 27} (29) ∃n∈M (n v c ∧ n v y) {(A ∃) 28} (30) c ◦ y {(M P )((DE)((2.5), 29)} contradiction {30, 10}

Moreover, the above presented assumptions lead us also to the conclusion that Sum is the function referring to the left argument that is it has the quality of uniqueness:

Theorem 2.5. ` ∀x∈M ∀y∈M ∀∅6=X⊆M (x Sum X ∧ y Sum X → x = y). Proof: (1) x Sum X {ass.} (2) y Sum X {ass.} (3) X ⊆ X {ref lexivity of incl.} (4) x Sum X ∧ y Sum X {(AC) 1, 2} (5) x Sum X ∧ y Sum X ∧ X ⊆ X {(AC) 4, 3} (6) x v y {(M P ) T h.2.4, 5} (7) y Sum X ∧ x Sum X ∧ X ⊆ X {(CON ), (COM M ) 5} (8) y v x {(M P ) T h.2.4, 5} (9) x = y {(M P )((AC) 6, 8 (M 2))} x Sum X ∧ y Sum X → x = y {(AI) 9}

It is obvious that in structures defined in this manner, the very monotonicity of the Sum relations does not impose the Strong Supplementation Principle.


53

Example 2.1. Let M = {x, y} and X = {x}, Y = {x, y} and x v y. Hence, from the definition of Sum ((2.10), p. 50) we obtain: x Sum X, y Sum Y and y 6v x, and it is not true that ∃z∈M (z v y ∧ z o x)26 .

Consequently, in mereological structures the very monotonicity of the Sum relation with the (M1)–(M3) assumptions for the relation v, is not sufficient for fulfilling the Strong Supplementation Principle. Let’s take the following example:

Example 2.2. Let M = {0, 1, 2, 3} and v be the linear order on M 27 , that is 0 v 1 v 2 v 3. In this model 3 Sum M . Moreover 3 6v 1 also takes place and for 3 there is no such z that z v 3 ∧ z o 1. z may assume values from the set {0, 1, 2, 3}, but 0 ◦ 1, 1 ◦ 1, 2 ◦ 1 and 3 ◦ 1.

In mereological structures, in order to have the possibility to form, out of any non-empty possible to define set of objects, the new whole composed of the sum of its elements, additional postulate of the existence of a sum – (M5) has been introduced:

∀∅6=Z⊆M ∃x∈M x Sum Z.28

(M5)

26 In some monographs, other version of supplementation principle is considered – (SSP◦ ), however, from the above-presented example, it stems that it is not equivalent to the classical principle (SSP). We should apply the notation by Pietruszczak [98, p. 75] and define the following sets: I(x) =df {y ∈ M : y v x}, O(y) =df {z ∈ M : z ◦ y}. The first S principle of monotonicity is satisfied – (MON1): ∀x∈M ∀y∈M ∀Z∈2M \{∅} (I(x) ⊆ O(Z) ∧ Z ⊆ I(y) → S x v y), as in the presented example: I(x) = X, O(x) = Y , O(y) = Y , I(y) = Y , O(Z) = Y, ∀Z∈2M \{∅} . The principle (SSP◦ ) defined in the following manner: ∀x∈M ∀y∈M (I(x) ⊆ O(y) → x v y) is also fulfilled, but the (SSP) is not. 27 We will say that the ordering relation v is the linear order on M , if it fulfils additionally the axiom of coherence, that is: ∀x∈M ∀y∈M (x 6= y → x v y ∨ y v x) [104, p. 126]. 28 The assumption that X 6= ∅ is redundant, as we reach contradiction otherwise.

54


This set of five postulates (M1)–(M5) defines the fundamental model of extensional Mereology (GEM – Ground Extensional Mereology), which we will call mereological structure. Numerous authors provided mathematical proofs [98], [27], that this type of model of mereological structures (GEM) hM, vi with additionally defined operations of a sum, product and supplementation and added zero element, creates the structure constituting the Boolean algebra, that is for any non-degenerated set of objects M : M > 2 the pair h2M \ {∅}, ⊆i is a mereological structure if we assume common definition of the set theoretic sum for the definition of the relation of sum. At this point, following Stone, any mereological structure is isomorphic with a certain non-empty family of sets ordered with the relation of inclusion [27].

2.3.2

Mereology founded on the relation of proper part

Due to the fact that the primary notion in Mereology may be the term of ’proper part’, which may be treated as synonymous to the expression ’fragment’ or ’piece’, it seems natural to propose axiomatisation of this theory with this notion. Let M be a non-empty set of objects. For any element from M , we assume that there are no two such objects out of which one would be a proper part of the second one and the second one a proper part of the first one:

∀x∈M ∀y∈M (x @ y → y 6@ x).

(L1)

The transitivity of this relation, similarly to the relation of a part, seems a natural quality, even though some challenge its fulfilment29 .

29

Pietruszczak in [98, p. 8] quotes the example by Rescher: a nucleus is a proper part of a cell, a cell – a proper part of an organ, but the nucleus is not a proper part of the organ. On the other hand, Simons explains the truthfulness of this postulate by comparing it to the space-time relation of inclusion.


∀x∈M ∀y∈M ∀z∈M (x @ y ∧ y @ z → x @ z).

55

(L2)

As a consequence, due to (L1), we may provide proof that the relation @ is irreflexive – no object constitutes its own proper part: Theorem 2.6. {(L1)} ` ∀x∈M x 6@ x. Proof: (1) ∀x∈M ∀y∈M (x @ y → y 6@ x) {ass.} (2) x @ x {N DP } (3) x @ x → x 6@ x {(D ∀) 1 − 2x} (4) x 6@ x {(M P ) 3, 2} contradiction {2, 4}

We introduce two additional postulates (L3) and (L4) assuring uniqueness and the existence of mereological sum:

∀x∈M ∀y∈M ∀∅6=Z⊆M (x Sum Z ∧ y Sum Z → x = y) ,

(L3)

∀∅6=Z⊆M ∃x∈M x Sum Z.

(L4)

At this point we may define the following:

∅= 6 Z ⊆ M → [x Sum Z ≡df ∀y∈Z y v x ∧ ∀t∈M (t v x → ∃w∈Z w ◦ t)]. (2.11) Because (L3) is true for every non-empty set Z, therefore also for Z = {x}. Consequently, pursuant to (M1) and (2.5) (p. 46) we receive that x Sum {x}, a condition stating that x is the only singleton whose sum is x – (L3*):

56


∀x∈M ∀y∈M (y Sum {x} → y = x).

(L3*)

It turns out that the uniqueness of relation Sum together with the postulate (L1) imposes the Weak Supplementation Principle. Theorem 2.7. {(L3∗), T h. 2.6, (2.5), (2.10), (2.12)} ` (W SP ). Proof: (1) ∀x∈M ∀y∈M (y Sum{x} → y = x) {ass.} (2) ∀x∈M x 6@ x {ass.} (3) ∃x∈M ∃y∈M (x @ y ∧ ¬(∃z∈M (z @ y ∧ z o x))) {N DP } (4) a @ b ∧ ¬(∃z∈M (z @ b ∧ z o a)) {(D ∃) 3 − 2x} (5) a @ b ∧ ∀z∈M (z @ b → z ◦ a)) {(N ∃), (N C), (2.6), (REP )} (6) a @ b {(DC) 5} (7) ∀z∈M (z @ b → z ◦ a) {(DC) 5} (8) z @ b → z ◦ a {(D ∀) 7} (9) z v b → z @ b ∨ z = b {(O ∀) − 2x, (DE) (2.12)} (10.1) z @ b {add. ass.} (10.2) z v b → z ◦ a {(T RAN ) 9, 8} (11.1) z = b {add. ass.} (11.2) b ◦ a {(M P )(a @ b → a ◦ b), 6} {a ◦ b ≡ b ◦ a} (11.3) b v b → b ◦ a {(AI)} (12) ∀z∈M (z v b → z ◦ a) {(AC) 10, 11} (13) b 6@ b {(D ∀) 2} (14.1) a 6= b {add. ass.} (14.2) a v b {(M P ) ((DE) (2.12)), 6, 14.1} (14.3) b Sum {a} {(DE)((AC) 14.2, 12; (2.10))} (14.4) b Sum{a} → b = a {(D ∀) 1} (14.5) b = a {(M P ) 14.4, 14.3} (14.5) contradiction {14.5, 14.1} (15.1) a = b {add. ass.} (15.2) b @ b {(LP ) 15.1, 6} (15.3) contradiction {15.2, 8} contradiction {(RN ) 14, 15}


57

It turns out that (L4) is equivalent to the postulate (M5), and (L3) together with (L1) results in (M4) (Theorem 2.13, p. 61). However, we will start with introducing the definition of an improper part and the remaining relations characteristic for mereological structures.

∀x∈M ∀y∈M (x v y ≡df x @ y ∨ x = y).

(2.12)

The intersection relation and the disjoint relation are introduced as in the previous section. We may notice, that the (M1) derives from (2.12); (M2) constitutes a conclusion from (2.12) and from (L1); and (M3) is a consequence of (L2). Formally we note down those propositions in the following manner:

Theorem 2.8. {(L1), (2.12)} ` (M 1). Proof: (1) ∀x∈M ∀y∈M (x v y ≡ x @ y ∨ x = y) {ass.} (2) ∀x∈M ∀y∈M (x @ y → y 6@ x) {ass.} (3) x v y ≡ x @ y ∨ x = y {(D ∀) 1 − 2x} (4) x @ y ∨ x = y → x v y {(DE) 3} (5.1) x = y {add. ass.} (5.2) x v x {(M P )(4, 5.1); (LP )} (6.1) x @ y {add. ass.} (6.2) y 6@ x {(M P )((D ∀) 2 − 2x, 6.1)} (6.3) y v x ≡ y @ x ∨ y = x {(D ∀) 1 − 2x} (6.4) y @ x ∨ y = x → y v x {(DE) 6.3} (6.5) y = x → y v x {(DA) 6.4, 6.2; } (6.6) x v x {(LP ) 6.5} x v x {(RW ) 5, 6}

Theorem 2.9. {(L1), (2.12)} ` (M 2).

58


Proof: (1) ∀x∈M ∀y∈M (x @ y → y 6@ x) {ass.} (2) ∀x∈M ∀y∈M (x v y ≡ x @ y ∨ x = y) {ass.} (2a) y v x {ass.} (2b) x v y {ass.} (3) x @ y → y 6@ x {(D ∀) 1 − 2x} (4) x v y ≡ x @ y ∨ x = y {(D ∀) 2 − 2x} (5) x v y → x @ y ∨ x = y {(DE) 4} (6) y v x ≡ y @ x ∨ y = x {(D ∀) 2 − 2x} (7) y v x → y @ x ∨ y = x {(DE) 6} (8) x @ y ∨ x = y {(M P ) 5, 2b} (9) y @ x ∨ y = x {(M P ) 7, 2a} (10.1) x @ y {add. ass.} (10.2) y 6@ x {(M P ) 3, 10.1} (10.3) y = x {(DA) 9, 10.2} (11.1) y @ x {add. ass.} (11.2) x 6@ y {(M P )((D ∀) 1 − 2x, 11.1)} (11.3) x = y {(DA) 8, 11.2} x = y {(RW ) 10, 11} Theorem 2.10. {(L2), (2.12)} ` (M 3). Proof: (1) ∀x∈M ∀y∈M ∀z∈M (x @ y ∧ y @ z → x @ z) {ass.} (2) ∀x∈M ∀y∈M (x v y ≡ x @ y ∨ x = y) {ass.} (3) x v y {ass.} (4) y v z {ass.} (5) x v y ≡ x @ y ∨ x = y {(D ∀) 2 − 2x} (6) y v z ≡ y @ z ∨ y = z {(D ∀) 2 − 2x} (7) x v y → x @ y ∨ x = y {(DE) 5} (8) y v z → y @ z ∨ y = z {(DE) 6} (9) x @ y ∨ x = y {(M P ) 7, 3} (10) y @ z ∨ y = z {(M P ) 8, 4} (11.1) x @ y {add. ass.} (11.1.1) y @ z {add. ass.} (11.1.2) x @ y ∧ y @ z {(AC) 11.1, 11.1.1}


59

(11.1.3) x @ z {(M P )((D ∀) 1 − 3x, 11.1.2} (11.1.4) x v z {(M P )((O ∀) 2 − 2x, (DE) 2, (DA) 2), 11.1.3} (11.2.1) y = z {add. ass.} (11.2.2) x v z {(LP ) 11.2.1, 3} (12.1) x = y {add. ass.} (12.2) x v z {(LP ) 12.1, 4} x v z {(RW ) 11, 12}

In addition, reverse relationships take place: Theorem 2.11. {(M 2), (2.7)} ` (L1). Proof: (1) ∀x∈M ∀y∈M (x v y ∧ y v x → x = y) {ass.} (2) ∀x∈M ∀y∈M (x @ y → x v y ∧ x 6= y) {ass.} (3) ¬(∀x∈M ∀y∈M x @ y → y 6@ x) {N DP } (4) ∃x∈M ∃y∈M (x @ y ∧ y @ x) {(N ∀), (REP ), (N N ), (N A) 3} (5) a @ b ∧ b @ a {(D ∃) 4 − 2x} (6) a @ b {(DC) 5} (7) b @ a {(DC) 5} (8) a @ b → a v b ∧ a 6= b {(D ∀) 2 − 2x} (9) b @ a → b v a ∧ b 6= a {(D ∀) 2 − 2x} (10) a v b ∧ a 6= b {(M P ) 8, 6} (11) a v b {(DC) 10} (12) a 6= b {(DC) 10} (13) b v a ∧ b 6= a {(M P ) 9, 7} (14) b v a {(DC) 13} (15) a v b ∧ b v a {(AC) 11, 14} (16) a v b ∧ b v a → a = b {(D ∀) 1} (17) a = b {(M P ) 16, 15} contradiction {12, 17}

