FOM: RE: FOM Arbitrary Objects

[Jay Halcomb]

Greetings,

Re (Charles Silver): "The trick is to make reasonable mathematical/logical
sense of this notion [arbitrary object] using just the resources of
first-order logic."

Hilbert used the epsilon symbol to elucidate the notion of an arbitrary
selection, which he axiomatized in the epsilon-calculus. Is this not
nowadays considered a useful and adequate treatment of 'arbitrary'? In what
respects is it deficient?

To my knowledge, some folks are working with an indexed epsilon calculus to
attack the problems of discourse analysis, context, and pronomial
cross-reference.( http://www-logic.stanford.edu/Abstracts/Lunch.html#D.Saren
ac )

A.C. Leisenring wrote (in 'Mathematical Logic and Hilbert's
Epsilon-symbol'):

"The epsilon-symbol is a logical constant which can be used in the formal
languages of mathematical logic to form certain expressions known as
epsilon-terms. Thus, if A is a formula of some formal language L and x is a
variable of L, then the expression epsilonxA is a well-formed formula of the
language. Intuitively, the epsilon-term epsilonxA says 'an x such that if
anything has the property A, then x has that property'. For example, suppose
we think of the variables of the language as ranging over the set of human
beings and we think of A as being the statement 'x is an honest politician'.
Then epsilonxA designates some politician whose honesty is beyond reproach,
assuming of course that such a politician exists. On the other hand, if
there are no honest politicians, then epsilonxA must denote someone [the
Fregean way -- Ed.] but we have no way of knowing who that person is.
Similarly, even if there are honest politicians, we have no way of knowing
which one of them epsilonxA designates.

Since the epsilon-symbol, or epsilon-operator as it is sometimes called,
selects an arbitrary member from a set of objects having some property, this
symbol is
often referred to as a 'logical choice function'. It is not surprising that
an investigation of the epsilon-symbol also sheds some light on the nature
of the axiom of choice. One of the main theorems of this book, Hilbert's
Second epsilon-Theorem, provides a formal justification of the use of the
epsilon-symbol in logical systems by showing that this symbol can be
eliminated from proofs for formulas which do not themselves contain the
symbol. That this theorem says intuitively is that the act of making
arbitrary choices is a legitimate logical procedure. However, it has been
shown by Cohen [1966] that an application of the axiom of choice in set
theory cannot in general be eliminated. It follows then the the real power
of the axiom of choice lies not in the fact that it allows one to make
arbitrary selections but rather in the fact that it asserts the existence of
a set consisting of the selected entities."

(Introduction, 'Mathematical Logic and Hilbert's Epsilon-symbol', A.C.
Leisenring, MacDonald Technical and Scientific Publishing, London, 1969)

Also, as Jan Mycielski previously remarked in FOM,
 http://www.math.psu.edu/simpson/fom/postings/0001/msg00009.html )

"Let fi be a quantifier-free formula. We define: (Ex)fi(x) = fi(epsilon x
fi),  and (Ax)fi(x) = fi(epsilon x (not fi)). Then, it is easy to check
that, the
rules of logic concerning quantifiers follow from Hilbert's axiom: fi(x)
implies fi(epsilon x fi)."

My thoughts:

'arbitrary' is not an absolute notion but can only be understood relative to
acts of choice, within some context. This is a psychological or
epistemological notion  at bottom. There aren't any objective classes of
'arbitrary objects' vs. 'non-arbitrary objects' -- only classes of
arbitrarily chosen objects relative to some specification. 'Arbitrary' is
elliptic for 'arbitrarily chosen'. It's not an absolute term, in the sense
of traditional logic.
English, though, allows this sort of ellipsis.

Jay Halcomb
Research interests: Logic and computer science
http://www.sonic.net/~halcomb