CC BY
0Логические исследования 2024. Т. 30. № 1. С. 27-40 УДК 16
Logical Investigations 2024, Vol. 30, No. 1, pp. 27-40 DOI: 10.21146/2074-1472-2024-30-1-27-40
Философия и логика
Philosophy and Logic
Murat Kelikli
On the structural properties of paradoxes: the distinction between formal language and natural language that comes with the use of the liar paradox
Murat Kelikli
Afyon Kocatepe University,
Ahmet Necdet Sezer Kampusu, Afyon Karahisar, 03200, Turkiye. E-mail: [email protected]
Abstract: This study delves into the intriguing realm of paradoxes that have long fascinated philosophers and logicians throughout history. It begins by discussing the nature and purpose of paradoxes, ranging from their role in entertainment to their capacity to reveal flaws within logical systems. This work emphasizes the challenge paradoxes pose to the completeness of systems and the subsequent development of axiomatic systems that aim to eliminate paradoxes. Rather than providing definitive solutions to paradoxes, the primary aim of this study is to defend the idea that systems containing paradoxes can coexist with completeness. The focus is on categorizing paradoxes, with special attention given to the group known as liar paradoxes, including Russell's famous variation. The text demonstrates how these paradoxes are intrinsically linked to the principle of non-contradiction and argues that the truth and contradiction of propositions are both the cause and consequence of these paradoxes, presenting a dilemma within the system. The work introduces the concept of Aristotelian Sets (A-Sets) and Empty Sets as potential solutions to these paradoxes. It explores the idea that these sets, when carefully defined, can provide a meaningful representation of individual substances and predicates without violating the principle of non-contradiction. By proposing the inclusion of non-existents within a naive set theory and introducing A-Sets, this work seeks to contribute to the ongoing discourse surrounding paradoxes and their resolution. Ultimately, this study offers a fresh perspective on the handling of paradoxes, emphasizing the importance of reevaluating the foundations of formal and natural languages in the pursuit of a more comprehensive understanding of logic and philosophy.
Keywords: paradoxes, empty set, liar paradox, aristotelian sets, non-contradiction
For citation: Kelikli M. "On the structural properties of paradoxes: the distinction between formal language and natural language that comes with the use of the liar paradox", Logicheskie Issledovaniya / LogicalInvestigations, 2024, Vol. 30, No. 1, pp. 27-40. DOI: 10.21146/20741472-2024-30-1-27-40
© Kelikli M., 2024
1. Introduction
Throughout the history of thought, paradoxes have emerged for many purposes such as entertainment, manipulation, and revealing the problems of the system. Solutions have been sought for paradoxes and new systems have been proposed to eliminate the existence of paradoxes. The expression of paradoxes in logic and the set theory based on it has troubled the completeness of the system and axiomatic systems without paradoxes have been produced.
In this study, our aim is not to provide a solution to paradoxes. My aim is to defend the belief that a system with paradoxes can be constructed in a way that does not violate completeness. It is possible to categorise paradoxes into groups according to their structure. In this paper, I will specially consider the group of liar paradoxes. This group is especially important for our purpose since it includes Russell's paradox, which led to the search for axioms of selection in set theory.
I will show why the paradoxes given in the first section can be grouped under the liar paradox group, and I will evaluate these paradoxes under some logic systems. I will show that the reason for this group of paradoxes is related to the principle of non-contradiction. Accordingly, it will be seen that the truth of a proposition and its contradiction is the cause and consequence of the paradox. This situation appears as a dilemma of the system. However, I cannot close the subject by saying that the thing that causes the paradox does not exist, because it continues to contradict in the system that does not exist. Thus, I will show that paradoxes are a problem in a system whose definition of the empty set is designed to prove the non-existent.
In the second part, I will give a new set definition using Aristotle's theory of predication and I will try to see why the liar paradox group does not pose a problem in the system with these additions. In order to create a new empty set structure to cover the non-existent, we need to define a new set. This new set definition is a completely new definition, but since I was inspired by Aristotle's theory of predicates in arriving at this definition, I will call it the Aristotelian Set. Of course, Aristotle and his followers do not have such a definition.
