MENTAL AS TRUTH

"Truth is indefinable" (Gottlob Frege).

"Truth is the all" (Hegel)

"Truth can be recognized by its beauty and simplicity" (Richard Feynman).

"Truth is not found in the heights, but in the depths of things" (Paul Twitchell).

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What is Truth?
Truth has been one of the fundamental subjects of reflection throughout history, mainly by theologians, philosophers, psychologists, linguists, logicians and mathematicians. It is a theme associated with the search for the essence and foundation of everything. As a concept, truth is presented as something diffuse, elusive, inapprehensible. That is why it has various definitions and interpretations, which are not necessarily mutually exclusive. But an integrative or unifying approach as knowledge, as value, as theory, as practice, as communication and as utility has not yet been achieved.

On the subject of truth, different questions arise, among them the following:

• Is truth absolute or relative? Is there only one (absolute) truth?
  
  
• Are there general truths and particular truths?
  
  
• Are there objective truths or are they all subjective?
  
  
• Are there deep truths and shallow truths?
  
  
• Are there a priori (prior to experience) and a posteriori (after experience) truths?
  
  
• Can truth be accessed? If so, how, through reason or intuition?
  
  
• Is truth the object of science? Since the characteristic features of scientific knowledge are rationality and objectivity, does truth fall into these two categories?
  
  
• Are there higher order truths? What is the truth of truth? This question is analogous to "What is the meaning of meaning?". A theory of truth, to be universal, should be applicable to itself.
  
  
• Is there a procedure, algorithm or criterion for deciding whether a statement is true? Leibniz dreamed of achieving this goal, but at a purely deductive level. He believed that a mechanism could be constructed that would generate all truths relative to a domain or context. Logical positivism advocated verificationism (at the physical level) to find out whether something was true or not.
  
  
• Is truth a concept? If it is a concept, is it primary or secondary?
  
  
• Is truth just a predicate?
  
  
• Can the predicate "truth" be defined by other terms, such that it can be substituted for its definition without loss of meaning in any context in which it appears?
  
  
• What kinds of things are true and false?
  
  
• Are there degrees of truth or just two values (true and false)?
  
  
• Is truth an abstraction or is it something concrete?
  
  
• Is truth the same as logical consequence?
  
  
• Are imaginary and possible worlds true?
  
  
• Is true equivalent to real?
  
  
• Is truth just a linguistic issue?
  
  
• What is the relationship between truth and meaning?

The Conceptions of Truth
Truth is conceived in many different ways, among them the following:

• Truth as correspondence.
  A statement is true if there is correspondence with a fact in the real world. This is Aristotle's original theory.
  
  
• Truth as deflation.
  The deflationary theory of truth holds that truth as a predicate of a statement adds nothing to that statement. For example, the statement "Madrid is the capital of Spain" has the same content as the sentence "It is true that Madrid is the capital of Spain".
  
  
• Truth as coherence.
  Truth is the coherent, the consistent, the non-contradictory. The truth of a proposition depends on the context of the set of all propositions and the relations between them.
  
  
• Truth is the absolute, the immutable, timeless.
  For Hegel, absolute truth is philosophy itself. "Truth is the whole."
  
  
• Truth is utility.
  To be true is to be useful. A proposition is true if it is useful. Utility is the essence of truth. William James, Peirce and Dewey were its main advocates, especially the former for his influential role in the spread of pragmatism. According to William James, an idea is true if it helps us to arrive at satisfactory relations with our experiences. Nietzsche identifies the true with what is suitable for the preservation of humanity.
  
  
• Truth is perfection.
  Truth is the perfect, the ideal, harmony and beauty. For example, a circle conceived as ideal and perfect, is truth. The cube, because of its perfection, is the symbol of truth. The cubic stone is the Masonic symbol of perfected man.
  
  
• Truth is the temporary and falsifiable.
  A theory is temporarily true if it satisfactorily describes a domain of reality and as long as no empirical evidence is found to refute that theory. This is Popper's falsifiability criterion. Theories must be able to be "falsified", not verified. The criterion of scientific status of a theory is its falsifiability. If a theory is falsifiable, it is scientific. Scientific and falsifiable imply each other.
  
  
• Truth is the universal.
  There is a long philosophical tradition that distinguishes between: a) universal, necessary and a priori truths; b) particular, contingent and a posteriori truths. Particular truths derive from universal truths, so that truth is not something associated with the particular but with the universal.
  
  
• Truth is possibility.
  The theory of possible worlds equates possibility with truth, for the possible is more important than the actual. The real is a particular case of the possible. The actual is associated with the material. The possible is associated with the mental, the imaginary and consciousness. It is impossible to understand this world without understanding the possible worlds.
  
