Dear Matthew, (01)
MW> None of this is remotely relevant to maximum allowable working
> temperatures or any idea of direct or indirect properties. (02)
But identity conditions are critical to this issue. (03)
MW> I don't have a problem with there being intensional definitions.
> The question is what do they define? (04)
They define the type, and every type determines a set. But it is
possible to have two types that happen to have exactly the same
set of instances. Plato used the two terms 'featherless biped'
and 'animal with speech', which happen to have the same extension. (05)
But that happens to be a coincidence. To emphasize the point,
Diogenes the Cynic plucked a chicken and threw it into the Academy
while shouting "Here is Plato's man." (06)
Therefore, the criteria for type identity are stricter (or finer)
than the criteria for set identity. That is the point that Church
made in the quotation of my previous note: (07)
Alonzo Church> 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. (08)
The usual distinction between types and sets is that the identity
criteria for types are based on the axioms and definitions for those
types. But the identity criteria for sets are based solely on their
members. Since we can't observe the future or any possible world,
we have to rely on the definitions and axioms because the entities
that constitute the sets are unobservable. (09)
MW> Well on the one hand I can quite agree that you can decide that
> two functions are different, even if they are extensionally the same.
> I choose not to, but I do not require that others make the same
> choice. On the other hand, the suggestion here is that the meaning
> is different, and here I think there is more of a problem. Meaning
> as I understand it is what links our terms to reality, and if two
> terms or definitions point to the same thing, I am struggling as
> to how the meaning can be different. Could you explain that specific
> point please? (010)
The word 'meaning' is problematical because it has too many different
meanings. Church was using it in an informal sense in the introduction
to his book. He avoided that word in his formalism by introducing
the terms intension and extension. For ontology, we can make similar
distinctions by distinguishing types and sets. (011)
As another example, every woman happens to be a daughter. Therefore,
the set of all women who work for IBM is identical to the set of
all daughters who work for IBM. Informally, one could say that
the words 'woman' and 'daughter' have different meanings. Formally,
we can say that the type Woman and the type Daughter have different
definitions. (012)
AC>> 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. (013)
MW> Could you be explicit about which ways please? (014)
A common criterion for intensional equivalence is having
provably equivalent definitions. However, this criterion can
be hard to use if the proof is as difficult as Fermat's Last
Theorem. Therefore, some people use the criterion of having
proofs that can easily be converted to one another by rules
that are simpler than a complete proof. (015)
To introduce a simpler method for defining the equivalence
of two definitions for functions, Church invented the lambda
calculus, which he presented in the book from which I extracted
those excerpts. Two functions that are equivalent according
to the lambda calculus are provably equivalent. However,
you might have other provably equivalent functions for which
the proof is a difficult as Fermat's Last Theorem. (016)
MW> Well I'm not an expert at lambda calculus, so could you explain
> how this works for equiangular and equilateral triangles please? (017)
Euclid showed that their definitions are provably equivalent. (018)
JFS>> But for large sets, such as the set of all people, cows, or
>> bacteria on the planet earth, a definition by extension is
>> impossible in practice. (019)
MW> I think this is one of the confusions. I am concerned about
> identity, not definition. So I am happy that two sets are the
> same if you can show that the membership is necessarily the same
> (if one rule that defines the membership can be converted to
> another as you state above would probably do the trick for me). (020)
For the future, for hypothetical situations, and for possible
worlds, all we have are definitions, axioms, and descriptions.
To determine identity we have to use some method of proof. (021)
Church's method of lambda conversion is a kind of *incomplete*
proof procedure: two definitions that are equivalent by
lambda conversion are provably equivalent. But there can
be provably equivalent function definitions that are not
convertible by lambda calculus. (022)
MW> On the other hand if you come across two sets with intensional
> definitions and they have the same membership, then I will deem
> that is a proof that the two definitions are equivalent. (023)
That is why it's important to distinguish sets and types. The
three types HomoSapiens, FeatherlessBiped, and AnimalWithSpeech
happen to have the same extension on planet earth at present.
But that's a coincidence that could be different in the future
or in some possible world. (024)
JFS>> The maximum temperature is similar: it's a role type that
>> can only be determined by something external. As Pat said,
>> look at the specifications. (025)
MW> I am quite happy that an extension might be determined by
> an intension, but when two definitions give the same extension,
> I will say that they define the same thing. (026)
By "same thing", I would be happy to say "same set", but not
"same type". (027)
MW> Take the sheep in a field, and I am interested in two sets,
> one the set of sheep with 2 eyes, the other the set of sheep
> with 4 legs. As it happens all the sheep have two eyes and four
> legs, so the sets are the same, and I would say that the two
> definitions have the same meaning because they point to the same set. (028)
That is why the highly ambiguous word 'meaning' should be avoided
as a formal term in ontology. A better distinction is 'type'
vs. 'set'. (029)
MW> All of which has nothing to do with extensional definitions,
> and everything to do with extensional identity. (030)
Extensional definitions are useful for sets. For example, we
can talk about the set consisting of three things: the moon,
the number 3, and the Statue of Liberty. That is a perfectly
fine set, but it's unlikely that anyone would consider it
a useful type. (031)
Extensional criteria are always used for set identity. But
for ontology, we are talking about general principles that
are likely to hold under a wide variety of circumstances.
That is why ontology specifies the criteria for types. (032)
John (033)
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