The History of Philosophical and Formal Logic: From Aristotle to Tarski

337 Pages·2017·4.13 MB·English

by  Alex Malpass

Th e History of Philosophical and
Formal Logic
i
Also available from Bloomsbury
Th e Bloomsbury Companion to Aristotle, edited by Claudia Baracchi
Th e Bloomsbury Companion to Bertrand Russell, edited by Russell Wahl
Th e Bloomsbury Companion to Philosophical Logic, edited by Leon Horsten
and Richard Pettigrew
Philosophical Logic, George Englebretsen and Charles Sayward
ii
Th e History of Philosophical and
Formal Logic
From Aristotle to Tarski
Edited by
Alex Malpass and Marianna Antonutti Marfori
Bloomsbury Academic
An imprint of Bloomsbury Publishing Plc
LONDON • OXFORD • NEW YORK • NEW DELHI • SYDNEY
iii
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© Alex Malpass, Marianna Antonutti Marfori and Contributors, 2017
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A catalogue record for this book is available from the British Library.
ISBN : HB : 978-1-4725-1350-2
e PDF : 978-1-4725-0525-5
ePub: 978-1-4725-0717-4
Library of Congress Cataloging-i n-Publication Data
Names: Malpass, Alex, editor. | Marfori, Marianna Antonutti, editor.
Title: The history of philosophical and formal logic from Aristotle to Tarski
/ edited by Alex Malpass and Marianna Antonutti Marfori.
Description: London, UK; New York, NY, USA: Bloomsbury Academic, an imprint
of Bloomsbury Publishing, Plc, [2017] | Includes bibliographical
references and index.
Identifi ers: LCCN 2016039911| ISBN 9781472513502 (hb) | ISBN 9781472505255
(epdf)
Subjects: LCSH: Logic--History.
Classifi cation: LCC BC15 .H57 2017 | DDC 160--dc23 LC record available at
https://lccn.loc.gov/2016039911
Cover design: Catherine Wood
Cover image © Dylan Griffi n
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iv
Contents
Preface vii
Introduction 1
Part I Th e Origins of Formal Logic
1 Aristotle’s Logic A driane Rini 29
2 Stoic Logic Katerina Ierodiakonou 51
3 Medieval Logic S ara L. Uckelman 71
Part II Th e Early Modern Period
4 Leibniz Jaap Maat 101
5 Bolzano J önne Kriener 121
6 Boole Giulia Terzian 143
Part III Mathematical Logic
7 C.S. Peirce Peter Øhrstrøm 165
8 Frege Walter B. Pedriali 183
9 Peano and Russell Alexander Bird 229
10 Hilbert C urtis Franks 243
Part IV Twentieth- Century Logic
11 Gödel P .D. Welch 269
12 Tarski Benedict Eastaugh 293
Index 315
v
vi
Preface
Students of introductory logic courses usually encounter the fundamental
concepts of formal logic – notions like valid argument, consistency, truth in a
model, soundness, completeness, and so on – for the fi rst time through those
courses. However, these concepts are oft en presented in an ahistorical way, which
can make them seem highly abstract, inaccessible, and set in stone, rather than
the result of many centuries of conceptual development and refi nement.
Understanding how the current shape of these concepts is the result of their
historical development can make them appear less abstract and artifi cial, and
thus more comprehensible. Furthermore, logicians of previous centuries
understood fundamental logical questions and concepts diff erently than we
typically do now. Th e richness of their ideas and approaches to logic have an
intrinsic interest, but they may also provide a valuable source for philosophically-
minded students who seek to challenge the currently dominant approach to
philosophical and formal logic.
Th e History of Philosophical and Formal Logic: From Aristotle to Tarski is a
partial history of these fundamental questions and concepts, beginning with the
work of Aristotle, the founder of the discipline, and ending with that of a modern
giant, Alfred Tarski. Each chapter has been written especially for this collection
and focuses on the work of a particular fi gure or school, with the fi rst part (‘Th e
Origins of Formal Logic’) concentrating on Ancient Greek logic (Aristotle and
the Stoics), together with the logical work of the scholastics in the Middle Ages.
In the second part (‘Th e Early Modern Period’) we see how Gottfried Leibniz’s
mathematizing approach to logic in the 17th century was followed by nineteenth
century thinkers with a similar cast of mind, Bernard Bolzano and George Boole.
Th ey can be seen as initiators of the mathematical revolution in logic that took
place in the late nineteenth and early twentieth centuries. Th e third part,
‘Mathematical Logic’, is concerned with fi gures whose work exemplifi es that
revolution: C.S. Peirce, Gottlob Frege, Giuseppe Peano, Bertrand Russell, and
David Hilbert. Th e fi nal part, ‘Twentieth-century Logic’, assesses the work of two
vii
viii Preface
