Traditional Formal Logic Research Articles

MARTIN-LÖF’S TYPE THEORУ: BETWEEN PHENOMENOLOGY AND ANALYTICAL PHILOSOPHY

Martin-Löf’s type theory stems simultaneously from Frege and Russell’s logic-ontological ideas and Husserl’s phenomenology. The article examines this intermediate status of type theory using as examples Martin-Löf ’s syntactical-semantic method and the role of evidence and canonical objects in his approach. Martin-Löf borrows the syntactical-semantic method from Frege and extends it drawing on Husserl’s theory of meaning. In type theory this method leads to the identity (isomorphism) of syntax and semantics (formal logic and formal ontology). Unlike traditional formal logic the type theory is an interpreted system from the very beginning. Being intuitionistic, Martin-Löf ’s theory is based on the notion of proof, not truth. From the meaning theory point of view, it is a variant of proof-theoretic semantics (Gentzen, Prawitz, Dummett) which understands meaning as an object constructed according to certain rules. So understood, the proof is based on evidence, which allows us to associate it with the theory of intentionality by Husserl. The article compares Martin-Löf ’s type theory with Husserl’s intentionality theory, especially with the latter’s noematic component. We may consider type-theoretical rules for constructing objects and operating with them as a concretization and formalization of the phenomenological notion of noema. Both are explications of the more general concept of meaning. The article discusses the interrelation between notions of sense and meaning (Sinn, Bedeutung) in Frege, Husserl and Martin-Löf. This reveals the uncertainty of Martin-Löf ’s position in relation to meaning theories of Frege and Husserl.

La Matriz de Pertenencia y Pertinencia como instrumento transdisciplinar para la detección de necesidades, una perspectiva desde la lógica del tercero incluido

El presente trabajo contiene como elemento medular de estructura y análisis epistémico, la propuesta emergente de la investigación que respalda la tesis doctoral “El sentido de pertenencia como resultado de la detección de necesidades y como detonante de motivación para la mejora en el desarrollo individual: Un aporte desde el pensamiento complejo”, defendida en 2019 y que sirve como punto de partida al sugerir un paradigma inicial de investigación, basado en la concepción de marcos epistémicos comunes con respecto a la determinación de la conducta humana. Tal propuesta, denominada La Matriz de Pertenencia y Pertinencia, busca establecer la diferencia transcompleja que existe entre uno de los principios de la Lógica Aristotélica conocida como la ley del tercero excluido, la cual establece que: Ley de la lógica, según la cual de dos enunciados, cuando uno niega lo que se afirma en el otro, uno de ellos es imprescindiblemente verídico. La formuló Aristóteles. Así, de los dos enunciados: “el Sol es astro” (A es B) y “el Sol no es astro” (A no es B), uno es imprescindiblemente verídico. Teniendo en cuenta los enunciados de este género, la lógica formal tradicional a menudo formulaba la ley del tercero excluido del siguiente modo: “A es B, o no es B” (lo tercero no está dado: tertium non datur). Esta ley a menudo se utiliza en el proceso de demostración, en particular, en las demostraciones por reducción al absurdo. Y la iniciativa compleja del pensamiento del sur que establece al tercero incluido como la alternativa capaz de integrar los supuestos de una realidad parcelaria, mediante un diálogo de saberes donde el resultado emerja directamente desde el seno de sus componentes y no como resultado de una disertación lógica, sesgada y determinista.
 Palabras clave: Sentido de pertenencia, pertinencia, conducta, tercero incluido, pensamiento complejo.

Tradicionālā loģika – arhaisms vai fenomens ar attīstības iespējām

The aim of the article is to show the specifics of traditional logic. To a large extent, logic initially operates as Aristotle's logic. The traditional formal logic appears as a combination of individual elements, which are obtained from logical systems of various historical eras and directions. The author believes that the subject of logic is objectively linked to the category of the form, but the difficulty in understanding the form lies in the complexity of the form’s nature. The author claims that the content, which differs from the form, is not included in it. There is a functional difference between the form and the content. It is assumed that the types of logic appear as a function of a certain form, and diversity of logic is a consequence of the various manifestations of the form. Logic is interested in the correct form. Such assessment of the form includes an idea of the form as a specific condition. The correct form is a condition for the safety of information included in the statements or their sets. The author argues that logic can be characterized as a science of formal conditions and ways to achieve information security of certain parameters. Supposedly, traditional logic is a formal (form-based) science, which however is limited to certain substantial conditions, and that is one of the factors that forms the specifics of science. The author emphasizes that the tools offered by traditional logic allow to find solutions related to novelty.

