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The Solution to Paradoxes
 THE SOLUTION
TO PARADOXES

"What they [the Greeks] call paradoxes, we call 'things that astonish'" (Cicero).

"On the back of all paradoxes one rides towards all truths" (Nietzsche).

"From paradox comes wisdom" (Masahiro Mori).

"All paradoxes can be reconciled" (The Kybalion).


The Paradoxes

The word "paradox" comes from the Greek (para and doxos), meaning "beyond belief, beyond credible, beyond common knowledge, beyond general opinion". A paradox is a statement or the result of reasoning that expresses something that defies common sense. There are many types of paradoxes: logical, mathematical, physical, linguistic, biological, economic, visual, and so on. We are especially interested in logical antinomian paradoxes and mathematical paradoxes because they are closely related to the topic of consciousness, since this type of paradoxes unite opposites.


Syntactic and semantic paradoxes

Syntactic (or formal) paradoxes are statements that are contradictory in their own structure, that is, in their syntax.

Semantic paradoxes are those that contain elements that require some interpretation. They have to do with the meaning or with the truth or falsity of what is being expressed. They have also been associated with opposite fuzzy concepts such as high-low, rich-poor, slow-fast, etc., where there is no clear boundary between the two poles.

According to Ramsey, semantic paradoxes −which he calls "epistemological− cannot be expressed with a formal logical system because they transcend logic by making use of terms such as "meaning," "define," "name," "truth," etc.

Tarski resolves the semantic paradoxes by introducing the concept of metalanguage. According to Tarski, the ordinary concept of truth is incoherent, so it must be replaced by a hierarchy of truth expressions, where at each level a language level must also be used. A simple statement such as "The leaves of trees are green" is expressed in an object language. But in order to be able to speak of truth or falsity of a statement, it is necessary to use a language that speaks of language, i.e., a metalanguage. Therefore, the above sentence should be expressed as "The statement 'The leaves of trees are green' is true", because it refers to the truth value of a statement of an object language. If this distinction is not made, it could be reasoned, for example, as follows: To avoid this error, you must use "Juliet" (in quotation marks) in the second sentence, to refer to the word and not the person.


Logical Paradoxes

Liar's Paradox

Logical paradoxes have a long history. Six centuries before Christ, legend has it that the Greek philosopher Epimenides of Crete stated in a public square, "All Cretans are liars." In the 4th century, Eubulides of Miletus expressed it in a more direct and simple way: "This sentence is false". It is the famous "liar's paradox". It is a paradox because if the sentence is true, then it is false; and if it is false, then it is true.

Logical paradoxes are usually associated with self-referential expressions or concepts. The paradigmatic example is precisely the liar's paradox, whose sentence refers to itself. It is also a paradigmatic example of semantic paradox, since it mixes language and metalanguage.


Zeno's paradox of Achilles and the tortoise. The paradox of infinity

Zeno of Elea −a disciple of Parmenides, an advocate that the sensations we obtain from the physical world are illusory and, specifically, that physical motion is pure illusion− was the originator of several paradoxes associated with the subject of infinity and which would demonstrate the impossibility of motion.

The best known of Zeno's paradoxes is the race between Achilles and the tortoise. If we assume that Achilles doubles the speed of the tortoise and d is the initial distance between the two runners, the distance Achilles travels in each step (when he reaches where the tortoise was in the previous step) is: d + d/2 + d/4 + d/8 +..., whose limit is 2d, but which requires infinite steps. Therefore, Achilles never catches up with the tortoise.


The paradox of vagueness

It is a sorites (Greek word meaning "heap or pile") type paradox:
  1. A grain of sand is not a heap, nor two grains, nor three.
  2. A million grains of sand is a heap.
  3. If we have nsand grains that do not form a heap, then neither do n+1 grains.
  4. If we have nsand grains that do form a heap, then so do n−1 grains.
Applying mathematical induction, it is shown that properties 1 and 3 imply that property 2 is not satisfied. Analogously, applying properties 2 and 4, it is shown that 3 grains is a heap, contradicting property 1. This is a semantic paradox because the meaning of "heap of sand" has not been clearly defined.


Russell's paradox

The liar's paradox is a simple, elementary paradox. A more complex paradox is the famous "Russell's paradox", a paradox he discovered when he was working on his work "The Principles of Mathematics" (1903).

This paradox was made known by Russell in a letter to Frege, who was a strong advocate of logicism, the thesis that mathematics is reducible to logic. For Frege, a set was a logical concept. At that time, Frege was working on the second volume of his work Grundgesetze der Arithmetik (The Basic Laws of Arithmetic), where he was trying to carry out his logicist project. In 1902, with the corrected proofs of the second volume already in print, he received a letter from Bertrand Russell warning him about a serious inconsistency in his logical system, which would later become known as "Russell's paradox". Frege then modified one of his axioms (the fifth), leaving a record of it in an appendix to his work.

