MENTAL, A FALSIFIABLE LANGUAGE

Is MENTAL a scientific theory?
"The essential feature of science is not its verifiability but its falsifiability" (Karl Popper).

"Insofar as a scientific statement speaks about reality, it must be falsifiable; and insofar as it is not falsifiable, it does not speak about reality" (Karl Popper).

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Popper's Falsificationism
The problem of induction

"Falsificationism," also called "refutationism" or the "falsifiability principle," is a theory of science developed by the philosopher Karl Popper. In essence, the theory is based on the problem posed by induction (or inductive inference) −problem already highlighted by Hume in the 18th century− which consists in inferring a universal statement from particular facts.

The example Popper gives is that of swans. If we repeatedly observe white swans, we can infer that "all swans are white". But it is enough that a black swan appears (or is not white) for this universal statement to be false. Therefore, according to Popper:

• We can never assert something universal on the basis of particular facts offered by experience, unless we are able to verify all those particular data. But that is almost never possible, because usually those particular facts are infinite.
  
  
• Hypotheses must be "falsified," not verified. Any scientific theory must be considered provisional, conjectural; it must be accepted provisionally until a counterexample (empirical evidence) is found to refute the theory. 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.
  
  
• Verificationism must be rejected, unless it is complete. Verificationism is the criterion of validity or demarcation of the logical positivism of the Vienna Circle. It consists in that every statement, in order to be scientific, must be experimentally verifiable or governed by the laws of logic.
  
  
• The central problem in philosophy of science is the "criterion of demarcation", between what is a scientific theory and what is not. This criterion separates science from non-science (metaphysics). This demarcation criterion must be falsifiability and not verifiability.
  
  The metaphysical is that which cannot be verified by experience, the non-falsifiable. Metaphysical ideas can serve as a guide, but not as the foundation of science. "God exists" is not a scientific statement because it is not falsifiable.
  
  
• The greater the field of application of a statement or a theory, the greater its value, but at the same time the greater the risk of being falsified. Examples:
  
  
  • The statement "all metals are electrically conductive" is more falsifiable than the statement "copper is electrically conductive", because to falsify the first one we can experiment with different metals (iron, lead, aluminum, etc.), while to falsify the second one we can only experiment with copper. And the first statement, because of its generality, has more value than the second.
    
    
  • Newton's theory of gravitation is a superior theory to Kepler's theory of planetary motion because it is more generic, but offers greater risk of being falsified.
    
    
  • "All planets move" is more falsifiable than "all planets move in elliptical orbits."
  
  
• Ideally, a theory should be based on as few hypotheses as possible to make it as generic as possible and maximize its falsifiability.
  
  Scientific theories should be as generic as possible, so that they explain as many phenomena as possible. The highest aspiration of science is to find the "theory of everything", the universal theory that explains everything. The more generic a theory is, the more falsifiable it is. Therefore, scientific theories should be as falsifiable as possible.
  
  The value of a hypothesis is measured by the falsifying potential it contains, that is, by the number of possible experiences that could lead to the rejection of the hypothesis.
  
  
• It is never possible to prove a theory; we only have the possibility of disproving it. Therefore there are no unquestionable theories or starting points, that is, that are always true or absolute. There are not, nor can there be, because scientific rationality demands it.
  
  
• Ambiguous sentences are not falsifiable. For example, "There is a green swan" is not falsifiable because it does not specify where or how.
  
  Statements that are equivalent to their opposites are also not falsifiable. For example, the statement "tomorrow it may rain" is not falsifiable because its opposite ("tomorrow it may not rain") is equivalent.
  
  
• Physics, chemistry, and non-instrospective (behaviorist) psychology are examples of sciences. In contrast, the social sciences, anthropology, sociology, and psychoanalysis are not scientific because they are based only on observations of particular cases and their theories are formulated in such a way that they are not falsifiable. Astrology and phrenology are pseudo-sciences.
  
  
• Falsificationism provides a basis for humanism, for the claim to possess the truth leads only to dogmatism and intolerance.
  
  
• Popper's theory is valid in the physical world, which is a dual world. But it does not make sense sometimes in the quantum world, which is a deep world, beyond the superficial physical world. Quantum theory can only be expressed in mathematical terms and describes matter as an abstraction.

Popper's falsificationism is really nothing more than a logical modus tollens. In a logical sentence of the type p→q (if p, then q), the modus ponens is: if p is true, then q is true. And modus tollens is: if q is false, then p is false.


