FOM: Arbitrary objects

[Dan Velleman]

I'm posting this informative message with permission of Professor Velleman.

-Martin
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Prof. Davis-
   In response to your posting to FOM, I thought you might find this brief
summary of Fine's book helpful:

Suppose M is a structure for a first-order language L.  Let I be the
universe of M.  Fine expands M to a generic model M+ by adding 3 things:
   1.  A set A, disjoint from I.  The elements of A will be the arbitrary
objects.
   2.  A relation < on A.  "a < b" is intended to mean "a depends on b".  The
idea is that if, for example, b is an arbitrary positive real number, and a
is an arbitrary square root of b, then a depends on b.  b can take on any
positive value.  Given a value for b, there are two possible values for a,
but different values for b will lead to different possible values for a.
   3.  A set V of partial functions from A to I.  These are the possible
assignments of values to the arbitrary objects.
There are some further restrictions on A, <, and V--for details, see Fine's
book.

Fine defines a semantics for generic models.  The basic idea is that if
P(a1, a2, ..., an) is a sentence referring to arbitrary objects a1, a2, ..,
an, then P is true iff for every v in V, P(v(a1), v(a2), ..., v(an)) is
true.  In other words, a statement about arbitrary objects is said to be
true iff it is true for every assignment of values to them.  Fine calls this
the "principle of generic attribution".

Fine then uses generic models to provide semantics and soundness proofs for
formal deduction systems.  Consider a system that uses letters a, b, c, ...
in rules like:  From P(a), you may infer (Ax)P(x), or:  From (Ex)P(x) you
may infer P(a).  (In such systems there are generally restrictions on the
use of the letters a, b, c, ... that occur in these rules, and these
restrictions are needed to ensure soundness.)  Fine interprets the letters
a, b, c, ... as denoting arbitrary objects, and proves that if the
denotations of these letters are chosen in a suitable way, then every line
in a deduction will be true in any structure in which the premises are all
true, where "suitable" is defined differently for different formal systems.
Fine analyzes systems of Hilbert, Gentzen, Quine, and Copi.  Soundness (in
the ordinary sense, in terms of nongeneric models) follows.

That's pretty sketchy, but it gives you some idea of what Fine has done.
Here's my opinion of it:  I don't find it very helpful to say that in
informal reasoning, when we say "Let a be arbitrary", we are using the
letter "a" to stand for an arbitrary object in Fine's sense.  I prefer to
say that "a" stands for an ordinary object, but an unspecified one.  And
therefore, I think that if a formal system accurately represents informal
reasoning, then it should not be necessary to use Fine's arbitrary objects
to analyze the soundness of the system.  But there are lots of formal
systems, and they don't all represent informal reasoning with complete
accuracy.  Fine's theory does seem to be useful for analyzing these formal
systems.  Furthermore, it gives a unified framework for analyzing the
semantics of formal systems, and it therefore makes it possible to make
interesting comparisons between systems.  For example, the suitable meanings
for the letters in Quine's system are different from the ones in Copi's
system, and that seems to tell us something interesting about the difference
between the two systems.

Fine's book "Reasoning With Arbitrary Objects" is pretty readable, although
parts of it are technical.  If you want to know more, I recommend it.
   -Dan Velleman