The considerations of Parmenides of Elea contained in the poem Περί φύσεως
were a revolution in the thinking of ancient Greeks about what is the subject of philosophy and what attributes it possesses. Probably Parmenides’s
theory of being and cognition was incomprehensible from an empirical point of
view for contemporary Greek. Plato’s dialogue Parmenides shows that Parmenides
of Elea’s theory was attacked. Zenon was a disciple of Parmenides who defended
his philosophy.
The subject of my research is the Zeno’s paradoxes formulated in defence Parmenides’s philosophy. I included the results of my research in a monograph Apagogic Zeno’s of Elea Arguments in Parmenides’s Apology. The book consists of four
chapters and an appendix. The appendix is a philological justification of the paradigm in the light of which I read passages of ancient texts. The main idea that
guided me at work was an adequate reading of Zeno’s paradoxes, firstly, within the
epistemological and ontological assumptions of Parmenides which are contained in
his epic poem Περί φύσεως, secondly, taking into account contemporary linguistic
research. They shed new light on the syntactic functions of the verb „be” in ancient
Greek and on the meanings in which it is occured. I mean the results of Ch. S. Kahn
contained in his monograph The Verb „Be” in Ancient Greek, in which he argues
that in ancient Greek there are no existential uses of this verb. Zeno’s arguments are
often read in an existential way as arguments against the existence of multiplicity
and against the existence of motion. For this reason I have reinterpreted paradoxes
in the light of Kahn’s research, and to reformulate traditionally understood Parmenides ontology.
The first chapter, entitled Parmenides’s Ontology, is focused, above all, on the
interpretation of the principle expressed in fragment B 2.3 and its influence on
the understanding of being. This text, however, tends to be interpreted diversely.
Referring to other fragments of the epic poem, I conducted a philological analysis of the fragment, trying to answer the question whether this is an idiomatic
expression or maybe this should be translated literally, interlinearly. What is (if
there is any) the grammatical subject of the sentence? Does ἔστιν in the first part
of this compound sentence have an implied predicative or does it constitute a separate predicate? What grammatical form does the whole sentence have? I present
a number of interpretations, distinguishing six groups of translations that can be
found in the literature on the subject. In my opinion, non-existential translations,
with the implied grammatical subject indicated by Parmenides in the prologue to
his works, comply with the philosopher’s intentions. The subject seems to be the
object of knowledge, which is the ontological truth. This choice really is essential
to Parmenides’s way of understanding the being.
Then, I performed a logical analysis of each type of translation, coming to believe that the fragment is first in history clear expression of the ontological principle of non-contradiction. Despite being non-idiomatic, existential and literal, all
other translations have a formal structure that corresponds with the principle. The
non-contradiction that we read about in the epic poem is absolutist – the aspects of
being are not distinguished in it. The object of knowledge (the ontic truth) cannot
be anything else than it is. The absolutism of non-contradiction of being reveals
the necessity of its identity.
Parmenides searched for truth that had attributes of immutability and non-complexity. Non-contradiction, which is a formal condition of being, does not admit
any change in Parmenides’s point of view and determines the immutability of Being. For every change would lead to a contradiction. The claim about an absolute
identity of being leads to a conviction that it is completely non-complex, absolutely
simple. Parmenides finds truth in Being that is immutable and non-complex. An
important attribute which results from such a perspective is an attribute of definiteness and finiteness of Being. In the book, I present that Parmenides was first in the
history of philosophy to introduce the concept of being in the form of the gerund
to eon, which expresses everlasting time. In the part devoted to the truthfulness of
Parmenides’s concept of being, I indicate that being belongs to another ontic plane
than the mutable, complex and indefinite world of material objects. Thus, I stand
in opposition to materialistic interpretations of Parmenides’s being.
Taking into consideration the apologetic motives that made Zeno write his book
of paradoxes, I present conclusions concerning the nature and attributes of Being
that are confirmed by the object of paradoxes – change and plurality. On the other
hand, in the light of these results, we get a clear background to answer the question why Zeno’s arguments are reasonings whose aim is to contradict plurality
and motion, and why the concept of endless sequence, which leads to a number of
theoretical difficulties, is emphasised in all of them.
The second chapter of the monograph includes texts, my own translations, linguistic as well as logical and philosophical analyses of all paradoxes that were handed over to us by ancient philosophers. It consists of four paradoxes of plurality, the
paradox of the Grain of Millet – that is sometimes classified as a paradox of plurality, but it seems that it can also be interpreted as a paradox of motion, which I justify
in the book – and then four paradoxes of motion: the Dichotomy, Achilles and the
Tortoise, the Arrow and the Stadium, and the paradox of Place. Ten paradoxes that
survived in the works of later philosophers such as Plato, Aristotle, Simplicius and
Diogenes were probably included in Zeno’s book compiled of forty arguments.