The two versions of falsacionism

The naive or dogmatic version of falsacionism is the one illustrated by the example of the swans: only one negative case is enough to reject the general statement. Moreover, it considers general scientific statements individually.

Popper later proposed a renewed version of falsificationism (methodological falsificationism), which is based on not rejecting the theory for a single case, but posing two possible causes:

1. Problems in the observation, since no observation is free from the possibility of error: Are our observation instruments reliable? Have we used them correctly? Have we really observed what we had to observe? Have we correctly applied the observation procedure?
   
   
2. The knowledge provided by the observation support may be incomplete or faulty.

Moreover, the renewed version does not consider individual sentences, but the entire theory. That a single sentence of the theory fails does not justify rejecting the entire theory. One possible "fix" for the theory in the face of a contrary observation is to include exceptions in the general sentence. In the case of the swan example, if a black swan appears in Australia, the universal statement would have to be modified (restricted, in this case) to take into account the new fact: "All swans are white except in Australia". Another alternative would be to introduce probability, based on statistical data; e.g., "The probability that a swan is white is 98%."


Evaluation of Popper's Theory
Popper has been a very influential author in 20th century philosophy. His falsificationist theory was born with the publication in 1934 of his work −today considered a classic− "The Logic of Scientific Inquiry" [2011] and it meant a strong turn and a certain questioning in the way of understanding the methodology of scientific theories.

There have been numerous criticisms of Popper's theory. It has even been claimed that falsificationism is the antithesis of inductivism. Popper has even been accused of calling everything into question, of doubting everything. The most common criticisms are the following:

• It is a paradoxical theory, because its aim is to avoid scientific dogmatism and yet the theory is too rigid and dogmatic. As every theory must be falsifiable, Popper's theory should also be falsifiable to be considered a true scientific theory (or metatheory). Since it is a dogmatic, non-falsifiable theory, it is not an acceptable scientific theory.
  
  
• It does not take into account that science is based mainly on induction, on inductive reasoning, so that to question induction is to hinder knowledge and scientific progress. Induction represents the natural tendency of the human mind to go from particular or singular facts to the general and universal, the natural tendency towards greater awareness, to unite or connect the dual: the particular with the general.
  
  Science, in general, is inductive, i.e., bottom-up. This aspect is more important than the deductive (or top-down) aspect. If inductive inferences are not valid, science loses its entire foundation.
  
  The advancement of science is based on evolving a theory to adapt it to observational facts.
  
  
• Since, according to falsificationism, all knowledge is hypothetical and conjectural, we can never be absolutely certain about the truth of a statement or theory. Therefore, we have to give up certainty.
  
  
• Many physical hypotheses and laws cannot be directly falsified by direct observation (such as the law of universal gravitation), but it is possible to verify the consequences of those hypotheses (the elliptical orbits of the planets).
  
  
• The social sciences, anthropology, sociology and psychoanalysis, although based on particular cases, their general conclusions represent advances in human knowledge.
  
  
• The subjectivity that is to be avoided is inevitable in scientific activity.
  
  
• It must be kept in mind that science is not reality, but only an approximation to reality. Reality is not limited to the phenomenal world, so other types of hypotheses should be allowed for. Falsificationism is reductionist, since it is limited to the experiential world (the physical world). Metaphysics and transcendental truths are excluded. It does not take into account that, according to the universal paradigm of descending causality, everything has its origin and foundation in the superior.
  
  
• Mathematical definitions, and the properties that result from the definitions, are not falsifiable. for example, the statement "all the points of a circle are equidistant from the center" is not falsifiable, so, according to Popper, it would not be a scientific statement. But, in this example, we are referring to an abstract element (the circumference), which does not exist in nature; it only exists in a Platonic world of ideas accessed by the human mind.
  
  
• The theorems of logic and mathematics are not falsifiable, since their truth is dependent on axioms and rules of inference, which are not subject to falsifiability. Although, according to Lakatos, mathematics is a quasi-experimental science, since axioms, definitions and demonstrations evolve in a manner analogous to the scientific theories of the positive sciences.
  
  To understand and explain mathematics it is not enough to analyze its logical structures or its language, but one must study its actual practice. The term "quasi-empiricism" is very close to the falsifiable conception of mathematics.

According to Thomas Kuhn, science is based on paradigms, conceptual schemes that condition the way of perceiving reality and elaborating scientific theories.

For Imre Lakatos, science progresses by the falsification of research programs, rather than by the naive falsification of universal sentences.

For Paul Feyerabend, scientific theories that gain general acceptance are based on social factors, rather than on the pursuit of pure rational method.

