"Universal logic is the general study of logical structures" (Jean-Yves Béziau).
"Logic is not a doctrine, but a reflection of the world. Logic is transcendental" (Wittgenstein, Tractatus 6.13).
"When our minds are fixed on inference, it seems natural to take 'implication' as the fundamental primitive relation" (Bertrand Russell).
Universal Logic
Antecedents
The past 20th century has witnessed the birth of a great variety of logics or logic systems. The list is almost endless [see Problematics - Limitations of Logic]:
Propositional (or zero-order), first-order, second-order, higher-order, abstract, adaptive, algebraic, categorical, combinatorial, quantum, decision, demonstration, deontic, dialectic, fuzzy, dynamic, Horn's clause, doxastic (of believing), equational, epistemic (of knowing), erothetic (the art of asking questions), IF, infinitary, intuitionistic, free, linear, matrix, mereological (that of the whole and the parts), modal (of necessity and possibility), nonmonotonic, paraclassical, paraconsistent, polyvalent (or plurivalent or multivalued), product (or tetravalent), relevant, substructural, temporal, topological, etc.
All these logics were born not only as philosophical theories or to formalize mathematical theories but also to cover needs in other fields where semantics plays a primary role: computer science, artificial intelligence, cognitive science, etc.
Faced with this proliferation of logics, there are two opposing positions. Some think that these systems are mere formal games without meaning, since there is only one logic, the only true logic: classical logic. Others believe that these systems provide value because, depending on the domain, a different logic must be used, that classical logic is not the only one that exists and that it is insufficient.
Traditionally, logic has been considered as the theory or science of the laws of valid or correct human reasoning. Just as there are laws that govern the physical world, our reason also functions according to certain laws. Therefore these laws should be fixed, immutable and universal.
We can ask ourselves several questions on the subject of human reasoning:
Do all human beings have the same reasoning capacity?
Do we reason differently in each situation or context?
Is there an absolute way to reason?
Does reasoning evolve or is it always fixed?
Currently (and surprisingly) there is no universally accepted notion of what logic or logical system is. We can also ask ourselves questions about logic:
What exactly is logic?
Is logic the same as logical system?
Is there an essence common to all logics (or logical systems)?
What is a logical structure?
What are the fundamental laws (or fundamental principles) of logic?
What is the relationship between logic and mathematics?
Is there only one logic and the other logics are particularizations of that one logic?
Do we use the same logic in everyday life as in science?
If there is only one logic, what role does this one logic play as the foundation of mathematics?
Do different logical systems reflect the diversity of reasoning?
Are there equivalent logics (or logical systems), as there are equivalent computational models?
Are there logical levels?
What is the relationship between logic and philosophy? Has the formalization of logic distanced it from philosophy?
What is the relation of logic to truth?
What is metalogic?
Have computers changed our conception of logic?
Classical (Aristotelian) logic has been considered for centuries the "only" possible logic −and, therefore, universal−, until other logics appeared. It happened the same as with Euclidean geometry, which was considered the "only possible geometry", until non-Euclidean geometries appeared, based on different axiomatic systems.
Aristotle was the first to create a logic as a theory of reasoning: the syllogism. Considered the founder of logic, he also established three major logical principles: the principle of non-contradiction, the principle of the excluded third party and the principle of identity.
Before Aristotle, the Greeks introduced a new form of reasoning: the reductio ad absurdum, whose main representative was Zeno of Elea. It is a reasoning that takes as a hypothesis the opposite of what is considered true. If a contradiction is reached, the assumption is true.
After Aristotelian logic, new logics appeared, such as binary logic, propositional logic and first-order predicate logic. Classical logic was formalized, becoming "mathematical logic".
Universal logic
There have been various attempts to elaborate a universal logic with which all kinds of reasoning could be formalized and all kinds of logics could be specified. Many creators of particular logics considered their logics as universal, since they claimed that they were useful for all kinds of situations, including the logic used in science and in everyday life. These were the cases, for example, of Stanislaw Lesniewski (mereological logic), Jean-Yves Girard (linear logic), Jaakko Hintikka (IF logic) and Ross Brady (relevant logic).
