"We live in a world with a non-classical logic" (Hilary Putnam).
"Paraconsistent logic inspired a new philosophy of science and extended the field of reason."
(Newton da Costa)
"The propositions p and ¬p have opposite senses, but correspond to one and the same reality" (Wittgenstein, Tractatus 4.0621).
Conceptos Previos
Classical Logic vs. Paraconsistent Logic
In classical logic governs the principle of non-contradiction, which in propositional logic takes the form ¬p∧¬p) or its equivalent p∨¬p (principle of the excluded third party). Another way of expressing contradiction is p ↔ ¬p. If the principle of contradiction (p∧¬p) −called "Scoto's principle"− were accepted, then any proposition would be true. Indeed, one has:
p∧¬p (premisa)
p (derived from 1)
¬p (follows from 1)
p∨q (is derived from 2 and from the law of introduction of disjunction: p → p∨q
q (derives from 3, from 4 and from the law of disjunctive syllogism: (p∨q)∧¬p → q
In this case, logical theory becomes trivial: every proposition is provable, is true, and is a theorem. What is called "logical explosion" takes place: in Latin, "ex contradictione sequitur quodlibet" (from a contradiction follows anything).
A paraconsistent logic is a non-trivial logic (i.e., it does not accept the principle of contradiction), but admits contradictory propositions with which one can reason and reach conclusions.
Examples
Situations in which we need paraconsistent logic are those that occur when we encounter logical and mathematical paradoxes, scientific contradictions or Kantian antinomies. They also occur in daily life when we face ethical dilemmas, legal problems or situations where the components of a group hold contradictory positions that must be harmonized and integrated.
An example of a logical paradox is the liar's paradox: "This sentence is false", which is both true and false: if it is true, then it is false; and if it is false, then it is true.
A variant of the liar's paradox is the sentence used by Gödel to prove his incompleteness theorem: "I am indemonstrable."
Another example of logical paradox is Russell's paradox: The set R (the sets that do not belong to themselves) belongs and does not belong to itself. For if it belongs to itself, then it does not belong to itself; and if it does not belong to itself, then it belongs to itself.
An example of a mathematical paradox is the definition of infinitesimal (ε): a positive real number that is less than any real number. Infinitesimal is not a real number because it would have to be less than itself (ε < ε), which is a contradiction.
Newton and Leibniz themselves-the inventors of infinitesimal calculus-admitted that the practical methods of calculus needed a logical-theoretical justification, although they saw an obvious geometrical-intuitive justification.
The infinitesimal calculus was put on a firm foundation with the definition with the definition of limit with Bolzano, Cauchy and Weierstrass. But it is a descriptive definition, not operationally applicable for the derivative calculus.
An example of scientific contradiction is Bohr's atomic model. In this model an electron orbits the nucleus of the atom without radiating energy. However, according to Maxwell's theory (expressed in his famous equations of electromagnetism), an accelerated electron radiates electromagnetic waves, with the consequent loss of energy, and would eventually collide with the nucleus. There is acceleration because a centripetal force causes the electron velocity vector to change direction continuously (without changing its magnitude) as it rotates around the nucleus. Therefore, Bohr's theory is inconsistent. Today we know that the inconsistency is due to combining laws of classical physics with specific laws of quantum physics.
A Kantian antinomy - Kant was the first to use the term "antinomy" - is a conflict between concepts derived from experience (synthesized by the mind) and concepts derived from transcendental ideas (such as God and the soul), which have no empirical basis (they are unscientific concepts).
Weak and strong paraconsistent logic
Weak paraconsistent logic is the logic commonly used in scientific theories, theories that are continually revised, mainly because of the appearance of new facts that contradict the theory in force at any given time. Its principles are:
Classical logic is always preferable.
Apparent contradictions are always due to human error.
No true theory should contain inconsistencies.
Strong paraconsistent logic is the logic that admits that contradictions exist. Its principles, on the contrary, are:
Classical logic is not always preferable because it is a limited and restrictive logic.
Some contradictions may not be due to human error.
Some theories may be inconsistent.
Traditionally, the presence of contradictions in theories, in mathematical systems and even in discourse itself, has always been considered a symptom of error or lack of rationality. The idea of rationality requires the absence of contradiction.
For Karl Popper, consistency is an indispensable condition of any good scientific theory.