  
• Truth is meaning.
  Truth resides in meaning, in the semantics of utterances.
  
  For Frege, the meaning of a proposition is its truth value.
  
  According to Quine, one cannot distinguish between the meaning of words and what is true by virtue of an objective fact. For example, the fact "cats are animals" is an analytic fact that is known simply in terms of the meaning of words, i.e., by the synthetic. Both concepts (analytic and synthetic) always go together. He also states that truths cannot be defended by definition. "Nothing is true by definition."
  
  For Jacques Derrida, truth is relative to the meaning of words. Language is not a window to the objective world, there is no direct access to the objective world because language imposes certain structures on the world. But language can be "deconstructed" by using words in new ways, creating new connections between words, etc., to show that language does not constitute a solid basis for truth.
  
  
• Truth is simplicity.
  Truth and simplicity always go hand in hand.
  
  "Truth is always expressed with the utmost simplicity" (Paul Twitchell).
  
  "You can recognize truth by its beauty and simplicity" (Richard Feynman).
  
  The simpler something is, the closer it is to the truth, and the more complex it is, the further it is from the truth. The clearest example is that of the models of Ptolemy (geocentric) and Copernicus (heliocentric). Copernicus' theory is truer than Ptolemy's because it is simpler. But it must be emphasized that both theories are valid.
  
  
• Truth is ineffable.
  Truth is something that cannot be reached or defined or expressed because it is at a deep level, superior to our capabilities. Truth transcends the real world. Truth resides in silence, stillness, the unmanifested. We only perceive its manifestations. For Frege, truth is indefinable, it is ineffable, it does not admit analysis. Pontius Pilate asked Jesus, "What is truth?" Jesus' answer was silence (John, 18:38).
  
  
• Truth is consensus.
  According to postmodern philosophers, truth is constructed by culture. Social consensus is truth. Truth is the result of a consensus (more or less explicit or implicit) of a social group regarding a set of shared common knowledge, i.e., a culture. A truth without consensus is sterile.
  
  
• Truth is the objective.
  Truth is that which is objective, that which is shared by all human beings. It does not depend on subjective beliefs or opinions. It includes objective scientific facts, scientific truths. For Kuhn and Feyerabend, there are no objective criteria of truth.
  
  
• Truth is the intuitive, the profound.
  Truth can only be intuited and can never be obtained by reason.
  
  
• Truth is consciousness.
  The ultimate approximation to truth is the archetypes of consciousness, which are common to the physical and the mental.
  
  
• Truth is abstraction.
  There is nothing concrete that is "truth".
  
  
• Truth is the plausible.
  For example, if we say that all swans are white, we are performing an induction, which is not complete because we do not verify this property in all swans. It is better to speak of plausibility of truth.
  
  
• Truth is a chimera.
  Some postmodern philosophers have made truth and its pursuit a chimera or fantasy of reason.
  
  
• Truth is revelation.
  Revelation is expressed in two directions: 1) superficially, immediately, directly, in sensations through the senses; 2) deeply, metaphysically, of the being of things, of their essence.
  
  
• Truth is common sense.
  "Common sense is the instinct of truth" (Max Jacob).
  
  
• Truth is the metaphorical.
  The metaphorical is capable of connecting all knowledge and achieving unity of differences. For postmodern philosophers all knowledge is metaphorical.
  
  
• Truth is philosophical categories.
  Philosophical categories are the supreme categories of reality.
  
  
• Truth are the archetypes.
  An archetype is a pattern from which objects, ideas or concepts are derived. In Plato's philosophy they are the ideal and general forms that are the models of all things. In Jungian psychology, archetypes are forms without content.
  
  
• Truth is a predicate.
  Truth is a predicate of a linguistic or metalinguistic type.
  
  
• Truth is unity.
  Truth is the unity of the world in its diversity. "Unity is variety and variety in unity is the supreme law of the universe" (Newton).
  
  
• Truth is security and confidence.
  True is that which provides security and confidence.
  
  
• Truth is an experience.
  "Evidence is nothing other than the "experience of truth" (Husserl).
  
  
• Truth is a paradigm.
  Truth is a way of looking at the world in general or a certain particular scientific domain, which is usually associated with the culture of a society. Therefore, it is something changeable in each cultural context and in each epoch of history.
  
  
• Truth as truthfulness.
  Truthfulness (from Latin "verus", true) indicates that there is a correspondence between what is believed, what is said and what is.
  
  
• Truth is the union or transcendence of opposites.
  This theory is that of mystical philosophers for whom the world is a unity, and where there is no separation between opposites or that truth is beyond duality. It is the union of ontology (what things are) and epistemology (what we know about them), between the mental and the physical world, between the subjective and the objective world, between the internal and the external, etc. The theory of correspondence is a particular case of the union of opposites.
  