The concept of "universal logic" was introduced in the 1990s by the French-Swiss logician Jean-Yves Béziau, although it has several antecedents, notably in the work of Tarski in the early 20th century.
For Béziau, universal logic is a general theory of the different existing logics or of the different logical structures. Therefore, universal logic is not a logic, neither particular nor universal. From the point of view of the theory of universal logic, the existence of a universal logic is not possible. Logic is not absolute or generic, it is relative or particular in each domain, which has its own laws.
The characteristics of universal logic are:
It studies the true nature of logic in general, trying to clarify its basic concepts. According to universal logic, logic is an autonomous branch of mathematics, with its own concepts and laws. Universal logic is the scientific study of rationality.
It studies the characteristics common to all logical systems. It seeks the reduction of all logics into only a few classes of logics, while preserving their diversity.
It seeks the fundamental structures or "mother structures" from which to derive all particular logical structures. It seeks to relate all logics from a higher point of view. It also seeks how one logical system can be transformed into another.
It studies the techniques used by the different logics and tries to elaborate general techniques and tools that can be applied to existing logical systems and that allow the construction of new ones.
It seeks to establish general or fundamental logical laws and theorems.
Studies the domain of validity or application of each logical system.
The first "World Congress and School on Universal Logic" took place in Montreaux (Switzerland) in 2005. In 2007 the journal "Logica Universalis" started publication.
The precedent of universal algebra
Universal algebra provides a guide to universal logic. Just as universal algebra is the general study of algebra, universal logic is the general study of logic.
While universal algebra is a recognized field, universal logic has only been recognized in recent times, most likely because of resistance to considering any other kind of logic than classical logic.
In algebra, 3 levels of abstraction are distinguished:
Specific algebra. It uses specific elements. For example, the group of integers under the operation of addition.
Abstract algebra. It abstracts the elements through the use of variables. It does not consider the nature of the objects and only considers the laws obeyed by those objects. For example, the definition of group, with a binary operation that complies with the associative, neutral element and inverse element laws.
Universal algebra. It admits any set of operations with the laws relating different operations. The laws are defined by equations.
According to the theory of universal algebra, there is no single algebra and there are no absolute laws of algebra. Similarly, according to the theory of universal logic, there is no single logic and there are no absolute laws of logic, not even the classical laws of Aristotelian logic: non-contradiction, excluded third and identity.
Following the analogy with algebra, there are 3 levels of abstraction in logic:
Specific logic. It uses specific logical elements. For example, truth values under logical operations.
Abstract logic. It abstracts logical elements by using variables. It does not consider the logical nature of objects and only considers the laws obeyed by those objects.
Universal logic. As in the case of universal algebra, universal logic is not a universal system but a universal systematization. It is a set of general concepts and techniques that make it possible to unify the treatment of different logical systems.
The relation between logic and mathematics
The relationship between logic and mathematics, reflected in mathematical logic, is ambiguous. On the one hand, we have that mathematical logic is the mathematical formalization of logic. On the other hand, mathematical logic can be considered to be the logic of mathematics. This dual aspect of mathematical logic has been reflected in the history of modern logic:
Fregean logic (by Gottlob Frege) was the attempt to reduce mathematics to logic. The Fregean approach corresponds to the logic of mathematics.
Boolean logic (by George Boole) tried to formalize mathematically (algebraically) the laws of thought. The Boolean approach corresponds to the mathematical formalization of logic.
Frege is considered the father of modern logic, for he was the originator of first-order predicate logic. He used a two-dimensional (graphical) notation, which cannot properly be considered a mathematical language.
Boole was the originator of propositional (or binary) logic, formalized as an algebra (the algebra of logic). The algebra of logic was introduced by Boole in his work "Mathematical Analysis of Logic" (1847) and further developed in his book "An Investigation of the Laws of Thought" (1854).
According to the intuitionist school, logic derives from mathematics. According to the logicist school, mathematics derives from logic. Today, logic is considered to be part of mathematics.
