Abstract
The paradoxes we have analyzed so far have a common property: they do not go beyond the language. The difficulties, which they expose, have intra-linguistic character. Their analysis, and all the more so their solution, does not require any confrontation of thought and reality. It is otherwise with paradoxes analyzed in this chapter. All of them result from such confrontation of the language and propositions we formulate in it with the real world which surrounds us. Ontology is said to be that discipline of philosophy, which shows the possible structure of being. Paradoxes of vagueness, motion, identity, sorites paradoxes and the problem of the many clearly show that no ontology, which uses the concepts of set, Euclidean point, rest, property or thing (understood as something defined by its properties) that are formed under the spell of language, has any logical basis. The use of all of these categories with reference to the world has been questioned by paradoxes. The image of the world they create is precise and static, and thus in disagreement with the reality itself, which by its nature is rich and dynamic in the variety of its phenomena. The effect of this disagreement is an inalienable contradiction, which appears every time we speak (think) of the real world by means of the language. For this reason, paradoxes sorites, motion, identity and the problem of the many deserve the name of ontological paradoxes. Just like Russell’s Antinomy showed naivety of Cantor’s set theory, ontological paradoxes show naivety of ontology and metaphysics of Aristotelian provenance.
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Notes
- 1.
Over a hundred years before Eubulides, Zeno of Elea (ca. 490–ca. 430 BC), presented an argument aimed at disqualifying the value of empirical evidence. This well-known paradox looks as follows: One grain of corn makes no noise when it falls on the ground. If x grains produce no sound, then x + 1 grains fall down soundlessly too. Consequently, no number of grains of corn makes any sound when falling on the ground. This, however, is in disagreement with reality.
- 2.
Diogenes Laertios, II 109.
- 3.
Usually, presentations of the Paradox of a Heap silently assume an appropriate form of aggregation of grains. However, 30,000 grains of wheat may not form a heap if they are spread on sufficiently large area. For this reason, both in this and all following arguments we assume silently that every considered aggregation of grains takes an “appropriate” form, i.e., one that is as similar to a heap as possible.
- 4.
An example given by Sorensen [1].
- 5.
Dummett [2], pp. 303–308.
- 6.
Russell [3]. Earlier Russell moved the question of vagueness in his 1913 manuscript Theory of Knowledge. He returned to the problem in the book The Analysis of Mind, London, Paul Kegan, 1921.
- 7.
Black [4].
- 8.
It is obvious that the analysis presented for the Paradox of a Chair can be reconstructed for any object of inanimate matter: a building, stone, pen, book, etc.
- 9.
Thomas Chippendale (1718–1779), a famous British furniture designer, the author of the book The Gentleman and Cabinet Maker’s Director, published in 1754.
- 10.
The expressions like “perfectly executed” or “ideal copy” have sense, since they are used in considerations, which are logically admissible mental experiments. We assume, moreover, that successive damages gradually destroy the chair without dividing it into two or more pieces.
- 11.
In Black’s understanding, a “normal” observer is anyone, who, in a situation similar to the one of the exhibition, does not expect to discover a precise boundary between the objects called “a” and objects called “non-a”; Black [4], p. 433.
- 12.
Williamson [5], pp. 79–80.
- 13.
Williamson [5], pp. 82–83.
- 14.
The abbreviation used here should be understood as follows: mothern + 1 = mother of mothern, for any natural number n ≥ 1.
- 15.
Naturally, the so-called moment of birth, or the moment of conception, is not a dimensionless time point either, but a non-zero time sequence.
- 16.
Kubiński [6], pp. 121–122.
- 17.
Kubiński also gives another version of this definition, in which the word “no one” is replaced by an expression “a visible majority”; Kubiński [6], p. 122.
- 18.
A boundary of the name a is the difference between the universum and the sum of both positive and negative scope of the name. In other words, the boundary of a name is formed by all objects which to not belong either to its positive or to its negative extension; Kubiński [6], p. 119.
- 19.
It is an interesting construction but it concerns partly defined names rather than vague ones.
- 20.
Sainsbury [7], p. 31.
- 21.
