Review of Suppes 1957 Proposals For Division by Zero

We review the exposition of division by zero and the definition of total arithmetical functions in ``Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to ``the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory. We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations are of Extension Type at least 1. Totalisations of the differential and integral calculus have Extension Type at least 2. In particular, Meadows have Extension Type 1, Wheels have Extension Type 2, and Transreal numbers have Extension Type 3. It appears that Suppes was the modern originator of the idea that all real numbers divided by zero are equal to zero. This has Extension Type 0 and is, therefore, arbitrary.

Detection of deception through saccadic movements in the psychological assessment and symptom monitoring of Covid-19

From the theory of the model of the mind's point of view, which indicates that in syllogisms there will be more eye movements from front to back between the terms of the premises that question some of Ford's predictions? (Espino & Santamaría, 1998) During the blinking process, sometimes the superior and inferior eyelids tend to completely or incompletely close. It intervenes notably in the comfort of vision when we perform surrounding vision tasks. Carrying out tasks such as conferences, paper, and electronic versions, it's linked to an increase in dry eye symptoms, therefore, among other factors, of altered blinking. (Rodríguez Montiel, 2015). This article intended to verify the phenomena of lying, deception, and self-deception that are directly related to psychological problems and their treatments (Porcel Medina & Gonzalez Fernandez, 2005). Among its defining features is the adoption of computational metaphor as a source of inspiration for the modeling of the structures and processes of the mind, and methodological functionalism, which legitimizes the study of mental processes and states (the "software") regardless of its physical installation base (the "hardware"). It is a computerized model of the human special abilities for reasoning and problem-solving skills. However, it is another modern language used in artificial intelligence (AI) and the logical extension that is based on first-order predicate calculus that includes an inference engine that uses backward chaining. It is a non-procedural language, which indicates that the instructions do not have to necessarily be executed in the order that they have been entered. (Pino Diez & Gómez Gómez, 2001) It is concluded that the specific causes of this ocular muscular activity constitute a true enigma located in the adjacent area of the midbrain. (García Alcolea, 2009). The article will be based according to the title, authors, year of publication, abstract, citations, and bibliographic references since it is an experimental design project.

The logical contingency of identity

I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

This paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

Un algoritmo en seudocodigo para el chequeo de la subsumicion en alc

En el presente artículo se describe un evaluador de satisfactibilidad para el chequeo de la subsumición en un lenguaje de atributos de conceptos (Attribute Language Concept, ALC). Los lenguajes de conceptos basados en las lógicas descriptivas (Description Logics, DLs) ofrecen servicios de razonamiento que permiten hacer clasificación y recuperación de la información dentro de la base de conocimiento. Los procesos de razonamiento de subsumición y de satisfactibilidad son equivalentes y se especifican por medio del Cálculo de Predicados de Primer Orden (First Order Predicate Calculus, FOPC) y el cálculo Tableaux. FOPC permite asociar cada expresión C de conceptos a una fórmula f c (x) de la lógica de predicados, de tal forma que un modelo de una fórmula f c (x) es un modelo del concepto C y viceversa. El cálculo Tableaux de primer orden siempre termina para las fórmulas asociadas a conceptos en el FOPC. El cálculo de terminación planteado permite una interpretación si la fórmula es satisfactible o se produce una contradicción si la fórmula es insatisfactible. Se plantea un algoritmo en seudocódigo para el chequeo de la subsumición.

On the Definitional Embeddability of Some Elementary Algebraic Theories into the First-Order Predicate Calculus

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus

In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.