For Charles Sanders Peirce, mathematics is more important than the positive sciences, for these sciences are founded on mathematical concepts, and mathematics transcends reality, for its aim is not to find out how things are, but how they might be, in this universe or in some other. In this sense, falsifiability is meaningless.


MENTAL and Falsifiability
In this topic we have the following features:

• Falsifiable language.
  MENTAL is an abstract language, so it does not refer directly to the concrete phenomenal world but to the general structure of internal and external reality; its primitives are archetypes. Therefore, it is not falsifiable in the sense that one cannot appeal to reality for its possible falsifiability.
  
  According to Popper, there are no untestable starting points. In the case of MENTAL, the primitives are the starting elements, which are semantic axioms. These axioms are not refutable. What is refutable is the language as a whole when a domain is found that needs a formal language and MENTAL is not applicable. Then the language will cease to be universal, limiting its field of application. To regain universality something would have to be done about it, such as adding another primitive, replacing some primitive with another primitive of higher level of abstraction, etc. Therefore, in this sense, MENTAL is a falsifiable language and is a scientific theory.
  
  
• Maximum falsifiability.
  Falsifiable applies to theories of reality. The broader the domain of reality to which a theory applies, the greater its falsifiability. Since the domain of MENTAL is very broad, since it applies to external and internal reality, its falsifiability is maximal.
  
  
• Maximum importance.
  MENTAL is more important than any particular positive science, for it transcends them. The broader the domain of reality to which a theory applies, the greater its importance. As the domain of MENTAL is very broad, since it applies to external and internal reality, its value or importance is of the highest degree.
  
  
• Supreme induction.
  MENTAL is the result of supreme induction. The search for general principles is the goal of science (and also of philosophy) and that goal can only be achieved by induction, abstraction and simplicity.
  
  
• Union of opposites.
  MENTAL connects opposites, including the generic and the specific, the deep and the superficial, the universal and the particular.

Coding

The example of the swans (inductive inference):

⟨( x/swan → x/white )⟩

The example of white swans with the exception of Australian swans:

⟨( x/swan → x/australian' → x/white )⟩

With MENTAL it is possible to apply to each expression x a factor f between 0 and 1 to specify the degree of certainty or validity of that expression, where 1 means to consider it in its integrity (1*x = x) and 0 to ignore it completely (0*x = θ). This factor f is subjective, although it is based on objective data. For example, the sentence s "All swans are white" we could encode it like this:

f*⟨( x/swan → x/white )⟩

If, for example, we assign to f the value of 0.9, we are considering the statement s to be 90% valid. But, in this case, it is more logical to apply the factor to the consequent of the condition:

⟨( x/swan → f*(x/white) )⟩

In this case, f=0.9 would indicate that the degree of truth that a swan is white is 90%.

We could also apply the factor to the white attribute:

⟨( x/swan → x/(f*white) )⟩

Assuming (0.5*white = gray) and (0*white = black), a value f=0.5 would indicate gray swan, and a value f=0 would indicate black swan.

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Addenda
Popper's 3 worlds

For Popper, there are 3 worlds:

1. The external world: the physical entities (the objective and corporeal reality).
   
   
2. The inner world: the non-corporeal of mental entities (subjective phenomena and states of consciousness).
   
   
3. The products of the human mind, which are entities that have their own existence. It is the world of culture, including all products of the human intellect (philosophical, scientific, artistic contents, etc.).

These 3 worlds interact with each other. Scientific theories and logical laws belong to the third world.

Popper did not believe in top-down causality, but in bottom-up causality: nature is creative, man being the supreme result of this creativity. Man is an emergent phenomenon, the result of a process of gradual evolution of nature. Mind and consciousness are epiphenomena of the brain.

For Popper there are 3 worlds. But according to the universal principle of downward causality, there is only one deep reality and all the rest are manifestations.

Popper's conception differs from Penrose's 3 worlds in that for Penrose, world 3 is the world of mathematics. According to Penrose, reality is a single unit classifiable into three worlds:

1. The physical world. It is the sensible and perceptible reality through sensations. The ontological foundation of the physical world is mathematical.
   
   
2. The world of psychic, personal and intersubjective experiences. It is the psychic world where consciousness takes place.
   
   
3. The mathematical world. It is a Platonic world: eternal, harmonious and perfect. The mathematical elements possess an existence that can only be discovered through intelligence.

Probability and falsificationism

There is a current of thought that argues that the logic used in inductive inference is not adequate, and proposes to use the concept of probability and probabilistic laws (specifically, Bayes' theorem) as the foundation of inductive inference, thus providing a safe, standard and universal methodology for science.