key fi gures in the development of contemporary mathematical logic, Kurt Gödel
and Alfred Tarski, who together shaped much of the course of logic in the
twentieth century. Each of the aforementioned fi gures was surrounded by other
logicians whose achievements, and their importance for the development of
logic, we were unfortunately unable to explore more extensively in this book.
Th e History of Philosophical and Formal Logic was conceived as a way for
undergraduate students with little training in formal logic to discover the roots
of logical concepts. However, we also hope that it provides a starting point for
anyone with an interest in this wonderful di scipline to become acquainted with
its history. By producing an introductory book whose chapters span the entire
history of the discipline, we aim to show how the elements of the standard
undergraduate logic curriculum took their current form only relatively recently,
emerging from a long development by many hands whose interests and
motivations varied widely. Moreover, by shedding light on a few topics that have
not been given as much attention in standard histories of logic, such as the
contributions of Leibniz and Bolzano, as well as the diagrammatic logics of C.S.
Peirce, we hope to play a part in encouraging a broader view of the history of
logic, which as a discipline surely has much to gain in being more inclusive of
other fi gures and traditions. We only regret not having been able to contribute
more in this regard ourselves.
Th is book was born from a series of lectures on the history of logic at the
University of Bristol, organised by Alex Malpass. Aimed at undergraduates in
philosophy as well as members of other departments and faculties, the lectures
were given by philosophers and logicians at Bristol, together with guest lectures
from researchers at other institutions. In order to bring the content of these lectures
to a wider audience, the speakers in the lecture series were invited by Alex Malpass
to contribute to this book. In the end, the chapters ‘Bolzano’ by Jönne Kriener,
‘Boole’ by Giulia Terzian, ‘Peano and Russell’ by Alexander Bird, and ‘Tarski’ by
Benedict Eastaugh were written by speakers in the lecture series, while the other
chapters were contributed by international experts invited by the editors.
Marianna Antonutti Marfori joined the project in 2014 to complete the book.
Th e writing of the Introduction refl ects the respective expertise of the two
editors: the fi rst half, from Aristotle to Boole, was written by Alex Malpass, while
the second half, from Peirce to Tarski, was written by Marianna Antonutti
Marfori.
Marianna Antonutti Marfori
Paris and Munich
March 2017
Introduction
Put simply, logic deals with the inference of one thing from another, such as the
inference of a conclusion from the premises of an argument. An example of this
would be the following:
Th e emission of certain chemicals that harm the ozone layer is likely to cause
global warming, and petrol- consuming vehicles emit these chemicals in large
quantities; therefore, global warming is likely.
Premises can either ‘succeed’ or ‘fail’ at ensuring that the conclusion holds, and
success means that whenever the premises are true, the conclusion is either also
true or at least more likely to be true. In the previous example, the premises
about the harmfulness of certain emissions and the fact that petrol- based
vehicles emit these chemicals in large quantities supports the conclusion that
global warming is likely. If the premises are indeed successful in establishing the
conclusion, then we call the resulting inference valid . Logic can be thought of as
the study, or perhaps even the ‘science’, of v alid inference . In this book we will be
looking at the core of this science of inference, by which we mean d eductive logic .
In a valid deductive argument, the truth of the premises ensures the truth of the
conclusion, such as in the following famous argument:
All men are mortal, Socrates is a man; therefore, Socrates is mortal.
In this example, one cannot grant the truth of the premises without also granting
that Socrates is mortal – the conclusion follows deductively from the premises.
Deductive logic is only one branch of logic, however, and there is also inductive
logic. Generally, inductive logic covers arguments where the conclusion is not
established conclusively from the premises, but the likelihood of its being true is
supported by the premises. Consider the following argument:
All observed swans have been white; therefore, the next swan to be observed will
be white.
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