Justifying the Self-Evident

In Fichte’s early views of the basic laws of traditional formal logic, primarily the law of identity, there is a tension that has gone surprisingly unexplored: While Fichte holds the statements of these laws to be self-evidently true and absolutely certain, he nevertheless claims that they remain to be justified by his “Science of Knowledge.” The aim of this article is to make sense of this tension and to explore how it translates into the dialectical structure and methodology of Fichte’s first Jena Wissenschaftslehre. This is done by, first, conjecturing—in a somewhat ahistorical, yet Fichte-based, fashion—a reason for Fichte’s justificatory demand. It is argued that the validity of the law of identity can be questioned because our belief in its absolute generality appears to be self-refuting in that it involves an antinomy akin to Grelling’s semantic antinomy of the heterological. This antinomy, when, secondly, related to Fichte’s purported justification of the law of identity, serves as a key to understanding why there is an antinomic conflict between Fichte’s supreme principle—namely, the self-positing pure I—and its adversary, the not-I, in the first place. Tracing their contradiction (whose synthetic resolution is the main goal of Fichte’s Wissenschaftslehre) back to that semantic antinomy inherent in our formal-logical certainties opens up a new way of seeing Fichte as radicalizing Kant’s critical philosophy, understood as the project of the self-preservation of reason against reason’s own antinomies.

The Erdõdy Castle in Vörösvár

The township of Rotenturm an der Pinka - ci-devant Vörösvár - lies along the Pinka in Burgenland, just 20 kilometres from Szombathely. In the middle of the village, in a deserted park stands an especially beautiful architectural creature of the Hungarian Romanticism, the Erdõdy Castle as a masterpiece of Antal Wéber. The building is historicising by mixing the styles from some architectural elements like the column of the carriage passage shown above to the composition itself. However, the building is basically differentiated from the style-mixing historicism of the close of the 19^th century by the fact that every architectural unit is still part of a suite with the traditional formal logic, with traditional proportions.

SPATIAL AND VISUAL REASONING: HUMANS VERSUS LOGIC

Glasgow's revival of the “imagery debate” in computational terms provides a renewed opportunity to review the role of logical reasoning in general problem solving. Of special interest is the long‐standing distinction between analogical or depictive problem representations, and the more abstract linguistic forms typified by traditional formal logic syntax.In our brief statement, we recall that logical reasoning rests on semantics not syntax, and that the concepts of soundness, completeness, and consistency are manifest in both depictive and linguistic representations. We emerge with two conclusions: (1) enduring confusion regarding computational aspects of the “imagery debate” arise from long‐standing confusion regarding key logical concepts, and related notions such as epistemological versus heuristic adequacy, logical versus probabilistic independence, and direct versus indirect representations; (2) the desire for depictive reasoning methods is ultimately motivated by human needs, not computational needs.

On the question ``do we need identity?''.

Sommers posed the question 'Do We Need Identity?' and answered in the negative. According to Sommers, the need for a special identity relation resulted from an arbitrary distinction between concept and object introduced by Frege and retained in modern predicate logic (MPL). This is reflected in the syntactic distinction between predicate and individual constant. Traditional formal logic (TFL) does not respect this distinction and, as a consequence, has no need for a special identity relation. But Sommers' position has not gained general acceptance. On the contrary, it has received considerable criticism. While it is conceded that TFL can express the identity of individual constants, it is quickly pointed out that this falls far short of providing the expressiveness of the logical identity relation. But the precise extent of the deficit in expressiveness, if indeed there is any deficit, has not been determined. It appears that Sommers' position on identity has not been adequately formalized to permit such a determination. This paper formalizes and extends Sommers' position on identity. This formalization is compared with MPL to define precisely the difference in expressive power. The conclusion is that it has less expressive power than MPL, but nonetheless does provide essentially all the expressiveness of the logical identity relation. The formal language defined for this investigation is similar to the language of MPL. The similarity will not only facilitate comparison, but perhaps will also make this formal language more palatable to readers whose experience and/or predisposition favors MPL.

