Intuitionistic Logic

The Philosophy of Alternative Logics

by Andrew Aberdein

Co-authored with Stephen Read. In Development of modern logic, L. Haaparanta, ed. (Oxford: Oxford University Press, 2009), pp. 613-723.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that... more This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals.

Keywords: classical logic, logical theory, intuitionistic logic, quantum logic, relevance logic, paraconsistent logic

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

On the Possibility of a Proof-Theoretical Solution to the Categoricity Problem for Logical Inferentialism

by James Kirkpatrick

BA Dissertation

An Intuitionistic Characterization of Classical Logic

by Ming Xiong

Journal of Philosophical Logic, Vol. 37 (4), pp. 299-317, August 2008.

By introducing the intensional mappings and their properties, we establish a semantical approach of characterizing... more By introducing the intensional mappings and their properties, we establish a semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_ complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ' weakest characterization property' of intensional mappings. In particular, we show that classical logic has the weakest characterization property cl, which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property cl.

Inferentialism and the categoricity problem: reply to Raatikainen

by Ole Hjortland

Analysis, July 2009, with Julien Murzi.

Una introducción al análisis categorista de la lógica

by Luis Estrada-González

Si ese viejo profesor no creyera que la teoría de categorías es una vieja moda francesa, diría que este artículo es... more Si ese viejo profesor no creyera que la teoría de categorías es una vieja moda francesa, diría que este artículo es una buena aproximación a lo que todo lógico educado debería saber de teoría de topos.

On A.A. Markov’s attitude towards Brouwer’s intuitionism

by Ioannis Vandoulakis

Abstracts of the 14th Congress of Logic, Methodology and Philosophy of Science, Nancy, July 19-26, 2011, 159

The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his... more The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his notes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.

Peirce and Brouwer

by Conor Mayo-Wilson

Although C.S. Peirce's logic has been studied extensively, few have noticed the remarkable resemblance between his... more Although C.S. Peirce's logic has been studied extensively, few have noticed the remarkable resemblance between his ideas on continuity and those of L.E.J. Brouwer. This oversight is especially surprising because Peirce explicitly denies that the law of excluded middle holds for propositions concerning real numbers. This paper provides a detailed comparison of C.S. Peirce and L.E.J. Brouwer's concepts of continuity and the logic of real numbers. I will trace three major themes in their respective work, which highlight the striking similarities in their views about the creation, composition, and logic of the continuum.

What is Wrong with Classical Negation?

by Nils Kürbis

The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on... more The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett’s outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan ‘meaning is use’. The second general assumption I see no need to question is Dummett’s particular breed of molecularism. Some of Dummett’s assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inferences in the Dummettian framework.

READING BETWEEN THE LINES IN CONSTRUCTIVE SET THEORY

by Raymond Turner

J Logic Computation (1997) 7 (2): 229-250.
doi: 10.1093/logcom/7.2.229

We formulate and investigate various ways of (conservatively) extending Martin-Löf's type theories with separation... more We formulate and investigate various ways of (conservatively) extending Martin-Löf's type theories with separation types and choice principles and demonstrate how these extenstions can be employed to formalize Bishop's mathematical practice of hiding and recovering witnessing information.

Review of Nicola Grana, Filosofia della logica, Sentieri della logica & Logica paraconsistente

by Lorenzo Peña

Theoria Nº 2, pp. 573-77. ISSN 0495-4548.
1985

This review goes into Nicola Grana's contributions to the study of logic, especially his comparative examination of... more This review goes into Nicola Grana's contributions to the study of logic, especially his comparative examination of different systems of paraconsistent logics, including transitive logic (i.e. contradictorial gradualism). Grana's scope is broad, ranging from quantum logic to intuitionism.

KEY-WORDS:
Nicola Grana, logic, paraconsistent logics, transitive logic, contradictorial gradualism, quantum logic, intuitionism.