Intermediate Quantities: Logic, Linguistics and Aristotelian Semantics

294 Pages·2017·41.28 MB·English

by  Philip L. Peterson

INTERMEDIATEQUANTIFIERS
Intermediate quantifiers express logical quantities which fall between Aristotle's
two quantities of categorical propositions - universal and particular. "Few", "many"
and "most" express the most commonly referred to intermediate quantities, yet this
book argues that an infinite number of additional quantities can be understood
through a deeper examination of the logical nature of all intermediate quantifiers.
Presenting and analyzing the logical and linguistic features of intermediate
quantifiers, in a fashion typical of traditional logic, Peterson presents the first account
to integrate the logic and semantics of intermediate quantifiers with the two traditional
quantities by traditional methods. Having introduced the basic idea of how to
approach the task in Chapter 1, with heavy emphasis on the linguistic meanings and
ordinary uses of English intermediate quantifier expressions, Peterson then
undertakes the task of completely integrating the three basic intermediate quantities
into traditional logic in Chapter 2. Drawing on the work of Robert Carnes and
taking a critical look at James McCawley's grammatical analysis, the author then
provides further revisions, extensions and explorations into logical inference,
linguistic meaning, algebraic methods, rules of infinite-quantity syllogism to reach
the conclusion that a new approach to foundations of mathematics, based on the
syllogistic logic of quantifiers, is possible to produce a new explanation of classical
distribution and an extension of "infinite-quantity" syllogistic to relations.
Considerable attention has been paid in recent decades to "generalized quantifiers",
and this original completion of an explanation for extending traditional syllogistic
logic to handle intermediate quantifiers offers invaluable insights for those studying
across areas of logic, linguistics, and the philosophy or semantics of natural language.
This book is dedicated to Laura Anne
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Intermediate Quantities
Logic, linguistics, and Aristotelian semantics
PHILIP L. PETERSON
Professor ofP hilosophy
Department ofP hilosophy, Syracuse University
New York, USA
First published 2000 by Ashgate Publishing
Reissued 2018 by Routledge
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Copyright © Philip L. Peterson 2000
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A Library of Congress record exists under LC control number: 99069528
ISBN 13: 978-1-138-70605-7 (hbk)
ISBN 13: 978-1-315-20201-3 (ebk)
Contents
Preface vii
Introduction 1
1 "Few", "Many", and "Most" 15
2 The Compleat Syllogistic: Including Rules
and Venn Diagrams for Five Quantities
(with Robert Carnes) 47
3 The Grammar of Some English Quantifiers 147
4 Complexly Fractionated Quantifiers 203
5 Distribution and Proportion 227
6 Reasonings About Relations 257
Bibliography 279
Index 283