The universalism of Boolean logic
Boole's work was extraordinary in its simplicity and affected not only logic. It had a great impact on all orders, taking an important step towards universality:
The algebra of logic.
With the algebra of logic, Boole provided a new conception of reasoning. Reasoning in general, including demonstrations, could be implemented as an arithmetic calculus, which was in line with Leibniz's idea of the Calculus Ratiocinator.
The laws of thought.
Boole contributed to theories of mind by attempting to formalize the laws of thought. Bool attempted to show that logic has its deep foundation in operations of the mind. The laws of thought reified logical thought, linking the deep with the superficial, a key aspect of consciousness.
Mathematical language.
Boole contributed to a new conception of mathematical language, thus giving impetus to modern mathematics. The Boolean method emphasized mathematical language as an instrument and manifestation of the laws of thought. He intimately related the theory of logic and the theory of language. By studying the laws of language we are studying the laws of thought. In language the operations of the mind are manifested, and the laws of the mind are expressed in the laws of language. Language is an instrument of thought. In this sense, Boole was a proponent of a formal language for mathematics.
Abstract algebra.
Boole contributed to the development of abstract algebra. He was the first to consider that algebra concerned not only numbers, but that variables represented indeterminate mathematical objects. The mathematical approach to logic, applied by Boole, was extended to other logics as well. Boolean logic was also instrumental in the development of universal algebra.
Computer science.
Boole is considered one of the founders of computer science, as binary algebra became the language of computers.
Structuralism.
Boolean logic is one of the origins of structuralism. Boole was the forerunner of the notion of mathematical structure. The spirit of mathematics, according to Bourbaki, is the study of abstract objects that form structures: "To be is to be an element of a structure". According to the ontology of modern mathematics, mathematical objects are not isolated, but are part of structures. To know an object is to know how it is related to the other objects of the structure.
The structure of Boolean algebra is a lattice, a structure that appears in many fields of mathematics. A lattice is a set with two dual operations (+ and *) that satisfy 4 properties:
Philosophy.
Boole broke new ground in the philosophy of science. For Boole, logic was not part of philosophy, but should be a branch of mathematics. He considered science to be concerned only with laws and not with the investigation of causes.
In short, Boole's achievement was extraordinary because its simplicity implied universality.
Paraconsistent logic and universalism
Classical logic is a logic in which the Aristotelian principle of non-contradiction governs. This principle states that a sentence is either true or false, a principle that has been the basis of reasoning for over 2000 years.
But paraconsistent logic questions this principle. Graham Priest [1987] claims that the invention of paraconsistent logic is the most important event ever produced in logic. Although it has precedents in Aristotle himself, the Russian Nicolai Vasiliev and the Pole Jean Lukasiewicz are considered the main precursors of paraconsistent logic.
Vasiliev presented in 1910 (in a lecture at Kazan University) an "imaginary logic" without the principle of non-contradiction. He named it so by analogy with Lobatchevsky's imaginary (non-Euclidean) geometry. He is also considered a precursor of polyvalent and intuitionistic logics. His style was informal but conceptually rich, and he did not go so far as to construct a formal system of paraconsistent logic:
The principle of non-contradiction is empirical, physical, superficial, accidental, independent of logic. It is not a true logical foundation. Logic remains logic without this principle. On the other hand, at the mental or deep level, everything is possible, so the principle of non-contradiction can be questioned.
Logic is founded on a deeper level which he called "metalogic".
We must consider the ontological aspect of logic. By changing the ontology we obtain different imaginary logics, thus opening up the possibility of experimentation in logic.
For his part, Lukasiewicz also considered the possibility of creating a logic that violated the principle of non-contradiction, but he did not elaborate any logical system reflecting his intuitions either. In 1920 he introduced trivalent logic, which, in a sense, overflowed the limits of classical Aristotelian logic.
In classical logic, a contradiction causes anything to be deduced: every well-formed logical expression is a theorem. A trivial system is that every well-formed formula is a theorem. This feature is called the "explosion principle", formally reflected in the so-called "Scoto's law": α, ¬α ⇒ β
Paraconsistent logic rejects the explosion principle. It studies nontrivial inconsistent systems.