According to Peter Vickers [2013], there are no inconsistencies in scientific theories; they are only "pragmatic inconsistencies", due to the use of approximations, idealizations or abstractions to approach the truth. There is no threat to classical logic from any scientific theory. Inconsistencies in science have been exaggerated by philosophers.
Characteristics of paraconsistent logic
Paraconsistent logic vs. propositional logic.
Paraconsistent logic is a weaker, more flexible, less strict and less limiting logic than propositional logic, since it considers some contradictory propositions valid. But it is not a generalization of propositional logic; that role belongs to intensional logic.
Being a flexible logic, paraconsistent logic is a more expressive logic than propositional logic.
Principles.
Paraconsistent logic abandons at least one of the principles involved in logical explosion:
Introduction of disjunction: A → A∨B.
Disjunctive syllogism: (A∨B)∧¬A → B
Paraconsistent logic vs. polyvalent logic.
Strong paraconsistent logic is a type of polyvalent logic, specifically a trivalent logic, since it admits three possible truth values: V (true), F (false), and V and F. But not every paraconsistent logic is polyvalent, and not every polyvalent logic is paraconsistent.
Paraconsistent logic vs. intuitionistic logic.
In intuitionistic logic the expression A∨¬A may not be equivalent to V. In paraconsistent logic the expression A∧¬A may not be equivalent to F. But both logics are not dual. Intuitionistic logic is a specific logic, whereas paraconsistent logic encompasses a wide variety of logics.
Paraconsistent logic vs. fuzzy logic.
Fuzzy logic is not paraconsistent logic. It is a logic in which truth values are affected by a factor between 0 and 1.
Paraconsistent mathematics
Mathematics has traditionally been the logical science par excellence. First logic was formalized mathematically, creating mathematical logic. Later mathematical logic was integrated into mathematics itself. Today logic is an inseparable part of mathematics.
An inconsistency is considered an anathema in mathematics. But mathematics evolves thanks to contradictions. Inconsistent expressions do not challenge mathematics. On the contrary, they enrich it, for they broaden our conception of what is mathematically possible.
In mathematics, truth is closely linked to logic. Truths (theorems) are considered to be a function of axioms and rules of inference. In this sense truth is not absolute.
Therefore, mathematics is not unique. There is a plurality of possible mathematics. A well-known example is that of non-Euclidean geometries, which are internally consistent, but incompatible with each other. The different non-Euclidean geometries arise from different conceptions of the axiom of parallels, an axiom that is independent of all other axioms.
There are also alternative set theories to the commonly accepted theory, the axiomatic theory of ZFC (Zermelo-Fraenkel with the axiom of choice). Some of these set theories are incompatible with each other and others are equivalent. One cannot claim that ZFC theory is the true set theory and that the others are false. They are all legitimate theories.
The conclusion is that mathematics is not absolute, but open and plural. There are many possible mathematics, but in all of them the same logic is used: classical logic, a logic that is free of inconsistencies. But if you use a paraconsistent logic, then mathematics becomes paraconsistent.
David Hilbert proposed at the beginning of the 20th century a project that is called "Hilbert's program". According to this project, mathematics should be based on a "metamathematics", a term coined by Hilbert himself, consisting of a small set of axioms (or self-evident truths) and a few rules of inference. With them it must be possible to derive all mathematical truths. This theory must be consistent (free of contradictions) and complete (it must be able to prove or disprove any statement).
However, Gödel's incompleteness theorem proved that Hilbert's claim was impossible, that in a formal axiomatic system (including the natural numbers) there are inaccessible sentences, so that it cannot be proved whether they are true or false. Gödel's theorem also states that we cannot have an axiomatic theory that is both complete and consistent. A theory is consistent if one statement cannot be proved and the opposite cannot be proved. A theory is complete if it can be proved for every possible statement whether it is true or false.
Faced with the dilemma posed by Gödel's theorem, what is usually done is to renounce completeness, that is, to accept that mathematics is incomplete, and not to renounce consistency, since consistency is considered sacred in mathematics. But there is also the possibility of renouncing consistency: adopting a paraconsistent mathematics and accepting that there may be contradictions, which should not imply that the complete system is inconsistent. A position along these lines would provide a simpler, more flexible and generic solution to paradoxes.
There is one reason to support mathematics including paraconsistency and it is of a historical type. Mathematics has worked with inconsistent theories, but they worked on a practical level, such as Cantor's set theory and infinitesimal calculus.