I am not proposing a completely new system here, but an extension of the existing set theory into a broader system. My proposal is to extend and renew the set theory on which the existing logic systems are based. Thus, the logic systems will also adapt to it. Thus, I believe that the logic systems that will be caused by a new extended set theory will produce solutions to some problems. One of these problems is paradoxes. As I hope, paradoxes will continue to exist in this extended system, but they will appear as natural structures.
In the paper I will show that many logic systems suffer from the same error due to the lack of set theory on which they are based.
2. Liar Paradoxes
Let's evaluate the Liar paradox as given by Fowler. Liar Paradox says that Epimenides the Cretan says that all the Cretans are liars,
1. but Epimenides is himself a Cretan; therefore, he is himself a liar. But if he is a liar, what he says is untrue, and consequently, the Cretans are veracious.
2. but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue.
Thus, we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful [Fowler, 1869, p. 163].
In main sentence 'says' must be as 'says truth'. Because 'says' is taken as its contradiction 'does not say', which does not bring the paradox, but 'says truth' is taken as its contradiction 'does not say truth', which is met with 'says liar'. Main sentence has two propositions, according to this we can take the terms as G: 'Cretan', F: 'Say truth', L: 'to be Liar'. So, we can give the problems as:
1. (Fx ^ Lx) A (Lx ^ ~ Fx) = Fx ^ ~ Fx,
2. (~ Fx Lx) A (~ Lx ^ Fx) = ~ Fx ^ Fx.
We found these equivalencies forms are Hypothetical Syllogisms. If we take 1 and 2 together, Fx ^ ~ Fx . This is a contradiction, but to have a contradiction is not necessarily to collapse into paradox. So, if we take 1 and 2;
Fx ^ ~ Fx = (F ^ Lx) A (Lx ^ ~ Fx) A (~ Fx ^ ~ Lx) A
(~ Lx ^ ~ Fx) =~ Fx A Fx.
We can see that the choice of L has no effect on the paradox. However, it is not possible to take L randomly to make sense. If L is 'red', our explanation is as follows:
1. (x says truth ^ x is Red) and (x is Red ^ x does not say truth);
2. (x does not say truth ^ x is not Red) and (x is not Red ^ x says truth).
Although this does not give a meaningful expression, it is sufficient to create a paradox. However, we cannot establish a meaningful main sentence. In the Cretan paradox, we can say that L is made contradictory with F and made
meaningful. In this case, if the main clause is designed as a set and evaluated with ~ Fx A Fx:
for x e G,x e F A G F . F F = {x} ,
but it must be F n ~ F = 0. Here, there is an impossible condition in set theory. Clearly, the paradox is that ~ Fx A Fx must be inconsistent, and an element F F = 0 is added to make it valid.
In this case, for an 'a' that makes Fx true, if it makes ~ Fx false; for any 'b' that makes ~ Fx true, if it makes Fx false, there seems to be no problem. However, if 'x' makes Fx and ~ Fx true, this reveals a contradiction. Then such a thing cannot make Fx true and cannot make ~ Fx true; that is, it will be false for Fx and ^ Fx •
If there is no such a Cretan, i.e. G = 0, then Vx (Gx — Fx) and Vx (Gx Fx) are co-truth. Because Vx (0x — Fx) then it is true and Vx (0x Fx) is true. That is, it is a co-truth that a Siren sings or does not sing. Because there is no existing Siren. Therefore, x, which causes the paradox, should not exist. The solution is to introduce a structure in which x does not exist.
