SUO: Re: Mapping from the Frying Pan to the Fire
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John, Rich, Tom, et al.
What's all this fuzz I here about nominalistic definitions?
The exoteric definition of a function as a set of ordered
pairs is far from being the sort of thing that extremists
in the ways of nominal thinking could possibly countenance,
since it does, before and after and above all, commit the
airy noting of referring to that divinest of abstractions,
just to name names, a set.
Now where I come from -- and no, it's not why I left --
the use of finite extension examples to witness a simple
matter of faith in rational inquiry does not rank a person
having to go about with a Scarlet "N" on his or her chest,
so let's have no more of that nominal name-calling, that
is to say, where it does not fit the author's intension.
The immanent question, the one that just keeps on remaining,
is whether we can possibly clarify the connotations and the
intensions, much less tell them apart, of ordinary language,
say, in the manner that the calculi of functional abstraction,
in particular, the combinator calculus and the calculus of
lambda converts, did for the language of recursive partial
functions. In effect, or in essence, whichever you prefer:
Can there be a "Calculus Of Semiotic Abstraction For Ordinary Language" (COSAFOL)?
Whatever the case, it's a far bat from the ballpark of concrete and simple examples
that might actually succeed in our lifetimes, ever growing briefer, to clear up what
is after all a very exoteric and almost 3-vial matter of basic discrete mathematics.
The next question is: Why do people with some reputed training in the 20th century
excuse for logic have so much trouble grasping a simple fact of simple mathematics?
Inasmuch as they seem to be otherwise healthy intellects, I have no choice but to
put it down to the deleterious effects of the particular logical tools they use.
And that may be something we can do something about.
Jon Awbrey
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John F. Sowa wrote:
>
> 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.
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