The great promoters of modern paraconsistent logic are the Brazilian mathematician, logician and philosopher Newton da Costa and the philosopher and logician Graham Priest.
The term "paraconsistent logic" was coined by the Peruvian philosopher and journalist Francisco Miró Quesada at the III Latin American Conference of Mathematical Logic in 1976. This term was quickly accepted and disseminated.
Paraconsistent logic is a generic or universal logic that is less restrictive than classical logic and encompasses a wide range of logics. It includes Heyting's intuitionistic logic, fuzzy logic, and other non-classical logics.
Paraconsistent logic is not restricted only to the realm of pure logic but has spread to many fields of application: artificial intelligence (automatic reasoning), robotics, cybernetics, control systems, quantum physics, etc.
Vasiliev considered himself a "logical heretic" for doubting an age-old conviction of mankind. Today this "heresy" is an essential part of modern logic. The principle of non-contradiction plays in logic what the postulate of parallels plays in Euclid's geometry.
MENTAL vs. Universal Logic
The term "universal logic" to refer to the general theory of the different existing logics is poorly chosen because it is an oxymoron. It implicitly implies that there is a universal logic: because it says that it is a logic and that it also has the qualifier of universal. To speak of universal logic makes as little sense as to speak of universal arithmetic or universal geometry. What does make sense is to speak of abstract logic, in which variables are used. Abstract arithmetic is algebra.
MENTAL's thesis.
The only thing that can be considered universal is the language that allows to express the different types of logics. The thesis of MENTAL affirms that everything expressible is expressible with MENTAL. Therefore, all logics derive from the expressive possibilities of MENTAL by applying and combining the degrees of freedom that are the universal semantic primitives. With MENTAL the underlying unity of all logics is revealed: there is only one logic and there are different logical systems, all of them elaborated with the same logical mechanisms. The universalistic approach of MENTAL brings order to the chaos of the multiplicity of existing logics, systematizes them and unifies them through a deep (or higher) approach.
Human reasoning.
All human beings have the same reasoning ability because the primary archetypes are always the same, which are absolute and unchanging. They are common to the internal (mental) world and to the external (physical) world. Another thing is their application in different situations, which give rise to different logical systems, but the logical mechanisms used are always the same.
The relationship between logic and mathematics.
Universal logic asserts that logic is an independent branch of mathematics. From MENTAL's universalist point of view it can be seen that this statement is not true. Logic cannot exist in isolation. Logic and mathematics are inseparable, they need each other. Logic needs mathematics and mathematics needs logic. Logic is a dimension of internal (mental) and external (physical) reality, represented by the primitive "Condition" (represented by the operator "→"), which is one of the universal semantic primitives and is therefore part of the foundation of mathematics. In this sense, logic is transcendental, as Wittgenstein claimed. The transcendental implies the universal. The primitive "Condition" cannot be formalized because it is an archetype of consciousness, because it belongs to the transcendental level, it is primary and inexpressible. Only particular expressions can be expressed.
The operator "→" is the only properly logical operator. The rest of the operators of the different logics are derived because they are expressible by means of the semantic primitives.
The operator "→" is an elementary consequence operator, in the sense that if in x→y, x exists, y is derived as a consequence. This operator has nothing to do with Tarski's "consequence operator". Tarski's operator is an operator formalized axiomatically without using logical operators.
The operator "→" is dual, it has two aspects:
Decision logic. As a pure, superficial or specific condition.
Logic of deduction. It is a logical, deep or generic implication that is expressed by a generic expression. In this case, the deduction is automatic.
Structures.
Except for atoms (or atomic expressions), all other expressions are structures. Structures can be pure or mixed. Pure structures are those in which only one primitive is involved, such as {a b c} or a→b. Mixed are those that include two or more primitives, such as a b→c} or a→{a b}.
