Foundations of Logico-Linguistics: A Unified Theory of Information, Language, and Logic

263 Pages·1978·9.65 MB·English

by  William S. Cooper (auth.)

FOUNDATIONS OF LOGICO-LINGUISTICS
SYNTHESE LANGUAGE LIBRAR Y
TEXTS AND STUDIES IN
LINGUISTICS AND PHILOSOPHY
Managing Editors:
J A A K K 0 H IN T I K K A • Academy of Finland and Stanford University
S TAN LEY PET E R S. The University of Texas at Austin
Editorial Board:
EM M 0 N B A C H • University ofM assachusetts at Amherst
J 0 A N B RES NAN. Massachuestts Institute of Technology
J 0 H N L YON S. University of Sussex
JU LIU SM. E. MORA VCSIK. Stanford University
PAT R IC K S U PPE S. Stanford University
DAN A S COT T • Oxford University
VOLUME 2
WILLIAM S. COOPER
FOUNDATIONS OF
LOGICO-LINGUISTICS
A Unified Theory 0/ In/ormation, Language, and Logic
D. REIDEL PUBLISHING COMPANY
DORDRECHT: HOLLAND / BOSTON: U. S. A.
Library of Congress Cataloging in Publication Data
Cooper, William S.
Foundations of logico·linguistics.
(Synthese language library; v. 2)
Bibliography: p.
Includes index.
1. Language and logic. 2. Information theory. I. Title.
II. Series.
P39.C68 410 78-552
ISBN·I3: 978·90·277·0876·2 e·ISBN·13: 978·94·009·9820·9
DOl: 10. I 007/978·94·009·9820·9
2-0876-0619·500·NC
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Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland
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FOREWORD
In 1962 a mimeographed sheet of paper fell into my possession. It had been
prepared by Ernest Adams of the Philosophy Department at Berkeley as a
handout for a colloquim. Headed 'SOME FALLACIES OF FORMAL LOGIC'
it simply listed eleven little pieces of reasoning, all in ordinary English, and all
absurd.
I still have the sheet, and quote a couple of the arguments here to give the
idea.
• If you throw switch S and switch T, the motor will start. There
fore, either if you throw switch S the motor will start, or, ify ou
throw switch T the motor will start.
• It is not the case that if John passes history he will graduate.
Therefore, John will pass history.
The disconcerting thing about these inferences is, of course, that under the
customary truth-functional interpretation of and, or, not, and if-then, they
are supposed to be valid. What, if anything, is wrong?
At first I was not disturbed by the examples. Having at that time consider
able personal commitment to rationality in general and formal logic in par
ticular, I felt it my duty and found myself easily able (or so I thought) to
explain away most of them. But on reflection I had to admit that my expla
nations had an ad hoc character, varying suspiciously from example to
example. Moreover, I had no idea whether these were isolated oddities or per
vasive problems: for all I knew there might be many more such examples
where those eleven came from. Under the influence of a temporary fit of
intellectual honesty I asked Adams if such was indeed the case. It was.
I then experimented with further argument forms generated more or less
randomly. It began to appear that once one got beyond the simple inference
patterns discussed in logic textbooks, English arguments for which the
classical rules oflogic failed to work right (when applied in textbook fashion)
were almost as common as those for which they did. One might almost as
well flip coins as use the classical logic to try to predict which English argu
ments would seem reasonable and which not!
vii
viii FOREWORD
My confidence in formal logic still shaken only slightly, I concluded that
either the English-to-Iogic 'translation' rules suggested in elementary text
books needed very substantial elaboration, or else that English conformed
better to one of the well-known nonclassical systems such as n-valued,
intuitionistic, or modal logic than to the classical two-valued system. The
latter possibility seemed especially real; after all, some of the founding fathers
of the nonclassical systems had expressed their dissatisfaction with the
classical conditional, explicitly mentioning its faults as a motivating factor in
the establishment of their systems. I found, however, that none of these sys
terns survived the test of actual experiment. Typically, they at first raised my
hopes by reversing the predictions of the classical logic for at least some of
the troublesome English examples for which the latter had failed. The prob
lem was, as further checking always seemed to show, that they reversed its
predictions in just as many cases where it had succeeded. The net gain in
fidelity to what seemed reasonable in English was approximately nil.
Well, there were still a number of lesser-known systems of logic to try -
systems proposed in the logico-philosophical journal literature as offering
conditional connectives more 'natural' in some senses that the standard con
ditionals. Discouragingly, none of those I examined fared much better than
their standard predecessors at capturing the properties of the colloquial
if-then and the other connectives. In fairness it must be noted that some
were never intended to do so. But I was surprised to discover that others,
which did seem to make some such claim, were typically based on only one
or two interesting but isolated examples of English usage - hardly a massive
body of evidence. Eventually I grew weary of checking out empirically unsup
ported systems and stopped. It seemed to me that a claim to have captured
the properties of the English if-then was essentially a linguistic claim, hence a
scientific claim calling for empirical justification. Anyone making such a
claim ought to accept the burden of presenting some evidence in support of
it, I thought, and not simply leave the empirical testing as an exercise to the
reader.
Of course, there is not necessarily anything wrong with a system of logic
which fails to conform to ordinary English usage. But it seemed nonetheless
legitimate to ask: If none of the usual systems of logic is the logic ofE nglish,
what is? To this question there appeared to be no convincing answer in either
the logico-philosophical or the linguistic literature. Conceivably the question
FOREWORD ix
was wrongly posed; for instance, for all I knew one of the standard systems
could still be the basic underlying logic of English, but with English-logic
translation rules far more subtle and elaborate than had been generally sup
posed. But if so, the question would merely arise in a different form, namely,
What are the translation rules?
The problem of how to discover the underlying logic of a natural language
seemed to me then a serious and important one, and it has left me without
excuse for idleness ever since. There are two general problems involved, both
formidable. The first is the matter of evidence. What sorts of empirical
observations are needed to lay bare a language's 'logical' structure? Is there a