We can see that in the Russell Paradox. It defines the set R of all sets that are not members of themselves, and,
1. if R contains itself, then R must be a set that is not a member of itself;
2. if R does not contain itself, then R is one of the sets that is not a member of itself, and is contained in R
here, F is 'contains itself' and L, i.e. ~ F is 'does not contain itself'. Thus, the element whose Fn = 0 is valid is taken as R, i.e. R e F and R F. There are more similar for liar paradox as 'I say truth that, I say untruth' or 'I am lying'. Russell thought that his paradox was of a kind with the paradox of the Liar, but Sainsbury finds this view controversial. Sainsbury said that "The Liar paradox has been of the utmost importance in theories of truth" [Sainsbury, 2009, p. 123]. Sainsbury wrote simplest version for liar paradox:
L1: L1 is false.
And this has two conditional claims:
a) If L1 is true, then it is false.
b) If L1 is false, then it is true.
Sainsbury assumes that anything that false is not true and anything that is true is not false; so (a) and (b) yield:
a') If L1 is true, then it is not true.
b') If L1 is not true, then it is true.
From A —> ~ A |= ~ A1, he found that L1 is not true and L1 is not false. So, he summarized that
G: L1 is neither true nor false.
And he says that 'Is this paradoxical? Not unless we have some independent reason to suppose that L1 is either true or false'. But similarly we can find that:
a") If L1 is not false, then it is false.
b") If L1 is false, then it is not false.
From ~ A ^ A |= A, So:
G': L1 is both true and false.
In this way, Sainsbury falls on a paradox through G and G'. However, Sains-bury tries to explain the possibility of G by giving 'neither true nor false' values of expressions such as 'question sentences, exclamation sentences'. However, by giving the values 'true and false', the effort to not fall into the paradox for the principle of non-contradiction with the G' status to be formed here reduces it to the paradox for the principle of excluded middle. If we include the fact that G and G' are true or false, wouldn't we be in paradox? This is not an escape from the paradox, but a fall to another paradox. Modal structures are close to answering us, however, if the assessment is carried out over G' and without escaping the fact that there is a proposition.
In the Kripke system, for liar paradox, we can evaluate an assessment in the direction of addressing the expression in the successor state. In wff and M = (W, R, h) let be ~ F A F ^ ±. Main sentence is w0, 1 is w1 and 2 is w2, such that w0,w1, w2 £ W.
wO
Fig. 1. Liar in the Worlds
1Consequentia Mirabilis.
As seen in Figure 1, it is concluded that2: w0 h (□Fx — ULx) A (ULx — □ ~ Fx) w0 h □[(Fx — Lx) A (Lx — Fx)] wl h (□Fx — □Lx) A (□Lx — □ ~ Fx) wl h □[(Fx — Lx) A (Lx — Fx)] w2 h (□Fx — □Lx) A (Lx Fx)] w2 h □[(Fx — Lx) A (Lx — Fx).
So,
h □(Fx — Lx) A (Lx — Fx)] h □[(- Fx — Lx) A (~ Lx — Fx)] h □(Fx Fx).
In this case, in addition to the choice of Fx and Lx, we see that the established structure is the necessarily cause of the paradox. We found that structurally, above, the paradox of the structure in which we take L as 'red' is independent of the choice of expressions. In other words, Lx =~ Fx is not the reason for the formation of the paradox. Taking this way made the paradox meaningful.
Therefore, paradox does not arise because of right and wrong. Paradox arises because of the truth of contradictory statements. So it is not a paradox to say that Fx is both true and false, the paradox is that Fx and ~ Fx are true at the same time.
Returning to the Russel paradox, it appears that this paradox came to us as a result of the examination of the sets of Cantor. The source of both Russell and Cantor paradox is said to originate from the expression 'a set is an element of itself'. To get rid of this paradox, Zermelo received Axiom of Pairing3.
Russell thinks that the occurrence of paradox is due to the fact that judgements try to make judgements about themselves. Accordingly, he wants to bring a solution by arguing that a proposition cannot be the predication of itself, so a set cannot be an element of itself. For this reason, for the axioms of selection, he rules that a set cannot be an element of itself. If the set containing itself as an element is the cause of the paradox;
(I) A1: Sentence A1 is false.
(II) A2: Sentence A2 is true. (III) A3: Sentence A3 is beautiful.
Why does a paradox occur in statement I but not in statements II and III?
2 a etatet.
