Table Of ContentTHEMES IN NEOPLATONIC AND
ARISTOTELIAN LOGIC
Were the most serious philosophers of the millennium A.D. 200 to A.D. 1200 just
confused mystics?This book shows that such was not the case. John Martin
rehabilitates Neoplatonism, founded by Plotinus and brought into Christianity by St
Augustine. The Neoplatonists devise ranking predicates like good, excellent, perfect
to divide the Chain of Being, and use the predicate intensifier hyper so that it
becomes a valid logical argument to reason from God is not (merely) goodto God
is hyper-good. In this way the relational facts underlying reality find expression in
Aristotle’s subject–predicate statements, and the Platonic tradition proves able to
subsume Aristotle’s logic while at the same time rejecting his metaphysics. In the
Middle Ages, when Aristotle’s larger philosophy was recovered and joined again to
the Neoplatonic tradition which was never lost, Neoplatonic logic lived alongside
Aristotle’s metaphysics in a sometimes confusing and unsettling way.
Showing Neoplatonism to be significantly richer in its logical and philosophical
ideas than it is usually given credit for, this book will be of interest not just to
historians of logic, but to philosophers, logicians, linguists, and theologians.
For
Jenefer
ATrue Intellective-Unintelligible
AWhole-Before-the-Part
Themes in Neoplatonic and
Aristotelian Logic
Order, Negation and Abstraction
John N. Martin
University of Cincinnati, USA
First published 2004 by Ashgate Publishing
Published 2016 by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
711 Third Avenue, New York, NY 10017, USA
Routledge(cid:3)is(cid:3)an(cid:3)imprint(cid:3)of(cid:3)the(cid:3)Taylor(cid:3)&(cid:3)Francis(cid:3)Group,(cid:3)an(cid:3)informa(cid:3)business
Copyright ©John N. Martin 2004
All rights reserved. No part of this book may be reprinted or reproduced
or utilised in any form or by any electronic, mechanical, or other means,
now known or hereafter invented, including photocopying and
recording, or in any information storage or retrieval system, without
permission in writing from the publishers.
Notice:
Product or corporate names may be trademarks or registered trademarks,
and are used only for identification and explanation without intent to
infringe.
John N. Martin has asserted his moral right under the Copyright, Designs and
Patents Act, 1988, to be identified as the author of this work.
British Library Cataloguing in Publication Data
Martin, John N.
Themes in Neoplatonic and Aristotelian logic : order,
negation and abstraction
1. Neoplatonism
I. Title
186.4
Library of Congress Cataloging-in-Publication Data
Martin, John N.
Themes in Neoplatonic and Aristotelian logic : order, negation and abstraction /
John N. Martin.
p. cm.
Includes bibliographical references and indexes.
1. Logic 2. Neoplatonism. 3. Logic, Ancient. 4. Syllogism—History. 5. Aristotle
I. Title.
BC57.M365 2003
160'.9—dc21
2003043684
(cid:11)(cid:21)(cid:4)(cid:16)(cid:561)(cid:351)(cid:349)(cid:350)(cid:342)(cid:349)(cid:347)(cid:346)(cid:348)(cid:342)(cid:350)(cid:343)(cid:343)(cid:342)(cid:561)(cid:507)(cid:145)(cid:139)(cid:148)(cid:508)
Contents
Preface vii
Introduction xi
1. Aristotle’s Natural Deduction Reconsidered 1
2. Ecthesis and Existence in the Syllogistic 19
3. Existence, Negation, and Abstraction in the Neoplatonic Hierarchy 25
4. ATense Logic for Boethius 53
5. Proclus on the Logic of the Ineffable 65
6. Proclus and the Neoplatonic Syllogistic 79
7. Ammonius on the Canons of Proclus 125
8. All Brutes are Subhuman: Aristotle and Ockham on Privative Negation 139
9. /Lukasiewicz’s Many-valued Logic and Neoplatonic Scalar Modality 167
Bibliography 195
Index of Names 201
Index of Topics 203
v
Preface
Neoplatonisim
The essays in this book are an effort to understand Neoplatonism. Relatively
speaking, Neoplatonism is a tradition ignored by Anglo-American philosophy. The
reasons are partly intellectual but also I think partly cultural. Neoplatonism is
viewed as essentially confused on an elementary level. Even a child would grant
that every multitude entails a whole, but only the muddled would conclude that
therefore there is a One prior in being to everything. Everyday counter-examples
easily refute the Neoplatonic claim that good, beautiful, real, spiritual, and knowing
are coextensive, and anyone who seriously thinks evil in unreal is morally naïve.