  
• Truth is the transcendental.
  Truth is beyond superficial appearances. It is the unmanifest and the generator of all manifestations.
  
  
• Truth is the spiritual.
  Truth is the spiritual, God or divine knowledge.
  
  
• Truth is emptiness.
  According to Buddhism, truth is emptiness, where the ultimate reality of all phenomena resides.

Truth as correspondence

According to the correspondence theory, a proposition is true if and only if it corresponds to a fact in the real world. Truth is a correspondence between what is said and what is. It is the most widespread and accepted interpretation of truth. It is present in pre-Socratic philosophy and is explicit in Plato: "True is the speech that says things as they are, false is the speech that says them as they are not". Here Plato implicitly affirms that truth is a relation between language (or thought), and that language is broader (has more possibilities) than reality.

Aristotle, in his Metaphysics, states, "To deny what is or to affirm what is not, is false, whereas to affirm what is and to deny what is not, is true." Here the Stagirite extends Plato's assertion by positing four alternatives, instead of the Platonic two:

Truth value	Plato	Aristotle extension.
True	Affirming what is	Denying what is not
False	To affirm what is not	To deny what is

Russell, at the time he defended the theory of logical atomism, held that truth is a correspondence between a proposition and a fact (or state of affairs): a true proposition and the fact to which it refers share the same structure.

The first Wittgenstein (the one of the Tractatus), with his figurative theory of meaning, establishes that there is a correspondence between language and reality, like that between a painting and reality. For the second Wittgenstein (that of Philosophical Investigations), the meaning of a linguistic expression is its use, so that the meaning of truth varies according to circumstances or cultural context.

For Davidson, facts, in themselves, are true sentences. And facts are not referenced by propositions.

Austin held that there need not be a structural or formal parallel between a true proposition and the state of affairs that makes it true. It is only necessary that the semantics of the language in which the proposition is expressed correlate the proposition with the state of affairs.

The correspondence theory has been the object of criticism, among them: 1) It does not clarify in what that relation or correspondence consists between what is said and what is; 2) It is a superficial theory that only tries to link or connect the mental world (thought, language) and the physical world.


The deflationary theory of truth

The deflationary theory of truth asserts that saying a statement is true is equivalent to asserting the statement itself. For example, "'Snow is white' is true" is equivalent to "Snow is white." The predicate "is true" assigned to a sentence does not add new knowledge, it is redundant, insubstantial. The predicate "truth" does not express a property. The belief that truth is a property is an illusion caused by the fact that we have the predicate "is true" in natural language.

This theory is called "deflationist" because the concept of truth loses its value. The deflationist theory is advocated by Frege, Wittgenstein, Ramsey, Rorty, Harwich, Quine, and Strawson.

• Frege was probably the first logician to express something very close to the idea that the predicate "truth", has no value.
  
  
• Wittgenstein, in "Remarks on the foundations of Mathematics" aligns himself with deflationism by stating: "For what does it mean that a proposition is true? p is true = p. That is the answer." And in "Philosophical Investigations" he states, "What engages the concept of truth (as in a cogwheel) that is a proposition."
  
  
• For Ramsey, truth is a redundant, superfluous concept. It is a problem that concerns only language. "...there is really no separate problem of truth, but simply a linguistic muddle."
  
  
• Rorty argues that the acceptance of the truth of a theory is the result of a communal decision based on consensus and consistency with other theories previously accepted by that community, and on its usefulness of the theory in adapting to the environment and surviving. In short, it is a deflationist theory of the coherentist, pragmatist, evolutionist and consensualist type of truth.
  
  
• Harwich establishes a variant of deflationism and describes his theory as "minimalist. He expresses it thus: 'p' is true if and only if p. This scheme is the definition of the predicate 'truth' as logical equivalence.
  
  
• Quine, in "Philosophy of Logic" states that the predicate "truth" is a deinterpreting mechanism. This mechanism allows what Quine calls "semantic ascent," the idea of being able to talk about language itself instead of talking about the things to which language refers.
  
  
• For Strawson, the predicate "truth" does not express a property. And the problem of truth is the problem of the use of the term "truth" in language.
  
  Some consider Strawson the first deflationist, but all admit his influence in the creation of this current.

Truth in Logic and Mathematics
Truth in Logic

Traditional logic is based on the assignment to every proposition of a truth value: T (true) or F (false).