3vx(x = 0 e x : y n x = 0).
However, the selection of the set x here prevents other expressions:
A Cretan says that all the Cretans are smart; L2: L2 is a good sentence
more seriously
A Cretan says that all the Cretans are honest (says truth); L3: L3 is true
and similarly. These expressions are meaningful, just like any other paradox. But they do not give us any paradox. The commentators saw paradoxes as defective expressions and aimed to establish reasonable general principles in which these statements were flawed.
Tarski started the semantic approach to reality by establishing metalanguage. However, Tarski's approach is not entirely semantic, but also includes elements of the axiomatic approach. I see that Tarski is basically trying to adapt 'axiom of pairing' to logic by establishing a meta-language. Here we evaluate the L1 expression for an object-language w0 and meta-language w1:
L1: L1 is false.
So,
L1 is existing in w0.
We have to assume that w0 does not go through the truth, only L1 in w0 depending on the object-language.
So rL1n and L1 must be in w1 and true. Now we have attributed truth value to L1. But since L1 is true, i.e. 'L1 is false' is true, so L1 does not exist in w0 and rrL1_n will be true in w2 (meta-language of w1) but L1 will not be included in w2. To better understand this, we have to look at the passage that Tarski used in Aristotle. In 1011b26-29:
To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false; but neither what is nor what is not is said to be or not to be.
Here 'to ov' means 'to be'. L1 is true in w1, meaning that L1 is loaded in w0, meaning 'w0 is L1'. But in w2, L1 is true in w1 and from here L1 is false w0. Thus, for w2, L1 will be both true (from w1) and false (from w0). Tarski seems to have exceeded paradox for a meta-language here, however paradox for
meta-meta-language reconstructs itself. Therefore, Tarski's axiomatic approach needs to be arranged for semantic theory [Tarski, 2006, p. 63]
Tarski says that false and not true are used interchangeably. Thus 'it is true that all cats are black' simply says 'all cats are black' and 'it is not true that all cats are black' simply says 'not all cats are black'. Let us try to simplify this statement; 'A is true' says 'A' and 'A is not true' says 'non-A'. So, A is true then non-A is false and A is false then non-A is true, and vice versa.
Aristotle evaluated beings ' as true', but he did not see right and wrong as an entity. Thus, it is out of the question to attribute truth to a substance. Similarly, accident, substance, force-act cannot be predicated. It can also be said that ' set' cannot be predicated in Russell and Cantor's paradoxes. Thus, there is no set of sets. Thus, in classical set theory, the sets of forces to be taken (especially on infinite sets) are problematic, but in Aristotelian sets, predications are realised smoothly.
On the other hand, we can say that such a co-truth statement cannot be realised in assertoric statements. That is, nothing makes its contradiction true, that is, something cannot belong both to the set and to the complement of this
However, we can talk about something that does not belong to both the set and its complement. This allows it to be something that does not exist (at least not defined in our universal set). Thus, we can talk about this thing not belonging to this set and not belonging to its complement. For example, we say that sirens are not human and not non-human. Even in a situation where our universal set is taken as animals, we say that apples will not be in the human set and will not be in the complement of the human set.
The biggest problem in paradoxes, then, arises from the fact that we cannot define that the set of the Cretan and other self-containing sets will not exist. Because although we say that they do not exist, they continue to exist. The definition of the empty set does not allow them not to exist.
The definition of non-contradiction is the objection that something can be both A and non-A at the same time. In paradoxes, since F F = 0, we can talk about something that is both F and non-F. In this case, we need to define such a set that its intersection with its contradiction yields nonexistence.
The problem that arises here is that the empty set is something that exists. The system is built on what exists. The inclusion of non-existents in the system leads to contradiction. For this reason, selection axioms are used to remove non-existents from the system. What we will do here will be to make arrangements in which non-existents can be taken in the naive set theory. For this, we need to introduce an additional set definition to sets.