According to the idea of Peter Unger, in this mental experiment damage can be minimalized to separation of a single atom from the whole; Unger [8], pp. 237–238.
- 22.
Sorensen [9].
- 23.
The expression “33-small(k)” is an abbreviated form of the sentence “k is 33-small”.
- 24.
Sorensen [10]. Sorensen does not notice that both ambiguity and universality are related to vagueness too. For instance, a boundary between an offspring and a young offspring is not precise, just like the boundary between being a boy and being a girl.
- 25.
Cf. Varzi [11].
- 26.
- 27.
- 28.
Naturally, the predicate 33-small(<33) being vague, it is not a case of a borderline instance. It was given here for better illustration of the sequence of predicates analyzed here.
- 29.
Hyde [14], p. 303.
- 30.
Russell [3].
- 31.
Dummett [2], p. 314.
- 32.
Lewis [16], p. 212.
- 33.
Williamson [17].
- 34.
Pawłowski [18], p. 72.
- 35.
Tye [19], p. 563.
- 36.
- 37.
Tye [20], pp. 535–536.
- 38.
Tye [20], p. 536.
- 39.
Tye [19], p. 563.
- 40.
Tye [20], p. 536.
- 41.
Tye [20], pp. 535–536.
- 42.
Following Russell, Tye argues for vagueness of even such seemingly precise properties as having the height of 2,000 feet. Indeed, one can safely assume that the standard from Trafalgar Square in London gives information which is as precise as the one given by the standard from Sèvres near Paris. Naturally, today neither the rod of Sèvres, nor the table from Trafalgar Square are standards for measuring length. Cf. a note above.
- 43.
Chibeni [22].
- 44.
Sorensen uses the word “vague” in parentheses, since he wants to show that vagueness of this ball is impossible; Sorensen [23], p. 275.
- 45.
Sorensen [23], p. 276.
- 46.
Fine [24].
- 47.
- 48.
Because of the fuzzy nature of the vagueness area this condition seems to be an idealization difficult to execute.
- 49.
Dialetheism is usually associated with the broad current of paraconsistent logics, i.e., ones, which tolerate contradiction. Priest and Tanaka [28].
- 50.
- 51.
Priest [32], p. 202.
- 52.
Hyde [33].
- 53.
- 54.
The author of the theory of partial definitions is Rudolf Carnap, who presented its outline in Testability and Meaning; Carnap [36].
- 55.
Kubiński [6].
- 56.
The assumption that \( K_{ 1} = \neg K_{ 2} \) means that the definition given by the two conditions becomes a complete definition, given in one of the two equivalent forms: either \( \forall \,x(K_{ 1} \left( x \right) \leftrightarrow Q\left( x \right)) \), or \( \forall \,x(K_{ 2} \left( x \right) \leftrightarrow \neg Q\left( x \right)) \) Przełęcki correctly observes that the same form of the definition does not mean that it defines a vague term. A vague term can also be defined with help of a complete definition, since it is enough for the predicate K 1 (i.e., also K 2), which occurs in it, to be vague. Przełęcki [37], p. 82.
- 57.
Przełęcki [37], p. 80.
- 58.
Bochwar [38].
- 59.
In Kleene’s book Introduction to Metamathematics, these conjunctions are called weak. He calls strong the conjunctions, which he introduced earlier, defined with help of the value U. In a straightforward way, Kleene states, however, that in case of partial predicates only strong conjunctions, i.e., the ones described in his 1938 work, can be applied; Kleene [39], p. 334. Naturally, all partial predicates are predicates, which are partly defined.
- 60.
Williamson [5], p. 288.
- 61.
Halldén [40].
- 62.
Williamson [5], pp. 105–106.
- 63.
Williamson [5], p. 107.
- 64.
- 65.
Williamson [5], p. 108.
- 66.
Tye [20].
- 67.