Preface
I did not begin to investigate the intermediate quantifiers with the aim of
applying or defending the traditional syllogism of Aristotle and his
medieval followers. Nothing was further from my mind. Rather, I began
(over 25 years ago) by wondering about the meanings of the "other"
quantifiers—those other than the universal and existential quantifiers of
predicate logic—because of assumptions used in then current discussions
in and about generative semantics. (For some early discussion about
generative semantics, see my Concepts and Language, Mouton 1973.) It
seemed to me that some of the arguments about examples depended on
misunderstanding the meanings of the natural language
quantifiers—especially, in contrast to what use of those quantifiers
implied. I return to this theme (basic meanings versus implications) in
Chapter 3 herein. It seemed to me then that no one could be clear about
what was meant by "many", for example, unless what was meant by its
denial and/or negation were understood. And, of course, understanding
that would require relating it to the uses of other quantifiers and to their
negations and denials. I found nothing in my mind, or in the
conversations and writings of others, about the meanings and logical
roles of these natural language quantifiers until I began to think of them
in the traditional manner—by examining appropriate affirmations,
denials, contraries, contradictories, and entailments that contained the
quantifiers.
The first results of the examination constitute Chapter 1 herein. So,
some use of traditional concepts from the syllogism paid off with respect
to getting the initial logical relations between some of these quantifiers
correctly described. I might well have stopped there and used this
achievement to re-examine the arguments about generative semantics
that had stimulated me. However, I did not do that—and haven't done it
yet. (And since generative semantics appears to be a moribund
alternative in current research on generative-transformational grammars,
I don't suppose I ever will re-examine those arguments.) Rather, what
happened next was that I talked to Robert Carnes about these quantifiers.
That conversation led to a full-scale syllogistic treatment of the three
new quantities—quantities expressible in English with "few", "many",
and "most", among other words and phrases. Again, what began as a
curious exercise turned out to show that the traditional syllogism was
very useful. For we quickly constructed a clear extension of the
syllogism to cover the three new quantities—entirely within the
traditional spirit. Distribution, for example, still plays a prominent role.
We changed the traditional syllogism from being a two-quantity system
(particular and universal) to being a five-quantity system (particular,
universal, and three intermediate quantities). The complete account—co-
authored with Robert Carries—constitutesChapter 2.
Speaking for myself (not for Carnes), I learned more about the basic
Aristotelian syllogism while carrying out this extension than I knew
about it before. I used to be a typical 20th century Anglo-American
teacher of symbolic logic, the kind who gives a 20 minute lecture each
semester on the syllogism—regarding it as, at best, an important
achievement whose time has passed. Today, I have changed my mind.
The syllogism isnot merely of historical interest. For Carnes and I have
shown, by extending it, that it has considerable value for current
research. We have not defended the syllogism by answering various
well-known objections to it (though I would be much more optimistic
about that now). Rather, wehave indirectly defended itby showing what
more it can do —when extended in the right way. It can provide a rich
foundation for ourunderstanding of the logical nature of all intermediate
quantifiers—not just "few", "many", and "most", but (as I show in
Section 5ofChapter 2,and inChapter 4) aninfinite number of additional
quantities. Only a little imagination is required to see that a new
approach to foundations of mathematics—based on the syllogistic logic
of quantifiers (via "fractionals")—might well be possible as an
alternative to the usual formal-system-with-set-theoretic-semantics
foundations. As shown in Chapter 5,this defense-by-extension produces
a new explanation of classical distribution (answering objections from
Geach, for example). Finally, how to extend the "infinite-quantity"
syllogistic torelations istreated inChapter 6.
The challenge that results from the syllogistic treatment of
intermediate quantifiers for empirical linguistic semantics is simply this:
confirm (or disconfirm if you can) via empirical observations and
analyses that sentences containing intermediate quantifiers do imply
other sentences in exactly the way that the 81 intermediate syllogisms
say they do (the 81 are the 105valid 5-quantity syllogisms minus the 24
valid ones of the traditional 2-quantity syllogism). It does not appear
that the meanings of natural language quantifiers have to be the way
these syllogistic entailments predict they are. But I claim (as an
empirical hypothesis) that they are—if all the relevant assumptions and
presuppositions of the extended syllogism are honored. (Honoring these
assumptions does not beg the empirical question, but only provides
legitimate ground rules for testing natural language examples. One
model for how to begin the empirical investigations occurs in my
"Event", with Kashi Wali, in Linguistic Analysis 1985b, reprinted in
Chapter 4 of my Fact Proposition Event, Kluwer 1997.) And the fact
that these logical meanings will be confirmed for the intermediate
quantifier words andphrases contributes to showing that the syllogistic
Vlll
framework is not only useful for characterizing the logical features of
natural languages, but that the syllogism is the most (not the least)
appropriate and fruitful approach to describing and explaining the "logic
of natural language" today—a conclusion reached in an entirely different
way by Fred Sommers in his The Logic of Natural Language (Oxford
1982).
Sommers defends the virtues of the traditional logic with ingenious
new devices and relates it to several topics in contemporary logical
research. We, on the other hand, do not defend the traditional doctrine
directly (but rather presume it), nor do we make use of Sommers'
innovations—not being aware of them until after we finished the basic
components of Chapter 2, before his book was published. Although our
account does not lean on Sommers' developments at all (and appears on
the surface tobe somewhat at odds with them), still the separation of our
extension from Sommers' analysis also contributes support for the
relevance of the syllogism today. For two entirely different and
unrelated projects of natural language analysis hit upon the syllogism as
exactly what was needed for their purposes. And they each
worked—even if much is left to be done, such as combining the insights
ofboth into acoherent super-syllogism for tomorrow.
I am very grateful to Robert Carnes for his major efforts in carrying
out the research herein. I could not have devised and completed Chapter
2 alone. On the other five chapters, his advice, criticisms,
encouragement, and hypotheses were also vital. I might have completed
those parts alone, but not nearly so well. Mark Brown has also helped
(see Chapter 2). Two linguists have been very helpful—Guy Carden in a
brief conversation years ago, and more recently James McCawley. I am
very pleased that McCawley could make use of Chapter 1 in his
Everything that Linguists have Always Wanted to Know about Logic
(University of Chicago Press, 1981). For it led to my thinking more
clearly about meanings, propositions, and inferences (as detailed in
Chapter 3)—as well as to certain criticisms of McCawley's
transformational grammar rule for "there" insertion.
I am also grateful to Syracuse University for providing me with
research leaves, and associated international travel grants, which were
absolutely necessary for completing this research and some related
investigations in epistemology and the philosophy of language. And I
am grateful for permission to reprint portions of my articles to Notre
Dame Journal of Formal Logic (in Chapter 1and sections 5.2 and 5.4 of
Chapter 2) and Journal of Philosophical Logic (in Chapters 4 and 5). I
thank Bret Garwood, Julio Lora, and Anne Sullivan Peterson for help
with the computer formatting ofthe text.
Some "modern logicians" have manifested an almost religious zeal in
trying toquash anything in logic research which favors the traditional
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