My Ontolog folder has become overstuffed, and I started deleting
things. But I found a note from January that is still relevant. (01)
Juan de Nadie asked a question:
> I'm reading "What is an ontology?" by Guarino, Oberle, & Staab.
>
> But I had some troubles in understand some mathematics.
> My doubts are regarding the definition 2.3 of the paper:
>
> "An intensional relation (or conceptual relation) ρ^n of arity n on <D,W> is
> a total function ρ^n : W → 2^D^n from the set W into the set of all n-ary
> (extensional) relations on D"
>
> I don't understand the 2^D^n, mainly this base 2. Why 2? (02)
I responded to this by complaining about the overuse -- I would even
say *abuse* -- of mathematical symbols. (03)
Leo responded:
> The point of using mathematical (or logical) notation is to make
> your statement precise and unambiguous. (04)
I completely agree. But the traditional notation for a function
from ρ to n is "f: ρ -> n". (05)
This notation is just as precise, it is far more readable, and it
avoids extraneous notions from Cantor's set theory in an introductory
paper that is *not* about set theory. (06)
Ed Barkmeyer
> I am strongly in favor of stating the fundamental principles in "plain
> language" and then following that with mathematical formulations, and I
> think that is the best practice for such papers. But I must confess
> that, a few steps into a theory, the "plain language" can become
> significantly less than "plain" and easily understood. (07)
I certainly agree. A good notation is extremely important, but
a notation that uses more symbols than necessary is not good. (08)
William Frank
> "One can use good English and formal definition together."
>
> Is the only reasonable one. What could really be argued about here? (09)
No quarrel from me. But adding unnecessary symbols and extraneous
notions (such as Cantor's set theory) is an *abuse* of formalism. (010)
At the end of this note, I quote the distinction between functions
in extension and intension by Alonzo Church. He knew mathematics,
he knew logic, and he certainly understood the need for precision.
But he also knew how to state his points clearly and precisely
in English and in formalisms. (011)
Fundamental principle: Mrs. Malaprop was a fictional character who
used big words inappropriately in a failed attempt to impress people.
An overuse of mathematical symbols does *not* impress mathematicians. (012)
John
_______________________________________________________________________ (013)
Source: http://www.jfsowa.com/logic/alonzo.htm (014)
2. Extension and Intension (015)
The foregoing discussion leaves it undetermined under what circumstances
two functions shall be considered the same. (016)
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_. (017)
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_. (018)
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. (019)
In the calculus of λ-conversion and the calculus of restricted
λ-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 λ-δ-conversion, where the notion
of identity of functions is introduced into the system by the symbol δ,
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 λ-δ-conversion are interpretable as denoting
functions in intension of an appropriate kind. (020)
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