Classical logic has been restricted by two laws concerning T and F:

1. The law of the excluded third party. Every proposition is T or F. There is no other intermediate value.
   
   
2. The law of no−contradiction. No proposition can be both T and F.

This duality of logical values has been generalized. The most complete generalization is the one that admits infinitely many intermediate values between 0 (representing F) and 1 (representing T).

We must distinguish between propositions and sentences (or statements):

• A proposition expresses a meaning and is associated with a logical structure of knowledge.
  
  
• A sentence or statement is an "embodiment" of a proposition, a materialization or manifestation in a given language (formal or natural).

A proposition need not be known or manifested in any given language to be true.

There are sentences that are disputed as to whether they have truth value. For example, "The present king of France is bald" is false for Russell, and neither true nor false for Strawson (so it would not be a proposition).

Expressions about the future (such as "It will rain tomorrow") are disputed as to whether they have truth value or whether it is unknown.

• For Frege, logical truth is only predicated of logical relations between mathematical objects in the Platonic logical−mathematical realm, the Third World, a realm that exists independently of the human mind. Reference and truth connect language with reality. Simple linguistic expressions refer to simple elements of reality (objects, individuals, etc.) and propositions correspond to facts, which may be true or false. The reference of a proposition is its truth value.
  
  
• For Wittgenstein, a tautology −a term introduced by himself− is a necessarily true proposition, i.e., one that is true in every possible world, and whose truth is a consequence of its mere form. Only the propositions of the empirical sciences have meaning. Logic consists only of tautologies. Every proposition about ethics or metaphysics is meaningless.
  
  
• For Poincaré, logic does not provide any truth because it all boils down to pure tautologies.

Truth in mathematics

In mathematics, truth is associated with proof. A statement is true if it is deducible from a set of axioms and the rules of derivation (or inference) of a consistent formal axiomatic system. A system is consistent if one cannot deduce one thing and the opposite. In mathematical expressions, "validity" is what counts, not "truth".

For Hilbert, the prime mover of the formal axiomatic method, truth and demonstrability are the same thing. According to this author, the formal axiomatic method is the "philosopher's stone" of mathematics, the source of all mathematical truths. But Gödel proved, with his incompleteness theorem, that in formal axiomatic systems there are indemonstrable truths.

Mathematics transcends the physical world. This became clear with the appearance of alternative geometries to Euclidean geometry in the 19th century, by modifying the axiom of parallels (the famous fifth postulate), as consistent as Euclidean geometry. Euclidean geometry was supposed to be the only "true" −evident, according to Kant−, since it expressed the reality of the physical world. But infinite possible alternative geometries can be constructed that are perfectly consistent. No geometry is truer than any other, nor can the truth of any of them be affirmed. At the mathematical level they are all valid.

• Bolyai created the geometry of the acute angle or hyperbolic geometry.
  
  
• Gauss discovered alternative geometries to Euclidean geometry, which he called "anti-Euclidean geometries", "astral geometries" and finally "non-Euclidean geometries".
  
  
• Riemann created obtuse angle geometry or elliptic geometry. It had its origin in the questioning of the infinity of physical space.
  
  
• Lobachevski also created a geometry of the acute angle, independently of Bolyai. He called it "imaginary geometry".
  
  
• Einstein used Riemannian geometry for his theory of general relativity, demonstrating that Euclidean geometry is not the geometry of the physical universe.

The discovery of non-Euclidean geometries opened the field of geometry to new horizons of freedom and creativity. And with them, imaginary mathematics and mathematical relativism were born, as geometric relativism spread to other branches of mathematics:

• The non-Aristotelian logics.
  They are logics that transcend even ordinary reasoning. Aristotle's logic was accepted as true and perfectly representative of the workings of the human mind. But it was shown that this simple logic of Aristotle was only one system of reasoning among an unlimited set of possibilities. The concept of logical truth ceased to be absolute, for what is true in one logical system may be false in another.
  
  
• The non-diophantine arithmetic. They are arithmetic with different operational rules. Traditional arithmetic is often called "Diophantine arithmetic", after Diophantus, for his important contributions to this branch of mathematics (including his famous treatise on arithmetic). This arithmetic has always been considered the "only true" one. However, it is perfectly licit to posit the existence of other alternative arithmeticians that are as consistent as the classical one.

This philosophy corresponds to that of an open mathematics, dependent only on the axioms to be established.

Model theory goes beyond specific formal axiomatic systems. A model is an interpretation of a formal axiomatic system among the various possible interpretations. In model theory, the notion of truth is relative to the structure of the formal axiomatic system. A proposition is true if its structure satisfies that system at the structural level, independently of its interpretations.

This view generalizes to all formal definitions to arrive at an "imaginary mathematics". Imaginary numbers, dual numbers, surreal numbers, and hypernumbers fall into this category.