According to Bourbaki, there are three fundamental structures in mathematics: algebraic, topological and order. Béziau looks for alternative "mother" structures of logical type. But there are no such mother logical structures, but the set of archetypes of consciousness with which all kinds of structures can be created.
Algebra of logic.
Boolean logic has been criticized as being more about algebra than logic. That with the algebrization of logic, logic is "diluted" in algebra and that logical concepts are hidden.
Indeed, in Boolean algebra, truth values (true, false) are represented by binary digits (1 and 0, respectively). And logical operations are implemented as arithmetic operations. For example, if x and y are binary values,
The logical conjunction is implemented as x*y.
Logical negation is implemented as 1−x.
The logical disjunction follows from the previous two by De Morgan's law and is x+y - x*y.
By means of these arithmetic operations we can verify tautologies or true expressions (the result is 1), false expressions (the result is 0) and verify laws (the result is the same algebraic expression). For example,
x∨¬x is implemented as x + (1−x) = 1. Then the law x∨¬x = 1 is satisfied.
x∧¬x is implemented as x(1−x) = 0. Then it is satisfied that x∧¬¬x = 0.
¬(x∨y) and ¬x∧¬y are implemented equal (1 - xy). Then the equality ¬(x∨y) = ¬x∧¬y (De Morgan's law) is verified.
Another way to implement logical calculations is by means of the minimum and maximum functions of binary values:
The logical conjunction is implemented as minimum(x, y).
Logical disjunction is implemented as max(x, y).
In MENTAL these dual operations can be expressed by the "Condition" operator:
⟨( min(xy) = (y ←' x<y → x) )⟩
⟨( max(xy) = (x ⟨' x<y ⟩ y) )⟩
These operations allow to generalize the digital values by any value between 0 and 1 and even to generalize the operands to any two real numbers. And the logical laws of binary logic still hold true. For example, De Morgan's dual laws:
(x∨y)' = x'∧y', then
1−max(x, y) = min(1−x, 1−y)
(x∧y)' = x'∨y', then
1−min(x, y) = max(1−x, 1−y)
You can also define a generalized operation ρ between conjunction and disjunction, between minimum and maximum:
⟨( ρ(xyf) = f*(rmax-rmin) + rmin) )⟩
where f is a value between 0 and 1,
⟨( rmin = min(x y) )⟩ ⟨( rmax = max(x y) )⟩
We must distinguish between logic (or logical sentences) and logical calculus. Logic is not reduced to algebra. Algebra only allows one to perform logical computations in certain logics, such as Boolean, polyvalent and fuzzy logic.
Dirchlet's motto is "Substitute ideas for calculus". But when an idea becomes a calculation, it is descending from the semantic level, it goes from the substance to the form.
The dual view of mathematical logic.
Neither of the two interpretations of mathematical logic (Fregean and Boolean) is correct:
Fregean logic because it ended in failure. Mathematics cannot be founded on logic.
Boolean logic because it is a "mathematization" of logic. This is not correct because logic is properly part of mathematics. It is more appropriate to say the "algebrization" of logic.
Therefore, the term "mathematical logic" has no sense in the strict sense, despite being a consolidated term since Giuseppe Peano coined it. It has historical meaning, when logic went from being a philosophical subdiscipline to being incorporated into mathematics.
Imaginary logics.
With MENTAL you can define imaginary logics based on imaginary expressions, making use of the degrees of freedom that are the primitives. For example,
(a→b = c)
⟨( x→y = x+3 )⟩
Imaginary logical expressions are substitutions in which a conditional expression appears on the left-hand side.
Imaginary logics can also be expressed (because the language allows it) by expressions such as:
x/{T F} // x is both true and false.
x/{f1*T f2*F} // x is both true with factor f1 and false with factor f2, where f1+f2 ≠ 1.
Logic vs. philosophy.
Just as mathematical logic can be interpreted in two ways, so can philosophical logic:
As the logic of philosophy. It is to apply the logical method to philosophy for its systematic study and to try to clarify its concepts. It is a bottom-up approach, since philosophy is on a higher level than logic.
As the philosophy of logic. It is trying to look for the philosophical foundations of logic. It is a top-down approach.