The object of paradoxes is constituted by things that belong to the changeable
and complex world, therefore the path that Zeno chose in order to substantiate
Parmenides’s theory of being is quite different from the one that its creator walked.
This is an indirect substantiation. It is supposed to prove the value of Truth and
also be an expression of critical stand on opinions that lead to contradictions and,
in consequence, do not provide valuable cognition. The polemic that Zeno had
with the opponents of Parmenides reveals that the cognition of material world is
imperfect since it results in contradiction and involves us in various theoretical
difficulties that are far more serious than those following the universally non-intuitive path of Being.
In the light of Kahn’s research results, I tried to demonstrate that the first of the
discussed arguments against plurality [Zeno, 19, B 3; Simplicius, In Aristotelis Physicorum, 140.28] could be interpreted in both locative and referential ways. Hence,
the phrase εἰ πολλά ἐστιν would mean that there are (are located) many things (scil.
in the world) and emphasise the semantic aspect – they are true. This hypothesis
on Zeno’s proof would emphasise both planes and it probably has a meaning that
connects both syntactic and semantic aspects. Thus, it would mean the following:
„there are many true things [scil. in the world]”. The referential interpretation („is”
means the truth) consists in presenting the argument and stressing, above all, the
semantic function of ἐστιν. The referential and truthful role of the verb „exist” also
appears in Parmenides’s epic poem. „Exist” designates the object of knowledge –
the true being. If we adopt such an understanding of ἐστιν, the phrase πολλά ἐστιν
would mean „many things are true”. A certain subtlety included in the text can
support this approach of Zeno. It is about using the concept of necessity. After the
analysis of the argument’s structure, it turned out that the logical structure of both
propositions that constitute the paradox was the same. These are conditional propositions in the form: if P, then Q. Formally speaking, the argument was conducted
in accordance to the scheme: if P, then R and if P, then not-R, thus not-P.
The second argument to be analysed was B 1 of Diels’s work [Zeno, 19, B 1,
Diels; In Aristotelis Physicorum, 140.34], which, similarly to the first argument, has
two parts. In the first part, the verb „is” can be understood in two ways: in the first
place, as a second-order veritative expression, which would translate the premise
of the argument in the following way: „if it is so that it is necessary…” (the second
part of condition follows); in the second place, as a phrase in which the subject of
the word „is” has not been stated explicitly just like it happens in fragment B 2.3.
Thus, the translation would assume an essential and identical form: „if it is, then
it is necessarily...” in which the phrase constitutes simultaneously the antecedent
of the conditional proposition, thus the premise of the argument. Historians of
philosophy usually interpret the text in an existential way: „if it exists, then it is
necessarily...”. Unlike the veritative interpretation, essential and existential propositions will have the same logical structure. A significant result of my research
was to observe that in the text, there are the so-called markers of truthfulness that
induce to choose the veritative interpretation. They include the words εἰπεῖν and
λέγειν. The veritative interpretation is different from other types of interpretation,
above all, at the level of formality. The functor „if..., then...” has different arguments
in the antecedent and the consequent position in both translations, thus it contains
a broader proof assumption.
The third argument, which is included B 2 in Diels’s work [Zeno, 19, B 2, Diels;
In Aristotelis Physicorum, 139.9], is aimed at substantiating the conviction that
complex things do not increase or decrease, which contradicts empirical experience. The paradox contains, above all, premises that constitute the result of a priori
cognition (for example, a complex thing can be divided infinitely, hence infinitely
small parts exist). It departs in its reasoning from connecting the concept of size
with only something that is concrete and measurable in the world. Thus, the idea
of infinitely small size, which does not have a measurable value, appears. It constitutes the result of infinite divisibility. If we divide something ad infinitum, we receive infinitely small parts. It seems that this idea would be later used in Eudoxus’s
mathematical method of exhaustion. This was a method of measuring a geometric
figure that consisted in extracting from it any part, bigger than the half of the figure,
whose size was known. The step was repeated with other parts of the figure, extracting every time more than a half of what was left. The infinite number of these
parts equals the area of the figure.
