SUO: RE: Mapping from one notation to another
John:
Thanks for correcting me on extension vs. intension. That was a careless
slip of the keyboard. Some easy examples proving the point (not original
with me, of course) are:
a) in the context of plane geometry, "equiangular triangle" and "equilateral
triangle" have different intensions (rules) but identical extensions;
b) Quine's distinction between "creature with a heart" and "creature with a
kidney": different intensions (meanings) but identical extensions.
When Church speaks of "functions in intension", is he doing anything more
than giving a name to examples like these? Or, at least, to examples like
(a)?
Also, a few weeks ago, Jon Awbrey made a plea for some fresh examples
whenever the topic of analyticity as a matter of degree came up. (b) is a
good one. For some of us, its truth might appear to be a contingent matter;
it might seem possible, however unlikely, that we might at some future date
happen upon a creature with a heart that lacks a kidney.
For others, a sketch of a background theory they have about living
multi-cellular organisms might be something like this: a fluid (blood) is
used to circulate nutrients around the system (proteins, sugars, carbs,
etc.), to circulate information around the system (hormones), and to carry
soluble-in-liquid waste out of the system. Next, there must be a pump in the
organism to keep the circulation of this fluid going. Also, in the latter
case, there must necessarily be a buffer area for storing the soluble waste
prior to excretion. Reason: a creature that was constantly dribbling a small
amount of waste liquid down its leg would be at an evolutionary disadvantage
(:>). Periodic episodes of excretion are evolutionarily advantageous.
This buffer area is what we call a kidney. Now the last piece of this
background theory needed is a reason why excretion needs to involve the
circulatory system at all. I can't imagine why, and think that we would have
to delve below the language of multi-cellular organisms and their organs, to
the language of molecular biology and even organic chemistry. Anyway,
suppose it to have been done, and incorporated into this background theory.
Now: for a person who holds this background theory, the sentence
"Necessarily, a creature that has a heart will have a kidney". We could say
that this necessity is physical necessity, not logical or semantic necessity
(both of the latter being matters of language, not of how the world is, with
the former being a matter of that subset of the language called the logical
operators). But physical necessity just IS semantic necessity, given two
conditions. First, that it is semantic necessity within the context of some
theory of physical nature which explains the necessity. Second, that the
statement in question is relatively close to the center of that theory's
semantic sphere, i.e. is one which those who accept the theory are going to
try like heck to hold true, come what may.
But how could one hold the statement in question true, if sustained
observation shows us no structure which buffers the waste products? In that
case, wouldn't the claim have to be given up? Well, the notion that it would
is the notion that single statements can be confirmed or disconfirmed by
experiment. So, to continue our gedanken experiment, consider the following:
Suppose we came across an organism that had a heart organ, but no apparent
kidney organ. How can this fail to disconfirm the statement "All creatures
with a heart are creatures with a kidney?" Well, one way is like this: "I
just know that the creature has to have a kidney. We just haven't recognized
it yet. Maybe the buffer (kidney) isn't a discrete structure within the
organism, but more like a function of existing structures. Perhaps the waste
concentration builds up to a certain level in the circulatory fluid, and
when that level is reached, the fluid is rapidly filtered through a
(non-buffering) mechanism which passes the waste component of the fluid to
an excretory mechanism. That mechanism, that function, then, would be the
creatures kidney. So we just have to realize that kidneys can be functions,
not necessarily structures."
Here is an example of how we accommodate recalcitrant experience in such a
way as to make a favored sentence immune from disconfirmation. So is the
sentence "All creatures with a heart are creatures with a kidney" now
analytic? Well, it has been treated as an analytic sentence on this
occasion. If the treatment is accepted by the relevant community as
legitimate, and if no other evidence is taken as evidence against the claim,
then the sentence is analytic for the entire community .... for the time
being. In the process of protecting the statement against prima facie
recalcitrant experience, the community has probably (unwittingly) accepted a
change in the definition of "kidney", from something like "discrete
structure within an organism that filters and buffers soluble waste
products", to "the function of filtering and buffering soluble waste
products produced by an organism".
Another example is the first law of thermodynamics: "heat cannot of itself
pass from one body to another body". Is this analytic or synthetic? It is
now highly analytic (analyticity being a matter of degree). Any graduate
student who proposed to write a dissertation on experiments which he claims
disprove the law would have his suggestion rejected. Patents for perpetual
motion machines (which violate the laws of thermodynamics) are routinely
rejected.
Yet one of the means by which the laws of thermodynamics achieved their
highly analytic, immune from disconfirmation by experimentation, status must
certainly have been that they explained experimental results that were
anomalous prior to their formulation, and also that concerted attempts to
find disconfirming evidence failed. During this period of initial hostility
and/or skepticism, the statements were surely synthetic, i.e. not yet immune
from disconfirmation by experience. So here we have a more robust example of
analyticity (which Jon requested some time ago) as a matter of degree. Quine
never bothered to cast his examples in a diachronic mode, showing how
sentences moved along the analytic/synthetic continuum over time. But
outside of mathematics and logic, analytic statements do not spring from the
semantic womb fully analytic.
So: "true no matter what" indicates that the statement in question is
analytic for the speaker in question (on the occasion in question). "true
because of what the expression means" indicates that the statement in
question is HIGHLY analytic for the speaker in question (on the occasion in
question). Generalize from occasions to speakers, and you get an ideolect.
Generalize from speakers to communities of speakers, and you get a dialect.
Generalize from communities to the larger community of educated speakers who
are mutually intelligible to one another, on most topics, most of the time,
and you get a language.