60


Theorem 2.12. {(M 3), (2.7), (L1)} ` (L2). Proof: (1) ∀x∈M ∀y∈M (x @ y ≡ x v y ∧ x 6= y) {ass.} (2) ∀x∈M ∀y∈M ∀z∈M (x v y ∧ y v z → x v z) {ass.} (2a) ∀x∈M ∀y∈M (x @ y → y 6@ y) {ass.} (3) ¬(∀x∈M ∀y∈M ∀z∈M (x @ y ∧ y @ z → x @ z)) {N DP } (4) ∃x∈M ∃y∈M ∃z∈M ¬(x @ y ∧ y @ z → x @ z) {(N ∀) 3 − 3x} (5) ∃x∈M ∃y∈M ∃z∈M (x @ y ∧ y @ z ∧ x 6@ z) {(REP ), (N N ), (N A) 4} (6) a @ b ∧ b @ c ∧ a 6@ c {(D ∀) 5 − 3x} (7) a @ b ∧ b @ c {(CON ), (DC) 6} (8) a 6@ c {(CON ), (DC) 6} (9) a @ b {(DC) 7} (10) b @ c {(DC) 7} (11) a @ b ≡ a v b ∧ a 6= b {(D ∀) 1 − 2x} (12) b @ c ≡ b v c ∧ b 6= c {(D ∀) 1 − 2x} (13) a v b ∧ a 6= b {(M P )((DE) 11, 9)} (14) a v b {(DC) 13} (15) b v c ∧ b 6= c {(M P )((DE) 12, 10)} (16) b v c {(DC) 12} (17) a v b ∧ b v c {(AC) 14, 16} (18) a v b ∧ b v c → a v c {(D ∀) 2 − 3x} (19) a v c {(M P ) 18, 17} (20.1) a = c {add. ass.} (20.2) c @ b {(LP ) 20.1, 9} (20.3) c @ b → b 6@ c {(D ∀) 2a − 2x} (20.4) b 6@ c {(M P ) 20.3, 20.2} (20.5) contradiction {20.4, 10} (21.1) a 6= c {add. ass.} (21.2) a v c ∧ a 6= c {(AC) 19, 21.1} (21.3) a @ c ≡ a v c ∧ a 6= c {(D ∀) 1 − 2x} (21.4) a @ c {(M P ) ((DE)21.3, 21.2)} (21.5) contradiction {21.4, 8} contradiction {(RN ) 20, 21} We should notice that in order to provide mathematical proof for the


61

Theorem 2.11 (p. 59) the principle (M2) was indispensable. Finally in the system (L1)–(L4) the Strong Supplementation Principle always takes place – (M4). Theorem 2.13. {(L1) − (L4), (2.12), (2.10), (M 1)} ` (M 4). Proof: (1) x 6v y ∧ ¬(∃z∈M z v x ∧ z o y) {N DP } (2) x 6v y {(DC) 1} (3) ¬(∃z∈M z v x ∧ z o y) {(DC) 1} (4) ∀z∈M (z v x → z ◦ y) {(N ∃), (REP ), (N A) 3, (2.6)} (5) x Sum {y : y v x} {(M P )((DE)((2.10), 4))} (6) y v x {(M P )((DE)((2.10), 5)(DC)(D ∀))} (7) y @ x ∨ x = y {(M P )((DE)(2.12), 6)} (8.1) y @ x {ass. dod.} (8.2) ∃m∈M (m @ x ∧ m o y) {(M P )((W SP ), 8.1)} (8.3) c @ x ∧ c o y {(D ∃) 8.2} (8.4) c v x {(M P )((DC)10, (DE)((DA)(2.12))} (8.5) c o y {(DC) 8.3} (8.6) c v x → c ◦ y {(D ∀) 4} (8.7) c ◦ y {(M P )(8.6, 8.4)} (8.8) contradiction {8.7, 8.5} (9.1) x = y {add. ass.} (9.2) x 6v x {(LP )(9.1, 2)} (9.3) contradiction {9.2, (M 1)} contradiction {(RN )(8, 9)} Mathematical proof has been also pesented by [63], [98], [27] showing that the system defined in this manner ((L1)–(L4)) is equivalent to the system (M1)–(M5) with primary relation v.

2.3.3

Mereology founded on the relation of disjointedness

In Chapter X On the Foundations of Mathematics [71, p. 142] Leśniewski provided axiomatisation of Mereology based also on the disjoint relation

62


– o (defined on M × M ), which he called the relation of being external. Independent of Leśniewski’s studies, Leonard and Goodman [57], provided similar axiomatisation founding their theory on the relation of overlapping – ◦ [111, p. 48] they called their calculus a calculus on individuals in order to emphasize that the discussed objects have the lowest logical type contrary to classes, functions, etc. [111, p. 10]. Pietruszczak in [98, p. 109–121] provides quite detailed analysis of axiomatisation of Leśniewski’s Mereology founded on disjoint relation. The first notion Leśniewski defines is the notion of a class:

∀x∈M ∀∅6=X⊆M (x Fu X ≡df ∀y∈M (y o x ≡ ∀z∈X z o y)).

(2.13)

An object x is a fusion of a non-empty distributive set X, if every element which is separate from x is separate from every element of the X set. Starting with the notion of being external, Leśniewski defines the two basic axioms (written formally by Pietruszczak) in the following manner:

∀x∈M ∀y∈M (x o y ≡ ∀z∈M ∃u∈M ((u o x ∨ u o y) ∧ ¬ u o z)).

(2.14)

Consequently, we will say that x is external to y if and only if when at any z, a certain object u is external to x or y but it is not external to z. Due to commutative quality of conjunction and alternative, we see that the relation o is symmetric (S o). Moreover, if we assume that x o x we conclude that ∃u∈M (u o x ∧ ¬ u o x), leading us to contradiction, therefore o is irreflexive (P Z o). Another axiom states that for any non-empty subset of the set M there exists exactly one element in M which is a fusion of elements of this subset;


63

the postulate joining the axiom of fusion’s existence with the uniqueness principle for the relation Fu:

∀∅6=ZM ∃x∈M (x Fu Z ∧ ∀y∈M y Fu Z → x = y).

(2.15)

Next, in terms of the relation o Leśniewski defines the relation of a part30 :

∀x∈M ∀y∈M (x v y ≡df ∃∅6=Z⊆M (y Fu Z ∧ x ∈ Z)).

(2.16)

The definition of the relation of a proper part is the same as def. (2.7) (p. 46). In the beginning we will provide mathematical proof for a certain auxiliary fact: Proposition 2.1. ` ∀x∈M ∀y∈M (y o x ≡ ∀z∈M (z v x → z o y)). Proof: (1) y o x {ass.} (2) z v x {ass.} (3) ∃∅6=Z⊆M (x Fu Z ∧ z ∈ Z) {(M P )((DE) (2.16), 2)} (4) x Fu Z1 ∧ z ∈ Z1 {(D ∃) 3} (5) z ∈ Z1 {(DC) 4} (6) x Fu Z1 {(DC) 4} (7) ∀m∈M (m o x ≡ ∀n∈Z1 n o m) {(M P )((DE) (2.13), 6)} (8) y o x ≡ ∀n∈Z1 n o y {(D ∀) 7} (9) ∀n∈Z1 (y o x ≡ n o y) {(R2 ≡)} (10) y o x ≡ z o y {(D ∀) 9, 5} z o y {(M P )((DE) 10, 1)} 30

Founding it on the definition (2.13).

64


We may verify that the relation of a part defined with the condition (2.16) (p. 63) fulfil postulates (M1), (M2) and (M3): Theorem 2.14. {(2.13), (2.16)} ` (M 1). Proof: (1) ∀x∈M ∀∅6=X⊆M (x Fu X ≡ ∀y∈M (y o x ≡ ∀z∈X z o y)) {ass.} (1a) ∀x∈M ∀y∈M (x v y ≡ ∃∅6=Z⊆M (y Fu Z ∧ x ∈ Z)) {ass.} (2) x Fu {x} ≡ ∀y∈M (y o x ≡ z o y) {(O ∀) 1 − 2x, (LP )} (3) ∀y∈M ((y o x ≡ y o x) → x Fu {x}) {(DE) 2, (S o)(R2 ∀)} (4) (y o x ≡ y o x) → x Fu {x} {(D ∀) 3} (5) x Fu {x} {(M P )((ABS) 4)} (6) x v x ≡ ∃∅6=Z⊆M (x Fu Z ∧ x ∈ Z) {(D ∀) 1a − 2x} (7) ∃∅6=Z⊆M (x Fu Z ∧ x ∈ Z) → x v x {(DE) 6} (8) x Fu {x} ∧ x ∈{x} → x v x {(D ∃) 7, (LP ) 5, 7} x v x {(M P )((AC) ( x ∈ {x}, 5) 8)}

Theorem 2.15. {(2.13), (2.16), (P rop. 2.1)} ` (M 3). Proof: (1) x v y {ass.} (2) y v z {ass.} (3) ∃∅6=Z1 ⊆M (y Fu Z1 ∧ x ∈ Z1 ) {(M P )((DE) (2.16)), 1} (4) ∃∅6=Z2 ⊆M (z Fu Z2 ∧ y ∈ Z2 ) {(M P )((DE) (2.16)), 2} (5) y Fu Za ∧ x ∈ Za {(D ∃) 3} (6) z Fu Zb ∧ y ∈ Zb {(D ∃) 4} (7) y Fu Za {(DC) 5} (8) x ∈ Za {(DC) 5} (9) z Fu Zb {(DC) 6} (10) y ∈ Zb {(DC) 6} (11) Z = Za ∪ Zb {ass.} (12) ¬ z Fu Z {N DP } (13) ¬(∀m∈M (m o z ≡ ∀n∈Z n o m)) {(2.13)} (14) ∃m∈M ¬(m o z ≡ ∀n∈Z n o m) {(N ∀), N ((DE) 13)}


65

(15) ∃m∈M (m o z ∧ ¬(∀n∈Z n o m)) {(N I) 14} (16) ∃m∈M (m o z ∧ ∃n∈Z n ◦ m) {(N ∀), (2.6)} (17) m1 o z ∧ ∃n∈Z n ◦ m1 {(D ∃) 16} (18) m1 o z {(DC) 17} (19) ∃n∈Z n ◦ m1 {(DC) 17} (20) ∀k∈M (k v z → k o m1 ) {(M P )((DE) (P rop. 2.1), 18)} (21) y v z → y o m1 {(D ∀) 20} (22) y o m1 {(M P ) 2, 21} (23) ∀l∈M (l o y ≡ ∀w∈Za w o l) {(M P )((DE) (2.13)), 7} (24) m1 o y ≡ ∀w∈Za w o m1 {(D ∀) 23} (25) ∀w∈Za w o m1 {(M P )((DE) 24, 22), (S o)} (26) ∀l∈M (l o z ≡ ∀v∈Zb v o l) {(M P )((DE) (2.13)), 9} (27) m1 o z ≡ ∀v∈Zb v o m1 {(D ∀) 26} (28) ∀v∈Zb v o m1 {(M P )((DE) 27, 18), } (29) ∀w∈Za ∪Zb w o m1 {sum of sets 25, 28} (30) contradiction {29, 19} (31) z Fu Z {N, 12} x v z {(M P )((DE)(2.16), (AC)8, 31)} Theorem 2.16. {(2.13), (2.15), (M 3)} ` (M 2). Proof: (1) x v y {ass.} (2) y v x {ass.} (3) ∃∅6=Z1 ⊆M (y Fu Z1 ∧ x ∈ Z1 ) {(M P )((DE)(2.13)), 1} (4) ∃∅6=Z2 ⊆M (x Fu Z2 ∧ y ∈ Z2 ) {(M P )((DE)(2.13)), 2} (5) x v y ∧ y v x {(AC) 1, 2} (6) Z = Z1 ∪ Z2 {ass.} (7) x Fu Z {5, see proof T h.2.15, 11 − 31} (8) y Fu Z {(RP R)5, see proof T h.2.15, 11 − 31} x = y {(M P )((D ∀)(2.15), ((AC)7, 8))}

Consequently, if postulates (M1)–(M3), are fulfilled, then conditions (L1)–(L2) take place. However, it turns out that the definition (2.13) (p.

66


62) and Proposition 2.1 (p. 63) are not sufficient to define the mereological structure. Pietruszczak provides the following example: Example 2.3. (Diagram 2.3) Let M = {1, 2, 3, 123, 321}. Vectors signify the direction of the relation v. We may notice that ∀Z⊆M \∅ exists Fu Z and 123 Fu M ∧ ¬ 123 Sum M because 321 6v 123 and (M4) does not take place for 321: 321 6v 123 ∧ ¬ ∃z∈M : (z v 321 ∧ z o 123).

123 r

1

r 321 H Y H I @ I @ 6 6 @ @ HH @ @ HH HH @ @ HH @ @ H @ @ HH@ @ H H @r r @r

2

Diagram 2.3

3

Consequently, we can see that the Strong Supplementation Principle (M4) shall be explicitly added to the introduced above postulates. Moreover, (M4) imposes certain dependency between the relation Sum and Fu. The following theorem takes place for any x, y ∈ M and any non-empty set X [98, see pp. 114–116]: Theorem 2.17. {(2.13), (P rop. 2.1), (M 4)} ` x Fu X ∧ m Sum X → x v m. Proof: (1) (2) (3) (4) (5) (7) (8)

x Fu X {ass.} m Sum X {ass.} x 6v m {N DP } ∃z∈M (z v x ∧ z o m) {(M P )((D ∀)(M 4), 3)} ∀w∈M (w o x ≡ ∀l∈X l o w) {(M P )((DE) (2.13), 1)} ∀y∈X y v m {(M P )((D ∀)(DE) (2.10), 2), (DC)} z1 v x {(D ∃) 4, (DC)}


67

(9) z1 o m {(D ∃) 4, (DC)} (10) ∀k∈M (k v m → k o z1 ) {(M P )((DE) (P rop. 2.1), 9)} (11) y v m → y o z1 {(D ∀) 10} (12) y o z1 {(M P )((D ∀) 7, 11)} (13) z1 o x ≡ ∀l∈X l o z1 {(D ∀) 5} (14) ∀l∈X (l o z1 → z1 o x) {(DE) 13} (15) y o z1 → z1 o x {(D ∀) 14} (16) z1 o x {(M P ) 15, 12} contradiction {8, 16} In addition, another theorem takes place: Theorem 2.18. {(2.13), (P rop. 2.1), (M 4)} ` x Fu X ∧ m Sum X → m v x. Proof: (1) x Fu X {ass.} (2) m Sum X {ass.} (3) m 6v x {N DP } (4) ∃z∈M (z v m ∧ z o x) {(M P )((M 4), 3)} (4a) z1 v m ∧ z1 o x {(D ∃) 4} (4b) z1 v m {(DC) 4a} (4c) z1 o x {(DC) 4a} (5) ∀y∈M (y o x ≡ ∀v∈X v o y) {(M P )((DE) (2.13), 1)} (6) z1 o x ≡ ∀v∈X v o z1 {(D ∀) 5} (7) ∀y∈M y v m {(M P )((DE) (2.10), 2), (DC)} (8) ∀v∈X v o z1 {(M P )((DE) 6, 4c)} (9) m o z1 {(D ∀) 8} contradiction {9, 4b} Consequently, pursuant to the postulate (M2) we receive the thesis that in mereological structures the relations Sum and Fu are equivalent. In Leonard and Goodman’s original system, the authors adopted both axioms by Leśniewski ((2.14), (2.15), p. 63) and the axiom stating the

68


existence of fusion (equivalent to (M5)). The relation of being external is the primary relation. The relation of fusion has been defined on the basis of the equivalence x ◦ y ≡ ¬ x o y in the following manner:

x Fu◦ X ≡df ∀z∈M (z ◦ x ≡ ∃y∈X y ◦ z).