In 1938, Stanislaw Lésniewski −Polish philosopher, mathematician and logician− showed that, despite this modification, Frege's system remained inconsistent. Russell's paradox meant the failure of logicism in Frege, although Russell, in his work Principia Mathematica, persevered in the idea of logicism, the idea that logic was the universal science and mathematics a branch of logic.

Russell's paradox is based on considering two types of sets:
  1. Those that do not belong to themselves. This is the case for most sets. For example, N (the set of natural numbers) is not a number, the set of dogs is not a dog, and so on.

  2. Those that belong to themselves. For example, the universal set U (the set of all sets) is a set, so it includes itself: U∈U. The set of abstract ideas is also an abstract idea. The set of things that are not men is also not a man.
He then defines the set R (from "Russell") consisting of all sets that do not belong to themselves. In classical notation, R={x | x∉x}. This set is contradictory. Indeed, R may or may not belong to itself:
Therefore, the logical equivalence R∉R ↔ R∈R.

Russell's paradox involved questioning the principle of set theory understanding, a principle introduced by Cantor, the originator of set theory. This principle states that a set can be defined by a property that all elements of the set satisfy. This principle is often expressed as "give me a property and I will give you a set". When the principle of understanding was invalidated, the validity of Cantor and Frege's naive set theory as a foundation of mathematics was questioned.

Russell's paradox has been discussed by philosophers and logicians ever since it became known, having been presented in a multitude of variants. One of these variants was provided by Russell himself in 1919 with the "barber's paradox": A generalized formulation of Russell's paradox is based on the concept of "impredicative property" (a term coined by Poincaré). A property is impredicative if it is not a property of itself. For example, the property "blue" is not blue, "round" is not round, "German" is not German, "long" is not long, and "monosyllabic" is not monosyllabic. A property is non-impredicative if it is a property of itself. For example, the property "polysyllabic" is polysyllabic, "written in Spanish" is written in Spanish ý "short" is short.

Now, the question arises whether the property "impredicative" is impredicative or not. Here the paradox appears. If it is, then it is not impredicative. And if it is not, then it is impredicative.

A similar approach is the Grelling-Nelson paradox (1908), based on the adjectives homological (that which can apply to itself) and heterological (that which cannot apply to itself). The question that leads to the paradox is: Is "heterological" heterological?

Frege's reply to Russell, after the latter communicated his paradox was, "Your discovery in any case is very remarkable and may perhaps lead to a great advance in logic, although it might seem undesirable at first sight" [Frege, 1980]. His estimate was even lower than it actually was, for Russell's paradox directly or indirectly produced great advances, in logic and mathematics. The paradox provoked a crisis in the foundations of mathematics during the first decades of the 20th century, but it constituted a fundamental factor towards the unification or conjunction of logic, mathematics, formal linguistics, philosophy (theory of truth, meaning and knowledge) and even theory of mind and consciousness.

The discovery of the paradoxes of set theory was one of the most profound discoveries of all time, somewhat comparable to the discovery of irrational numbers. Russell's paradox (and its variants) continues to this day to be a source of inspiration for those interested in these disciplines. Its importance in the history of thought has been such that, on the occasion of the centenary of its discovery (2001), an international conference was organized at the University of Munich [Link, 2004].


Non-circular logical paradoxes. Yablo's paradox

For a long time it was thought that all logical paradoxes had a circular structure. But in 1993, Stephen Yablo published a short paper showing a paradox that was not circular, but consisted of infinite statements [Yablo, 1993] [Uzquiano, 2011]: This series of statements is paradoxical because they are all both true and false. Indeed: This paradox is not circular because an infinite descending chain of contradictory statements is generated.


The Solution Attempts of Logical Paradoxes

In order to solve the issue of logical paradoxes in set theory, several systems were tried. The most important were: intuitionism, the axiomatization of set theory, Russell's theory of logical types, and Russell's classless theory.


Intuitionism

The first proposal to solve the problem of paradoxes came from a Dutch mathematician named Brouwer, who proposed a radical redefinition of all mathematics in order to solve the conflict. Brouwer's program was based on basing mathematics on intuitive concepts. This philosophy rejected many fundamental principles of mathematics, but on the other hand, it satisfactorily solved the problem of paradoxes. In particular, Brouwer rejected the principle of the excluded third, which states that the elements of a set either have a property or they do not. This school of thought was called "intuitionism".


