SUO: Re: Mapping from one notation to another
Tom,
I'd like to cite one of my favorite sources for the definition
of intension and extension, namely the first three pages of
Church's little book on the lambda calculus:
http://www.jfsowa.com/logic/alonzo.htm
The Calculi of Lambda Conversion
Church defines the distinction in terms of functions, but
you can generalize his definition to relations and other
mathematical structures.
He starts by defining a function as a rule, rather than
a set of tuples:
A function is a rule of correspondence by which when anything
is given (as argument) another thing (the value of the function
for that argument) may be obtained. That is, a function is an
operation which may be applied on one thing (the argument)
to yield another thing (the value of the function).
I very much prefer this definition to the nominalistic definitions,
which identify a function (or relation) with a set of tuples.
Later, Church goes on to make what I believe is the clearest
and best definition of the distinction to be found in the 20th
century literature (in clarity and precision, it even rivals
the writings of Peirce and the medieval logicians):
The foregoing discussion leaves it undetermined under what
circumstances two functions shall be considered the same.
The most immediate and, from some points of view, the best
way to settle this question is to specify that two functions
f and g are the same if they have the same range of arguments
and, for every element a that belongs to this range, (fa) is
the same as (ga). When this is done we shall say that we are
dealing with functions in extension.
It is possible, however, to allow two functions to be different
on the ground that the rule of correspondence is different
in meaning in the two cases although always yielding the same
result when applied to any particular argument. When this is done
we shall say that we are dealing with functions in intension.
The notion of difference in meaning between two rules of
correspondence is a vague one, but, in terms of some system of
notation, it can be made exact In various ways. We shall not
attempt to decide what is the true notion of difference in meaning
but shall speak of functions in intension in any case where a
more severe criterion of identity is adopted than for functions
in extension. There is thus not one notion of function in intension,
but many notions; involving various degrees of intensionality.
Then Church defines his version of the lambda calculus as a method
of defining one family of intensional definitions while leaving
open the possibility of having other, equally useful definitions
for other purposes:
In the calculus of ?-conversion and the calculus of restricted
?-K-conversion, as developed below, It is possible, if desired,
to interpret the expressions of the calculus as denoting functions
in extension. However, in the calculus of ?-?-conversion, where
the notion of identity of functions is introduced into the system
by the symbol ?, it is necessary, in order to preserve the finitary
character of the transformation rules, so to formulate these rules
that an interpretation by functions in extension becomes impossible.
The expressions which appear in the calculus of ?-?-conversion are
interpretable as denoting functions in intension of an appropriate
kind.
For such reasons, I object to identifying the intension of a relation
with the set of tuples:
TJ> My own thoughts about the difference between a row of a table
> (a tuple) and the table itself (a relation) is that the former is
> part of the extension of the relation, while the latter (more
> specifically, the set membership conditions which define it)
> represents the intension of the relation. Next, that one cannot
> always infer the intensional rules from the extensional instances
> because, at any given moment, the set of all those instances may
> not define the boundary conditions of all those rules.
This is one approach, but Church's definition is more general
because it allows the possibility of different intensional rules
for generating or selecting the elements of the set.
John
PS: You might also like to see another of my favorite excerpts
from Church, which I copied from Cathy Legg's old web site:
http://www.jfsowa.com/ontology/church.htm
Alonzo Church on Women and Abstract Entities
PPS: Note that Church uses the capital letter Sigma for the
existential quantifier in the lambda calculus. That is Peirce's
notation, which is still used by logicians who want to have a
different kind of quantifier for one reason or another. In his
famous paper on undecidability, Goedel uses Peirce's notation,
capital Pi, for the universal quantifier.
PPPS: And if you have difficulties getting the Greek letters
to display properly, you are probably using an obsolete browser,
such as Internet Explorer. Please upgrade to Mozilla, Opera,
or something more modern and less susceptible to viruses.