(2.17)

Consequently, it is easy to provide proof that the F u◦ defined in this manner is equivalent to the definition of Fu by Leśniewski. Breitkopf and Eberle focused on similar issues. Breitkopf [9] founded his system on the relations of overlapping and Eberle [23] defined the relation ’part-whole’ and formalised the calculus on individuals.

2.4

Extensionality principle

The very notion of extension, as it was presented in Chapter 1.3 (p. 25), is associated with Frege, who wanted to present that it is possible to found mathematics on logic. Frege defined the postulate, nowadays known as the Frege’s Basic Law V which states that every propositional function indicates a class of objects fulfilling it. Whitehead and Russell modified in this manner the Frege’s Basic Law V and defined the notion of extensional function as the propositional function whose true or false value depends only on the value (extension) of its arguments [140, p. 74]. This ’range of values’ is treated as a new object and called: a class. Whitehead and Russell assume that every function of one variable, on all its values is equivalent to a certain propositional function for the same values (the so-called axiom of classes [140, p. 167–169]). In this context they define the notion of identity of objects in the following manner:

x = y. =: (φ) : φ!x. ⊃ .φ!y

(2.18)

This notation signifies that x and y are identical, if every propositional function fulfilled by x is also fulfilled by y. Today this expression would

2.4. EXTENSIONALITY PRINCIPLE

69

be noted down in the following manner: x = y ≡ ∀φ (φx =⇒ φy) – it is nothing else than the principle of identity noted down with the use of a language of the second order logic31 . This statement may be generalised to any function on the condition that we adopt the axiom of classes mentioned previously. At this point this principle has the following form:

`: x = y. ⊃ .ψx ⊃ ψy 32

(2.19)

signifying that if x = y and x fulfils any function (propositional or not33 ), then y as well fulfils this function. In the context of the theory of classes, those postulates adopt the following form:

`: .ψx. ≡x .χx :≡ zˆ(ψz) = zˆ(χz)

(2.20)

Two classes are identical if and only if functions defining them are formally equivalent, that is when they have the same range of values. Due to the fact that an object is a member of a class if and only if it fulfils the function defining a class, the postulate (2.20) is equivalent to the axiom

31 Let M = (D, R1 , ..., Rk ) be the relations’ structure, where the relation of identity ’=’ not necessarily has to be the primary relation. Let L be adequately the language of the first order logic. We may expand M to a standard structure of the second order (M, S), where S = P (D) and consider the L2 language as the language of the second order logic after adding quantifications and variables. Then the formula ∀ φ (φx =⇒ φy) defines the relation of identity on the structure M , that is for any a, b ∈ D, a = b if and only if (M, S) ∀ φ (φx =⇒ φy)[a, b] [36]. 32 Equivalent notation: ` (x = y) → (ψx → ψy). 33 Whitehead and Russell distinguish two types of functions: extensional and intensional. They describe them in the following manner: ”the truthfulness of propositions with the function φ may depend either on the very function or on its extension. In the first case we will call this function intensional, in the second – extensional. For example the expression (x).φx or (∃x ).φx is an extensional function φ, because φ is formally equivalent to ψ, that is if φx. ≡x .ψx, then (x).φx. ≡ .(x).ψx and (∃x ).φx. ≡ .(∃x).ψx. On the other hand, the expression ’I think that (x).φx’ is an intensional function because even if φx. ≡ .ψx, takes place, there is no reason to give grounds for the proposition ’I think that (x).ψx’, because the proposition ’I think that (x).φx’ [140, p. 187] took place.

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of equivalence of sets ((2.22), p. 72) defined for classes. Whitehead and Russell presented that every proposition concerning a class expresses the extensional quality of a function defining this class, therefore it does not depend on truthfulness or false of this specific function selected to specify a class, but only on its extension. Analogically, propositions concern binary relations if we treat them in an extensional sense that is as a class of pairs designated by a certain function of two variables [140, p. 191–204]. Let’s move to mereological extensionality principle – (ZE). Here, we have to mention that during in-depth analysis of Principia Mathematica, Leśniewski also refers to the extensionality problem. He believes, that a certain inconsistency is present in Principia (and it is not an only one inconsistency, as we have already mentioned in the Chapter dedicated to Leśniewski). For Leśniewski, intensional functors are placed out of the reach of logic at all; for him, logic means extensional logic [141, p. 137]. Therefore the very notion of non-extensional Mereology for Leśniewski would be inacceptable. On the other hand, this monograph will not focus on the analysis of non-extensional Mereology, but rather the analysis of non-antisymmetric Mereology. Consequently there is no reason to apprehend inconsistency of the theory with the intuition of the author. Mereological extensionality principle has been defined for complex objects that are objects having proper parts34 . Let’s assume that we have a mereological structure described with axioms (M1)–(M5). This principle may be written down in the following manner:

∀z∈M (z @ x ≡ z @ y) → x = y.

(ZE)

If we take a closer look at (M1)–(M5) postulates, we may notice that (ZE) is a direct consequence of the axioms of the theory.

34

For objects not having proper parts, this principle does not have to take place. Pietruszczak [98, p. 70] provides the example of a structure h{1, 2}, @i, where the postulates (L1)–(L3), are fulfilled and the empty set is the only intersection of sets {1} and {2}, but 1 6= 2.


71

Theorem 2.19. In any mereological structure such that card(M) >2: ` ∀x∈M ∀y∈M ((∀z∈M (z @ x ≡ z @ y)) → x = y). Proof: (1) ∀z∈M (z @ x ≡ z @ y) ∧ x 6= y {N DP } (2) x 6= y {(DC) 1} (3) ∀z∈M (z @ x → z @ y) {(DC), (DE) 1} (4) y @ x → y @ y {(D ∀) 3} (5) ∀m∈M m 6@ m {T h. 2.6} (6) y 6@ y {(D ∀) 5} (7) y 6@ x {(T OLL) 4, 6} (8) ∀a∈M ∀b∈M (a @ b ≡ a v b ∧ a 6= b) {(2.7)} (9) y @ x ≡ y v x ∧ x 6= y {(D ∀) 8} (10) y v x ∧ x 6= y → y @ x {(DE) 9} (11) ¬(y v x ∧ x 6= y) {(T OLL) 10, 7} (12) y 6v x ∨ x = y {(N C) 11} (13.1) x = y {add. ass.} (13.2) contradiction {13.1, 2} (14.1) y 6v x {add. ass.} (14.2) ∃w∈M (w v y ∧ w o x) {(M P )((M 4), 14.1)} (14.3) c v y ∧ c o x {(D ∃) 14.2} (14.4) ∀z∈M (z @ y → z @ x) {(DC) 1, (DE)} (14.5) c @ y → c @ x {(D ∀) 14.4} (14.6) c v y {(DC) 14.3} (14.7) c o x {(DC) 14.3} (14.8) c 6= y {ass., because card(M ) > 2} (14.9) c @ y {(M P )((DE)(2.12), (AC)14.6, 14.8)} (14.10) c @ x {(M P )(14.5, 14.9)} (14.11) contradiction {14.7, 14.10} contradiction {(RN )13, 14}

In mereological structures the reverse implication does not have to take place.

72


¬(x = y → ∀z∈M (z @ x ≡ z @ y)).

(2.21)

The interval [0, 1] will always be the same interval irrespective of the fact if we perceive it as a sum of segments [0, 1/2] and [1/4, 1] or as a sum of [0, 1/2], [1/4, 3/4] and [1/4, 1]. To sum up: when we have two identical objects x and y and due to indistinguishability of identical objects, what is true for x should be also true for y, for example, a lump of clay and a monument of clay. It seems natural to believe that they have the same proper parts, but it is not true that all that is true for the lump is also true for the monument; e.g. the lump can be flattened and the monument cannot. Due to those reasons some authors reject the principle of extensionality [10]. On the foundations of exact sciences, in the set theory, the notion of identity of objects is expressed by the so-called Axiom of Extensionality , that is the axiom of equality of sets stating that every set is uniquely designated by its elements. It means that two sets are equal (that is identical in mathematical sense, that is they are the same set) if they have the same elements [48], [104]. The principle of equivalence of sets may be formally noted down in the following manner:

A = B ≡ ∀x∈A (x ∈ A ≡ x ∈ B).

(2.22)

From this principle we may deduce that sets {a}, {a, b}, {a, b, c} etc. are uniquely designated by its elements, therefore for any a there exists exactly one set whose only element is a35 , etc.

35

We have to mention here that the classical set theory is not the only one operating in sciences. We have an alternative set theory, called the theory of multisets, where the occurrence of the elements is counted: {(a, m(a)) : a ∈ A}. Hence, there is a difference between sets: {a, a} and {a}.


73

Above we have presented that in mereological structures the principle of extensionality always takes place. Pietruszczak in [98, p. 73], presented that if we wrote down the extensionality principle in e.g. terms of disjoint relations, in the following manner:

x = y ≡ ∀z∈M (z o x ≡ z o y),

(2.23)

then it is equivalent to the (L3) postulate. Among others also Breitkopf [9] provides this version of extensionality principle (in the language of first order logic) on his list of 48 theories stemming from the axiomatisation by Goodman, among which there is also the formula called by Leśniewski weak identity.

x = y ≡ ∀z∈M (z v x ≡ z v y) .

(2.24)

On the foundation of the calculus of classes, two classes K and L are identical if the class K is a sub-class of the class L and vice versa, the class L is a sub-class of the class K that is all elements are shared.

K ⊆ L ∧ L ⊆ K → K = L.

(2.25)

The fact that this condition is analogical to (M2), provides an incentive for us to focus on detailed analysis of such a model of mereology, which skips or even negates the (M2) postulate, that is we admit the case where x v y ∧ y v x ∧ x 6= y, in other words we admit a case where we have two different objects, which are their parts. We will verify if theoretically it may happen that even third element does not exist, an element distinguishing them. Moreover, we will check how the rejection of antisymmetry influences the extensionality principle, that is whether it excludes this principle or maybe in some cases the principle may be conserved. In extensional mereology, as we have noticed, anti-symmetry imposes extensionality.

Chapter 3

Non-antisymmetric mereology

3.1

Basic qualities

Let E signify a binary relation and M is any non-empty set of objects having at least two elements. Let the relation E be a reflexive and transitive relation1 , that is it fulfils two conditions:

∀x∈M x E x, ∀x∈M ∀y∈M ∀z∈M (x E y ∧ y E z → x E z).

(NAM1) (NAM3)

Based on tautology (T3)2 and the commutative quality of conjunction, the formula (NAM3) is equivalent to the formula ∀x∈M ∀y∈M ∀z∈M (y E z → (x E y → x E z)), which, on the other hand, founded on principles of logic concerning divisibility of the general quantifier, is equivalent to the following notation:

1 2

The relation of divisibility ’|’ in a set of integers is such a relation. See the Annex.

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CHAPTER 3. NON-ANTISYMMETRIC MEREOLOGY

∀y∈M ∀z∈M (y E z → ∀x∈M (x E y → x E z)).

(3.1)

With the use of the relation E being ’non-antisymmetric’ equivalent of mereological relation of a part, we will define other relations having their equivalents in extensional mereology (Chapter 2). A relation of intersection or overlapping is a relation ◦ fulfilling the following assumption: ∀x∈M ∀y∈M (x ◦ y ≡df ∃z∈M (z E x ∧ z E y)).

(Def. ◦)

disjointedness relation is a relation o where: ∀x∈M ∀y∈M (x o y ≡df ¬ x ◦ y).

(Def. o)

Definition of a proper part will be provided later, when we will discuss the principles of supplementation, as it is more complex situation than in extensional mereology. Now we will analyse NAM qualities one by one at notions defined in the above-presented manner (an outline of the theory in an abbreviated form may be found in [95]). Proposition 3.1. For any elements x, y, z ∈ M , the following takes place: (i) x ◦ x, (ii) x ◦ y → y ◦ x, (iii) x E y → x ◦ y, (iv) x E y → ∀z∈M (z ◦ x → z ◦ y), (v) x ◦ y → ∀z∈M (x E z → z ◦ y), (vi) x ◦ y → ∃z∈M ∀u∈M (u ◦ z → (u ◦ x ∧ u ◦ y)), (vii) x o y → y o x, (viii) x o y → ∀z∈M (z E x → z o y), (ix) (z E x ∧ z o y) → x 5 y.

3.1. BASIC QUALITIES

77

Proof: (i) ` ∀x∈M x ◦ x (1) ¬ x ◦ x {N DP } (2) ∀x∈M ∀y∈M (∃z∈M (z E x ∧ z E y) → x ◦ y) {(DE) Def. ◦} (3) ∃z∈M (z E x ∧ z E x) → x ◦ x {(D ∀) 2 − 2x} (4) ¬(∃z∈M z E x) {(T OLL) 3, 1; (ABS)} (5) ∀z∈M z 5 x {(N ∃)} (6) x 5 x {(D ∀) 5} contradiction {6, (N AM 1)}

(ii) ` ∀x∈M ∀y∈M (x ◦ y → y ◦ x) The mathematical proof results from the commutative law for conjunction and from the Def. ◦ (p. 76).