The probability of an event or phenomenon is the ratio of the number of observed favorable cases (a finite number) to the number of possible cases. In the case where the number of possible cases is infinite, the probability is zero.

The mathematician Thomas Bayes, in the 18th century, devised a simple mathematical formula for calculating the probability of a conditional hypothesis:

P_f(h/e) = P _i(h)·P(e/h)/P(e)

• P_f(h/e) is the final probability of a hypothesis h, known an observational evidence e.
  
  
• P_i(h) is the initial probability assigned to the hypothesis h.
  
  
• P(e/h) is the probability assigned to the evidence e under the assumption that h is true.
  
  
• P(e) is the probability assigned to the evidence e ignoring the hypothesis h.

That is: conditional probability = unconditional probability × predictive power.

Bayes' theorem combines inductive and deductive reasoning through the common nexus of the concept of probability:

• The inductive process takes place when new evidence appears: P_f is gradually transformed into P_i.
  
  
• The deductive process is realized in the generalization of P(e/h).

However, the concept of probability and Bayes' theorem also do not serve as a foundation for induction:

• The probabilistic approach does not serve to capture the essence of causation. There is no pure causal relationship.
  
  
• Bayes' formula is applied on subjective grounds (degree of belief in the hypothesis). There is no objective criterion that avoids the subjectivism of researchers, who cling to their preferred theories.

Goldbach's conjecture

An example of mathematical falsifiability is Goldbach's conjecture, one of the oldest open problems in number theory: every even number can be written as the sum of two prime numbers (equal or different). For example, 6=3+3, 12=5+7, 14=3+11, and so on. So far no example has been found that makes the conjecture falsifiable. It has been proved by computers for all even numbers less than 1018. Most mathematicians believe it to be true.


Bibliography

• Antiseri, Dario. Karl Popper: Protagonista del siglo XX. Unión Editorial, 2002.
  
  
• Artigas, Mariano. Lógica y ética en Karl Popper. Eunsa, 2001.
  
  
• Lakatos, Imre. La metodología de los programas de investigación científica. Alianza Universitaria, 1988.
  
  
• Martínez Solano, José Francisco. El problema de la verdad en K. R. Popper. Netbiblo, 2005.
  
  
• Moya, Eugenio. Ciencia, sociedad y mundo abierto. Homenaje a Karl R.Popper. Editorial Comares, 2004.
  
  
• Moya, Eugenio. Conocimiento y verdad. La epistemología crítica de K. R. Popper. Editorial Biblioteca Nueva, 2001.
  
  
• Popper, Karl. La lógica de la investigación científica. Tecnos, 2011. [Versión española del clásico “The Logic of Scientific Discovery”, de 1934 y revisada en 1968.]
  
  
• Popper, Karl. Realismo y el objetivo de la ciencia. Postscriptum a “La lógica de la investigación científica”. Vol. I, Tecnos, 2010.
  
  
• Popper, Karl. El universo abierto, un argumento a favor del indeterminismo. Postscriptum a “La lógica de la investigación científica”. Vol. II, Tecnos, 2010.
  
  
• Popper, Karl. Teoría cuántica y el cisma en física. Postscriptum a “La lógica de la investigación científica”. Vol. III, Tecnos, 2010.
  
  
• Popper, Karl. Conjeturas y refutaciones. El desarrollo del conocimiento científico. Paidós Ibérica, 2008.
  
  
• Popper, Karl. Búsqueda sin término. Una autobiografía intelectual. Tecnos, 2011.
  
  
• Popper, Karl. Conocimiento objetivo. Un enfoque evolucionista. Tecnos, 2010.
  
  
• Popper, Karl. El mito del marco común. En defensa de la ciencia y la racionalidad. Paidós Ibérica. 2005.
  
  
• Popper, Karl. Los dos problemas fundamentales de la epistemología. Tecnos. 1998.
  
  
• Queraltó, Ramón. Karl Popper: de la epistemología a la metafísica. Universidad de Sevilla. Secretariado de Publicaciones, 1996.
  
  
• Rivadulla Rodríguez, Andrés (coord.). Hipótesis y verdad en ciencia. Ensayos sobre la filosofía de Karl R. Popper. Editorial Complutense, Facultad de Filosofía, 2004.
  
  
• Suárez Íñiguez, Enrique. La fuerza de la razón. Introducción a la filosofía de Karl Popper. UNAM-Miguel Ángel Porrúa, 1998.