Intuitionistic Proof Versus Classical Truth
Intuitionistic Proof Versus Classical Truth

Author: Enrico Martino

Publisher: Springer

Published: 2018-02-23

Total Pages: 173

ISBN-13: 3319743570

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This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers – both new and previously published – it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer’s idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer’s work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics.

What Truth is
What Truth is

Author: Mark Jago

Publisher: Oxford University Press

Published: 2018

Total Pages: 369

ISBN-13: 0198823819

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Mark Jago offers a new metaphysical account of truth. He argues that to be true is to be made true by the existence of a suitable worldly entity. Truth arises as a relation between a proposition - the content of our sayings, thoughts, beliefs, and so on - and an entity (or entities) in the world.

The Boundary Stones of Thought
The Boundary Stones of Thought

Author: Ian Rumfitt

Publisher:

Published: 2015

Total Pages: 369

ISBN-13: 0198733631

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Classical logic has been attacked by adherents of rival, anti-realist logical systems: Ian Rumfitt comes to its defence. He considers the nature of logic, and how to arbitrate between different logics. He argues that classical logic may dispense with the principle of bivalence, and may thus be liberated from the dead hand of classical semantics.

A Short Introduction to Intuitionistic Logic
A Short Introduction to Intuitionistic Logic

Author: Grigori Mints

Publisher: Springer Science & Business Media

Published: 2000-10-31

Total Pages: 130

ISBN-13: 0306463946

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Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs to make the material more accessible. The presentation is based on natural deduction and readers are assumed to be familiar with basic notions of first order logic.

Proof Methods for Modal and Intuitionistic Logics
Proof Methods for Modal and Intuitionistic Logics

Author: M. Fitting

Publisher: Springer Science & Business Media

Published: 1983-04-30

Total Pages: 574

ISBN-13: 9789027715739

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"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics (many introduced by Lemmon). We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions (generally with and without the Barcan formula). Finally, we present the full variety of proof methods for Intuitionistic logic (and of course Classical logic too). We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature.

Logical Pluralism
Logical Pluralism

Author: JC Beall

Publisher: Oxford University Press

Published: 2006

Total Pages: 152

ISBN-13: 0199288402

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Consequence is at the heart of logic, and an account of consequence offers a vital tool in the evaluation of arguments. This text presents what the authors term as 'logical pluralism' arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them.

Constructivism in Mathematics, Vol 1
Constructivism in Mathematics, Vol 1

Author: A.S. Troelstra

Publisher: Elsevier Science

Published: 1988-07-15

Total Pages: 355

ISBN-13: 9780444702661

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These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.

Kurt Gödel and the Foundations of Mathematics
Kurt Gödel and the Foundations of Mathematics

Author: Matthias Baaz

Publisher: Cambridge University Press

Published: 2011-06-06

Total Pages: 541

ISBN-13: 1139498436

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This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Logical Options
Logical Options

Author: John L. Bell

Publisher: Broadview Press

Published: 2001-03-30

Total Pages: 313

ISBN-13: 1551112973

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Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy students and professional philosophers who have learned some classical first-order logic and would like to learn about other logics important to their philosophical work.