(iii) ` ∀x∈M ∀y∈M (x E y → x ◦ y) (1) ∀x∈M ∀y∈M (x ◦ y ≡ ∃z∈M (z E x ∧ z E y)) {Def. ◦} (2) x E y ∧ ¬ x ◦ y {N DP } (3) ∃z∈M (z E x ∧ z E y) → x ◦ y {(D ∀) 1 − 2x, (DE)} (4) x E y {(DC) 2} (5) ¬ x ◦ y {(DC) 2} (6) ¬(∃z∈M z E x ∧ z E y) {(T OLL) 3, 5} (7) ∀z∈M z 5 x ∨ z 5 y {(N ∃), (N C) 6} (8) x 5 x ∨ x 5 y {(D ∀) 7} (9.1) x 5 x {add. ass.} (9.2) contradiction {9.1, (N AM 1)} (10.1) x 5 y {add. ass.} (10.2) contradiction {4, 10.1} contradiction {(RN ) 9, 10}

(iv) ` ∀x∈M ∀y∈M (x E y → ∀z∈M (z ◦ x → z ◦ y))

78


(0) x E y ∧ ¬(∀z∈M (z ◦ x → z ◦ y)) {N DP } (1) x E y {(DC) 0} (2) ∃z∈M (z ◦ x ∧ z o y) {(DC) 0, (N ∀), (REP ), Def. o} (3) c ◦ x ∧ c o y {(D ∃) 2} (3a) c ◦ x {(DC) 3} (3b) c o y {(DC) 3} (4) ∃z∈M (z E c ∧ z E x) {(M P )(Def. ◦, 3a)} (5) d E c ∧ d E x {(D ∃) 4} (6) d E x {(DC) 5} (7) d E c {(DC) 5} (8) d E y {(N AM 3) 6, 1} (9) ∃m∈M (m E c ∧ m E y) {(A ∃), (AC), 7, 8} (10) c ◦ y {(M P )((DE)Def. ◦, 9)} contradiction {10, 3b}

(v) ` ∀x∈M ∀y∈M (x ◦ y → ∀z∈M (x E z → z ◦ y)) (0) x ◦ y ∧ ¬(∀z∈M (x E z → z ◦ y)) {N DP } (1) x ◦ y {(DC) 0} (2) ∃z∈M (x E z ∧ z o y) {(DC) 0, (N ∀), (REP ), Def. o} (3) x E c ∧ c o y {(D ∃) 2} (3a) x E c {(DC) 3} (3b) c o y {(DC) 3} (4) ∃m∈M (m E x ∧ m E y) {(M P )((DE)Def. ◦, 1)} (5) d E x ∧ d E y {(D ∃) 4} (5a) d E x {(DC) 5} (5b) d E y {(DC) 5} (6) d E c {(N AM 3) 5a, 3a} (7) ∃n∈M (n E y ∧ n E c) {(A ∃), (AC), 5b, 6} (8) y ◦ c {(M P )((DE) Def. ◦, 7)} contradiction {8, 3b}

(vi) ` ∀x∈M ∀y∈M (x ◦ y → ∃z∈M ∀u∈M (u ◦ z → (u ◦ x ∧ u ◦ y)))

3.1. BASIC QUALITIES

79

(1) x ◦ y {ass.} (2) ∃z∈M (z E x ∧ z E y) {(M P )((DE)Def. ◦, 1)} (3) c E x ∧ c E y {(D ∃) 2} (4) c E x {(DC) 3} (5) c E y {(DC) 3} (6) ∀m∈M (m ◦ c → m ◦ x) {(M P )((iv), 4)} (7) ∀n∈M (n ◦ c → n ◦ y) {(M P )((iv), 5)} (8) ∀m∈M (m ◦ c → m ◦ x) ∧ (m ◦ c → m ◦ y) {(AC) 6, 7; (R ∀)} (9) ∀m∈M (m ◦ c → (m ◦ x ∧ m ◦ y)) {(T 2)} ∃z∈M ∀m∈M (m ◦ z → m ◦ x ∧ m ◦ y) {(D ∃) 9} (vii) Mathematical proof is directly from the Def. o (p. 76) and on the basis of (ii). (viii) ` ∀x∈M ∀y∈M (x o y → ∀z∈M (z E x → z o y)) (1) x o y ∧ ¬(∀z∈M z E x → z o y) {N DP } (2) x o y {(DC) 1} (3) ∃z∈M (z E x ∧ z ◦ y) {(DC) 1, (N ∀), (REP ), (N A), Def. o} (4) c E x ∧ c ◦ y {(D ∃) 3} (5) c E x {(DC) 4} (6) c ◦ y {(DC) 4} (7) ∃m∈M (m E c ∧ m E y) {(M P )((DE)Def. ◦, 6)} (8) d E c ∧ d E y {(D ∃) 7} (9) d E c {(DC) 8} (10) d E y {(DC) 8} (11) d E x {(N AM 3) 9, 5} (12) d E x ∧ d E y {(AC) 11, 10} (13) ∃z∈M (z E x ∧ z E y) {(D ∃) 12} (14) x ◦ y {(M P )((DE)Def. ◦, 13)} contradiction {14, 2} (ix) ` ∀x∈M ∀y∈M ∀z∈M (z E x ∧ z o y → x 5 y)

80


(1) z E x ∧ z o y ∧ x E y {N DP } (2) z E x ∧ x E y ∧ z o y {(CON ), (COM M ) 1} (3) z E y ∧ z o y {(M P )((N AM 3), (DC) 2)} (4) z E y {(DC) 3} (5) z o y {(DC) 3} (6) z ◦ y {(M P )((iii), 4)} contradiction {5, 6}

3.2

Negation of Antisymmetry

In the previous section we described mereology without antisymmetry, examining the case where we skip this postulate. However, what result will we achieve if we impose a stronger condition that is if we negate the (M2) postulate? Let’s start with simple assumptions. Let the relation E fulfil the following postulates:

∀x∈M ∀y∈M (x E y ≡ ∀z∈M (z E x → z E y)), ∀x∈M ∃y∈M (x E y ∧ y E x).

(NAM6) (NAM2)

(NAM6) is closely related to transitivity for the relation E3 , (NAM2) and it states that for every x I will find such y that x and y are quasi-parts4 of themselves. The case where x = y is of course trivial. We will take such a case when there exists at least one y such that y 6= x. We will call this new postulate (NAM21).

∃x∈M ∃y∈M (y 6= x ∧ x E y ∧ y E x).

3

(NAM21)

See the comment on p. 76 concerning formula (NAM3). We will use the term ’quasi-part’ for the relation E and ’quasi-proper part for the relation C. 4

3.2. NEGATION OF ANTISYMMETRY

81

We may notice, that (NAM21) is a negation of (M2). If there are no other elements in M (Card(M)=2) fulfilling (NAM21), then x and y may be called twins. On the other hand, why couldn’t there exist more than one pair of twins or every element has its twin? Of course this thesis does not stem from (NAM2). Therefore we will introduce additional assumption – (NAM22).

∀x∈M ∃y∈M (y 6= x ∧ x E y ∧ y E x).

(NAM22)

We know that the extensionality principle ((2.4), p. 42) was defined for proper parts and the antisymmetry axiom (M2) – for improper parts. For those philosophers who reject antisymmetry, the extensionality principle is not only the postulate of identity of objects, but it is also recognized even as ’mereological equivalent’ in the sense that is specifies conditions where two objects are fully not distinguishable from the perspective of purely mereological descriptions. In such cases even ’mutual parts’ [20] are mentioned and the constitution of objects which requires those mutual parts [132]: „clay is an improper part of a monument (made of clay), and a monument is an improper part of clay” and clay is not a monument. We will gather all those postulates and we will attempt to analyse them: (NAM1) ∀x∈M x E x, (NAM2) ∀x∈M ∃y∈M (x E y ∧ y E x), (NAM21) ∃x∈M ∃y∈M (y 6= x ∧ x E y ∧ y E x), (NAM22) ∀x∈M ∃y∈M (y 6= x ∧ x E y ∧ y E x), (NAM3) ∀x∈M ∀y∈M ∀z∈M (x E y ∧ y E z → x E z), (NAM6A) ∀x∈M ∀y∈M (x E y → ∀z∈M (z E x → z E y)).

For further needs, we will discuss the case where the following system of axioms is fulfilled: (NAM1), (NAM21) and (NAM3) as a general model of non-antisymmetric mereology. Moreover, the choice of definition of quasi-proper part is a very important issue. Its impact will be presented in consecutive Sections.

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Definition 3.1. Under the term non-antisymmetric mereology – NAM we will understand a theory where the following postulates are fulfilled: (NAM1), (NAM21) and (NAM3).

Example 3.1. Let M = Z where Z means a set of integers and x E y ≡ x|y, then x ◦ y ≡ ∃z∈Z (z|x ∧ z|y) and we have ∀x,y∈Z x ◦ z (assuming z = 1 is sufficient) and the postulate (NAM21) is fulfilled, it is sufficient to take a pair (2, −2) : 2| − 2, −2|2, but 2 6= −2.

3.3

Relation of sum

Another relation characteristic for NAM is the relation of sum. The definition is analogical as for the extensional mereology EM. Sum is defined on the Cartesian product M × 2M and it is an explication of mereological definition of a class by Leśniewski [63].

∀x∈M ∀∅6=X⊆M x Sum X ≡df ∀y∈X y E x ∧ ∀z∈M (z E x → ∃w (w ∈ X → w ◦ z)). (Def. Sum) The object x is a sum (class) of objects which are elements of the set X if every element of X is a quasi-part of x and every quasi-part of x has quasi-intersection with a certain element of the set X. Certain qualities of the relation Sum result from this definition (similarly as in EM): every class is a sum of their quasi-parts and every object is a class. Formally we express it in the following manner: Proposition 3.2. Let x be any element of M. Then: (i) x Sum {y ∈ M : y E x}, (ii) x Sum {x}.

3.3. RELATION OF SUM

83

Proof : (i) ` x Sum {y : y E x} (1) X = {y : y E x} {ass.} (2) x Sum X {ass.} (3) ∀m∈X m E x ∧ ∀z∈M (z E x → ∃w (w ∈ X → w ◦ z)) {(M P )((DE)Def.Sum, 2)} (4) ∀z∈M (z E x → ∃w (w ∈ X → w ◦ z)) {(DC) 3} (5) y E x → ∃w (w ∈ X → w ◦ y) {(D ∀) 4} (6) ∀m∈X m E x {(DC), 3} (7) y E x {(D ∀) 6, 1} ∃w (w ∈ X → w ◦ y) {(M P ) 5, 7}

(ii) ` x Sum {x}. It is enough to take X = {x : x E x} = {x} in (i).

If we only skip the postulate of antisymmetry we may encounter the following situation: Example 3.2. (Diagram 3.2) Let M = {1, 2, 12, 21} Vectors are symbols of the relation E, and the direction of inclusion is compliant to arrows. We have the following model: 1 E 12, 2 E 12, 1 E 21, 2 E 21, 12 5 21, 21 5 12. 12 r

r 21 6

6 I@ @ @ @ @ @

1 r

@

@r 2

Diagram 3.2

84


The example 3.2 presents a case of non-extensional mereology: objects 12 and 21 have the same proper parts, but they are not identical (more detailed analysis will be presented in section concerning the extensionality principle), however, it is not a case of non-antisymmetric mereology – the postulate (NAM21) is not fulfilled here. We will modify a bit the example 3.2, to make it fulfil the NAM postulates:

Example 3.3. (Diagram 3.3) Let M = {1, 2, 3, 123, 321}. The principle (NAM21) is fulfilled: 2 E 3 and 3 E 2 and 2 6= 3. Moreover 1 E 123, 2 E 123, 1 E 321, 2 E 321, 3 E 123 and 3 E 321. We have 123 Sum {1, 2, 3} and 321 Sum {1, 2, 3}, but 123 5 321 nor 321 5 123. Moreover we notice that 2 Sum {3}, because 3 E 2 and z assumes values from the set {2, 3} and the assumptions of the Def. Sum (p. 82) are fulfilled. In addition the following takes place: 2 Sum {2, 3}, because 2 E 2 and 3 E 2 and we may assume w = z, because then z ◦ z. On the other hand, it is not true that 321 Sum {2, 3}, because 1 E 321, but ¬ 1 ◦ 2, and ¬ 1 ◦ 3. 123 r

1

r 321 H Y H I @ I @ 6 6 @ H @ @ HHH @ HH @ @ H @ @ HH @ @ H HH @ @ H r @ r @ -r

2

Diagram 3.3

3

Conclusion 3.1. In the model NAM the relation Sum does not have to be a monotonic relation.

Conclusion 3.2. In non-antisymmetric mereology, the relation Sum is not a function in reference to the left argument: ¬(∀x∈M ∀y∈M ∀∅6=X⊆M x Sum X ∧ y Sum X → x = y).

3.3. RELATION OF SUM

85

As we have noticed, in extensional mereology the very monotonicity of relation of a part does not impose the Strong Supplementation Principle (Example 2.2, p. 53). Moreover, it is quite obvious that if the diagram of relations is complete (Example 3.6, p. 86), the monotonicity principle is conserved, as not every element is a quasi-part of every other element. In EM we assume the existence of a sum as the biggest element, even though sometimes it may be inconsistent with intuition (in the Example3.3 the sum of the whole set M does not exist). A counterpart of the axiom of a sum in NAM would be the following postulate:

∀∅6=X⊆M ∃x∈M x Sum X.

(NAM5)

Example 3.4 illustrates the existence of a sum for every subset M .