The fourth paradox of plurality is the reasoning which Plato included in the
Parmenides – this is called the argument by similarity and dissimilarity [Plato,
Parmenides, 127e]. On the basis of comparative analysis of Plato’s texts with fragments from Simplicius’s work, I substantiate in the monograph that the argument
is original and that Plato presumably did not change it significantly. In comparison
to other paradoxes, this one is different because the concepts used in the paradoxes
handed over by Simplicius (related to plurality) and Aristotle (related to motion)
are measurable and refer to the category of quantity, whereas those used in Plato’s
reasoning cannot be described quantitatively (at least it was not possible in antiquity).
The fourth paradox that I discuss is called the paradox of Place [Aristotle,
Physics, 209b; In Aristotelis Physicorum, 562.3–6]. The purpose of substantiation
is a conviction that space cannot be a being or a being cannot be located in space.
It can be understood as some form of a modern question whether a set of all sets
is possible to exist, or, historically speaking, as a polemic with Melissus about the
nature of being.
The Grain of Millet [Physics, 250a; In Aristotelis Physicorum, 1108.18], the sixth
paradox described in the book, can be understood in two ways. It is usually thought
that Zeno’s intention is to reject plurality by stating that if a hundredth part of
millet does not make a sound and a grain, which is the measurement of a sack of
millet, does not make any sound either, then the sack itself will not make a sound.
The lack of sound induces us to accept the conviction that there is no plurality that
is called the sack. The Greek text, however, does not speak about the sack that does
not make a sound (to which the translators often pay attention), but literally, it is
about the fact that when a hundredth part of millet is falling on the ground, the air
is not moving (μὴ κινεῖν τὸν ἀέρα). The argumentation is opposed to the possibility
of motion, but from a different perspective that it is presented in other paradoxes
of motion. In the Grain of Millet, it is the cause of motion, not the path, which an
object has to travel, that is divided. If as a result of the division the part of the grain
is so small that it does not have a mass, then it will not disturb anything. The paradox also tends to be interpreted as an argument undermining the value of sensory
cognition and its inconsistency with intellectual cognition.
Another class of arguments that I explore and analyse after the paradoxes of
plurality are the paradoxes of motion. They are mainly presented by Aristotle in the
Physics [239b], which also contains his own commentaries. The first paradox is the
Dichotomy, which consists in the endless division of a stretch of path. In order to
break Zeno’s argumentation, Aristotle develops the theory of potential and actual
infinity. He resolves the paradox using a paraphrase, in which the concept of time
and path is employed. At the same time, he acknowledges, however, that if we read
the text literally, taking into consideration only the path, the difficulties will remain.
The second argument against motion that I discuss in the book is Achilles and the
Tortoise, which was constructed according to a similar idea related to the infinite
division of path. Aristotle, however, expounded it in a perfunctory and incomplete
way. In the monograph, I paraphrased and supplemented the argument, coming
to believe that it was conducted in accordance with the scheme modus ponendo
ponens. It indicates that the argumentation which Aristotle presented was probably
reformulated by the philosopher himself. The arguments: Dichotomy and Achilles
and the Tortoise complement one another. The Achilles shows that a runner cannot
reach a certain place even if he has started his race, whereas the Dichotomy states
that he cannot even start running.
The third paradox of motion is the Arrow. Unlike Achilles and the Tortoise
and the Dichotomy, it considers time dependence, indicating that the flying arrow
is always in a now, which is related to space. Being in a now means being at rest.
Therefore, rest is being motionless in space. I also pay attention to the fact that
contrary to other Zeno’s reasonings, the concept of infinity does not play a significant role in argumentation. The author refers to, above all, the concept of finiteness
and indivisibility of periods of time. Following the concept of non-complexity and
simplicity, it results that change is impossible if it is located in non-complex and
simple points. The last argument that I discuss is the Stadium. It causes a number
of difficulties that result from an ambiguous message of Aristotle. A very detailed
analysis of interpretation problems can be found in Ross’s work. In the monograph,
I present his analysis and interpretation of this paradox, which is different from
a traditional interpretation based on Alexander’s scheme.
I try to analyse these paradoxes in an exhaustive and complementary way,
examining every time the syntactic structure of argumentation and the semantic form. Defending Parmenides, Zeno stands out against plurality and change,
indirectly substantiating that from the empirical perspective, objections against
the attributes of non-complexity and immutability of being lead to even greater
difficulties – falling into contradiction as a result of acknowledging plurality and
change. The majority of paradoxes has the form of inference based on the modus
tollendo tollens, sometimes called dialectic, which laid foundations of the future
procedure of indirect proof. Paradoxes use the concept of infinity, which is an antinomic concept. In his paradoxes, Zeno uses a certain general scheme of reasoning,
which consists in making an assumption that in order to be a specific part (of the
body, length, path, time, place, quantity), something must have the definite whole.