The axiomatization of set theory

The aim was to establish a set of axioms that would avoid circularity. Zermelo's axiomatization of set theory eliminated the principle of comprehension, but included in exchange the axiom of separation which states that given the property P and a set C, there exists a subset of C whose elements have the property P (it can be the empty set). On this basis, Zermelo proved that there does not exist the universal set U, the set of all sets, for if we suppose that there exists U, then with the property of not belonging to itself, there would exist the subset R, which is contradictory. The separation axiom, together with six others (among them the axiom of choice and the axiom of infinity) constituted the initial axiomatics of set theory, which was later extended.


Russell's theory of logical types

The theory of logical types −or doctrine of logical types− was developed by Russell in "The Principles of Mathematics" (1903) and expanded in the article "Mathematical Logic as Based on the Theory of Types" (1908) and in Appendix B of Principia Mathematica (PM) (1910-1913), the three-volume work written jointly with Whitehead with which they intended to found mathematics by means of logic. In the 1903 work he introduced "simple type theory", and in the other two he introduced "branched type theory".

Logical type theory was intended to establish a conceptual and operational framework for avoiding logical paradoxes. Since the essential problem with logical paradoxes is that they refer (more or less explicitly) to themselves, Russell established different levels of types: The so-called "vicious circle principle" attempts to avoid extraneous loops. It states that no concept c (in particular, classes) can refer to concepts of equal or higher rank than c. According to this type theory: So much for simple type theory. Branched type theory arose when Russell realized the need to restrict the range of a bound variable in a general proposition to exclude the possibility that the proposition itself was included in the range of that variable would lead him into a vicious circle. Russell applied the process of restriction to the arguments of a propositional function in general, rather than to specific membership relations.

Branching type theory consists of two different hierarchies: that of types of classes (introduced in simple type theory) and that of orders of propositional functions. A propositional function is "something that contains a variable x and expresses a proposition as soon as a value is assigned to x". For example, "x is a man" is a propositional function and "Socrates" a possible argument, which would produce a concrete proposition ("Socrates is a man"). A class is defined as "all objects that satisfy a propositional function" (PM).

The hierarchy of orders of propositional functions is: Therefore: Simple type theory resolves logical paradoxes such as Russell's. In contrast, semantic paradoxes (such as the liar's) require the branching theory, although Russell applied the latter theory to both types of paradoxes, seeing them as examples of the same vicious circle.

The branching type theory blocks semantic paradoxes, but at a high cost: mathematical induction is not applicable. So Russell added in PM the "axiom of reducibility" to relax the scope of the vicious circle principle and thus to be able to legitimize induction and other mathematical reasoning.

The axiom of reducibility states that any propositional function (of any order) can be formally expressed by an equivalent first-order predicative function. This axiom reduces hierarchy of orders to a single level, i.e., if a predicate occurs at a certain level, it occurs already at the first level. According to Russell, the axiom of reducibility is a generalization of Leibniz's law of identity of indiscernibles (two objects having the same properties are the same object).

Russell thought of abandoning this axiom in the second edition of PM (1925). He doubted that the axiom was truly logical, which distanced him from his goal of constructing mathematics as a branch of logic, the universal science. Wittgenstein rejected, in the Tractatus, the axiom of reducibility and even set theory itself as superfluous.

Russell recognized (at the end of the aforementioned Appendix) that type theory was not the final and definitive solution to all problems concerning logical objects. Although it resolved the particular paradox he had communicated to Frege, he had not been able to find a complete solution.

Ramsey showed in 1925 that simple type theory resolved logical paradoxes. But that it was of no use (not even the branching theory) for semantic paradoxes, since they belong to a higher level: to a meta-theory. Ramsey claimed that the axiom of reducibility had no place in mathematics because it was neither self-evident nor philosophically justified, and that "what cannot be proved without it cannot be regarded as proved at all". The theory of types, in its two versions (the simple and the ramified) suffered great attacks. For some it was insufficient because it did not resolve all the logical paradoxes. For others it was too strong a theory because it limited many useful and consistent mathematical definitions, but violated the principle of the vicious circle.


A classless theory

Since the paradox Russell discovered had arisen from the concept of class, Russell advocated a "no-classes theory" that would put an end to the problem of paradoxes. He described this theory in a 1906 paper, which was never published because he withheld it to correct some paradoxical aspects he had discovered. The original paper and its revised version were finally published posthumously in 1973.

In his work "Introduction to Mathematical Philosophy" (1918) he said, "classes are logical fictions" and "classes are incomplete symbols". Russell abandoned classes and remained only with propositional functions. This meant a process of "deplatonization" of mathematics (by not considering classes as real entities), a process that had its antecedent in his "theory of descriptions".