Universal algebra vs. universal logic.
The analogy Béziau draws between universal algebra and universal logic fails on the issue of its foundations. Universal algebra is a mathematical theory based on a relatively small number of concepts. Universal logic, on the other hand, does not have a commonly accepted conceptual basis and is scattered over several theories, each supported by mathematical structures that share a non-essentialist view of the logical phenomenon. With MENTAL, algebra and logic share the same foundations.
Logic.
Metalogy can be interpreted in two ways:
As logic of logic. An example of a metalogical expression is a rule of rules, defined by two hierarchical generic expressions.
That which is beyond logic. It would be the universal language itself.
The "contrary" operator.
The contrary operator is a universal meta-operator because it affects not only logic but also dual operations such as subtraction (+'), division (*'), and so on.
Paraconsistent logic.
Paraconsistent logic is a deep logic, a logic of mind and consciousness. It is a general logic that brings us closer to the deep, where everything is possible, where there are no limitations. Its merit has been to free us from the immovable and unquestioned assumptions of the Western tradition and to offer us greater degrees of freedom.
Paraconsistent logic is a logic fully assumed in the MENTAL paradigm. From the mental (inner) point of view, true and false can happen at the same time because true and false belong to the relationship between the physical and the mental world. This is reflected in Tarski's famous definition of truth expressed in the example
"'The snow is white' is true if and only if the snow is white."
From the mental point of view we cannot speak of true or false but of existence or not of an expression.
Classical logic is associated with the macroscopic physical world. Paraconsistent logic is linked to the mental world, where there are no limitations and everything is possible. In this sense, paraconsistent logic is superior and more general than classical logic.
There is a logic of the physical world, where space and time are considered, and a logic of the mental world. That is why, in Zeno's famous paradox, at the physical level Achilles catches up with the tortoise and at the logical level he never catches up with it.
Paraconsistent logic has been questioned by the fact that the very definition of the concept of negation already implies a clear distinction and separation from affirmation, so it must necessarily comply with the principle of non-contradiction. But one thing are the concepts and another thing are the laws that govern these concepts. The degrees of freedom are the primitives, the primary archetypes, which allow us to express imaginary relations.
The application of paraconsistent logic to quantum physics is explained because, at the deep physical level, matter behaves like the mental level, where the macroscopic logic of the physical world is broken, and where everything is possible because space and time are blurred or disappear, and non-classical features such as indeterminism, holism, non-locality, etc. appear: being in several places at the same time, having several superimposed states, performing instantaneous actions at a distance (quantum entanglement), etc.
Duality and contradiction.
Western thought has been governed by the principle of non-contradiction. It rejects contradiction because it involves undermining the foundation of science. "Acceptance of inconsistency would mean the complete collapse of science" (Popper). On the other hand, in the East, opposites are contemplated and admitted, a philosophy reflected in the symbol of yin-yang. They also believe in non-duality, the transcendence of opposites from where the totality is contemplated, without distinctions. The West represents the superficial and dual consciousness, the consciousness of the external (physical) world. The East represents the deep and unified consciousness, the consciousness of the internal (mental) world.
In the West, thought tends to be kept always alive. In the East, every thought issued automatically becomes old or dead. The real intellectual work is the continuous confrontation with the new. "In the mouth thought dies" (Wittgenstein).
Every primitive concept has its dual, which is often expressed as contrary or complementary. The union of a concept and its dual is the key to consciousness. At the logical level this union is called a contradiction.
The dual is the rational and immanent. The non-dual (or the union of opposites) is the intuitive and transcendent. Beyond duality does not govern the principle of non-contradiction, where consciousness dwells.
The propulsive force of reason is contradictions. Thanks to them, we rise to a higher level of consciousness by the dialectic triad thesis-antithesis-synthesis. Thanks to them the human being makes science and philosophy. Thanks to them we rise to intuition, which is a higher plane than the rational one.