The infinite division (of the body, length, path, time, place, quantity) indicates,
however, that such a whole does not exist in the world. If it does not exist, then its
parts do not exist either. All arguments are supposed to prove that cognition based
on empirical experience and the use of principles that are typical of knowledge,
leads to insurmountable difficulties.
The third chapter of the monograph concerns epistemological and ontological
assumptions made by Zeno. I make an attempt to reconstruct his philosophical
opinions on the basis of works by later writers, for whom the content of the paradoxes played the most vital role. I divide the chapter into three subsequent paragraphs, which are focused on the dialectic, the ontology of paradoxes and the theory of cognition. Zeno’s dialectic is the reductio ad impossibile (διά τοῦ ἀδυνάτου),
based on the rule that is today called modus tollendo tollens. In the part devoted
to the ontology, I try to reconstruct Zeno’s opinions on the subject of the unity of
being and the structure of time and space. In the light of infinite divisibility, we
can think, on the one hand, that the structure is continuous because between any
two points on the time axis or segment of space one could indicate a point that is
located in between them. The majority of paradoxes is constructed in such a way
that a certain whole exists which is divided endlessly. Since things (solids in space)
with size and thickness, as well as segments that constitute a model representation
of path in space can be divided endlessly, their continuous structure should be
recognised. Zeno’s arguments, however, also include the idea of discrete structure
(like in the paradox of Achilles and the Tortoise).
One could not say much about Zeno’s theory of cognition. In the monograph,
I assume, above all, that he was under Parmenides’s strong influence. I present him
as a moderate apriorist, who perceived, above all, rational cognition as valuable,
but to some extent appreciated empirical experience (for example, he based his
arguments against plurality and motion on empirical premises). We should also
remember about Diogenes’s words. According to him, Zeno thought that in nature,
everything came into existence out of the hot and the cold, the dry and the moist
which changed from one into another and interpenetrated. People proceeded from
earth and the soul was a harmonious mixture of warmth, cold, dryness and humidity. Zeno was also a naturalist and he rather did not reject empirical experience radically; yet, he perceived it as less valuable than reason. Assumptions concerning the
boundaries of cognition should be understood differently in ancient Greece than in
the post-Cartesian times. The only question regarded the possibility of surpassing
in cognition that which was material and mutable in order to definitely and in an
adequate way capture the immutable, incorporeal and non-complex reality, which
constituted the source of immutable truth. Zeno is a philosopher whose indirect
aim was to substantiate that such cognition was possible. The immanent and corporeal world is the object of opinions, whereas transcendent being constitutes the
object of true cognition.
The paradoxes in which Zeno acted against the generally recognised truths,
such as motion and the plurality of things, persuaded philosophers from Pyrrho’s
school to admit him as a sceptic. It is unlikely, however, that he really rejected the
possibility of true cognition. He was indeed opposed to the truthfulness of opinions, but not knowledge.
The fourth chapter, entitled Two ancient tendencies of interpretation of Parmenides’s nature of being is aimed at presenting Plato’s and Aristotle’s influence on
the interpretation and understanding of Parmenides’s thought in later centuries.
I also accept the hypothesis that the way of understanding Parmenides’s ontology
and theory of cognition had a greater or lesser impact on the way paradoxes, or
more broadly the philosophy of Zeno of Elea would be interpreted. In the monograph, I substantiate that Plato interpreted Parmenides’s concept of being in an
idealistic way, whereas Aristotle understood it hylomorphically.
For Plato, Being–the One was the essence (εἶδος), being one idea (οὖσαν ἰδέαν),
the true being. Its attribute was that it was thought of as one and as always the same
in. The One was transcendent, which meant that it surpassed the world of corporeal particulars. Aristotle, on the other hand, interpreted Parmenides’s being as
a type of compound that consisted of the mutable material (which made up being)
and the form. Transformations of the mutable material would be cyclical, so in
terms of quantity, the matter did not change. Thus, this was a moderately realistic
interpretation that ascribed attributes of quantitative immutability and unity to
a certain whole composed of everything in a similar way as a whole was constituted by particulars with the difference that it did not concern immutability in
terms of quantity. This was some conception of being as the world. The theoretical
framework of Aristotle’s metaphysics made it possible to describe this conception
from the point of view of the theory of form and matter and talk about it rationally
within the scope of the conceptual framework of Metaphysics.
The last part of my monograph is an appendix in which mainly the summary of
Kahn’s work, The Verb „Be” in Ancient Greek, is presented.