practical informant technique, as might be hoped on the basis of experience
in conventional linguistics, or is some radically different experimental
methodology indicated? The second problem, really an elaboration of the
first, is that of constructing a foundational theory within which to interpret
or even motivate the observations. It is recognized these days that just as
scientific theories need to be tested out against observation, so one needs a
prior theoretical framework (or 'paradigm') before one can draw interesting
conclusions from the data or even know what data to gather. What then
should be the paradigm within which to explore natural language logic?
Issues relating to these questions have of course been illuminated by many
prominent thinkers, usually from the point of view of one particular disci
line. Yet the problem as a whole remains elusive. The trouble seems to be that
when the matter of the logical structure of natural language is examined from
the perspective of anyone of the existing disciplines (with the possible
exception of philosophy), the problem either cannot be perceived at all or
else appears already solved. Nor is it clear that if the problem as a whole were
to be solved, the solution would be recognized as a solution within any pres
ently existing tradition. (One is reminded of the man who saw a jigsaw puzzle
before and again after it had been assembled. When an attempt was made to
point out to him the beautiful picture which had emerged he remarked, "But
what has been accomplished? There are no new pieces there at all!")
It may be that most of what is needed to construct a unified theory of
language and logic is already at hand, lacking only the connecting links. Most
of the pieces of the jigsaw puzzle may already be in plain sight, in other
words, in which case it is high time to start fitting them together. I personally
suspect this to be the case, and hope that the present work may contribute to
x FOREWORD
the fitting-together process. To be specific, I believe that it is now possible to
start to integrate parts of mathematical logic, descriptive linguistics, the phil
osophy of language and logic, automata theory, Bayesian probability theory,
and certain areas of artificial intelligence research, into a coherent logico
linguistic theory of human communication.
Having admitted to unification as my ulterior motive, I must quickly beg
for charity. Tolerance is needed on the part of the reader of a work con
cerned with unification. Since each reader is apt to be acquainted with some
but not all of the disciplines involved, a certain amount of introductory
material must be included for every field touched upon - to the annoyance of
those already familiar with it. A related problem is that of mathematical level.
The goal of unification demands clarity as to which are the primitive terms of
the theory, which the definitions, and which the theorems - in other words a
formal mathematical development. Such a development is bound to be too
technical for some and not detailed enough for others. The compromise
adopted here is to assume familiarity with elementary logic and set theory,
the rudiments of probability theory, and a nodding acquaintance with
descriptive linguistics. To ease the mathematical burden all formal develop
ments are presented in the sans-serif typeface in which this sentence is set.
Readers interested only in the intuitive drift of the argument may wish to
skip or skim much of this material. The mathematical proofs can in any case
be omitted by those willing to take the theorems on faith.
In an area as difficult as the foundations of language and logic, the only
certainty is that any large new theory will tum out to be wrong in at least
some points, and will eventually be superseded. Theories should therefore be
stated as precisely as possible, not because their proponents are sure they are
correct, but because an exact statement makes it easier for others to discover
wherein they err and to improve them. It is in the spirit of this dialectic pro
cess that the present theory is set forth in rigorous mathematical dress (in the
sans-serif type) as well as by looser intuitive arguments.
Colleagues have commented that Chapter 8, in which the previously
developed abstract theory is applied in a case study of an actual linguistic
construction (the English conditional), makes everything else coalesce and
ought on all sound heuristic principles to come at the beginning. They agree,
however, that logically it has to remain near the end because it treats of
hypotheses which cannot even be stated precisely, let alone tested scientifically,
FOREWORD xi
without benefit of the preceding theoretical development. I think the advice
they would have me convey to you is this: If the going gets uncomfortably
abstract, don't give up until you reach Chapter 8, and if necessary skip to it.
Many friends and colleagues influenced the ideas in this book and
encouraged the writing of it. While I don't suppose they would go so far as to
share the blame for it, still I'd like to mention some of them. Victor Yngve of
the University of Chicago was kind enough to comment on the first draft. It
was Professor Yngve who first stimulated my interest in scientific language
study as early as 1958, and who several years later drew to my attention the
possible significance of automata theory as a vehicle for serious pragmatic
language investigations. Phyllis Baxendale, then at the !.B.M. Research Lab
oratory in San Jose, made possible some early computer experimentation in
an area of overlap between logic and linguistics. Don Swanson of the Univer
sity of Chicago commented on an early draft, after having deaned into exist
ence the favorable conditions which allowed it to be written. My indebted
ness to Ernst Adams, particularly in the matter of if-then, should be obvious.
I have also benefited from conversations on that topic with Brian Skyrms of
the University of Illinois. M. E. Maron of the University of California at
Berkely offered some valuable expository advice on the early chapters. Patrick
Wilson of the same institution supplied specific criticisms and general
encouragement in exquisitely balanced proportions. The exposition has also
profited from a number of specific suggestions by J. L. Kuhns of Operating
Systems, Inc. lowe much to Paul Huizinga of the University of California and
Ian Carlstrom, now of Case Western Reserve University, for checking the
mathematical proofs and in one instance suggesting a stronger theorem. I was
assisted by the thoughtful commentary of the publisher's reviewer. I am also
grateful for the extensive and very useful notes of another reviewer whose
identity is unknown to me but is presumably known to him.
Berkeley, April 1977 W.S.C.