These discoveries highlighted that mathematics transcended physical reality. The physical universe is only a particular case of the mathematical universe. Mathematics is open, infinite and belongs to a realm superior to the physical, an abstract realm that is the foundation of the mental world.

• For Plato, mathematical truths are eternal truths, timeless, immutable, universal, necessary and not dependent on the minds of men.
  
  
• For Aristotle, mathematical objects belong to an intermediate world between the sensible world and the world of ideas. The foundation of mathematics is based on method, not on ontology. Demonstration is the cornerstone of the mathematical edifice. All new knowledge always derives from previous knowledge.
  
  There are two essential foundations: first principles or axiomatics and the process of demonstration or deductive logic. Aristotle shaped the axiomatic demonstrative method, but it was Euclid's Elements, the "magnum opus" of geometry, that enshrined this method as the philosopher's stone of rigor in the construction of knowledge. The Elements consist of 13 books. Book I begins with 23 definitions of terms (point, line, surface, angle, parallel lines, etc.), 5 postulates and a set of "common notions" or logical axioms (such as "the whole is greater than the part", "two things equal to a third are equal to each other", etc.). Hilbert [1996] refined Euclid's geometry in his work "Fundamentals of Geometry".
  
  
• Leibniz distinguished between truths of fact and truths of reason. The truths of fact are those referring to the external, phenomenal and sensible world, which are contingent. The truths of reason are necessary and true in all possible worlds. Mathematical truths belong to the second type.
  
  
• For John Stuart Mill, mathematics is an empirical, experimental science. Therefore, mathematical truth has the same character as the truth of the empirical sciences, i.e., subject to abstractions, generalizations, and inductions.
  
  
• For Imre Lakatos, mathematics has a quasi-−empirical character. He equates the truth of mathematical propositions with the truth of the propositions of the empirical sciences of nature. Science is incapable of attaining truth, but each new theory is capable of explaining more things than the previous one, and of predicting new facts. There are no certainties, only fallible knowledge. "We never know, we only conjecture."
  
  
• For Brower's intuitionistic school, the law of the excluded third is not a valid law for infinite domains. For example, the question, "In the decimal expansion of π, is there a digit that appears more frequently than others?" is neither true nor false. Accepting the principle of excluded third is equivalent to accepting the principle of solving all mathematical problems, and there are mathematical problems that are unsolvable. Mathematical objects are constructions of the intellect from basic intuitions, so the truth comes from the construction made in the human mind. Unknown truths that are not given by direct intuition or through a mathematical construction do not exist. Contradiction is not a property of mathematical constructions, it is not a mathematical problem but a logical problem associated with language. The absence of contradiction in a demonstration is not a sufficient guarantee for the validity of a mathematical argument. A demonstration must be an effective construction.
  
  
• For Saunders Mac Lane mathematics is not a science, for its results cannot be falsified by facts or experiments. In mathematics there are no ontological commitments. Therefore, it makes no sense to speak of mathematical truth. Mathematics must be concerned with formal rigor and logical consistency. "Mathematics has access to absolute rigor because rigor concerns form, not facts." Validity is associated with form, and truth with substance.

Formal Theories of Truth
Tarski's semantic theory of truth

Tarski's theory of truth is the most widely accepted and well-known formal theory of truth, although it is not without its critics. It is considered the successor theory of correspondentism. He disclosed it in his essay "The Semantic Conception of Truth and the Foundations of Semantics" (1930) and "The Concept of Truth in Formalized Languages" (1935).

His theory is based on the central idea that in order to define truth it is necessary to distinguish between object language and metalanguage (a language that speaks of the object language), or else a single language that distinguishes between linguistic and metalinguistic aspects. The metalanguage must contain sufficient expressive means to be able to refer to the object language, i.e. it must be at least as rich as the object language. And both languages must be formalized.

• If this distinction is not made, it could be reasoned, for example, as follows:
  
  
  Romeo loves Juliet. Juliet is a seven-letter word. Therefore, Romeo loves a seven-letter word.
  
  To avoid this error, you should put "Juliet" in quotation marks in the second sentence, to refer to the word and not the person.
  
  
• A simple statement such as "Snow is white" is expressed in object language. But in order to be able to speak of truth or falsity of that 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 'Snow is white' is true", it refers to the truth value of an object language statement and the predicate "truth" belongs to the metalanguage.
  
  
• To avoid the liar's paradox ("this sentence is false"), Tarski proposed to reduce the expressive power of language, so that in it the predicate "truth" could not be expressed because this predicate belongs to the metalanguage. But a metalanguage cannot express its own truth predicate either. This leads to an infinite hierarchy of languages L, L₁, L₂,..., where in each one a truth predicate applicable to the previous level of the hierarchy is defined. Thus, there are as many truth predicates as there are languages in the hierarchy except for the first one. An object language cannot be semantically closed.
  