In both cases (true no matter what, and true by meaning), analyticity is a
matter of dispositions to linguistic behavior. Such dispositions are
constantly changing, and from the micro-level of one speaker on one
occasion, currents of change in the same direction arise. These are the
semantic currents of a language, and dictionary revisions (specialized
dictionaries and general ones both) occur periodically when the currents
become large and constant enough to become definitive of correct linguistic
behavior. Kidneys, we learn, can be functions; they don't have to be organs
(given our hypothetical example). There just aren't any exceptions to the
first law of thermodynamics; we describe any putative exceptions in a way
which deflects their force onto other parts of our background theory.
I'm sure, of course, that Peirce said all this before, and said it much
clearer! (and 8>) as well.) But it's as well to emphasize it, because it
bears on what we are doing. If the ontological framework we are creating,
along with other groups, is not as flexible as this semantic mutability of
real language requires, then our framework will fail of its purpose. In
working with real-world databases in a dozen plus industries, I assure you
that the definitions of tables in those databases changes frequently. Those
definitions (as the set membership conditions enforced by schemas and code,
regardless of what the verbal definitions are -- the two often being
significantly different!) are the analytic statements definitive of the
semantics of those databases (along with the cardinality of relationships
and the domains of attributes). Those analytic statements, for a given
database, change much more frequently than do the dictionary definitions of
natural languages, or even the specialized definitions of a specific
community (Newton's dictionary of telecommunications, for example).
Since our objective, as I take it, is to facilitate the automatic
recognition of as much of the semantics of the world's databases as we can,
so we can query across databases we've never queried before and know what
the results mean, we must have enough flexibility in our ontology frameworks
to keep up with this extremely rapid rate of change in the semantics of real
world databases. I'd like to see, besides axiomatizations of various
proposals, a discussion of how our work will support what I've described
here -- or else a statement that I've got the ultimate objective wrong, and
what the real objective is.
So I offer this, along with several of my earlier diatribes, as cashing in
my claim, back in June, I think, to explain in my own words why I believe
that Quine demolished the analytic/synthetic distinction, replacing it with
an analytic/synthetic continuum. I believe earlier comments of mine also
explain why I think that definition by essential properties (necessary and
sufficient conditions) must be supplemented with definition by paradigm
examples (family resemblances), and why I think this is just a reformulation
of the claim of analyticity as a matter of degree.
But the bottom line is this: the semantics of natural languages, and
specialized dialects of it (manufacturing, health care, law, physics, etc.),
is constantly changing. Formalizations of any piece of any of these
languages distill out a subset of the semantics of that piece of language,
and freeze it. The formalism is frozen, but the real living language, in
which we think and talk to others, is never frozen. The formalism,
therefore, becomes more and more of an anachronism. Formalisms of linguistic
focii near the center of our conceptual sphere, become anachronisms very,
very slowly. I am talking here about logic and mathematics. Formalisms of
linguistic focii which are "elongated", stretching from the periphery to
near the core of our linguistic sphere, morph more frequently because they
have extensions onto which pretty direct experience pretty directly
impinges. The impact of such changes may leave the front-line troops, i.e.
the statements which are the most direct reports of experience we have,
unchanged and instead force changes further away from the periphery. But in
these linguistic focii, as a whole, semantic changes are occurring. Those of
their statements which reside in our upper level ontology will be pretty
stable (almost by definition of "upper level ontology"), but mid-level and
low-level (at the surface) statements will not be. Linguistic focii
represented by real world relational databases have a higher rate of
semantic change than anything else I have talked about. How will an
ontological structure accommodate that rapid change?
I'm thinking along these lines, right now: for each ontological category
(node in a tree structure), definition in terms of necessary conditions (the
ones we think are likely to remain relatively stable). Then a list of
optional further conditions which, for any registered table of any database,
can be checked off when that table is registered into the ontology. For that
table, and each of those conditions, the checking off is optional. If it is
done, the condition can be marked as included in the definition, or excluded
from it.
Then, queries across a world of databases can specify, for each ontological
category, which tables registered into that category are to be included. The
specification goes like this:
1. The necessary conditions must be met, of course, or else the table would
not be registered into that category.
2. Zero, one or more optional conditions are marked as inclusive or
exclusive conditions. For each condition, null (unmarked) is interpreted as
one way or the other.
3. Optional conditions not listed for the query are treated as irrelevant.
The query is then executed. A result set is returned. Based on 1-3, the
semantics of the result set are pretty darn clear.
What makes this flexible enough to record rapidly changing semantics,
without requiring the hierarchy of ontological categories to change their
meaning frequently, is that the categories are defined in terms of a set
(perhaps a relatively small one) of conditions, the ones not likely to need
to be changed. Associated with each of those categories is an open-ended
list of optional conditions. So as the meaning of my Customer table changes
from year to year, it remains registered under the Customer category, which
category does not change its necessary conditions.
The result is that all the Customer tables registered under the Customer
node in my ontology have a minimal, stable set of semantic features in
common, but then constitute a flexible group of tables with Wittgensteinian
family resemblances to one another.
The flexibility of this approach is that the volatile semantics are now
recorded in an open-ended list of optional semantic conditions, clustered
together around a minimal set of stable necessary conditions which alone
define the semantic category.
This has been pure stream of consciousness for the last 45 minutes. If it's
off base, by being either irrelevant or incorrect, I'd like to hear the
argument. If it's not, what should we do about such reflections to
materialize any practical value they might have?
Tom
-----Original Message-----
From: John F. Sowa [mailto:sowa@bestweb.net]
Sent: Friday, September 12, 2003 11:00 PM
To: Tom Johnston
Cc: Jon Awbrey; rich@valutech.com; SUO; cg@cs.uah.edu
Subject: 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.