The principle of explosion reveals that the union of opposites produces total consciousness, where everything is true. The union of opposites (as in logical paradoxes) produces consciousness. Paradox frees the mind from the limits of reason, producing a kind of mental "cramp" that broadens consciousness and increases mental energy. The middle point between the opposites is a privileged place from where the opposites are contemplated and where a third category is produced. That is why the number 3 is the symbol of consciousness.
In MENTAL, the union of opposites occurs in language, from which all dualities are contemplated and integrated. With MENTAL, everything (including logic) is clarified, resolved or simplified. The key to everything lies in language as the unifying element. The key to universality is simplicity.
Some logical results obtained with MENTAL are the following:
The consideration of truth as a qualitative magnitude: f*T, being f a value between 0 and 1, and T the quality of truth. This concept unifies many logics, including binary logic, multipurpose logic and fuzzy logic.
The solution to the problem of implication in propositional logic.
It makes no sense to assign a truth value to the logical implication (p→q) depending on the truth values of p and q because there is no explicit relation between p and q. The correct interpretation is to use existence instead of truth: If p exists, then q exists and evaluates. If p does not exist, q is not considered. [See Applications - Logic - The Implication Problem].
The formalization, in a very simple way, of modal logic. [See Applications - Modal Logic].
Addenda
Myths and paraconsistency
Myths are essentially paraconsistent or inconsistent. The "mytho-logic" is a kind of pseudo-logic where contradictory and paradoxical beings appear: incarnate gods, virgin mothers, etc. In myths there is no subordination to any rule, there is no space or time, every conceivable relationship is possible because everything is interrelated, everything is on the same plane, anything can happen and creativity is total. All myths are potentially the same myth, for they evoke a transcendental world, the world of all possibilities, the world of pure consciousness, where there are no restrictive laws.
Myths face a double opposition. On the one hand they oppose the real (all myth is fiction). On the other hand, they are opposed to the rational (every myth is contradictory or absurd). Myth and logic appear as antonyms.
The mythical image does not refer to something outside of itself; it simply refers to itself. The mythical image is tautogorical. In the universe of myth the whole has no parts because the part is the whole, where cause and effect are two aspects of the same thing, where no dividing line can be drawn between the real and the imaginary, between the true and the false, between waking and dreaming, between life and death.
Bibliography
Barnes, D.W.; Mack, J.M. An Algebraic Introduction to Mathematical Logic. Springer-Verlag, 1975.
Béziau, Jean-Yves. Logic may be simple. Logic, Congruence and Algebra. Logic and Logic Philosophy, vol. 5, pp. 129-147, 1997. Disponible online.
Béziau, Jean-Yves. 13 questions about universal logic. Bulletin of the Section of Logic, 35-2/3: 133-150, 2006. Disponible online.
Béziau, Jean-Yves (ed.). Universal Logic: an Anthology. From Paul Hertz to Dov Gabbay. Birkhäuser, 2012.
Béziau, Jean-Yves (ed.). Logica Universalis: towards a general theory of logic. Springer, 2007.
Béziau, Jean-Yves. From Paraconsistent Logic to Universal Logic. Sarites no. 12, pp. 5-32, May 2001. Disponible online.
Béziau, Jean-Yves. The future of paraconsistent logic. Disponible online.
Béziau, Jean-Yves (ed.). Studies in Universal Logic. Springer.
Birkhoff, G. Lattice Theory. AMS, 1940.
Brady, Ross. Universal Logic. CSLI (Center for Study of Language and Information) Publications, 2006.
Church, Alonzo. Introduction to Mathematical Logic. Princeto University Press, 1996.
Curry, H.B. Foundations of Mathematical Logic. McGraw Hill, 1963.
Curry, H.B. Leçons de logique algébrique. 1952.
Epstein, R.L. Classical Mathematical Logic. The Semantic Foundations of Logic. Vol 1, Princeton University Press, 2006.
Girard, Jean-Yves. On the unity of logic. Annals of Pure and Applied Logic. 59, 1993.
Koslow, A. et al (eds.). The Road to Universal Logic. Birkhäuser, 2015.
Koslow, A. A Structuralist Theory of Logic. Cambridge University Press, 1992.