A paraconsistent mathematics may or may not be complete, depending on what is held to be true and what rules of inference are used. Allowing inconsistencies in mathematics would open up many fields that were previously closed to this discipline.
Set theory
The first set theory was that created by Cantor in 1874, when studying the subject of the different types of infinities. Because of its general character, it was soon adopted as the foundation of mathematics. Years later, paradoxes began to appear, such as Russell's paradox, the Burali-Forti paradox and several others. Precisely because it gave rise to contradictions, Cantor's theory is called "naive". It seems that it was Paul Halmos who called it so in his 1940 work [Halmos, 2011].
The crisis of Cantorian set theory was the third great crisis in the history of mathematics. The first crisis was the discovery of irrational numbers by the Pythagoreans. The second crisis was the lack of theoretical foundation for differential and integral calculus. And the fourth has been Gödel's incompleteness theorem concerning formal axiomatic systems.
Cantor's theory was a non-formalized and non-axiomatic theory. It did not use formal logic, but was defined informally by natural language. It captured the intuitive essence of set by means of three principles:
The principle of abstraction, comprehension or intension. Every property determines a set: the set formed by the elements having that property.
The principle of extension. Two sets are equal if their members are the same.
The principle of hierarchy. A set is a mathematical entity, so it in turn can be part of another set.
Frege attempted to formalize Cantor's set theory as the logical foundation of arithmetic, but his purpose was in vain when Russell communicated to him the paradox that today bears his name.
For their part, Russell and Whitehead attempted to found mathematics on logic, like Frege. To avoid Russell's paradox, they developed "type theory," a theory based on hierarchies of sets to avoid the self-references that occur in paradoxes.
In the early 20th century, Ernst Zermelo devised an axiomatic system that prevented Russell's paradox. Sets were constructed hierarchically.
Zermelo's ideas were later refined by Thoralf Skolem and Abraham Fraenkel and John von Neumann, resulting in the first axiomatic set theory, known in short as ZF (Zermelo-Fraenkel) theory. Subsequently, the axiom of choice was added, resulting in the ZFC theory (ZF and choice, choice), the axiomatic set theory considered standard.
Quantum logic as paraconsistent logic
Quantum logic was proposed by Garrett Birkhoff and John von Neumann in a 1936 paper [see Bibliography] as a modified version of propositional logic. Its most prominent feature is that the distributive property is not valid, due to Heisenberg's indeterminacy principle:
p∧(q∨r) ≡ (p∧q)∨(p∧r)
Quantum logic is a paraconsistent logic because it admits the conjunction of opposite states. For example, a spin has two opposite states in superposition, although only one of the two states is manifested when an observation is made.
According to Putnam, quantum logic is the most appropriate logic for making inferences at the general level. He states this in his essay "Is Logic Empirical?" [1969]. However, the most widely accepted position today is that quantum logic provides a formalism that allows one to relate quantum quantities (momentum, position, state, etc.), but that it should not be considered a deductive system.
MENTAL and Paraconsistent Logic
The Liar's Paradox
The most commonly accepted solution to the liar's paradox is Tarski's hierarchy. To establish whether a sentence is true or false, it must be done, not from within the sentence, but from a higher level: from a metalanguage. In turn, to establish the truth of a metalanguage sentence, it must be done from the meta-metalanguage. And so on.
However, it is not necessary to turn to a metalanguage. Truth or falsity is an attribute (extrinsic predicate). Indeed, the liar's paradox can be expressed as follows: (p =: p/F);which represents the infinite fractal expression ((((p/F)/F)/F)/F...
then we have an infinite oscillating proposition between the states p/V and p/F. Assuming that the abstract transition time between the two states is zero, then the liar's sentence would be both true and false: {VF}.
The paradox of the infinitesimal
The paradox of the infinitesimal is solved by defining the infinitesimal by means of the imaginary expression ε*ε = 0), which is simpler than the concept of limit and which also has the advantage of being operational. Indeed, in the derivative of xn, which is ((x+ε)n - x< sup>n)/ε, the expressions in which εn appears are simply eliminated, with n>>>1. The calculation of the derivative is straightforward and its result is nxn−1.