  
• The definition of truth is not absolute, but relative to a language (since the same sentence can be true in one language and false in another) and has to be formulated in a metalanguage, a language that speaks of the object language.
  
  
• The metalanguage must be of a higher type and contain the object language. It must include metalinguistic expressions to be able to refer to expressions of the object language, among them, the predicate "truth" and contemplate metalinguistic variables.
  
  A metalinguistic variable represents an indeterminate element of the object language. For example, x in the open expression "x is a man". A closed expression would be "Socrates is a man". Open expressions are neither true nor false, but are satisfied or not by a set of sequences of elements. For example, the set formed by the sequences (Madrid, Spain) and (Paris, France) satisfy the open expression "x is the capital of y".
  
  
• The concept of truth is defined by the concept of satisfiability: a sentence is true if the sequence of elements corresponding to the metalinguistic variables satisfies the open expression, and false if it does not. In this definition of truth, a sequence may contain subsequences.
  
  Satisfiability expressions can also contain quantifiers. For example, "There exists a city x such that it is the capital of y" is satisfied by (Madrid, Spain). And "All cities of x have more than y million inhabitants" is satisfied by (Madrid, Barcelona), 1).

Tarski proved that the concept of truth is indefinable in a formalized language: "A language strong enough to express arithmetic and in which classical logic is valid is inconsistent, because in such a language the liar can be formalized" ('this sentence is false')". It is a theorem that he included in his 1935 paper. Tarski was the first to show clearly that there could never be a formal definition of the predicate "truth" in a language, because its definition leads to contradictions.

Tarski's theory is a semantic theory of truth. It relates the syntactic level (purely formal) and the epistemological level (of interpretation). And it is a correspondent theory between language and reality, which follows Aristotle's criterion.

Criticism:

• The definition of truth depends on set theory. This dependence is termed by Hintikka as "Tarski's curse".
  
  
• In natural language we have a single veritative predicate ("true", T) and its opposite ("false", F). Tarski proposes, instead, an infinity of veritative predicates: T, T₁, T₂, T₃, . .. associated to each metalanguage level. What is desirable is uniqueness of truth and not diversity.
  
  
• It has no philosophical foundations, although Tarski believed that his theory was a contribution to the philosophical problem of truth. But he later claimed that his conception was "philosophically neutral."

Kripke: truth as a "fixed point"

Tarski's hierarchical approach was the dominant answer to the problem of semantic paradoxes. But in 1975, Kripke in his famous and influential paper "Outline of a Theory of Truth" [1975] proposed a semantic theory of truth that represented an advance over Tarsky's theory.

For Kripke, the construction of the hierarchical truth predicates of Tarsky's theory is a particular case of a more general approach: the "fixed point approach". The main points of this theory are:

• Unlike Tarski, Kripke attempts to construct a language that contains its own veritative predicate and also allows self-reference, the two things that Tarski avoided in order not to fall into logical paradoxes.
  
  
• He uses a third truth-value ("indefinite") in his theoretical apparatus to allow the treatment of paradoxes, for he holds that "pathological" sentences are those that are neither true nor false, such as the liar's sentence (its truth-value is "indefinite").
  
  
• It recursively constructs a succession of formal languages, each of which represents a stage in the progressive acquisition of the veritative predicate or truth as a partially defined property. Each language has its own veritative predicate, but it applies to only some sentences of the language.
  
  The process starts with a minimal language (initial fixed point or minimum) in which it assigns the veritative predicate to ∅ (the empty set). The process ends when it reaches a language (final fixed point) whose veritative predicate applies to all its sentences.
  
  
• A grounded sentence is one that has a veritative value at the minimum fixed point. A sentence is paradoxical if it has no truth value at any fixed point.
  
  
• There is more than one fixed point. They all serve as supporting points for the construction of the notion of "truth". If none in particular is selected, then the concept of truth admits as many possible interpretations as there are fixed points.
  
  
• If in a language there is a predicate that is interpreted as a fixed point, then that language contains its own truth predicate.

Kripke thus showed that a language can consistently contain its own truth predicate, which Tarski considered impossible, since truth belongs to the metalanguage. Kripke's method allows the definition of the predicate "truth" within the language itself, but at the cost of extending the logic to three truth values, where the laws of the excluded third and of non-contradiction are not fulfilled.


Davidson's theory of meaning

Davidson attempted to apply Tarski's definition of truth to natural languages to create a theory of meaning:

• The theory should be able to derive statements of the form "O means p", with O a sentence of natural language (object language), and p the meaning of O (p belongs to the metalanguage). p must be derived from O. p is the "translation" of O into the metalanguage.
  