Opinions concordant with Williamson’s theory were expressed even earlier. In his 1948 paper Zmiana i sprzeczność (Change and inconsistency), Ajdukiewicz states: “We shall lack the means to decide about a man in a given age whether he is young or not young. This is an obvious fact. But some people are ready to conclude therefrom that for a man who grows old it is neither true that he is young nor that he is not young. This attack on the principle of the excluded middle errs fundamentally, because it mistakes the situation, in which we are unable to determine between two contradictory sentences for the situation, in which none of the two contradictory sentences is true. That we cannot determine the truth of either “he is young” or ‘he is not young” for fundamental reasons, not merely because of technical difficulties, is no proof at all that none of them is true”. Ajdukiewicz [45], pp. 105–106.
- 68.
- 69.
Let us assume here that the crowd in question is composed of 1019 people. From the assumption, someone who looks at the crowd knows that there are not 1018 people in it. Moreover, it is the biggest number, of which he knows that it does not express the number of people in the crowd. He must know that, because with respect to the next number, 1019, he does not know that it does not express the number of people in the crowd. This means that so far our observer knows that the area of vagueness (impreciseness) starts with the number 1019. Only up to 1018, he has certitude that those numbers do not express the number of people in the crowd. Naturally, the number, which expresses the actual number of people in the crowd, 1019, belongs to that area. Consequently, our observer knows that the number of people in the crowd equals 1019 or is greater than 1019. However, it enough for our observer to get to know Williamson’s reasoning to discover that this is precisely 1019. This contradicts not only the whole reasoning but also, and most of all, the fundamental assumption of epistemicism. What is worst is that the assumptions he accepts are in disagreement with the evident state of affairs.
- 70.
It should be mentioned here that natural science also distinguishes between homo sapiens and homo sapiens sapiens accepting without hesitation that we are representatives of the latter.
- 71.
The problem whether someone is aware of the knowledge he possesses or not seems to belong rather to psychology of mental processes. Moreover, this question is most probably connected with a person’s mental faculties, intellectual development, etc., and cannot be globally determined by means of sentence calculus, especially if the solution were to apply to any person.
- 72.
“A margin for error principle is a principle of the form: ‘A’ is true in all cases similar to cases in which ‘It is known that A’ is true”, Williamson [5], p. 227.
- 73.
“In effect, knowledge that one knows requires two margins of error [in the case described here], More generally, every iteration of knowledge widens the required margin.”, Williamson [5], s. 228.
- 74.
It should be noted that proving a rather commonly accepted thesis about impreciseness of knowledge concerning facts, which are essentially related to vagueness, Williamson paradoxically accepts that such knowledge is more precise than it is generally accepted. Usually, no one assumes that the area of vagueness is precisely delimited. This way, Williamson, who tries to persuade us to accept the thesis about impreciseness of knowledge assumes that knowledge is much more precise than we think.
- 75.
Although we shall not mention it later, we have in mind only “appropriately arranged quantities of grains”.
- 76.
Williamson [5], pp. 232–233.
- 77.
After all, for most logicians Z 0 is a manifestly false sentence.
- 78.
The ridiculous and pompous sound of that name is intended.
- 79.
Wright [48].
- 80.
Tolerance is explained in the beginning of this chapter.
- 81.
Sorensen [49].
- 82.
The research of cloning showed the falsehood of the second of these assumptions.
- 83.
That we usually do not do it does not mean at all that we never do it. Using vague expressions outside the range of their preciseness may mean e.g., a joke or irony.
- 84.
Williamson [5], p. 113.
- 85.
Zadeh [50].
- 86.
Williamson [5], p. 123.
- 87.
- 88.
Unger [52], pp. 178–182.
- 89.
“For, if there are no heaps, we can define the word ‘heap’, for example, so that a heap may consist, minimally, of two items: for example, beans or grains of sand, touching each other”, Unger [8], p. 250.
- 90.
Unger [51], p. 147.
- 91.
Unger [51], pp. 147–148.
- 92.
He confesses that he blushes recalling his works of the period. Unger [53], p. 1.
- 93.
- 94.
- 95.
The problem raised by Geach and Unger is not new. Actually, it has an ancient pedigree. The true author of the idea, which is the core of the paradox, is Chrysippus. In the paragraph devoted to paradoxes of identity below, we mention the Paradox of 1001 cats, which is both a repetition of Chrysippus’ paradox and the paradox of the many. Chrysippus’ paradox as a dilemma of identity is discussed in detail in the same paragraph.
