[FOM] explicit variables

[Jim Hardy]

I received some offlist emails asking for a précis and references on the
semantics of variables.  I'm replying onlist as the interest seems to be
more than an isolated individual.

First two caveats, my knowledge of this area stems from my dissertation and
subsequent work.  I'm not up to date with what may have happened in the area
in the last 5 years.  Anyone who would like a copy of said dissertation
"Instantial Reasoning, Arbitrary Objects and Holey Propositions" can view it
at http://www.cold.isu.edu/jim/dissertation.pdf .  If you'd like the .bib
file, please email me offlist.  Also, I sketch the views of several people
who may be members of this list.  Obviously my sketches are intended as
thumbnails of the views, not as full presentations.  If anyone feels my
thumbnails are misleading or incorrect, please chime in.

There are several basic positions regarding free variables.

The first claims that free variables have no semantic significance.
Generally this is couched in the claim that open formula are really just
shorthand for quantified formulas.  To be fair, I don't know anyone who has
really run with this view, but it's the view one tends to encounter as a
kind of background when one starts asking people about free variables.

A second view is that free variables function as names for 'arbitrary' or
'variable' objects.  This is Locke's view with respect to what he calls
'abstract' objects (objected to by Berkeley, it seems to have been Czuber's
view (objected to by Frege), and most recently Kit Fine's view in
_Reasoning_with_Arbitrary_Objects and related work.  I must say that Fine
seems to have laid to rest the formal objections to this view.  Fine, along
with Locke, argues that free variables are best taken as referring to
arbitrary objects.  Arbitrary object range over normal objects and have all
and only those (normal) properties which are shared by the object in their
range.  As Berkeley pointed out, this means that the arbitrary triangle has
the disjunctive property of being scalene-or-isosceles-or-equilateral but is
not scalene, is not isosceles, and finally is not equilateral.  Fine takes
some of the teeth out of this objection by showing that it is possible to
build a consistent formal theory of it.  Philosophical worries still remain
however and in my dissertation I argued that Fine's theory does not
adequately capture common forms of reasoning.

A third view takes Berkeley's side against Locke.  The view here is that
free variables refer arbitrarily to normal objects rather than, as Locke and
Fine would have it, referring normally to arbitrary objects.  The only full
treatment of this view of which I am aware is by B.H. Slater (sorry I seem
to have misplaced the full reference) who takes free variables to be epsilon
terms.  As such their semantic significance is given by a choice function.
Somewhat oddly, I didn't find much secondary literature on this view.
Though I must admit it was the view I opted not to cover in my dissertation
when I needed to cut something out.

A fourth view has been championed by Jeffrey King.  A good starting point is
his article "Instantial Terms, Anaphora and Arbitrary Objects" or perhaps
"Pronouns, Descriptions, and the Semantics of Discourse."  King maintains
that free variables are, at least in many circumstances, 'context dependent
quantifiers'.  The basic idea is that a free variable functions much like a
bound variable except that the nature of the quantifier is given by the
discourse context rather than made explicit in the formula.  As a result the
quantifier can change according to certain discourse rules or even be taken
out of the discourse and be made explicit in the formula.  It is this fact
that allows us to in to infer (x)Fx from Fx in certain discourse situations
and to infer Fx from (Ex)Fx in others.

A fifth view, my own, is that free variables are indicators for holes in the
propositional structure.  The basic idea here is that to get the
significance of 'Fx' consider first the significance of 'Fa'.  If you now
extract the significance of 'a' from the significance of 'Fa' you are left
with a propositional structure that has a hole.  This 'holey proposition' is
the significance of 'Fx' and the hole itself is the significance of 'x'.  It
is important in this view to distinguish between reference and indication.
'x' is not a name of the hole, 'Fx' is not a about the hole as 'Fa' is about
a.  Instead 'x' serves to indicate that there is a hole.  Along with King I
maintain that there are important discourse rules that govern the filling
and vacating of these holes.  (He agrees about the discourse rules, not
about the holes.)

These are the major view of which I'm aware.  Additionally, it is quite
common to see free variables painted as the formal equivalent of pronouns.
'Bx' is the equivalent of 'He/she/it is bald'.  There is a vast literature
on pronouns, but surprisingly little of it tries to make the connection to
free variables in formal systems.  The literature which directly and
primarily addresses the question of the semantics of free variables is
rather small and quite manageable.  On the other hand, once one begins
digging into the issues, one see that the context to a vast array of
philosophical problems, each of which has its own vast literature.  Among
the most obvious of these is the problem of universals.  Marco Santambrogio
has written on the problem of generic objects, a problem which seems
situated partway between that of arbitrary object and that of universals.

I'll stop for now.  There is of course a lot more to say, but this should
serve as introduction.

Cheers,

Jim Hardy