Abstract

Neste artigo, eu revisito a teoria do sentido e referência de Frege no cenário construtivo das explicações de significado da teoria dos tipos, estendendo e aprimorando a análise em termos de programa-valor da teoria de sentido e referência proposta por Martin-Löf, baseada em trabalhos anteriores de Dummett. Eu proponho um critério de identidade computacional para senti-dos e defendo que ele valida o que considero a interpretação mais plausível do princípio de equipolência de Frege, tanto para sentenças quanto para ter-mos singulares. Antes de fazer isso, examino a implementação de Frege de sua teoria dos sentidos e referência no quadro lógico de Grundgesetze, sua doutrina dos valores da verdade, e sua interpretação de igualdade de sentido como equipolência de afirmações.

Highlights

  • Frege was one of the first to dispute that all mathematical truths are based on intuition, the cornerstone of the dominant Kantian philosophy of mathematics, and put forward the thesis that all arithmetical truths are reducible to logical truths.2 The success of his logicism would eventually depend on the plausibility of the development of a new system of logic in which all the objects and concepts of arithmetic could be defined

  • I would like to focus on Frege’s theory of sense and reference, the doctrine that singular terms and complete sentences have a mode of presentation and denote an object, whose implementation consists in the second substantial change made to the ideography

  • Frege on sameness of sense In Frege’s works we find the two forms of judgment that are current in modern logic, the judgment form that asserts that a proposition is true, which is written in turnstile notation as

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Summary

Introduction

Frege was one of the first to dispute that all mathematical truths are based on intuition, the cornerstone of the dominant Kantian philosophy of mathematics, and put forward the thesis that all arithmetical truths are reducible to logical truths. The success of his logicism would eventually depend on the plausibility of the development of a new system of logic in which all the objects and concepts of arithmetic could be defined. That was the original purpose of his Begriffsschrift, the book which lays down the first systematic treatment of modern logic, the ideography, and applies the system to the mathematical theory of sequences Frege hints at his logicist program in the preface to his book, announcing that his investigations will be continued with the logical elucidation of the concepts of number, magnitude, and so forth in what he describes as an immediately following publication. Namely, the problem of explaining the apparent difference in cognitive value between true identity statements of the form a = a and a = b My goal in this paper is to revise some of Martin-Löf’s main views, a task that will be carried out in Section 4, where I carry the meaning explanations to their ultimate logical conclusions

Frege on sameness of sense
The doctrine of truth values
The equipollence principle
Constructive semantics in type theory
Judgments
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