Important issues in knowledge representation

This paper discusses a number of important issues that drive knowledge representation research. It begins by considering the relationship between knowledge and the world and the use of knowledge by reasoning agents (both biological and mechanical) and concludes that a knowledge representation system must support activities of perception, learning, and planning to act. An argument is made that the mechanisms of traditional formal logic, while important to our understanding of mechanical reasoning, are not by themselves sufficient to solve all of the associated problems. In particular, notational aspects of a knowledge representation system are important--both for computational and conceptual reasons. Two such aspects are distinguished--expressive adequacy and notational efficacy. The paper also discusses the structure of conceptual representations and argues that taxonomic classification structures can advance both expressive adequacy and notational efficacy. It predicts that such techniques will eventually be applicable throughout computer science and that their application can produce a new style of programming--more oriented toward specifying the desired behavior in conceptual terms. Such "taxonomic programming" can have advantages for flexibility, extensibility, and maintainability, as well as for documentation and user education.

Evert W. Beth. Preface. English translation of XL 256(1). Science a road to wisdom, by Evert W. Beth, D. Reidel Publishing Company, Dordrecht, Holland, 1968, pp. XI–XIII. - Evert W. Beth. Science as a cultural factor. English translation of XL 256(2). Science a road to wisdom, by Evert W. Beth, D. Reidel Publishing Company, Dordrecht, Holland, 1968, pp. 1–10. -

Evert W. Beth. Preface. English translation of XL 256(1). Science a road to wisdom, by Evert W. Beth, D. Reidel Publishing Company, Dordrecht, Holland, 1968, pp. XI–XIII. - Evert W. Beth. Science as a cultural factor. English translation of XL 256(2). Science a road to wisdom, by Evert W. Beth, D. Reidel Publishing Company, Dordrecht, Holland, 1968, pp. 1–10. -

On the Situation in Logic and its Place in University Education (1)

It becomes more obvious each year that the appeals of the party and government to strengthen the ties between philosophy and modern science are not being adequately implemented by the chairs of logic in the universities. In a number of institutions, a situation has persisted for a long time in which only a few professors and instructors deal with the subject area of dialectical and mathematical logic at a high professional level. Most of the teaching staff continue to conduct their work essentially within the confines and at the level of the old, arbitrarily designated traditional formal logic, primarily in the form in which it took shape in the middle of the last century. It is clear that this does not accord with the demands of the party and of our philosophical and scientific public to raise the theoretical level of all instructional and research work in higher educational institutions. The erroneous orientation toward the old formal logic was, in particular, one of the reasons why the fundamental pr...

Concerning the method of logical schemes, the notion of logical calculus and the role of consequence relations

I intend to formulate here some general principles of the method of logical schemes, being an essential procedure in formal logic. It consists in considering logical schemes (or forms) of expressions, of sentences and of inferences. First applications of it may be found at the origin of formal logic, i. e. in the Aristotle's logic and in that of the Stoic school. The method of logical schemes had been applied in the traditional formal logic under the form of a "theory of forms of thought". In the modern time, there are QuiNE's logical calculi built by means of "dummy letters" [1], [2] and BERNAY's logical calculi determined by "Schlussschemata" [3] that may be considered as applications although not in its full extent, of the method of logical schemes. There are semantical and syntactical principles of the method of logical schemes. The former will rather not be considered in this paper, however. I am interested here in some syntactical principles of this method, especially in its connection with the relations of consequence. Simoultaneously, I will present a kind of logical calculi corresponding to so called "lower" logical cal? culi and clearly opposed to the formalized theories. The last section 13 of this paper contains five appendices (A, B, C, D, E) including some definitions and proofs omitted in the text.

A Note on Traditional Formal Logic

A Note on Traditional Formal Logic

The Symbolization of Traditional Formal Logic

The Symbolization of Traditional Formal Logic