The multitude of gods, the levels of causation, the priority of soul and the heavens
to bodily events are doctrines that shade into religion, mysticism, and magic. As a
result, few graduate students read Plotinus and Proclus, to say nothing of lesser
figures in Hellenistic philosophy. Many primary texts are not available in English,
and although there is a handful of first-rate English-speaking scholars, a large part
of the secondary literature is in foreign languages. Neoplatonic scholars are very
often members of religious orders and interested in philosophical theology, or are
schooled in continental philosophy and use its idioms. Thus, typical American
philosophy students, and their teachers, could not read much of the material even if
they wanted to, and in any case are generally sceptical of the schools of philosophy
that generate it.
But Neoplatonism dominated philosophy for a millennium. Could it be that for
1000 years the best minds in philosophy were elementarily confused, linguistically
incompetent, and morally naïve? Mysticism and religion are just unfortunate human
tendencies, one is told, that surface even in serious philosophy if we are not careful.
But I had my doubts. Priority, existence, coextension, and negation are topics in
logic and semantics, and as the history of scholarship in mediaeval philosophy has
shown, it is possible without training in modern mathematics and logic to study a
period for generations without seeing what is there. I wanted to see for myself, and
these papers describe the results.
The papers in this collection are presented in the historical order of the
philosophers they discuss: Aristotle before Plotinus, Plotinus before Proclus, and so
on. But that is not the order in which they were written. “A Tense Logic for
Boethius” (Chapter 4) was my initial effort at applying the methods of modern logic
and formal semantics to ancient logic. It gave me confidence to pursue Plotinus, the
patriarch of the tradition. The result was “Existence, Negation, and Abstraction in
the Neoplatonic Hierarchy” (Chapter 3). In this paper I argue that the mysticism in
the Enneadsis accompanied by some logically interesting theory. Plotinus exploits
properties of scalar adjectives only recently studied in linguistics, and marshals
vii
viii Themes in Neoplatonic and Aristotelian Logic
them in structures of an algebraic character by means of negation operations
characteristic of the Neoplatonic tradition. When formulated in this way, the views
amount to serious positons in modern logic and the philosophy of language.
My excursion into the “logic” of Plotinus made it clear to me that I needed to
work out in a satisfactory way Aristotle’s syllogistic, the theory that lies at the
background of all ancient logic and that was bent to new purposes by Neoplatonists.
I found that the standard account of the syllogistic as developed in the 1960s by
Smiley and Corcoran could be abstracted to a lattice theoretic semantics and given
a more standard Henkin style completeness proof. These results are outlined in
“Aristotle’s Natural Deduction Reconsidered” (Chapter 1) and “Ecthesis and
Existence in the Syllogistic” (Chapter 2).
I was ready then for attempting to understand Proclus, who is without question
the most logically sophisticated of Neoplatonic thinkers. Proclus is a bit off-putting
to the modern reader. He wrote prayers to the Sun. The academy under his
leardership was a religious institution adhering to a mystical creed, and as its head
he officiated at religious rituals. Indeed, when Adrian the Apostate was restoring the
Pagan gods he was not so much righting fallen statues as he was promulgating the
sort of Platonic theology represented by Proclus. What could be more foreign to
modern sympathies than the theology of the henads?