  
• A sentence O is true if and only if p. For example, "'Snow is white' if and only if snow is white". p can be interpreted as "the truth conditions" of O. The meaning of a sentence is determined when its truth conditions (or circumstances) are specified. The theory of meaning is a theory of truth.
  
  
• The theory must be recursive by using a set of rules for the derivation of infinite statements.
  
  
• The theory must be complete. Every statement must have a corresponding meaning.
  
  
• Frege's principle of compositionality must govern: the meaning of a compound expression is determined by the meaning of its components and the relations between them.

The search for truth in Descartes

Descartes developed the hypothesis of the evil god (or genius) in his work "Metaphysical Meditations" in which he culminates his system of the search for truth through methodical doubt:

• Perhaps we have been created by a god who forces us to systematically deceive ourselves to make us believe that we are in the truth, when we are really in error.
  
  
• Perhaps our recognition of something as true is a consequence of our own nature. If we had a different nature, perhaps our knowledge would be different. Beings that have undergone a different evolution might have other knowledge than ours.
  
  
• Since this hypothesis is possible, we need to question the strongest knowledge. We need to question the legitimacy of propositions that seem to have the strongest evidence and question even the veracity of mathematics itself.

Descartes' aim was to investigate the possibility of finding something that is absolutely indubitable, not subject to possible deception or impossible to be deceived.


MENTAL as Truth
MENTAL provides a general framework of truth:

• Proposition and sentence.
  In MENTAL, proposition and sentence (or statement or sentence) coincide in "expression," which is a manifestation of language that has structure and meaning.
  
  
• The integration of conceptions of truth.
  All conceptions and theories of truth are partly right, for they are different aspects or visions of truth. Here we can evoke the famous story of the elephant and the blind men, in which each blind man saw only a particular aspect of the elephant, but not the elephant as a whole. MENTAL includes all aspects of truth as experience, utility, consciousness, etc.
  
  
• Truth and the two modes of consciousness. There are two modes of consciousness: 1) the deep, theoretical and intuitive; 2) the superficial, the practical and rational. To these two modes of consciousness correspond respectively two types of truth: 1) necessary, universal, intuitive and a priori, which are the primary archetypes; 2) contingent, particular, rational and a posteriori.
  
  
• Deep truth cannot be formalized, but superficial truth can.
  Meaning cannot be formalized. Semantics is ineffable, it is inexpressible. The idea of the ineffability of semantics has been upheld by several authors, most notably Tarski and Hintikka. All attempts to formalize semantics have failed because semantics belongs to the deep and cannot be explained, brought to the surface. We can only show its manifestations. And truth, which is a kind of meaning, cannot be formalized either.
  
  Just as a language cannot express its own semantics, neither can it express truth or falsity. Deep truth cannot be expressed, it cannot be explained, like life, consciousness, information and semantics. Deep truth is unmanifest. We only perceive its manifestations.
  
  
• Inner or abstract truth.
  Everything that is expressible in MENTAL is truth at the internal or abstract level. In this case, truth is equivalent to existence (in abstract space).
  
  
• Truth vs. existence.
  We can conceive of truth as correspondence and interpret the expression x/T as meaning that x exists in the real (external) world. At the internal level, truth is equivalent to existence in the environment, Therefore, at the internal level we have the meta-expressions α (existence) and θ (non-existence). And at the external level (the real world) we have T (true) and F (false).
  
  

  Inner level	Outer level
  α (existencia)	T (verdadero)
  θ (no−existencia)	F (falso)

  
  For example "The king of France is bald" exists at the internal or abstract level:
  king(France)/bald.
  If we want to express that there is no king of France in the real world:
  rey(Francia)/F.
  
  
• MENTAL as a fixed point.
  MENTAL can be considered a fixed point in the following sense. If we consider epistemology (what we know) as a function applied to ontology (what is), at the deep level ontology and epistemology coincide, i.e., we are at a fixed point. Truth is the fixed point ontology=epistemology. In general, truth resides in the union of opposites, where there is no duality.
  
  
• Truth as a qualitative magnitude.
  A qualitative magnitude is a magnitude formed by the product of a factor f (between 0 and 1) and a unit. In this case, the unit is T (truth), so the magnitude truth is f*T.
  
  The generalization of the contrary operator is based on the complementary to 1 of the factor f. The contraries are the complementaries, as the motto of Bohr's coat of arms (Contraria Sunt Complementa), who adopted it to reflect the principle of wave−particle complementarity, says.
  