Example 3.4. (Diagram 3.4) Let M = {1, 2, 12, 21}, 1 E 12, 2 E 12, 1 E 21, 2 E 21. Moreover, let 12 E 21 and 21 E 12 and 12 6= 21. For any element of the set M , x Sum {x}, 12 Sum {12, 21}, because due to Def. Sum (p. 82): 1 E 12 and 12 ◦ 1, 2 E 12 and 12 ◦ 2, 21 E 12 and 12 ◦ 21. It is obvious that this formula is true also for the very 12 as from the reflexivity of the relation E, 12 E 12 and from Def. ◦ (p. 76), 12 ◦ 12. By analogy, we may show that 12 Sum {1, 2}, 21 Sum {1, 2}, 12 Sum {1, 2, 21} and 12 Sum M and 21 Sum M . As an example we will show that 12 Sum {1, 2}. From the Def. Sum, 1 E 12 and 2 E 12. In order to verify the second part of the conjunction, the set of all those possible z which are quasi-parts 12 agrees with the set M . We have to show that for each of them we will find an element in the set {1, 2}, which overlaps with them. Consequently, we have: 1 ◦ 1, 2 ◦ 2, 12 ◦ 1 and 21 ◦ 1. Therefore we have shown that in fact 12 Sum {1, 2}.

Lack of antisymmetry for the relation of quasi-part in NAM results in lack of uniqueness of sum. In the Example 3.4, 12 is the sum of the whole M and 21 is the sum of the whole M .

86


12 r

-r 21 6

6 @ I @ @ @ @ @

@ @r 2

1 r

Diagram 3.4

We will consider another example. On the diagram of the Example 3.5 the postulates (NAM22), (NAM6A) and (NAM5) are fulfilled. Example 3.5. (Diagram 3.5) M = {1, 2, 12, 21}. 12 is a twin for 21 and 1 for 2. The postulate (NAM5) is conserved: ∀x∈M x Sum {x} and 12 Sum {12, 21}, because: 1 E 12 and 12 ◦ 1, 2 E 12 and 12 ◦ 2, 21 E 12 and 12 ◦ 21, 12 E 12 and 12 ◦ 12. By analogy, we may show that 1 Sum {1, 2}, 12 Sum {1, 2} and 21 Sum {1, 2}. Moreover, 12 Sum {1, 2, 21} and 21 Sum {1, 2, 12}, 12 Sum {12, 2, 21} etc. Finally, also 12 Sum M and 21 Sum M takes place. 12 r

-r 21 6

@ I 6 @ @ @ @ @

1 r

@ -r 2 @

Diagram 3.5

Next example (3.6) presents a case where the postulates (NAM6A), (NAM22) and (NAM5) are conserved and the diagram of relations is complete, as every element is a quasi-part of every element from M and constitutes the sum of the remaining ones. Example 3.6. (Diagram 3.6) M = {1, 2, 12} and 12 Sum {1, 2}, 1 Sum {2, 12} and 2 Sum {1, 12}. Moreover, 12 Sum M , 1 Sum M and 2 Sum M . As we have noticed in the examples above, it seems that (NAM22) imposes the existence of a sum, which cannot be said about (NAM21),

3.4. QUASI-PROPER PARTS

87

r 12 I @ @ @

1 r

Rr 2 @ -

Diagram 3.6

but it is not the case. We will take another example (3.7): the postulates (NAM22), (NAM1), (NAM3) are fulfilled, but ¬(NAM5). Example 3.7. (Diagram 3.7) M = {1, 2, 3, 4} and 1 E 2 and 2 E 1, 3 E 4 and 4 E 3. Moreover, 1 Sum {2}, 2 Sum {1}, etc., but the sum of the whole M does not exist.

r

1

-r

r

2 3

-r

4

Diagram 3.7

In every case where the diagram of relations is not compact5 , the postulate (NAM5) will not take place of course. Despite the fact, for the purpose of this analysis we will not assume the existence of sum. As you can see from the above-presented examples, antisymmetry ’glues’ all pairs of mutual vertexes.

3.4

Quasi-proper parts

Now we will define the relation of quasi-proper part by analogy to the definition of this relation in extensional mereology. Under relation C – a counterpart of the relation of a proper part, we understand the following relation:

5 A diagram may be called compact if for every pair of vertexes there is a ’path’ joining them.

88


∀x∈M ∀y∈M (x C y ≡df (x E y ∧ x 6= y)).

(Def.1 C)

Consequently, x is a quasi-proper part of y if and only if x is a quasipart of y and x is not an y. The second definition of quasi-proper part looks like this:

∀x∈M ∀y∈M (x C y ≡df (x E y ∧ y 5 x)),

(Def.2 C)

x is a quasi-proper part of y if and only if x is a quasi-part of y and y is not a quasi-part of x. We know that in EM Def.1 C and Def.2 C overlap (Th. 2.1, p. 47), and in NAM ? In the Example 3.4 (p. 85), 12 E 21 and 21 E 12, therefore, compliant to the Def.2 C, 12 6 21, and due to Def.1 C (p. 88), 12 C 21, because 12 6= 21. Consequently, those definitions do not overlap in NAM, they even lead to completely different conclusions. Which definition shall be taken into consideration? Another consequence of rejecting antisymmetry is the fact that in NAM the postulate (L1) is not fulfilled, and the postulate is obligatory in EM (Chapter 2). In the Example 3.4, 12 E 21 and 21 E 12. Because there is no antisymmetry, we do not have equality of objects 12 and 21, but from the Def.1 C, 12 C 21 and 21 C 12. Consequently, the condition (L2) is fulfilled, but the condition (L1) is not (at Def.1 C). Now we will check the qualities of the relation of quasi-proper part at formulas derived in the mentioned manner. We know that the relation of a proper part in EM is irreflexive, asymmetric and transitive, and in NAM ?

Theorem 3.1. In NAM: (i) {Def.1 C} ` ∀x∈M x 6 x, (ii) {Def.1 C, (N AM 21)} ` ∀x∈M ∀y∈M ¬(x C y → y 6 x).

3.4. QUASI-PROPER PARTS

89

Proof : (i) ` ∀x∈M x 6 x (1) ∀x∈M ∀y∈M (x C y ≡ x E y ∧ x 6= y) {ass.} (2) x C x ≡ x E x ∧ x 6= x {(D ∀) 1 − 2x} (3) x C x → x E x ∧ x 6= x {(OE) 2} (4) x C x {N DP } (5) x E x ∧ x 6= x {(M P ) 3, 4} (6) x 6= x {(DC) 5} contradiction {6, reflexivity ’=’ }

(ii) ` ∀x∈M ∀y∈M ¬(x C y → y 6 x) (1) ∀x∈M ∀y∈M (x C y ≡ x E y ∧ x 6= y) {ass.} (2) ∃x∈M ∃y∈M (x E y ∧ y E x ∧ x 6= y) {ass.} (3) b E c ∧ c E b ∧ c 6= b {(D ∃) 2 − 2x} (4) b E c {(OK), (CON ) 3} (5) c E b ∧ c 6= b {(DC), (CON ) 3} (6) c E b ∧ c 6= b → c C b {(D ∀) 1 − 2x, (DE) 1} (7) c C b {(M P ) 6, 5} (8) b E c ∧ b 6= c {(COM M ), (DC), (CON ) 3} (9) b E c ∧ b 6= c → b C c {(D ∀) 1 − 2x, (DE) 1} (10) b C c {(M P ) 9, 8} (11) b C c ∧ c C b {(AC) 10, 7} (12) b C c ∧ ¬ c 6 b {(N N ) 11} (13) ¬(b C c → c 6 b) {(REP ) 12} ∀x∈M ∀y∈M ¬(x C y → y 6 x) {(A ∀) 13}

On the other hand, if the relation of quasi-proper part is defined with the use of Def.2 C, then this relation is irreflexive, asymmetric and transitive.

90


Theorem 3.2. In NAM: (i) {Def.2 C} ` ∀x∈M x 6 x, (ii) {Def.2 C} ` ∀x∈M ∀y∈M (x C y → y 6 x), (iii) {Def.2 C, (N AM 3)} ` ∀x∈M ∀y∈M ∀z∈M (x C y ∧ y C z → x C z).

Proof : (i) ` ∀x∈M x 6 x (1) ∀x∈M ∀y∈M (x C y ≡ x E y ∧ x 6= y) {ass.} (2) x C x → x E x ∧ x 5 x {(D ∀) 1 − 2x, (DE)} x 6 x {(T OLL) (ZSA), 2}

(ii) ` ∀x∈M ∀y∈M (x C y → y 6 x) (1) ∀x∈M ∀y∈M (x C y ≡ x E y ∧ y 5 x) {ass.} (2) x C y ∧ y C x {N DP } (3) x C y {(DC) 2} (4) y C x {(DC) 2} (5) x C y → x E y ∧ y 5 x {(D ∀), 1 − 2x, (DE)} (6) y C x → y E x ∧ x 5 y {(D ∀), 1 − 2x, (DE)} (7) x E y ∧ y 5 x {(M P ), 5, 3} (8) y E x ∧ x 5 y {(M P ), 6, 4} (9) x E y ∧ y 5 x ∧ y 5 x ∧ x 5 y {(AC), 7, 8, (CON )} (10) x E y ∧ x 5 y {(COM M ), (CON ), (DC) 9} contradiction

(iii) ` ∀x∈M ∀y∈M ∀z∈M (x C y ∧ y C z → x C z) (1) ∀x∈M (2) x C y (3) y C z (4) ∀x∈M → x E z)

∀y∈M (x C y ≡ x E y ∧ y 5 x) {ass.} {ass.} {ass.} ∀y∈M ∀z∈M (x E y ∧ y E z {ass.}

3.5. SUPPLEMENTATION PRINCIPLES

91

(5) x C y → x E y ∧ y 5 x {(D ∀) 1, (DE)} (6) y C z → y E z ∧ z 5 y {(D ∀) 1, (DE)} (7) x E y ∧ y 5 x {(M P ) 5, 2} (8) y E z ∧ z 5 y {(M P ) 6, 3} (9) y 5 x ∧ x E y ∧ y E z ∧ z 5 y {(AC) 7, 8, (COM M )} (10) x E y ∧ y E z {(COM M ), (CON ) 9, (DC) 9} (11) x E z {(M P )((D ∀) 4 − 3x, 10)} (12.1) z 5 x {add. ass.} (12.2) x E z ∧ z 5 x {(AC) 12.1, 11} (12.3) x E z ∧ z 5 x → x C z {(D ∀) 1 − 2x, (DE) 1} (12.4) x C z {(M P ) 12.3, 12.2} x C y ∧ y C z → x C z {(AI) 12}

3.5

Supplementation Principles

In order to examine supplementation principles, we will return for a while to the formula defined beforehand – ((3.1), p. 76). With the use of the transposition principle, we receive the following dependency: ∃z∈M (z E x ∧ ¬ z E y) → x 5 y. Consequently the condition in the form x 5 y → ∃z∈M (z E x ∧ z o y) is also a condition not resulting from axioms (NAM1) and (NAM3) (like in the case of extensional mereology). It is a counterpart of the Strong Supplementation Principle (SSP) which in case of EM implies the Weak Supplementation Principle (WSP) and it imposes the condition of antisymmetry (M2) (Th. 2.3, p. 49). In case of NAM, such dependencies do not take place, as we will see in consecutive Sections. Consequently we have two supplementation principles (SSP) and (WSP) (Chapter 2.3.1), which, defined for the relations E and C look as follows:

∀x∈M ∀y∈M (x 5 y → ∃z∈M (z E x ∧ z o y)),

(SSP)

∀x∈M ∀y∈M (x C y → ∃z∈M (z C y ∧ z o x)).

(WSP)

92


A. Varzi [139], provides a very good example when he discusses mereological proper parts. This example suits our context very well. When we take two words composed of the same letters: (1) F=’FALLOUT’, (2) O=’OUTFALL’. It is reasonable to believe that those two words differ despite the fact that they are composed of the same letters. Varzi distinguishes two perspectives of perceiving those words depending on whether F and O are treated as objects of word type or as word-tokens. In the first case lack of identity – F 6= O is undeniable, if we think about the word object as about the object formed with adequate letters. In the second case, F and O have different proper parts, so their difference is also obvious. At this point, a question arises – what type of constitution we have in mind? If we think that the word object is a set of letters, then relations taking place are: relation of membership of an element in a set (and not the relation of being a part); however, then we have to forget about the order of elements and it is not a good perspective. If we think in mereological categories and we assume that letters are mereological atoms, (as all language types have to be formed out of certain atom blocks), then we may think about the word object as about an object formed out of letters. However, on this level (in extensional mereology) it is commonly believed that the Week Supplementation Principle (WSP) should take place because it is a crucial element of the relation of a proper part, and in this case (WSP) does not take place. The diagram of this relation would be similar to the Diagram 3.2 (p. 83), with the difference that we would have 7 proper parts (identical) equally for 12, and for 21, but 12 6= 21 (F 6= O). Maybe this example perfectly suits the category of non-antisymmetric mereology: we have objects with the same quasi-parts which are not identical. Now, we will analyse the relationship of both supplementation principles with definitions of quasi-proper part in NAM 6 .

6

Short paper on this issue was published in RSL [96].