Modern readers may need to be reminded, however, that religious commitments
are compatible with good philosophy. We are used to pulling the philosophy apart
from the religion in the writings of Augustine and Aquinas, indeed even those of
Cantor and Gödel. Proclus, who lectured regularly on mathematics and on the full
spectrum of works by Plato and Aristotle, including the Organon, made interesting
contributions to geometry and logic.
His logical ideas are explored in “Proclus on the Logic of the Ineffable” (Chapter
5), an introductory essay, and more fully in “Proclus and the Neoplatonic
Syllogistic” (Chapter 6). Proclus uses Neoplatonic scalar negations, I argue, to
convert the Platonic tree of diairesis (Porphyry’s version of Aristotle’s hierarchy of
genera and species) into the Chain of Being (a total ordering under a causation
relation), and then reasons about this ordering using the syllogistic in combination
with special negations. It proves possible to explain why the syllogistic works in this
structure by appealing to the lattice semantics of Chapter 1. In “Ammonius on the
Canons of Proclus” (Chapter 7) I show how Proclus uses Neoplatonic negations to
replicate in his logic the Aristotelian law of obversion.
The collection concludes with two studies of Neoplatonic ideas in later logic. In
“All Brutes are Subhuman: Aristotle and Ockham on Privative Negation” (Chapter
8) two closely related notions of privative negation are distinguished, one
Neoplatonic and one more Aristotelian. The Neoplatonic operation consists of
moving down an ordering, in this case the ontic hierarchy, from a higher to a lower
node. The second is the notion of privation that mediaeval logicians abstract from
Aristotle and which forms the core of the modern sense of privative negation in
linguistics. It is a variety of the non-natural relative complementation of a species
within a genus. The two notions are closely related in their theoretical roles and are
interdefinable. Lastly, in “/Lukasiewicz’s Many-valued Logic and Neoplatonic
Scalar Modality” (Chapter 9) I argue that Neoplatonism provides the key to
Preface ix
understanding the modal interpretation of the truth-values that /Lukasiewicz puts
forward for his many-valued logic. He is serious when he reads the values as
necessary, true, andpossible, but he intends these in the senses familiar from the
traditional philosophy he learned as a student. This tradition incorporated
metaphysical uses of modal and semantic terms that derive from Christian
Neoplatonism.
Publication Data
The original publication details for those papers in the collection that have been
published previously are as follows: “Aristotle’s Natural Deduction Reconsidered”,
History and Philosophy of Logic 18 (1997) 1–15; “Existence, Negation, and
Abstraction in the Neoplatonic Hierarchy”, History and Philosophy of Logic 16
(1995) 169–196; “ATense Logic for Boethius”, History and Philosophy of Logic10
(1989) 203–212; “Proclus and the Neoplatonic Syllogistic”, Journal of
Philosophical Logic 30 (2001) 187–240; “Proclus on the Logic of the Ineffable”,
Proceedings of the Society for Medieval Logic and Metaphysics 2 (2003); “All
Brutes are Subhuman:Aristotle and Ockham on Privative Negation”, Synthese134
(3) (2003) 429–461; “/Lukasiewicz’s Many-Valued Logic and Neoplatonic Scalar
Modality”, History and Philosophy of Logic 23 (2002) 95–120. As required by
copyright, these are reproduced unaltered, apart from minor changes of style/design.
That the previously published papers must be reproduced in their original form
necessitates a certain amount of repetition. Chiefly affected is the lattice semantics
for the syllogistic, which is set out in varying detail in Chapters 1, 6, 7, and 8, and
the linguistics of scalar predicates, repeated in Chapters 3, 5, 6, 8, and 9. Reprinting
the papers in full, however, has the virtue of keeping each self-contained, and allows
narrow technical ideas, which depending on context occasionally vary slightly in
definition, to be adapted to the subject matter. Footnotes that have been added for
this volume are distinguished by an asterisk and are enclosed in square brackets.
The author wishes to express his gratitude to John Bickle and the Philosophy
Department at the University of Cincinnati for arranging his teaching schedule so as
to allow for the preparation of this volume, and to Jeff Wismann for his help in
proofreading.