3. Aristotelian Sets (A-Sets)
In 133a32-34 Aristotle argues that identical things have the same properties, and in 152a33-37 he states that identical things have the same accidents. However, in 152b25-29, Aristotle claims that there is a sameness for all predicates without making a distinction between accidents and attributes. In 152a33-37, Aristotle argues that identical things must share the same accidents and that things with the same accidents must be identical. Similarly, in 133a32-34, he puts forward an analogous argument based on adjectives. Then, at 152b25-29, he extends his previous explanation to its most general form.
Aristotle's principle of the indiscernibles of identicals is made explicit at 152b27-29, where it is associated with the term 'predication' [Barnes, 1977, p. 49]. Barnes argues that this term is the same one previously used by Aristotle. As a result, Barnes concludes that this law can be derived from Aristotle. In contrast, White argues that Aristotle later revises his position at 152b27-29 and expresses the concept of identity as 'A and B are identical and each is true' [White, 1971, p. 179]. Barnes also equates the claim 'z is true of x' with 'z is a property of x', which coincides with White's interpretation of Aristotle's identity. Thus, the relation between identity and predication becomes the focus of this discussion.
In this case, in accordance with Aristotle's predicate, with the opposite evaluation of being an element, 'a belongs to A'; 'A is the predicate of a'. If all predicates of 'a' are A, B, C, then a = [\A,B,C|}. For example:
A = B for A = {a,b,c} and B = {a,b,c}
A & B for A = {\X,Y,Z\} and B = {\X,Y,Z\}.
It can be seen that identity and equality do not entail each other. What is equal is identical, but the converse is not always true, i.e. if A = B then A & B. Does saying that A and B are the same mean that 'A is B'? It can be seen that the structure Aristotle defines with predicates is not on the set to which it belongs, but on the predicates.
Let us look at how the predication 'A is B' should be realised in set theory. If the phrase 'A is B' is said, is Ba, that is A £ B. In this case, it becomes B = {a,b,c,A}, which gives us Aa, that is A £ A, because of A = B. In the same way, it will be A = {a, b, c, A, B}, and B will expand to B = {a, b, c, A, B}.
In this case, we need to extend our set to infinity for all subsets. So, all sets must be infinite sets. If the expression 'A is B' is associated with A c B, that is yx (x £ A ^ x £ B), then the statement 'a student of the philosophy department is a student of the university' or 'the philosophy department is a subset of the university' holds true. However, it cannot be said that 'the
philosophy department is the university'. Thus, taking the expression 'A is B' as Vx (Ax — Bx) is meaningless. Thus, there is no question of taking predicate as an equality. So, identity notion and Aristotle's passage in Prior Analytics 24b27-30 will guide us. What I understand here is that Aristotle focused on the subject while examining the predicate. Consequently, it can be said that when something is predicated of something else, it is predicated by means of the predicate of what they are predicated. Thus, it should be expressed 'every A is B as VX (XB — Xa)'. Also, as we saw from Categories 1b10, predicates as said of subject will comply with this condition and this display will be provided for all predications.
In accordance with Aristotle's predicate, with the opposite assessment of being an element, 'a belongs to A'; 'A is the predicate of a'. If all the predicates of 'a' are A, B, C, then we will show as a = {|A, B, C|}.
Definition 1. We will refer to the set of predicates attributed to an existing being as the Aristotelian Set, abbreviated as A-Set.
Islamic philosophers interpret Aristotle's subject-predicate relationship as 'S, P exists' rather than 'S is P'. Consequently, the statement 'S is P' is understood as ' S exists and is P'. Back, however, contends that Aristotle's statement should be construed as ' S exists and P is predicated of S'. Nonetheless, if there exists a P predicated of S, then S must already exist, and this predication serves as evidence of S's existence [Back, 2000, p. 2-3]; [Farabi, 1990, p. 46]; [Ibn Sina, 2006, p. 35].