The classless theory is also called "substitutional theory", since it is really a calculus based on the operation of substitution. In this theory there is only one kind of variable: the unrestricted variable, a variable whose rank is that of all entities whose values are all of the same type. This approach attempts to reconcile the restrictive view of type theory and the non-restrictive view, because the substitutional theory is the means of generating the type hierarchy.

Russell also failed to eliminate paradoxes with this theory, for contradictions also appeared.


Mathematical Paradoxes

There are countless mathematical paradoxes. Here we are going to refer to the most interesting and representative ones:
Specification of Paradoxes in MENTAL

The Liar's Paradox

"This sentence is false" can be expressed in MENTAL like this: The expression s represents the fractal expression Assuming the following axioms, where code>s represents a statement and T indicates truth:

⟨( (s/F)/F = s/T )⟩ // the false of the false is true

⟨( (s/F)/T = s/F )⟩ // the true of the false is false

⟨( (s/T)/F = s/F )⟩ // the false of the true is false

⟨( (s/T)/T = s/T )⟩ // the true of the true is true

then, the expression s represents the time sequence That is, there is an indefinite oscillation between the states "s is False" and "s is True". This can be interpreted, not as a contradiction, but as a dual dynamic system, an oscillating temporal system, a logical equivalence or oscillator between two contradictory expressions: (s/F ↔ s/T). Here, in this endless dynamic, the infinite appears.

If the (abstract) computational time required to switch between one state and another is assumed to be zero, then the statement s can be considered to be both true and false:

⟨( (s =: s/F) → s/{T F} )⟩

Another version of the liar's paradox is:

"The following sentence is false" "The previous sentence is false."

In MENTAL coding:

((s1 =: s2/F) (s2 =: s1/F))

The expressions s1 and s2 represent, respectively, the expressions

s1 s2/F (s1/F)/F ((s2/F)/F)/F (((s1/F)/F)/F)/F)/F ...

s2 s1/F (s2/F)/F ((s1/F)/F)/F (((s2/F)/F)/F)/F ...

If we assume the above axioms, then the expressions s1 and s2 represent, respectively, the oscillating time sequences In this case, what you get are two loops: one between "s2 is False" and "s1 is True" and the other between "s1 is False" and "s2 is True".

As in the previous case, if we assume that the (abstract) computational time required to switch between one state and another is zero, then s1 and s2 can be considered to be both True and False:

⟨( ( ((s1 =: s2/F) ∧ (s2 =: s1/F)) → {s1/{T F} s2/{T F}}} )⟩

A disguised form of the liar's paradox is the sentence "I am indemonstrable", used by Gödel in his famous incompleteness theorem.

The sentence "I am undemonstrable" can be expressed in MENTAL as follows:

(s =: s/I) // s is an undemonstrable sentence (I)

The expression s represents the fractal expression Assuming the following axioms, where s represents a sentence and D indicates provable:

⟨( (s/I)/I = s/D )⟩ // the indemonstrable of the indemonstrable is provable

⟨( (s/I)/D = s/I )⟩ // the demonstrable of the indemonstrable is indemonstrable

⟨( (s/D)/I = s/I )⟩ // the indemonstrable of the demonstrable is indemonstrable

⟨( (s/D)/D = s/D )⟩ // the demonstrable of the demonstrable is demonstrable

then, the expression s represents the oscillating time sequence (or loop) That is, an indefinite oscillation between the states "s is unprovable" and "s is provable". Again we have a logical oscillator: (s/I ↔ s/D).

If the computational time required to switch between one state and another is assumed to be zero, then s can be considered to be both provable and undemonstrable:

⟨( (s =: s/I) → s/{D I} )⟩


Russell's paradox

In MENTAL, Russell's paradox can be expressed as follows:

( R = {⟨( C ← C/conj ← C∉C )⟩} ) // set of Russell
being
⟨( C/conj =: {C↓}=C )⟩ // condition that C is a set

The paradox is expressed by the logical equivalence: (R∈R ↔ R∉R). This expression is a logical equivalence between two contradictory expressions, a "logical oscillator", as in the case of the liar's paradox, in this case between two relations: R∈R and R∉R. Assuming that the computational (or abstract) oscillation time between the two expressions is zero, we have the concurrent space-time expression (at the abstract level) {R∈R R∉R}.


The solution to Zeno's paradox of Achilles and the tortoise

The solution is based on distinguishing two levels: This paradox is detailed in the next chapter.