Russell's paradox
Russell's paradox is expressed as follows:
(R = {⟨(C ← C∉C)⟩})
(R∈R ↔ R∉R)
This last expression is an imaginary expression and specifies a relation of logical equivalence between two expressions contradictory to each other, a contradiction that derives from the very definition of the set R. We also have the equivalence between two generic expressions:
(⟨(y ←' R∈R → x)⟩ ≡ {x y})
since the condition R∈R is satisfied and not satisfied at the same time.
The two modes of consciousness and logics
The West and the East deal differently with contradictions. The West is rationalistic and rejects contradictions. The East, on the contrary, accepts them, and the symbol of this philosophy is the yin-yang, which represents the union, integration and interconnection of opposites in a superior unity. For the Orient nothing is isolated, everything is interrelated.
East and West represent the two modes of consciousness. The East represents the synthetic, intuitive and deep consciousness. The West represents the analytical, rational and superficial consciousness.
These two modes of consciousness are reflected in two types of logics:
A superficial and analytical logic, which is the classical logic, valid for the macroscopic world, the superficial world.
A deep and synthetic logic, which is valid for the microscopic world, the deep world.
At the deep level there is only one logic, the deep logic. At the superficial level there are many possible logics, the superficial logics. Deep logic is the source and foundation of all logics. Classical logic is not the only logic that exists, nor is it the only true logic. It is one of the possible logics and the most superficial and restrictive of all.
Deep logic is a generic logic based on the following:
On the primitive "Condition".
In the conjunction of opposite concepts: true and false, rich and poor, high and low, etc.
In the possibility of using degrees of truth or concepts in general.
So rather than distinguishing between consistent and paraconsistent logics it is assumed that there is no inconsistent logic, it is better to speak of a single deep logic and different "surface logics". Paraconsistent logics can be considered intermediate logics between deep logic and surface logics.
In quantum physics traditional logic does not work. Deep logic must be applied. In this sense, deep logic is not intended to replace classical logic, but to complement it for special situations, when there is no duality. It is the same as with relativity, which has not eliminated Newtonian mechanics, but in special situations, when the speed is close to the speed of light, the theory of relativity must be applied. At the physical level, deep (quantum) physics uses a deep and synthetic logic.
The quantum world is a border world between the transcendent and the immanent, between the deep and the superficial. Actually the quantum world is archetypal because it connects both worlds.
Feynman said that the quantum world cannot be understood. It is not understood from the pure rational mode. But it is understood from the intuitive mode of consciousness.
Newton da Costa, in 1958, proposed that paraconsistent logic should be the foundation of mathematics. Two things must be said to this proposal: 1) paraconsistent logic cannot be the foundation of mathematics because its definition is ambiguous and a foundation of mathematics must necessarily be simple; 2) the logic on which mathematics must be based must be a deep or transcendental logic (as Wittgenstein said).
MENTAL and paraconsistency
MENTAL offers a solution to the problem of inconsistencies:
Naive language.
MENTAL is a naive language in the sense that its foundations are simple, clear and consistent, without considering a priori possible inconsistencies in its manifestations, inconsistencies that are expressible because the language allows it. Language has no restrictions.
Complete language.
Language is complete because every well-formed expression (those that respect syntactic rules) is part of language, it potentially exists. The limits of MENTAL are the limits of the expressible, based on the degrees of freedom (the universal semantic primitives). But completeness implies that some expressions may be inconsistent. So consistency and incompleteness are closely related. For there to be completeness there must inevitably be inconsistency. But contradictions are localized and do not lead to the inconsistency of the whole language and of all the systems that can be constructed with it.
Inconsistent expressions are not a source of problems and to be avoided; on the contrary, they are a source of reflection and awareness, for in consciousness the opposites or duals are united. Contradictions must be harmonized or reconciled.
Imaginary expressions can also be specified. Imaginary expressions play an essential role in mathematics, paradigmatic examples of which are the definitions of infinitesimal and imaginary unity.
Interpretable language.
The primitives have a single interpretation, but the names admit of many interpretations, depending on the model under consideration.
Mathematical universes.
With language one can construct different mathematical universes or possible worlds (with different laws and theorems), but all with a common root, which are the primitives. According to Cantor, the essence of mathematics lies in its complete freedom. MENTAL is the most evident example of expressive freedom. MENTAL is the Magna Carta of possible worlds.
Mathematics, at a deep level, is unique. But at the surface level, there are many possible mathematics.
The question of truth.