3.5.1

93

Principles of supplementation and Def.1 C

We will start with the analysis of examples provided previously (p. 84 – 87). The Example 3.3 depicts a case where the postulate (NAM21) is fulfilled, and the sum (NAM5) does not exist and the Strong Supplementation Principle (SSP) is not fulfilled: 123 5 321 and ¬ ∃z∈M (z E 123 ∧ z o 321). Moreover, the Weak Supplementation Principle (WSP) is not fulfilled: 2 C 3 and ¬ ∃z∈M (z C 3 ∧ z o 2). In the Example 3.4 the postulates (NAM21) and (NAM5) are fulfilled and the (SSP) takes place, and the (WSP) does not take place. We have: 12 5 1 and 2 E 12 ∧ 2 o 1, inversely: 21 5 2 and 1 E 21 ∧ 1 o 2. Moreover: 12 5 2 and 1 E 12 ∧ 1 o 2, 21 5 1 and 2 E 21 ∧ 1 o 2, 1 5 2 and 1 E 1 ∧ 1 o 2 and similarly for the example when 2 5 1 then 2 E 2 ∧ 2 o 1. For (WSP) we have: 12 C 21, but there is no such z being a quasi-proper part of 21: z C 21 ∧ z o 12, consequently (WSP) is not fulfilled. The Example 3.5 depicts the case where the postulates (NAM22) and (NAM5) are fulfilled and neither (SSP) nor (WSP) are fulfilled: 12 5 1 and the z ∈ M does not exist: z E 12 ∧ z o 1. Similarly 1 C 12, but again z ∈ M does not exist: z C 12 ∧ z o 1. On the other hand, the Example 3.6 presents a case where postulates (NAM22) and (NAM5) are kept and the (SSP) principle is conserved, because every element from M is a quasi-part of every element of M , and the (WSP) principle is not fulfilled – 1 C 12 and there is no quasi-proper part in 12 which would be disjoint with 1. In the Example 3.7 the postulate (NAM22) is conserved, but the (NAM5) is not. The (SSP) principle is fulfilled, because e.g. for 1 5 3 the following takes place 1 E 1 ∧ 1 o 3. Analogical reasoning is applied for cases when 1 5 4 and 3 5 1 and 4 5 1, etc. The (WSP) is not conserved because 1 C 2 and ¬ ∃z∈M (z C 2 ∧ z o 1). The Example 3.8 illustrates a case when the axioms (NAM21) and

94


(NAM5)7 , are fulfilled, but neither (WSP) nor (SSP) takes place: 1 C 2 and there is no quasi-proper part in 2 – of such z, that: z o 1, because 3 ◦ 1. Similarly 3 5 1 and again there is no such quasi-part in 3 – of such z that: z o 1, because 2 ◦ 1.

Example 3.8. (Diagram 3.8) M = {1, 2, 3, 23} and 1 E 2, 1 E 3, 2 E 3, 3 E 2, but 2 6= 3, 23 Sum M . r 23 I @ 6 @ @

2 r I @ @ @

@r 1

@ -r 3

Diagram 3.8

The last example (modification 3.3, p. 84) illustrates the case when the postulate (NAM21) is fulfilled – 2 E 3 and 3 E 2 and 2 6= 3, but neither (NAM5) nor (SSP) or (WSP) take place.

Example 3.9. (Diagram 3.9) Let M = {0, 1, 2, 3, 123, 321}. Moreover 1 E 123, 2 E 123, 1 E 321, 2 E 321, 123 5 321, 321 5 123, 3 E 321, 3 E 123, 2 E 3 and 3 E 2. In addition, we have 0 E 2 and 0 E 3. From the transitivity of relation E we receive that 0 E 321. The (SSP) does not take place: 3 5 0 and ¬ ∃z∈M (z E 3 ∧ z o 0) and (WSP) does not take place: 0 C 2, ¬ ∃z∈M (z C 2 ∧ z o 0).

Analysis of those examples leads us to the conclusion that (NAM21), that is the negation of antisymmetry, imposes the rejection of the (WSP)

7 It is easy to verify that for every subset of M its sum exists, e.g. 23 Sum {2, 3}, because: 1 E 23 ∧ 1 ◦ 2, 2 E 23 ∧ 2 ◦ 3, 3 E 23 ∧ 3 ◦ 2 and 23 E 23 ∧ 23 ◦ 2.


123 r

95

321

r H Y 6 6 I @ K A I @ H @ HH A@ @ HH @ @ H AA @ HH @ AH@ H @ @ A HH @ @ A H r @ r @ -r H A 1 63 2 I @ A @ A @ A @ @ A @ A @A Ar 0 @

Diagram 3.9

principle. As (NAM22) is a condition stronger than (NAM21), this proposition should be true also for (NAM22). Such result seems reasonable if we bear in mind (Section 2.3.1), that in extensional mereology the principle (WSP) together with axioms (M1) and (M3) (reflexivity and transitivity of the relation of part) impose antisymmetry – (M2) (Th. 2.3, p. 49). Due to the fact that we assume that in every NAM model the relation of quasi-part is reflexive and transitive, we have to reject the (WSP) principle. To sum up: in non-antisymmetric mereology NAM due to Def.1 C of the quasi-proper part, the rejection of antisymmetry imposes the rejection of the Weak Supplementation Principle [96]. Theorem 3.3. {(N AM 3), Def.1 C} ` (N AM 21) → ¬(W SP ). Proof : (1) (2) (3) (4) (5) (6) (7) (8) (9)

∃x∈M ∃y∈M (x E y ∧ y E x ∧ x 6= y) {ass.} ∀x∈M ∀y∈M (x C y ≡ x E y ∧ x 6= y) {ass.} ∀x∈M ∀y∈M (x E y ∧ y E z → x E z) {ass.} ∀x∈M ∀y∈M (x C y → ∃z∈M (z C y ∧ z o x)) {N DP } a E b ∧ b E a ∧ a 6= b {(O ∃) 1 − 2x} a E b ∧ a 6= b → a C b {(D ∀) 2 − 2x, (DE)} a E b ∧ a 6= b {(OK) 5, (COM M ), (CON )} a C b {(M P ) 6, 7} a C b → ∃z∈M (z C b ∧ z o a) {(O ∀ 4 − 2x)}

96


(10) ∃z∈M (z C b ∧ z o a) {(M P ) 9, 8} (11) c C b ∧ c o a {(D ∃) 10} (12) c C b {(DC) 11} (13) c C b → c E b ∧ c 6= b {(D ∀) 2 − 2x, (DE)} (14) c E b {(M P ) 13, 12, (DC)} (15) b E a {(OK) 5, (COM M )} (16) c E b ∧ b E a {(AC) 15, 14} (17) c E b ∧ b E a → c E a {(D ∀) 3 − 3x} (18) c E a {(M P ) 17, 16} (19) c o a {(DC) 11} contradiction {18, 19}

It is a well-known fact that in extensional Mereology the Strong Supplementation Principle imposes the Weak Supplementation Principle (Th. 2.2, p. 49), and in NAM, when we look at examples, we do not see such dependency; the Week Supplementation Principle (WSP) seems to be deductively stronger than the Strong Supplementation Principle.

3.5.2

Principles of supplementation and Def.2 C

We will start from the same examples (p. 84 – 87, 94), as previously, but this time we will apply the second definition of the relation of quasi-proper part – Def.2 C. In terms of the Strong Supplementation Principle the definition of the quasi-proper part does not have any impact on it; therefore its status remains the same. Consequently, we will focus on the analysis of the Weak Supplementation Principle only. In the Example 3.3 ((NAM21) and ¬(NAM5) take place), the (WSP) principle is conserved ((SSP) – does not take place8 ). At this definition 2 6 3 and 3 6 2. We will show that (WSP) is conserved: 1 C 123 ∧ 2 C 123 ∧ 2 o 1. 8 In the context of Strong Supplementation Principle which takes place in various examples, we should return to Section 3.5.1, p. 93.


97

Similarly 2 C 123 ∧ 1 C 123 ∧ 1 o 2, 3 C 123 ∧ 1 C 123 ∧ 1 o 3, 1 C 321 ∧ 2 C 321 ∧ 2 o 1, 2 C 321 ∧ 1 C 321 ∧ 1 o 2 and 3 C 321 ∧ 1 C 321 ∧ 1 o 3. In the Example 3.4 ((NAM21) and (NAM5) take place) the (WSP) principle is also fulfilled ((SSP) also takes place). The only case where it could not take place is when 12 E 21, but due to the definition – Def.2 C, 12 6 21 and 21 6 12, therefore (WSP) is fulfilled: 1 C 12 ∧ 2 C 12 ∧ 2 o 1, 1 C 21 ∧ 2 C 21 ∧ 2 o 1, 2 C 12 ∧ 1 C 12 ∧ 1 o 2 and 2 C 21 ∧ 1 C 21 ∧ 1 o 2. Example 3.5 ((NAM22) and (NAM5) take place), but the (WSP) does not take place (nor does the (SSP)): 1 C 12, but there doesn’t exist such z, that z C 12 and z o 1, because at Def.2 C, the only possibility is z = 2, but 2 E 1, consequently 2 ◦ 1. Example 3.6 ((NAM22) and (NAM5) take place). Def.2 C results in the fact that there is no such element which would be a quasi-proper part of any other element and all elements are quasi-parts of other elements. Consequently, the (WSP) principle is fulfilled (in this example also the (SSP) takes place). Example 3.7 ((NAM22), takes place, but the (NAM5) doesn’t). Similarly to the Example 3.6, there is no such element which would be the quasi-proper part of any other element. Consequently, the (WSP) is conserved here (also the (SSP) is conserved here). Example 3.8 ((NAM21) and (NAM5) take place). The (WSP) principle is not conserved (neither is the (SSP)): 1 C 2, but 2 6 3 and 3 6 2 and ¬ ∃z∈M (z C 2 ∧ z o 1), as in this model only 1 C 2. Example 3.9 ((NAM21) takes place, but (NAM5) doesn’t). The (WSP) principle is not conserved (neither is the (SSP)): 0 C 2 and ¬ ∃z∈M (z C 2 ∧ z o 0), because 2 6 3 nor 3 6 2 and only 0 C 2. Let’s take one more example:

98


Example 3.10. (Diagram 3.10) M = {0, 1, 2, 3, 4, 5} and 0 E 1 and 1 E 0 and 0 E 2, 0 E 3, 1 E 2, 1 E 3, 1 E 4, 1 E 5, 2 E 4 and 4 E 2 and 3 E 5 and 5 E 3. 4 r

r5 A6 K 6 A A 2 ? r A r?3 A I @ K A A@ A A @A A @Ar 1 A 6 A A r 0 A?

Diagram 3.10

Consequently, the sum for the whole set M does not exist – the postulate (NAM5) is not fulfilled, every element has its twin – the postulate (NAM22) is fulfilled. Moreover, for both definitions of quasi-proper part, the (WSP) does not take place: for Def.1 C, 0 C 2 and ¬ ∃z∈M z C 2 ∧ z o 0, because 0 C 2, 1 C 2, but 1 ◦ 0. For the Def.2 C, 1 C 4, but 0 C 4 and 0 ◦ 1. Similarly, (SSP) does not take place: 2 5 3 and for 0 E 2 we have 0 E 3, for 1 E 2 we have 1 E 3, for 4 E 2 we have 4 ◦ 3, because 1 E 4 and 1 E 3. To sum up, it may seem that in non-antisymmetric mereology, with consideration of the relation of quasi-proper part – Def.2 C, if the postulate (NAM22) is fulfilled, then either both complementation principles take place or none of them takes place. We will modify a bit the Example 3.3:

Example 3.11. (Diagram 3.11) Let M = {1, 2, 3, 4, 5, 6, 7, 8}. The principle (NAM22) is fulfilled: 1 E 2 and 2 E 1 and 3 E 4 and 4 E 3. Moreover: 5 E 6 and 6 E 5 and 7 E 8 and 8 E 7. In addition 6 5 7 and it is not true that there exists such z: z E 6 ∧ z o 7, because 5, 6, 7, 8 as their quasi-parts have the following elements: 1, 2, 3, 4 and adequately, their twin (to make the picture clear, relations resulting from the transitivity of the relation E, are


99

not indicated with arrows) for example 5 E 6 and 5 ◦ 7. Consequently (SSP) does not take place. On the other hand, the (WSP) takes place, as due to the Def.2 C, where we have twins, those elements are not quasi-proper parts of their twins. Consequently, the diagram of relations is adequately bigger and it resembles the diagram from the Example 3.2.

5 r

-r6 I @ 6 @

I @ 6 @ @

@ @ @

r

r -8 6 @ @

@ @

@

1

7r @ I 6 @ @ @

@ @r

2

@ @ r

3

@ @

@ -r

4

Finally we have the theorem [96]: Theorem 3.4. {(Def.2 E), (N AM 3)} ` (SSP ) → (W SP ). Proof : (1) ∀x∈M ∀y∈M (x 5 y → ∃m∈M (m E x ∧ m o y)) {ass.} (2) ∀x∈M ∀y∈M ∀z∈M (x E y ∧ y E z → x E z) {ass.} (3) ∀x∈M ∀y∈M (x C y ≡ x E y ∧ y 5 x) {ass.} (4) a C b → a E b ∧ b 5 a {(D ∀) 3 − 2x, (DE)} (4a) a C b {ass.} (5) a E b ∧ b 5 a {(M P ) (4, 4a)} (5b) b 5 a {(DC) 5} (6) ∃m∈M (m E b ∧ m o a) {(M P )((D ∀) 1, 5)} (7) c E b ∧ c o a {(D ∃) 6} (8) c E b {(DC) 7} (9) c o a {(DC)7} (10.1) b 5 c {add. ass.} (10.2) c E b ∧ b 5 c {(AC) 8, 10.1} (10.3) c E b ∧ b 5 c → c C b {(D ∀) 3, (DE)}

Diagram 3.11

100


(10.4) c C b {(M P ) 10.3, 10.2} (11) c C b ∧ c o a {(AC) 10.4, 9} (12) ∃w∈M (w C b ∧ w o a) {(A ∃) 11} a C b → ∃w∈M (w C b ∧ w o a) {(AI) 12}

On the other hand, when the diagram of relations is complete (examples 3.6, 3.7, p. 86), for Def.2 C, the (WSP) always takes place.

3.6

Relation of fusion

Now we will see what happens to the relation of fusion in NAM structures compliant to its definition by Leśniewski (Section 2.3.3, (2.13), p. 62). Following Leonard and Goodman, we will write down the Fu definition in terms of relation of overlapping because it will make it easier for us to analyse examples provided in Section 3.3:

∀x∈M ∀∅6=X⊆M (x Fu X ≡df ∀y∈M (y ◦ x ≡ ∃z∈X z ◦ y)).

(3.2)

We know that in extensional mereology, the Strong Supplementation Principle is the condition equivalent to the dependency Fu ⊆ Sum (Th. 2.17, p. 66). We can rather expect that in NAM structures it will not be the case. We will start with examples. The diagram of relations 3.3 (pp. 84 – 87, 94, 98 – 98) describes the case where (SSP) does not take place and the sum for the whole M does not exist and the 123 Fu M and 321 Fu M , as every element x in M overlaps with z 321 and 123, consequently, it is enough to take z = x pursuant to the definition of fusion. As a result, fusion is not a relation uniquely designated in NAM structures and Fu Sum.