- 96.
Lewis [56], p. 164.
- 97.
Those interested in the attempts at solving the paradox with help of nihilism, over-population, brutalism, relative identity, partial identity, may consult Weatherson [57].
- 98.
Diogenes Laertios, op.cit., X, 125.
- 99.
From this point of view, the so called natural species look rather funny, since they owe their naturality only to the exceptional slowness of evolutionary changes. A living being A is a representative of a natural species, because we look at it. A discovered relic of a being B is also a representative of a natural species, because we can go describe it and give it a name. But all beings of the evolutionary chain joining A and B are apparently representatives of non-natural species.
- 100.
- 101.
Plutarchus, [Life of Theseus].
- 102.
Deutsch [58], p. 2.5.
- 103.
Theseus’s ship has its modern equivalents, e.g., in Buddhist temples in Japan. One of the oldest and most important of them is the Horyuji temple, built by Shotoku-Taishi in the old capital of Japan, Nara, in 607 AD. The temple survives thanks to successive replacement of its wooden parts.
- 104.
- 105.
Cf. Wiggins [62].
- 106.
Kripke [59], p. 114.
- 107.
Cf. Chisholm [63].
- 108.
A detailed presentation of the discussion on that topic can be found in: Deutsch [58], p. 2.5.
- 109.
Deutsch [58], p. 2.2.
- 110.
- 111.
- 112.
Whitehead [65], p. 95.
- 113.
Ajdukiewicz [66], p. 140.
- 114.
Ajdukiewicz [66], p. 93.
- 115.
Aristotle, Physics, 232b–234a.
- 116.
- 117.
Weyl [69].
- 118.
As a matter of fact, similar questions were analyzed already in the Middle Ages. Kilvington considers the possibility of cutting a cone in infinitely many parts. We cut a cone of a finite height h with a plane parallel to the base of the cone at the half of its height. A new cone of the height h/2 is similarly cut at the half of its height, and so on infinitely. Kilvington wonders whether it is possible to go on cutting the successively formed cones infinitely. He comes to a conclusion that for technical reasons the cutting will be no longer possible at some point. He observes, nevertheless, that theoretically it should be possible to perform infinitely many cuttings in finite time, because every successive cutting requires less and less time. Podkoński [70].
- 119.
- 120.
Placek [68], p. 68.
- 121.
An interesting problem is considered by Priest [72], p. 1–2. He calls it Bernadete Paradox, from the name of the author of this construction, Bernadete [73], p. 259. In fact, the paradox is an example of an infinity machine, which suspends the laws of logic. Let us assume that a person A moves successively along a line from −∞, through 0, to +∞. However, her every move beyond the point 0 causes that in a double distance from the point 0, there appears an obstacle which A cannot overcome. Thus, if A reached the point x, an obstacle appeared in the point 2x. Priest proposes a following formalization of this construction: (1) (Rx ∧ y < x) → Ry, (2) (By ∧ y < x) → ¬Rx, (3) ¬∃x(x < y ∧ Bx) → Ry, (4) x ≤ 0 → ¬Bx, (5) x > 0 → (Bx ↔ Rx/2); where Rx and Bx mean respectively: “the person A reached the point x” and “an obstacle appeared in the point x”. He also notes that the set of premises is non-contradictory {1,2,3}. To be true, the set {1,2,3,4} is non-contradictory too. But a contradiction does follow already from the set {1,2,3,4,5}: the person A cannot go beyond the point 0 by any non-zero magnitude a (obviously, the point a is identified here with its distance from 0, which is a) because if she did it, she would have to cover the distance of, say a/4, much earlier; but that would mean that in the point a/2 an obstacle appeared, which A would not be able to overcome; consequently, A would not be able to reach the point a. If so, then the person A cannot go beyond the point 0, which means that no obstacle will appear—contradiction. Naturally, the infinity machine is defined by the premise (5). One can clearly see that Bernadete Paradox is inspired by Zeno’s Dichotomy. It attempts to define the infinity machine taking the example from motion. There is, however an important difference between the two paradoxes: Bernadete’s machine is contradictory, motion is not.