The traditional true and false are replaced by the concepts of existence or non-existence of an expression in abstract space.
Truth is an attribute (extrinsic predicate) that can be applied to any expression, logical or otherwise. No metalanguage is needed. Possible attributes are V (true), F (false) or {VF} (true and false). These logical values are generalizable and are expressed as logical magnitudes, i.e., affected by a factor between 0 and 1. For example,
x/(0.7*V) x/(0.3*F) x/{0.7*V 0.3*F} x/(0.7*{VF})
One could even specify imaginary expressions, where V and F were not complementary, e.g., x/{0.7*V 0.6*F}
MENTAL vs. axiomatic set theory
When the paradoxes in Cantor's original theory were discovered, especially Russell's paradox, two possible solutions were possible:
Keep the classical logic and discard (or, at least, modify) the principle of understanding, because not all properties are valid, since some are contradictory.
Keep the principle of comprehension and change the underlying logic: admit a paraconsistent logic that allows contradictory sentences in set theory. This was the position of authors such as Newton da Costa, Graham Priest and Ross Brady. The latter author defended a naive set theory in his work "Universal Logic" [2006].
The solution adopted was the first one, which gave rise to the axiomatic set theory ZFC, a consistent but complex theory. It discards the abstraction principle and replaces it with several axioms, created ad-hoc to avoid paradoxes. The collection of sets that can be formed is a hierarchy and they are constructed bottom-up. With this system, Russell's set cannot be constructed, so the paradox disappears.
From the conceptual point of view, it would have been more logical to keep the principle of comprehension and invent a new logic (or make the existing logic more flexible) so that it would assimilate these paradoxes and develop all their expressive possibilities. One cannot reject what is expressible. Contradictory expressions are not to be avoided, but assimilated because in them lies precisely consciousness, in the union of opposites.
MENTAL provides obvious advantages over the ZFC theory:
ZFC is an axiomatic theory that includes the natural numbers, so it is bounded by Gödel's incompleteness theorem. The true meaning of this theorem is that mathematics cannot be grounded in itself. It must be grounded on something higher (or deeper).
ZFC is a complex theory. But the foundation of mathematics must necessarily be simple, and not only simple, but of supreme simplicity. The complex is built from the simple. We must apply the principle of Occam's razor: among several alternative theories, we must choose the simplest, because it is the one that is closest to the truth.
ZFC is based on the concept of set. But the concept of set is only one of the dimensions or degrees of freedom of reality. The set is one of the 12 archetypes of consciousness. The concept of set is a primary archetype and it is not possible to define it without falling into similar concepts, such as group, collection, gathering, etc. Therefore, mathematics cannot be founded only on the notion of set. It needs the other primary archetypes.
It is often asserted that sets are sufficient for mathematics, since they allow us to define numbers, functions, relations, etc. This is not true, since other additional concepts are needed that have not been formally defined, such as the concepts of function, predicate, equality and relation. The only relation that is defined is that of membership of an element to a set.
ZFC uses first-order predicate logic, which is a limited, non-generic logic. MENTAL allows to express predicates of any order and not only in logical expressions, as well as quantifiers of any order.
ZFC was created primarily to avoid paradoxes. The motivation for creating MENTAL was to try to discover the underlying language that relates the ultimate foundations of reality, which are the philosophical categories, the archetypes of consciousness, the universal semantic primitives. Moreover, from the deep level, problems in general, including paradoxes, disappear. Problems arise from the superficial level, where there is duality.
ZFC views the natural numbers as sets, an idea of Frege and continued by Russell. According to von Neumann's interpretation, the natural numbers (0, 1, 2, ... ) correspond to the sequence of sets
∅,{∅},{∅,{∅}},...
where each number n is a set containing all the previous ones, such that the successor of n is
n∪{n} = {0,1,2,...,n}
From this definition we obtain the Peano axioms, which are theorems of the ZFC theory.
But this does not conform to reality. Natural number is another primary archetype, different from set, though related to it: numbers are properties of sets, but they are not sets. Natural numbers are not formally definable. They are only grasped by intuition, like sets and the rest of the primary archetypes.
The axioms of a formal system must be self-evident. But this is not the case in ZFC, especially because of the controversial axiom of choice. This axiom states that in any collection of non-empty sets, there exists another set that contains an element of each of those sets. This axiom is obviously fulfilled in finite sets, but in infinite sets it can only be fulfilled when the sets follow a pattern.