3.6. RELATION OF FUSION

101

In the Example 3.4 the conditions (SSP) and (NAM5) take place and there exists fusion for every subset M . If all elements overlap, compliant to (3.2), (p. 100), it is obvious that such fusion exists (as in the Example 3.3). Let’s take a subset which is more “dubious”, e.g. X = {1, 2}. We will notice that 12 Fu X, because 1 ◦ 12, 2 ◦ 12, 21 ◦ 12 and of course 12 ◦ 12. Moreover, from the second part of the definition of fusion we have: 1 ◦ 1, 2 ◦ 2, 1 ◦ 21 and 1 ◦ 12. Consequently, in fact 12 Fu {1, 2}. Another example (3.5), is a case where sum exists, but the principle (SSP) is not fulfilled. In this case the fusion exists for every subset M as well. Example 3.6 describes a case where there exists a sum for every subset M (every element is a sum of the remaining ones) and the postulate (SSP) is fulfilled, as every element is a quasi-part of the remaining ones. Also in this case for any subset M fusion exists. It might seem that fusion always exists, but let’s take the Example 3.7 and let X = {1, 2, 3}. It turns out that ¬ 3 Fu X, because pursuant to (3.2), ¬ 1 ◦ 3, but 2 ◦ 1. Similarly ¬ 4 Fu X, because ¬ 1 ◦ 4, but 1 ◦ 1. By analogy, ¬ 1 Fu X, because ¬ 3 ◦ 1, but 3 ◦ 3. We have already noticed on the diagram of relations 3.3, that the relation of fusion is not an indisputable relation. Let’s look at the Example 3.3 once more: 123 Fu M and 321 Fu M , but 123 5 321 nor 321 5 123, consequently, the relation Fu is not a monotonic relation. Conclusion 3.3. In NAM structures, the relation Fu is neither monotonic relation nor uniquely designated one.

The fact that in certain situations neither Fu Sum, nor Sum Fu results from the lack of uniqueness. Let’s take the Example 3.5. We have: 12 Fu {1, 2}, as for 1 ◦ 12 we will assume z = 1 (pursuant to (3.2)), for 2 ◦ 12 we will assume z = 2, for 21 ◦ 12 we will assume z = 1 and for 12 ◦ 12 we will assume z = 1. Moreover 1 Sum {1, 2} because pursuant to Def. Sum

102


(p. 82) for 1 E 1 we assume w = 1 and for 2 E 1 we assume w = 2, but 12 5 1, consequently Fu Sum, but Sum ⊆ Fu. By analogy, we may verify that 12 Sum {1, 2} and 1 Fu {1, 2}, consequently Sum Fu, but Fu ⊆ Sum. Therefore, as a result of introducing the postulate (NAM21) ambiguities of this type occur.

3.7

Extensionality principle and quasi-proper parts

First we will see the impact of both definitions of quasi-proper part on the extensionality principle. We will start with writing down the extensionality principle with the aid of the relation C:

∀x∈M ∀y∈M (∀z∈M (z C x ≡ z C y) → x = y).

(NAMZE)

Let’s take into examination the Def.1 C (p. 88). In the examples 3.4, 3.5, 3.6 postulates (NAM5) and (NAMZE) are conserved (the mutual parts have different proper parts.). Instead in examples 3.7, 3.9, (NAM5) is not fulfilled but (NAMZE) is conserved. In the Example 3.3 neither (NAM5) nor (NAMZE) is conserved. One can suppose that (NAM5) imposes (NAMZE) but let us consider the following example:

Example 3.12. (Diagram 3.12) M = {0, 1, 2, 3, 23} and 0 E 2, 0 E 1, 1 E 0, 0 E 3, 2 E 23, 1 E 23, 3 E 23, 23Sum M . r 23 @ I 6 @ @ r -r 3 @ 61

2 r I @ @ @

@r?0

Diagram 3.12

3.8. ALGEBRAIC ASPECTS OF NAM

103

We have: 0 C 2, 1 C 2, 0 C 3, 1 C 3 but 2 6= 3. At the Def.2 C the situation is similar. In the Example 3.4, 12 6 21 and 21 6 12, consequently, 12 and 21 have the same quasi-proper parts and they are different; extensionality is not conserved. In Examples 3.6 and 3.7 – since the predecessor of the implication in (NAMZE) is false, the extensional principle is fulfilled. It could seem that the principle (NAM22) imposes extensionality at the Def.2 C, but the Example 3.5 is its counterexample. Hence, the following conclusion: Conclusion 3.4. Non-antisymmetric mereology may be (but it doesn’t have to be) a non-extensional mereology.

To sum up, in general, extensionality cannot be “exchangeable” with antisymmetry and non-antisymmetric mereology may be fully an extensional theory. Everything depends which definition of the proper part we will take into consideration. Consequently, the term ’non-extensional mereology’, as it is used [20], should refer only to selected cases. It is obvious that if the diagram of relations is complete (Examples 3.6, 3.7) the extensionality principle is always conserved.

3.8

Algebraic aspects of NAM

Due to the fact that the relation E is not partial order, therefore we will not take into consideration notions typical for partially ordered structures such as e.g. supremum. We will try to see what will happen if we assume that the postulate (NAM5) is fulfilled. Moreover, for the operation of a product to be always feasible, we assume that every two elements of the universum M overlap, that is ∀x,y∈M x ◦ y. It signifies that we restrict our domain of reasoning only to such structures as presented, for which certain

104


algebraic operations are always feasible. With this assumption, we may define certain algebraic notions for non-antisymmetric mereology: notions which are identical like in the case of mereological structures9 . F According to tradition, we introduce the unary relation : 2M \{∅} −→ M , allowing to define additional algebraic operations and the notion of mereological space in the following manner:

∀X∈2M \{∅}

G

X := (ix) x Sum X.

(3.3)

The description operator (ix) introduced in this manner, originating in Russell’s theory, is a one-argument name-forming operator interpreted as “the only x such that...” [90, p. 126]. Due to the fact that in the NAM theory we cannot guarantee the uniqueness of this x we have to modify the notation of the unary relation in the following manner:

∀X∈2M \{∅}

G

X := {x ∈ M : x Sum X},

(3.4)

F where , as it is presented in the Example 3.13 (p. 105), assigns to collective sets exactly element which is a set according to the classical set theory, F one M that is : 2 \{∅} −→ 2M \{∅}. With this unary operation we will try to define the mereological space and two other calculations. Due to the fact that every two elements of M overlap (x ◦ y), the relation Sum ((Def. Sum), p. 82) and the definition (3.4) have the following form:

∀X∈2M \{∅} (x Sum X ≡ ∀y∈X y E x),

9

See [95].

(3.5)


∀X∈2M \{∅}

G

105

X := {x ∈ M : ∀y∈X y E x}.

(3.6)

Let m ∈ M and m = {z ∈ M : z E m}. Then ∀x,y∈M

Λ :=

G

M = {x ∈ M : ∀z∈M z E x},

G (x ∪ y) = {w ∈ M : ∀z∈{u∈M : u E x G x u y := (x ∩ y) = {w ∈ M : ∀z∈{u∈M : u E x

x t y :=

(Def. Λ)

∨ u E y}

z E w},

(Def. t)

∧ u E y}

z E w}.

(Def. u)

Consequently, mereological space is a set of those objects which are classes10 of all their quasi-parts. The sum (t) of two objects is a set of those classes whose quasi-parts are included either in the first or in the second object. Due to the fact that two objects overlap, their product (u) is a set of those classes whose quasi-parts are included both in the first and in the second object. We will consider the following example: Example 3.13. (Diagram 3.13) F Let M = {0, 1, 2, 12, 21, 3} and let X = {0, 1, 2}. We will notice that {0, 1, 2} = {2, 12, 21, 3}. Let a = 2 and F b = 1, then a t b = {0, 1, 2} = {2, 12, 21, 3}. If the set {2, 12, 21, 3} was a collective set, then it would be a set identical to the set {0, 1, 2, 12, 21, 3}, because 0 and 1 are quasi-parts of 2, but 2 5 0.FTherefore we may draw the conclusion that sets formed as a result of the operator’s influence are distributive sets that is classical set theoretical objects.

10 We have already mentioned that this notion is equivalent to the notion of mereological sum.

106


12 r @ I @

r3 @ I@

@r 21

@r 62 r1 6 r?0

Diagram 3.13 Conclusion 3.5. Object created as a result of the influence of the operator F are distributive sets.

We will consider also one-element sets. We notice that as the primary relation is the relation of quasi-part, pursuant to (NAM1) every element of the set (any set) is its quasi-part. Consequently, every element of the distributive set is its quasi-part, but it is not true that every quasi-part of the distributive set is its element, what has been presented in the example above. Consequently, we have the following definition of a quasi-part of any set: Definition 3.2. Let X ∈ 2M \ {∅} and z ∈ M . We will say that z is a quasi-part of X (z E X) if and only if ∀m∈X z E m.

For example in the Diagram 3.13, 1 E{12, 21, 3}, because 1 E 12, 1 E 21, 1 E 3, but 1 ∈ / {12, 21, 3} from Cantor’s perspective. This definition is not required for collective sets, but it is important for distributive sets. When we look at distributive sets created as a result of algebraic operations, we may suppose that their elements become atoms in reference to the relation E; otherwise, we would reach a paradox case described in the Example 3.13.


107

Moreover: Conclusion 3.6. If z ∈ X, then ∀w∈F X z E w. Proof: Let w ∈

F

X, then, pursuant to (3.6), ∀y∈X y E w.

But z ∈ X, consequently z E w.

The above-presented definitions provide the basis for the following features of t, and u: Proposition 3.3. For any x, y, z ∈ M operations t, u have the following qualities: (i) x E Λ, (ii) xFE x t y F ∧ y E x t y, (iii) {x} =F {z ∈ M : z E x}, (iv) x t x = {x}, (v) x t y = yF t x, (vi) x u x = {x}, (vii) x u y = y u x. Proof:: (i) `F∀x∈M x E Λ Λ = MF= {w ∈ M : ∀z∈M z E w}. Let w ∈ M . Then ∀z∈M z E w, F but x ∈ M , consequently x E w. Pursuant to the Definition 3.2, x E M , therefore x E Λ. (ii) ` ∀x∈M ∀y∈M (x E x t y ∧ y E x t y) x = {z ∈FM : z E x} and y = {z ∈ M : z E y}. x t y = (x ∪ y) = {w ∈ M : ∀z∈{u∈M : u E x ∨ u E y} z E w}, but pursuant to (NAM1), x E x, consequently x ∈ {u ∈ M : u E x ∨ u E y}, therefore x E w, where w is any element of the set x t y. Consequently, pursuant to the Definition 3.2, x E x t y. By analogy for y.

108


F F (iii) ` ∀x∈M {z ∈ M : z E x} = {x} F {x} = F {z ∈ M : x E z}. Let a ∈ {z ∈ M : z E x}, then ∀y∈{z∈M : z E x} y E a. F As a result, pursuant to (NAM1), x E a, that is a ∈ {x}. Let x E a and let y ∈ {z ∈ M : z E x}. Then y E x. F Consequently, from (NAM3), y E a. As a result a ∈ {z ∈ M : z E x}. F (iv) ` ∀Fx∈M x t x = {x} F x t x = {z ∈ M : z E x ∨ z E x} =(ABS) {z ∈ M : z E x} =(iii) F {x}. (v) ` ∀x∈M ∀y∈M (x t y = y t x) The mathematical proof results from the commutative quality of the set theory sum. F (vi) ` ∀x∈M F x u x = {x} x Fu x =(Def. u) {z ∈ M :Fz E x ∧ z E x} =(ABS) {z ∈ M : z E x} =(iii) {x} (vii) ` ∀x∈M ∀y∈M (x u y = y u x) The mathematical proof results from the commutative quality of the set theory product.

We should notice that the unary operation that is:

F

is inversely monotonic,


Proposition 3.4. If x E y, then

F

{z ∈ M : z E y} ⊆

109

F

{z ∈ M : z E x}.

Proof : F Let a ∈ {z ∈ M : z E y}, then ∀n∈{z∈M : z E y} n E a. But x E y, so x ∈ {z ∈ M : z E y}, that is ∀n (n ∈ {z ∈ M : z E x} → n ∈ {z ∈F M : z E y}). As a result ∀n∈{z∈M : z E x} n E a, that is a ∈ {z ∈ M : z E x}.

F Moreover, the element M is in a sense an element which is the largest in reference to the relation E (in a sense, because E is not a relation of order on M ). We will add to our system the set ∅:

G

∅ = {x ∈ M : ∀y∈∅ y E x} = {x ∈ M : ∀y (y ∈ ∅ → y E x)} = M. (3.7)

F Due to the fact that the operation , is a particular operation of sum, we will examine what will happen if we introduce a certain small improvement, that is instead of the relation E we will apply the relation C, compliant to the Def.1 C (p. 88). We will do this replacement in order to introduce the possibility to separate elements in a manner providing sense to algebraic formulas. Then:

\

X := {x ∈ M : ∀y∈X y C x},

(3.8)

T where : 2M → 2M (we expand the M with the set ∅). We will notice that:

\

∅ = {x ∈ M : ∀y (y ∈ ∅ → y C x)} = M,

(3.9)

110

CHAPTER 3. NON-ANTISYMMETRIC MEREOLOGY \

M = {x ∈ M : ∀y (y ∈ M → y C x)} = ∅.

(3.10)

The last equality results from the fact that in NAM ∀x∈M x 6 x (Th. 3.1, p. 88). Unary relations defined in this manner have a series of qualities: Proposition 3.5. For any sets A, B ∈ 2M F (i) ∀x∈A x E A,F F (ii) (A ⊆ B) → ( FB ⊆ A), F F (iii) (A ⊆ B) → T ( (A ∪ B) = A ∩ B), (iv) ∀x∈A / A, Tx∈ (v) A ∪ TA ⊆ M , (vi) A ∩ A = ∅, T T (viii) T(A ⊆ B) →T( B T ⊆ A), (ix) T (A ∪TB) = T A ∩ B, (x) A ∪ B ⊆ (A ∩ B).