Aristotle does not discuss absolute absence but rather presents the concept of 'non-being (p] ovtoq)' in 13b16-18. When X is not a being, the statements 'X is Y' and 'X is not Y' become invalid. Consequently, nothing can be predicated to X, and we have S = Q. In set theory, an empty set is defined as 'a set that contains no elements'. However, it is important to note that this empty set is not regarded as 'non-being'; there is a structural understanding associated with it. Thus:
Vx(x e 0)
and
VY(Q e Y).
In contemporary terms, when confronted with the statement 'Socrates does not exist', we must express it as ~ (3x) (Sx). Consequently, it can be asserted that S corresponds to an empty set, as there are no elements belonging to S. The concept of the empty set is defined as having everything predicated to it, yet nothing predicated of it. This aligns with Aristotle's definition of individual
substance. In fact, Aristotle's individual substances align with this description. Hence, from Aristotle's perspective, there exists a shared representation of individual substances with 0.
Definition 2. If 3yVx ~ (xy), then y does not exist and is denoted by Q and called as A-empty Set.
Note that the properties of the empty set will not apply to Q. For the empty set, the {0} set is created and becomes 0 G {0}. This indicates that the empty set exists. However, we cannot find a set of {q} for Q, which is contrary to the definition of Q.
If the general predication is taken as Vx (Ax — Bx), the expression Vx ($x — Bx) is valid. However, we take general predication as VX (Xb — XXa), the expression VX (XB — Xq) is inconsistent. Thus, a problem that arises semantically is solved.
Vx ($x — Bx) is valid because, for Vx G 0 then x G B statement will be 0 C B. Therefore, all sets predicate to 0, ensuring that every predicate holds true for 0. For instance, if we consider the Siren, which does not exist, both propositions, 'Every Siren is mortal' and 'Every Siren is immortal', are simultaneously true. According to Aristotle, this implies the existence of sirens, and in this context, sirens become conceivable. What I gather from this is that by embracing 0 as a universal notation, it can be demonstrated that there must exist an individual substance to which every predicate is ascribed.
4. Conclusion
In this case, as we have seen, the inadequacy of the existing set theory and the logic systems developed accordingly is revealed. When Aristotelian sets, which will eliminate this deficiency, are combined with the existing structure and applied, we have seen that we have overcome the contradictory structure of paradoxes (which is contrary to the principle of non-contradiction). More importantly, it became clear why paradoxes are problematic in logic systems, even though they are found semantically. We have seen that this problem lies in the definition of the empty set.
This is because the contradictory P and ~ P cannot be true at the same time. When "non-existent", taken as an empty set, is true for P, it makes ~ P true and ensures that the contradictions are true at the same time. On the other hand, if "non-existent" is an A-empty set, P and ~ P are false at the same time and act in a meaningful and systematic way that does not cause problems with the principle of non-contradiction. Therefore, set theory needs Aristotelian sets to be added to extend itself.
Aristotle gives the principle of non-contradiction as that a thing cannot be taken as belonging to both A and non-A [Aristoteles, 1831, 1005b19-20]. Thus, it would be true that what exists is A and false that it is non-A, or true that it is non-A and false that it is A. But something that does not exist can neither belong to A nor to non-A, that is, it is false to be A and false to be non-A. What the principle of non-contradiction tells us is that it is impossible for anything that exists or does not exist to be true to be A and true to be non-A.
It would be false for something that does not exist to be A and false for it to be non-A. However, it is true that A is true and non-A is true for empty sets. In other words, when empty sets are considered as non-existent, they will reveal a contradiction, and as we have seen, they are the paradox itself. Thus, the contradiction arising from the non-existence of A and non-A is eliminated. For the new A-empty sets we have defined, they will not belong to A and non-A together by nature, which does not contradict the principle of noncontradiction. In this case, it is false for a non-existent Cretan to lie or not to lie.
Since empty sets do not denote something that does not exist, they cannot act like non-existent sets, since they denote something that actually exists but is not a set, they can belong to both A and non-A together. However, nonexistent sets such as Russell's set will be A-empty set (R = Q). Thus, the existence of these sets will not disturb the structure of the system.