ZFC axioms are simply a more or less arbitrary selection of definitions, properties or descriptions. MENTAL is grounded in universal concepts that are necessarily present in all possible worlds.
ZFC does not provide a complete language for all mathematics. In fact, the definitions (union of sets, power set, etc.) are in natural language and describe how new sets are obtained, but without an operational type formalization. MENTAL is a formal operational and descriptive language for mathematics and for all formal sciences in general.
MENTAL and the Hilbert program
With MENTAL, Hilbert's program of foundational mathematics is possible, but with modifications:
Instead of formal axioms there are semantic axioms (the universal semantic primitives), which are dimensions or degrees of freedom. MENTAL is the metamathematics that grounds mathematics. Mathematics cannot be grounded in formal axioms because they belong to mathematics, and mathematics cannot be grounded in itself.
The system is complete. Every well-formed expression is a "theorem", but not in the sense of being true or false, but in the sense of being constructible, of being brought into existence. All well-formed expressions potentially exist.
All expressions have different interpretations, depending on the model under consideration. The interpretation of primitives is unique. What are interpretable are the names that appear in the expressions.
The language is consistent, but inconsistent and imaginary sentences can be expressed. Self-referential expressions represent fractal expressions.
In addition to grounding mathematics, it clarifies the nature of mathematics: mathematics is the study of the relationships between manifestations of primary archetypes.
MENTAL unites theory and practice. It unites the descriptive and the operative. In general, it unites the opposites or duals. It harmonizes the two modes of consciousness through universal semantic primitives.
Generalization
Inconsistencies occur not only in logic, but in all imaginary expressions, which are constructed using the condition, substitution or equivalence symbol. Some examples of imaginary expressions, besides those of Russell's paradox and the definition of infinitesimal, are:
(i*i = −1) (definition of the imaginary unit)
⟨( x→y → x=y )⟩
(a+a = 1.8*a)
Addenda
Origins of paraconsistent logic
Paraconsistent logic can be traced back to Aristotle himself, the originator of logic. Aristotle knew about paradoxes. He called them "sorites" (Greek for "heap") because the main paradox refers to when a few grains of sand become a heap. Today this paradox is formalized by degrees of truth with fuzzy logic.
In the middle of the 20th century an attempt was made to integrate contradictions into a logical theory that was broader and more flexible than traditional classical logic: paraconsistent logic. This new logic was formally and independently proposed in 1948 by the Pole Stanislaw Jaskowski, a student of Lukasiewicz, the creator of polyvalent logic, and by the Brazilian Newton da Costa in the 1950s.
Jaskowski gave three criteria that a paraconsistent logic must meet:
It must not be trivial.
It must permit practical inferences.
It must have an intuitive justification.
To illustrate these criteria, Jaskowski gave the example of a group of people who have opposing views. Some think that wealth should be distributed equally. Others think that each should have what he earns from his work. The group as a whole has a state with inconsistent information.
Da Costa is considered the father of paraconsistent logic. The term "paraconsistent logic" was suggested to da Costa by the Peruvian philosopher Francisco Miró Quesada. Other terms suggested by him were "ultraconsistent logic" and "metaconsistent logic". Da Costa chose "paraconsistent logic".
Da Costa was interested in the nature of knowledge, especially scientific knowledge, as well as the role of mathematics and logic as instruments for achieving that knowledge. He was also interested in paraconsistent mathematics (non-Euclidean geometries) and its applications to physics. According to Da Costa, scientific knowledge is quasi-true and justified.
At present there is not a single paraconsistent logic, but a great variety of them, including adaptive logics, relevance logic, and deontic logic.
Adaptive logic is a logic in which the rules of inference can change over time. The logic is dynamic. The rules of inference change depending on what has been derived at each point in time, and other sentences that had been derived no longer exist.
A logic is relevant if and only if it satisfies the following condition: "If A→B is a theorem, then A and B share a logical expression.
Deontic logic −from the Greek, "due, necessary"− is the logic of norms and normative ideas. It is a type of modal logic.
Paraconsistent logic has acquired great importance because of its numerous applications: artificial intelligence, machine learning, knowledge management, belief revision, databases, foundations of mathematics, software engineering, economics, control systems, etc.