Proof: F (i) ` ∀ x E A. x∈A F A = {w F∈ M : ∀n∈A n E w}. Let w ∈ A. Then ∀n∈A n E w, but x ∈ A. F As a result, x E w, so from the Definition 3.2, x E A. F F (ii) ` ∀F B ⊆ A. A,B∈2M (A ⊆ B) → Let x ∈ B, that is ∀n∈B n E x, but ∀ Fn (n ∈ A → n ∈ B). As a result ∀n∈A n E x, therefore x ∈ A. F F F (iii) ` (A ⊆ B) → ( (A ∪ B) = A ∩ F B). F If A ⊆ B, then ∪ B = B, result F(A ∪ B) = B. F AF F as a F From (ii), B ⊆ A, so A ∩ B = B.


111

T (iv) ` ∀x∈A x ∈ / A. T From the Theorem 3.1, ∀x x 6 x, as a result if x ∈ A, then x ∈ / A. T (v) ` A ∪ A ⊆ M . The mathematical proof is obvious. T (vi) ` A ∩ A = ∅. T Let’s assume that there is such x that: x ∈ A ∩ A, so x ∈ A T and x ∈ A. T Because x ∈ A, then from (iv), x ∈ / A, consequently – contradiction. T T (viii) ` T(A ⊆ B) → ( B ⊆ A). Let x ∈ B, that is ∀y∈B y C x, T but ∀y (y ∈ A → y ∈ B). As a result ∀y∈A y C x, so x ∈ A. T T T (ix) ` (A ∪ B) = A ∩ B. T T We should notice that A ⊆ T A∪B, as a result from (viii), (A∪B) ⊆ A T and B ⊆ T A ∪ B, therefore T T(A ∪ B) ⊆ B, that is (A ∪ B) ⊆ A ∩ B. T T T Inversely. Let’s assume ∩ B * (A T that A T T ∪ B), consequently ∃x (x ∈ A ∧ x ∈ B ∧ x ∈ / (A ∪ B), that is ∀m∈A m C x and ∀n∈BTn C x, so ∀z∈A∪B T z C x, that is x ∈ (A ∪ B), but x ∈ / (A ∪ B), as a result we obtain contradiction.

112


T T T (x) ` A ∪ B ⊆ (A ∩ B). T T We should notice that A ⊆ (A T T A ∩ B ⊆ A,Tso T T ∩ B). By analogy B ⊆ (A ∩ B), so A ∪ B ⊆ (A ∩ B).

T Consequently, the operation introduced in this manner has certain features similar to the classical operation of supplementation on sets. Let’s 3.13 and T let A = {0, 1, 2, 12}, B = {0, 1, 2, 21}. T take the Diagram T Then: A = {3}, B = {3}, A ∪ 6 M, A ∪ B = T T AT= {0, 1, 2, 12, 3} = {0, 1,T2, 12, 21}, (A ∪ B) = {3},T A ∩T B = {3}, but A ∩ B = {0, 1, 2} and (A ∩ B) = {12, 21, 3} and A ∪ B = {3}. We could continue this reasoning and try to define new operations on elements of M . However, the problem we encounter is lack of possibility F to realise those operations on more than two elements, as operations and T transfer us to the space of distributive sets. Consequently, F the notation of the type (x t y) t z is not defined (the element x t y ∈ X, and x ∈ M ). The only possibility is to define operations directly on elements of 2M , but at this point we may use classical operations on sets. Still, it remains an interesting fact that with the aid of unary operators introduced in this manner we realise the transfer from the collective sets space to the distributive sets space. Consequently, those two concepts do not have to be treated as two “equivalent” in a sense approaches, but it seems that Nonantisymmetric Mereology may be recognized as more fundamental theory than the classical theory of sets.

3.9

NAM and other axiomatisations of extensional mereology

At the end we would like to pay closer attention to potential relationship of the NAM model if we were to take into consideration other axiomatisations

3.9. NAM AND EM

113

of extensional mereology. We know that in extensional mereology (Chapter 2), the system of axioms (M1)–(M5) is equivalent to the system (L1)–(L4) (Th. 2.11 – 2.13, pp. 59 – 61) and definitions of proper part are equivalent (Th. 2.1). We also noticed that in NAM those definitions are fully divergent and lead to different conclusions. We shall start with analysing several examples from the mentioned perspective. Example 3.4: for Def.1 C, the postulate (L1) is not fulfilled, because 12 C 21 and 21 C 12 and the postulate (L3) is not fulfilled – there is no uniqueness of the relation of sum: 12 Sum M oraz 21 Sum M . On the other hand, if we take into consideration the Def.2 C, then the postulate (L1) is conserved, because 12 6 21 and 21 6 12, and the postulate (L3) still doesn’t take place. Example 3.6: The postulate (L3) is not fulfilled as every element is a sum of the remaining elements; therefore we do not have uniqueness of a sum. Moreover, for the Def.1 C the axiom (L1) is not fulfilled, for any element from M , every element is a quasi-part of the other one and in case of Def.2 C any object is not a quasi-part of the other, therefore (L1) is conserved. Consequently, if x E y ∧ y E x ∧ x 6= y, persuant to Def.2 C, we receive the following: x 6 y and y 6 x, what has been presented in the Theorem 3.2. The following question arises: can we axiomatise non-antisymmetric mereology in yet one other way with the use of e.g. the term of primary relations of a quasi-proper part as it has been done for extensional mereology? Simons [111] instead of taking into consideration the postulate (L3), analyses the Weak Supplementation Principle (WSP). In case of NAM of course it is not possible, as – we have already presented it in the Theorem 3.3 (p. 95) – rejecting antisymmetry excludes (WSP) at the Def.1 C. We can certainly take the axiom (L2) defining the transitivity of quasiproper part:

∀x∈M ∀y∈M ∀z∈M (x C y ∧ y C z → x C z).

(L2)

114


If we define the relation of quasi-proper part – E by analogy, as in EM, that is:

∀x∈M ∀y∈M (x E y ≡df x C y ∨ x = y).

(3.11)

Then, pursuant to (3.11), for x = y, we receive the postulate (NAM1). Pursuant to (L2) and from the transitivity of the relation ’=’ we receive (NAM3). We are left with the (NAM21) condition. We have noticed that (NAM21) imposes the negation of (L3); therefore we could take the postulate in the following form:

∃x∈M ∃y∈M ∃∅6=Z⊆M (x Sum Z ∧ y Sum Z ∧ x 6= y).

(3.12)

(L1) is conserved in NAM only in case of Def.2 C. Moreover, we know that in EM (Chapter 2), (L1) together with the definition (3.11) (Th. 2.9, p. 57) adapted to the relation E imposes antisymmetry of the relation E, as a result, we have to skip (L1). Our analyses concerning the extensionality principle show that extensionality is conserved or not. Consequently, finally, our new system of axioms could be made of the following propositions: (L2) and (3.12) (p. 114) with definitions: (3.11) (p. 114) and Def. Sum (p. 82). However, the Example 3.2 (p. 83) shows that it is not sufficient: 12 Sum {1, 2} and 21 Sum {1, 2}, but it is not non-antisymmetric Mereology, as here the postulate (M2) is conserved. As a result, we have to add explicite the (NAM21) condition. In addition, following Leśniewski (Section 2.3.3) we may also adopt the disjointedness relation as a primary term. We have the notion of fusion by Leśniewski ((2.13), p. 62), but we have to reject the principle of extensionality for this relation ((2.15), p. 63). The relation of a part is defined according to Leśniewski, as follows:

3.9. NAM AND EM

∀x∈M ∀y∈M (x E y ≡df ∃∅ 6= Z⊆M (y Fu Z ∧ x ∈ Z)).

115

(3.13)

In Theorems 2.14 – 2.16 (pp. 64 – 65) we have presented that the relation of a part is a partial order. Due to the fact that we have rejected the uniqueness of fusion, we have only the quasi-ordering relation, therefore our relation fulfils the postulates (NAM1) and (NAM3). The postulate (NAM21) has to be imposed. We will verify if it is reasonable in a theory introduced in this manner. From the condition (3.13) (p. 115) we receive a postulate in the following form:

∃∅6=Z1 ,Z2 ⊆M ∀x∈M ∃y∈M (y Fu Z1 ∧ y ∈ Z2 ∧ x Fu Z2 ∧ x ∈ Z1 ∧ x 6= y). (3.14) The Postulate (3.14) is the explication of the rejection of the uniqueness condition for the relation of fusion (equation (2.15), p. 63):

∃∅6=Z⊆M ∀x∈M ∃y∈M (y Fu Z ∧ x Fu Z ∧ x 6= y).

(3.15)

Consequently, lack of uniqueness of Fu is the direct consequence of the (NAM21) condition. As a result, a system of postulates for the primary relation of disjointedness – o would be made of the following formulas: (2.13) (p. 62) with the definition (3.13) (p. 115) and the condition (NAM21).

Appendix Let ψ, φ, χ be the formulas of a classical propositional calculus (CPC). We will apply the following abbreviations for the applied principles and rules: (MP) detachment ((φ → ψ) ∧ φ) → ψ (TOLL) tollens ((φ → ψ) ∧ ¬ ψ) → ¬ φ (AA) attachment of alternation φ → (φ ∨ ψ) ψ → (φ ∨ ψ) (DA) detachment of alternation ((φ ∨ ψ) ∧ ¬ φ) → ψ (DC) detachment of conjunction (φ ∧ ψ) → ψ

(φ ∧ ψ) → φ

(AC) attachment of conjunction φ, ψ →(φ ∧ ψ) (DE) detachment of equivalence (φ ≡ ψ) → (φ → ψ) (φ ≡ ψ) → (ψ → φ) (NN) double negation ¬ ¬ φ → φ φ → ¬ ¬ φ (TRAN) transitivity (φ → ψ) ∧ (ψ → χ) → (φ → χ)

118


(NA) negation of alternation ¬(φ ∨ ψ) ≡ (¬ φ ∧ ¬ ψ) (NC) negation of conjunction ¬(φ ∧ ψ) ≡ (¬ φ ∨ ¬ ψ) (REP) replacement (φ → ψ) ≡ (¬ φ ∨ ψ) (COMM) commutation (φ ∨ ψ) ≡ (ψ ∨ φ) (φ ∧ ψ) ≡ (ψ ∧ φ) (CON) connectivity (φ ∨ ψ) ∨ χ ≡ ψ ∨ (φ ∨ χ) (φ ∧ ψ) ∧ χ ≡ ψ ∧ (φ ∧ χ) (ABS) absorption (φ ∨ φ) ≡ φ (φ ∧ φ) ≡ φ (SEP) separation (φ ∧ (ψ ∨ χ)) ≡ (φ ∧ ψ) ∨ (φ ∧ χ) (φ ∨ (ψ ∧ χ)) ≡ (φ ∨ ψ) ∧ (φ ∨ χ) (LP) Leibnitz’s Principle (ψ ≡ χ) ≡ (φ(ψ/χ)) (ZSA) contradiction principle ¬(φ ∧ ¬ φ) (AI) attachment of implication (RW) branched direct math. proof (RN) branched math. proof by contradiction (NDP) non direct proof

(D ∀) detachment of general quantifier (∀α φ(α)) → (φ(γ/α)

3.9. NAM AND EM

119

(A ∀) attachment of general quantifier φ → ∀α φ(α) If α is not a free variable. (N ∀) negation of general quantifier ¬(∀α φ(α)) → ∃α ¬ φ(α) (A ∃) attachment of existential quantifier φ(γ/α) → ∃α φ(α) (D ∃) detachment of existential quantifier ∃α φ(α) → φ(c) Where c is a name constant depending on all free variables of the formula ∃α φ(α). (N ∃) negation of existential quantifier ¬(∃α φ(α)) → (∀α ¬ φ(α)) (R1 ) (∀α φ(α) ∧ ∀α ψ(α)) ≡ ∀α (φ(α) ∧ ψ(α)) (R2 ) (∃α φ(α) ∨ ∃α ψ(α)) ≡ ∃α (φ(α) ∨ ψ(α)) (R2 ∀) (ψ ∧ ∀α φ(α)) ≡ (∀α (ψ ∧ φ(α))) (R2 ∃) (ψ ∧ ∃α φ(α)) ≡ (∃α (ψ ∧ φ(α))) If ψ does not contain free occurrences of α. (R2 ≡) (ψ ≡ ∀α φ(α)) ≡ (∀α (ψ ≡ φ(α))) (T2) (φ → ψ) ∧ (φ → χ) → (φ → (ψ ∧ χ)) (T3) (φ ∧ ψ → χ) ≡ (φ → (ψ → χ))

Contents Introduction

5

1 Stanisław Leśniewski 1.1 Lvov-Warsaw School (1895-1939) . . . . . . . . . 1.2 Warsaw School of Logic (1918–1939) . . . . . . . 1.2.1 Lvov period . . . . . . . . . . . . . . . . . 1.2.2 Warsaw period . . . . . . . . . . . . . . . 1.3 Stanisław Leśniewski (1886–1939) and his works . 2 Leśniewski’s Mereology 2.1 The relation of being an element . . . . . . 2.2 The relation of composing a whole . . . . . 2.3 Axiomatisation of EM . . . . . . . . . . . . 2.3.1 Mereology founded on the relation of 2.3.2 Mereology founded on the relation of 2.3.3 Mereology founded on the relation of 2.4 Extensionality principle . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . part . . . . . proper part . disjointedness . . . . . . . .

3 Non-antisymmetric mereology 3.1 Basic qualities . . . . . . . . . . . . . . . . . . . . 3.2 Negation of Antisymmetry . . . . . . . . . . . . 3.3 Relation of sum . . . . . . . . . . . . . . . . . . . 3.4 Quasi-proper parts . . . . . . . . . . . . . . . . . 3.5 Supplementation Principles . . . . . . . . . . . . 3.5.1 Principles of supplementation and Def.1 C 3.5.2 Principles of supplementation and Def.2 C 3.6 Relation of fusion . . . . . . . . . . . . . . . . . . 121

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . .

11 11 14 14 16 20

. . . . . . .

35 35 38 45 45 54 61 68

75 . 75 . 80 . 82 . 87 . 91 . 93 . 96 . 100

3.7 3.8 3.9

Extensionality principle and quasi-proper parts . . . . . . . 102 Algebraic aspects of NAM . . . . . . . . . . . . . . . . . . . 103 NAM and EM . . . . . . . . . . . . . . . . . . . . . . . . . 112

Appendix

117

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