FOM: Re: Arbitrary Objects
charles silver
silver_1 at mindspring.com
Sat Feb 9 10:26:46 EST 2002
Fine, Kit, _Arbitrary Objects_ (Aristotelian Society Series, Vol 3,
Blackwell, Oxford, 1985).
TABLE OF CONTENTS:
Preface
Introduction
I... The General Framework
II...Some Standard Systems
III..Systems in General
IV..Non-Standard Systems
The Preface begins:
"This book had its origin in the classroom. I was teaching natural
deduction to a group of students and ahd come to the point at which the rule
of universal generalization is introduced. I had wanted to give an
explanation of the rule in terms of arbitrary objects. But my sense of
rigour got in the way, and I gave instead an explanation in terms of
schematic names.
When I left the classroom, I gave the matter some more thought. I hold
it as a general methodological principle that when there is a clash between
intuition and rigour, when one's sense of rigour prevents one from saying
what, from an intuitive point of view, it seems that one can say, then it is
rigour and not intuition that should give way. Applying this principle to
the case at hand, it seemed that there should be an account of arbitrary
objects upon the basis of which a satisfactory explanation of the rule of
universal generalization could be given. It was the attempt to develop such
an account that led to the present work."
The Introduction begins:
"This book deals with certain problems in understanding natural
deduction and ordinary reasoning." He singles out two rules for special
consideration: universal generalization (UG) and existential instantiation
(EI).
Part I, The General Framework, begins with Chapter 1, Arbitrary Objects
Defended. In this section, he says (p. 5) "An arbitrary object has those
properties common to the individual objects in its range. So, an arbitrary
number is odd or even, an arbitrary man is mortal, since each number is odd
or even, each individual man is mortal...."
"Such a view used to be quite common, but has now fallen into
disrepute." Fine says that "Frege led the way" in this, and "[w]here Frege
led, others have been glad to follow. Among the many subsequent
philosophers who have spoken against arbitrary objects, we might mention
Russell, Lesniewski, Tarski, Church, Quine, Rescher, and Lewis. [Fine
provides references for each.]..."
"In the face of such united opposition, it might appear rash to defend
any form of the theory of arbitrary objects. But that is precisely what I
intend to do. Indeed, I would want to claim, not only that a form of the
theory is defensible, but also that it is extremely valuable. In
application to a wide variety of topics -- the logic of generality, the use
of variables in mathematics, the role of pronouns in natural language -- the
theory provides explanations that are as good as those of standard
quantification theory and sometime" (p.6)
He then mounts a defense of arbitrary objects by replying to criticisms
of them, and in the process of responding to criticisms reveals what exactly
he takes arbitrary objects to be.
After he has dealt with criticisms of arbitrary objects, a theory of
them emerges, for which he then provides a technical account in Chapter 2
(of Part I), The Models. Whereas the earlier discussion was exclusively
philosophical, this chapter is quite technical. He develops the model
theory for arbitrary objects and proves a number of lemmas pertaining to
them. One aspect of this which deserves to be singled out is his account,
both philosophical and mathematical, of how certain arbitrary objects come
to depend on others. This leads to "dependency relations" among arbitrary
objects, which, later on, is more fully developed (and applied to various
standard natural deduction systems) in terms of "dependency diagrams." Fine
provides fully convincing reasons (philosophical and technical) that a
satisfactory account of arbitrary objects must take their dependency
relationships into consideration.
The "A-model," explained in Chapter 2, then has various conditions
applied to it in Chapter 3, The Conditions. Four types of conditions on
A-models are considered: i) the extendibility of value assignments, ii) the
existence of A-objects (i.e., arbitrary objects), iii) their identity, and
iv) their multiplicity. Incidentally, he also calls A-objects "generic
objects," and A-models "generic models," which may be of interest to
category theorists (though I am not qualified to say whether there are
important comparisons or not.)
You may be wondering what this is leading up to. Among other things
Fine does with A-models, he uses them to prove a kind of "generic
completeness theorem," and similarly, generic soundness. His view and
treatment of arbitrary objects thus leads to a way of evaluating each
natural deduction system not only in terms of its completeness and
soundness, but also for appraising it on the basis of its "naturalness" and
"intuitiveness." Thus, the arbitrary-object approach is pressed into
service for helping us to see which natural deduction systems seem
preferable over others, and on what basis.
Fine considers several well-known systems of natural deduction, among
them systems of Hilbert, Gentzen, Quine, Copi (as made sound by Kalish),
Kalish & Montague, and numerous others. He offers variants of these
systems, compares them, and sometimes creates hybrid systems by combining
two distinct ones. In each case, he employs his notions of genericity in
evaluating them.
There is much, much more to Fine's book, but I hope the above gives some
idea of the book. He ends with a short Chapter on Inclusive and
Intuitionistic Systems. Though this chapter is brief, it is very
provocative.
At any rate, I hope FOMers will read Fine's book, because I would like
to know their opinions of Fine's treatment of the issues raised in it.
Charlie
P.S. To Thomas: I hope this short summary suffices to take your "God-given"
graduate students off the hook.
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