Dialetheism
Paraconsistent logic has led to the establishment of the philosophical school of dialetheism (dialetheism in English), whose main contemporary advocate is the Australian philosopher Graham Priest. This school admits strong paraconsistency: there exist truths that are contradictory. A dialethia is a sentence such that it and its negation are both true, so it does not fulfill the principle of non-contradiction. Logical dialethia is polyvalent and paraconsistent, but the converse is not true.
Dialtheia logic is a trivalent logic: a sentence can be true (V) or false (F) or true and false (V and F) at the same time. The negation of a statement V and F is also V and F (it is a fixed point with respect to the negation). The truth tables corresponding to conjunction, disjunction and negation are the same as Lukasiewicz's trivalent logic, where the third value is interpreted as "indeterminate" or "neither true nor false".
According to Priest, the need to accept contradictions as true comes from logic (the liar's paradox), set theory (Russell's paradox), and social reality, such as legal contradictions and opposing group positions on sensitive issues. Priest has made several contributions to paraconsistent logic. Priest and da Costa are the most prominent figures in paraconsistent logic.
Paraconsistent logic and myth
Myth has a logical structure that can be qualified as paraconsistent. Indeed, the myth has the following characteristics:
Paradoxical.
In essence, myth is contradictory, inconsistent, paradoxical, pre-logical or pseudo-logical. It is "mytho-logical." It does not follow a rational and superficial logic. The logic is rather deep and intuitive. It is open to all possibilities. At any moment the unexpected can happen.
Myth does not include the principle of non-contradiction, for that would be tantamount to asserting nothing and everything would be trivial.
Imagination versus logic.
Imagination and emotion replace logic and make inconsistencies imperceptible. Imagination (as a faculty of the soul) is above the rational mind. Imagination connects with the intuitive mind (intuitions are messages from the soul).
Mysterious.
Mythical thought is mysterious because it is not conceivable by reason. Despite its mysterious character, myth has meaning because it refers to a higher, metaphysical reality. It does not refer to a superficial physical reality where normal logic rules.
Beliefs.
Myth concerns beliefs, rather than knowledge. Ordinary logical statements concern physical reality and beliefs concern metaphysical reality.
Union of opposites.
The paradoxical character of the myth is part of its profound message. It seeks to overcome and transcend dualism in order to develop intuition and the journey to the higher, the supernatural, to the soul. "The paradoxical nature of mythical stories is part of their message" [Edmund Leach, 1969].
Myth offers solutions to dichotomies or dual categories (living-dead, dream-waking, reality-fiction, truth-appearance, rational-absurd, etc.) by means of a third mediating element that connects the opposites or duals. It is in this intermediate region where the entities of the mythical universe sprout: contradictory, anomalous, supernatural beings, such as virgins-mothers and gods-men. In the union of opposites is consciousness. "The function of myths in culture is to provide a logical model to resolve a contradiction" [Claude Lévi-Strauss, 2009].
Rhetoric.
Mythical messages are redundant and rhetorical. These characteristics play an important role, since they allow us to contemplate the vagueness of the messages from different points of view and thus reinforce the essential meaning.
Cohesion.
The myth provides a worldview, a model that integrates and coheses a culture. It is a Magna Carta of beliefs, values that structure a society.
Symbolic language.
Mythical language is symbolic and metaphorical.
Experience.
A myth is not a narrative, but a profound reality that is lived and experienced. Mythic experience is halfway between two worlds without a clear boundary between them: between life and death, between the world of the dream and the world of objective reality. Opposing symbolic contents interpenetrate, as in the symbol of yin-yang.
Hidden meaning.
In the mythical world there is no such thing as chance. Everything has a meaning and obeys a purpose. The meaning of the mythical message resides in what is hidden, in what is not explicit, in what is not revealed. The message is a code intelligible only to those initiates who know how to see the deep meaning of the symbolic and metaphorical, beyond the literal.
Tautegoric image.
The mythical image −as the symbol− refers to itself, it does not represent something else. "The mythical image is tautegoric" [Cassirer, 1995].
Globality.
In the universe of myth, the whole has no parts because the part is the whole.
Freedom.
In myth there are no rules or limiting conditions. That is why there is no (rational) logic, because logic limits.
Equivalence of myths.
At a deep level, all myths are potentially the same myth.
Reproduction.
Myths are reproduced, sometimes with the same details, in various regions and cultures of the world.