- 122.
Placek [68], p. 68.
- 123.
Achilles and the Tortoise Argument is commonly ascribed to Zeno of Elea. Diogenes Laertios believes Zeno to be the author of the dilemma too, but adds that according to the opinion of Favorinus, presented in his Historiae diversae, it was Parmenides who discovered the paradox, Diogenes Laertios, op.cit., IX.5, 28, p. 532.
- 124.
Aristotle Physics, VI, 239b10-33, 405.
- 125.
Ibidem.
- 126.
Ibidem.
- 127.
Euclid’s Elements begin with a statement that “We call a point something that has no parts”.
- 128.
A non-zero number x is an infinitesimal number if and only if its absolute value is greater than zero and smaller than any positive real number. Infinitesimal numbers belong to the so called hiperreal numbers. The author of the theory of hiperreal numbers is John Conway. He had, however, many forerunners: Archimedes, Isaac Newton, Gottfried Leibniz, Leonhard Euler, Augustin Louis Cauchy. Conway [74], Conway and Guy [75], Robinson [76].
- 129.
This claim becomes even more evident, when we look at the example of a shutter in film photo cameras. The opening of the shutter decides about the length of exposition time of the film, which is equivalent to the length of “looking” at the photographed object by the camera. The time is zero if the shutter is completely closed. But then nothing can be seen, not even stillness.
- 130.
Russell [77].
- 131.
Russell [77], p. 122.
- 132.
Nicod [78].
- 133.
Russell [77], pp. 122–123.
- 134.
Whitehead [79].
- 135.
Whitehead [80].
- 136.
Whitehead [65].
- 137.
- 138.
- 139.
Tarski [86].
- 140.
Putnam [87], pp. 5–6.
- 141.
This name is not used in the literature of the subject but has been proposed by the author of this study.
- 142.
Putnam [87], p. 4.
- 143.
The supposed preciseness of philosophical thinking often assumes a naive, and sometimes a ridiculous form. This great and diverse problem can be richly illustrated by various casus. Let us remind just a few. 1. Using the term “natural species”, which is natural only because we look at it: other phases of development of the species are transitional phases. 2. Continual failure in defining man. 3. Ideas such as physicalism: a “precise”, physical description of an emotional state of a man, e.g. disappointment, hope or anger, enumerates a sequence of properties, such as flexing “appropriate” muscles, but finishes up with the magical expression “etc.”.
- 144.
Bergson stated straightforwardly that there are changes but there are no objects which change “under” those changes; change does not require a substrate. There are motions but there is no motionless, unchanging object that could be in motion: motion does not imply something which moves.
- 145.
Cf. Russell’s comments on a fading, sc. changing color, wallpaper. Russell [77].
- 146.
After all, the different speed of change for a cloud and a stone cannot testify to the difference between the two objects qua beings.
- 147.
Wittgenstein [88], p. 58.
- 148.
Wittgenstein [88], p. 43.
- 149.
Wittgenstein [88], p. 44.
- 150.
Wittgenstein [88], pp. 44–45.
- 151.
Wittgenstein [88], p. 45.
- 152.
This story, rather sarcastic towards numerous conclusions in ontology, takes its idea from a fragment of Bergson’s 1907 book, L’évolution créatrice, where human mind is compared to a cine-camera; Bergson [89], pp. 257–260.
- 153.
Bergson tries to show the way in which the paradox can be avoided. In his opinion, our mind can go another way. It can place itself in a mobile reality, assimilate to its ever changing direction and finally comprehend it intuitively. To do so, it must force itself to reverse the direction of the actions which constitute its thinking, it must constantly change, or rather alter, all of its categories. But in this way, it will finally reach the fluid concepts, which can follow reality in all of its tides and ebbs and take on motion of the very inner life of things. Only in this way one can create a philosophy, which will be freed from disputes between schools and capable of solving all problems in a natural way because it will liberate itself from the shackles of artificial terms that were selected in the times those problems were posed.
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Łukowski, P. (2010). Ontological Paradoxes. In: Paradoxes. Trends in Logic, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1476-2_5
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