  

  Bohr Coat of Arms

  
  (f*T)' = f*T' = f*F = (1−f)*T

(f*F)' = f*F' = f*T = (1−f)*F
  
  For the case f=1, we have:
  (T' = F) and (F' = T).
  
  When f=0.5, you have:
  (0.5*T)' = 0.5*T' = 0.5*F = 0.5*T
  
  Therefore, from the point of view of the negation function, this value is a fixed point that also unites and balances the opposites (T and F) and represents "the undefined".
  
  
• The four laws of truth. The Aristotelian expression mentioned above, included in his Metaphysics, can be expressed thus in MENTAL:
  
  ( (T/T) = T )
  (preaching the truth of the truth is truth)
  
  ( (T/F) = F )
  (to preach the falsehood of the truth is false)
  
  ( (F/T) = F )
  (to predicate the truth of falsehood is false)
  
  ( (F/F) = T )
  (predicate falsehood of falsehood is true)
  
  We can qualify these expressions as Aristotle's "four laws of truth". These laws have their analogy in the laws of the signs of arithmetic:
  
  (+×+ = +)   (+×− = −)   (−×−+ = −)  (−×− = +)
  
  With truth as a qualitative magnitude, the four laws of truth" generalize as follows:
  
  ( (f1*T)/(f2*T) = (f1*F2)*T )

( (f1*T)/(f2*F) = (f1*T)/((1−f2)*T) = (f1*(1−f2))*T )

( (f1**F)/(f2*T) = ((1−f1)*T)/f2*T) = ((1−f1)*F2)*T )

( (f1*F)/(f2*F) = (1−f1)*(1−f2)*T )
  
  
• Correspondence theory.
  With MENTAL it becomes clear what is the nature of the relationship between a declarative statement and the fact to which it refers. The same archetypes are manifested at the physical and mental (or linguistic) level. The relation between statement and fact is not a direct, horizontal relation, but an indirect, vertical relation, through the primary archetypes. If we assume that in nature there is a more or less hidden or explicit linguistics, then for example the fact that "snow is white" is itself a true proposition. In MENTAL there is correspondence between language and reality, for both are manifestations of the same primary archetypes.
  
  
• Generalization of formal theories of truth.
  Formal theories of truth can be generalized by considering truth as a qualitative magnitude:
  
  

  Theory	Expression	Generalized
  Deflationary	⟨("p"/T ≡ p )⟩	⟨("p"/(f*T) ≡ f*p )⟩
  Minimalist (Harwich)	⟨( "p"/T ↔ p  )⟩	⟨( "p"/(f*T) ↔ f*p )⟩
  Tarski	⟨("p"/T ← p )⟩	⟨("p"/(f*T) ← f*p )⟩
  Kripke (fixed point)	⟨( p/T = p )⟩	⟨( p/(f*T) = f*p )⟩
  Davidson	⟨("O"/T ↔ p )⟩	⟨("O"/(f*T) ↔ f*p )⟩
  Quine	⟨("p"/T = p )⟩	⟨("p"/(f*T) = f*p )⟩
  Frege	v(p) = T or F	v(p) = f*T
  MENTAL	p/T	p/(f*T)

  
  where: v(p) is the truth value of p; f*p is interpreted as p with existence degree f. For example,
  
  (Juan/alto)/(0.7*T) = 0.7*(Juan/alto)
  
  
• Relative truths.
  MENTAL allows us to express truths relative to possible or imaginary worlds. For example,
  
  x/(V/world1)
  (x is T in world1)
  
  x/((f*T)/mundo1)
  (x is f*T in world1)

Truth is the essence of all that exists

What is truth? Truth is the absolute, the permanent, the permanent, the stable, the necessary, that which cannot be questioned, that which is always present in all things and cannot be avoided, the essence of everything. The essence of reality is abstraction in its highest expression, the deepest, that which is common to everything, the simplest: the philosophical categories, the primal archetypes, the universal semantic primitives. The true nature of a model of reality encoded with MENTAL is precisely its code, which conceptually connects the internal and the external.

At the deep level the truth is contacted. At this level the concepts of truth, maximum abstraction, maximum simplicity, maximum freedom, maximum consciousness, maximum power and maximum creativity are integrated.

MENTAL is the solid and unquestionable truth sought by Descartes, for it is the foundation of possible worlds. Every possible world is governed by the primitives of MENTAL.

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Addenda
The cube, symbol of truth

The cube is the symbol of truth, stability, wisdom, perfection and totality. It shows all its faces equal, without hiding anything. It is also the symbol of the multiple and harmonious union of opposites. Truth is eternal and not subject to change. The unfolded (open) cube forms a cross, the symbol of the universal, eternal and prototypical man, the human connection with the universal truth.


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