The solution to the paradox of vagueness

The solution is based on using fuzzy logic with a truth value equal to a real number between 0 and 1, in order to overcome the truth-false dualism, in this case, the conceptual duality between heap and non-heap. Since MENTAL handles qualitative quantities, in this case, the form is f*heap, with f being a factor between 0 and 1. We establish a function between n grains of sand and a factor f (between 0 and 1) indicating the degree to which these grains form a heap. If we consider that from n1 grains a heap begins to form, and with n2 (>n1) grains the heap is already fully formed, we have:

( f = (0 ← (n<n1) →' (1 ← (n>n2) →' (n−n1).÷(n2−n1).) )

That is, f is 0 if n<n1 and 1 if n>n2. In the case, n1≤n≤n2, f is a proportional value.

We can also establish the equivalence (f*heap ≡ heap/(f*T)), i.e., the equivalence between the degree of heap formation and the degree of truth of heap existence.


Formalization of Yablo's paradox in MENTAL

Original sentences: The expression of the paradox is:
MENTAL and the Philosophy of Paradoxes

Paradoxical expressions have traditionally been thought to be avoided because they posed problems: they reflected contradictions or challenged common sense. Paradoxes were a device used by some Greek philosophers to demonstrate the impossibility of knowing the truth through reason or to highlight the limits of reason. Although historically, paradoxes have been associated with crises in thought, in the end they produced revolutionary advances. Paradoxes not only do not constitute a problem, but offer solutions to problems, and also present many aspects of great interest:
Paradoxes as imaginary expressions

Imaginary expressions are syntactically correct expressions but express something that goes against common sense. They are atypical and strange relations between expressions. All paradoxes converge at the end in an imaginary expression.

Examples of imaginary expressions are:

(3 = 7) // an imaginary substitution expression

(∞ =: ∞+1) // definition of infinity

(i*i = −1) // definition of the imaginary unit

(ε*ε = 0) // definition of infinitesimal

⟨( (b ←' x → a) = {a b} )⟩ // an expression of the imaginary logic

⟨( c+v = c ⟩ // an expression of the imaginary algebra

One way to look at the expression (i2 = −1) is (dividing both terms by i), which is a recursive expression. Assuming an initial value of 1 for i, we obtain the time sequence 1, −1, 1, −1,.... That is, we have in this case a numerical oscillator, which can be expressed by the expression (1 = −1), an expression of union of opposite arithmetic units, reminiscent of the basic pattern of logical paradoxes. If we consider null (abstract) time between two successive values, we have that i={1 −1}, i.e., the spatio-temporal concurrence of the arithmetic unit and its opposite.

The simpler the imaginary expressions are, the greater their power and creativity. In this regard, we should mention dual numbers. Clifford invented the dual numbers D, a generalization of the real numbers, by the general expression r = r1 + ε< i>r2, where r1 and r2 are real numbers and ε defined by three possible imaginary expressions: The usefulness of many imaginary expressions in MENTAL is yet to be discovered. The field of imaginary expressions is a field that has hardly been explored, but offers enormous possibilities. Imaginary expressions can help simplify or relate expressions and link or connect different domains.

MENTAL, through the combinatorics of primitives, sets the limits of what is expressible, whether they are normal (real) or imaginary expressions. There are inexpressible entities, such as most irrational numbers. The only expressible (describable) irrational numbers are those that can be defined by recursive expressions or by appealing to infinity in a descriptive way.


Characteristics of paradoxes
The solution to paradoxes
MENTAL vs. Russell's logical type theory

Addenda

Self-referential expressions

Self-reference appears in a multitude of disciplines:
The physical paradoxes

The two great physical theories of the 20th century, the theory of relativity and quantum mechanics, are riddled with paradoxical results that defy common sense. This is because both theories are of a deep (archetypal) nature. The theory of relativity is difficult to understand. In fact, it was said that only a few people had come to understand it. Eddington, when asked by a journalist, stated that only Einstein and himself understood it. And with respect to quantum mechanics, Feynman said: "I think I can safely say that no one understands quantum mechanics".

In relativity theory, the invariance of the speed of light in vacuum (c) with respect to all inertial systems leads to paradoxes such as: In quantum mechanics the union of opposites rules because it is a world close to consciousness: But all these physical paradoxes are better understood if we consider that consciousness is the foundation of all things [Goswami, 1998]:
Gestalt images

Rubin's
glass
The gestalt images are a good metaphor for logical paradoxes. They are images that offer two ways of looking at it. Our consciousness collapses successively in one of them. It is an equivalence of opposite or complementary views in which we pass from one view to the other, as in the liar's paradox or Russell's paradox.


Bibliography