However, paradoxes are not semantically defective sentences. These are a natural result of natural languages. This situation can be moved by accepting an axiomatic condition for algorithmic language, but it is problematic for human language processing and the connection between the two. Especially in today's artificial intelligence and Search engine, I can say that it will lead to troublesome results. Thus, examination of self-reference situations and determination of foundations are required within the framework of formal language.
It is the non-existents that give rise to paradoxes, and these are intended to be thrown out of the system with axioms. Thus, a formal language was created in which non-existents are not used and non-existents cannot be talked about. One of the main differences between natural language and formal language is that natural language allows paradoxes by talking about non-existents, which does not cause language problems. Formal language does not allow talking about non-existents and tries to get rid of paradoxes by excluding them, because paradoxes break the system.
References
Aristoteles, 1831 - Aristoteles, Aristotelis Opera, (I. Bekker, Ed.), apud G. Reimerum, 1831.
Barnes, 1977 - Barnes, K.T. Aristotle on Identity and Its Problems, Phronesis, 1977, Vol. 22/1, pp. 48-62.
Back, 2000 - Back, A.T., Aristotle's Theory of Predication, Brill, 2000.
Farabi, 1990 - Farabi, Farabinin Peri Hermeneias Muhtasari, (M. Turker-Kuyel, Trans.). Atatürk Kultur Merkezi Yayinlari, 1990.
Fowler, 1869 - Fowler, T., The Elements of Deductive Logic, (3rd ed.), Clarendon Press, 1869.
Ibn Sina, 2006 - Ibn Sina, Yorum Uzerine, (O. Turker, Trans.), Litera Yayincilik, 2006.
Sainsbury, 2009 - Sainsbury, R.M., Paradoxes, Cambridge University Press, 2009.
Tarski, 2006 - Tarski, A., "Truth and Proof", Scientific American, 1969, Vol. 220, June, pp. 63-77.
White, 1971 - White, N.P. "Aristotle on Sameness and Oneness", The Philosophical Review, 1971, Vol. 80/2, pp. 177-197.
М. КЕликли
О структурных свойствах парадоксов: различие между формальным языком и естественным, возникающее при использовании парадокса лжеца
Мурат Келикли
Университет Афйон Коджатепе
Ахмет Недждет Сезер Кампюс, Афйон Карахисар, 03200, Турция. E-mail: [email protected]
Аннотация: Это исследование посвящено интригующей области парадоксов, которые издавна привлекали внимание философов и логиков на протяжении всей истории. Оно начинается с обсуждения природы и назначения парадоксов, от их развлекательной роли и до способности выявлять недостатки в логических системах. В этой работе подчеркивается сложность, которую парадоксы представляют для полноты систем и последующего развития аксиоматических систем, направленных на устранение парадоксов. Вместо того, чтобы предлагать окончательные решения парадоксов, основная цель данного исследования - защитить идею о том, что системы, содержащие парадоксы, могут сосуществовать в полной мере. Работа фокусируется на классификации парадоксов, причем особое внимание уделяется группе, известной как парадоксы лжеца, включая знаменитую вариацию Рассела. В тексте демонстрируется, как эти парадоксы неразрывно связаны с принципом непротиворечия, и утверждается, что истинность и противоречивость утверждений являются одновременно причиной и следствием этих парадоксов, представляя собой дилемму внутри системы. В работе вводится концепция аристотелевых множеств (А-множеств) и пустых множеств как потенциальных решений этих парадоксов. В ней исследуется идея о том, что эти множества, при надлежащем определении, могут обеспечить осмысленное представление индивидуальных субстанций и предикатов, не нарушая принципа непротиворечия. Предлагая включить несуществующие элементы в теорию множеств и вводя А-множества, эта работа стремится внести свой вклад в продолжающийся дискурс, касающийся парадоксов и их разрешения. В конечном счете, это исследование предлагает новый взгляд на работу с парадоксами, подчеркивая важность переоценки основ формального и естественного языков в стремлении к более полному пониманию логики и философии.
Ключевые слова: парадоксы, пустое множество, парадокс лжеца, аристотелевы множества, непротиворечие
