[Inquiry] Re: Questions Involving Pure Symbols -- Discussion

[Jon Awbrey]

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QUIPS.  Discussion Note 49

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BU = Ben Udell

Re: QUIPS-DIS 38.  http://stderr.org/pipermail/inquiry/2005-May/002728.html
In: QUIPS-DIS.     http://stderr.org/pipermail/inquiry/2005-May/thread.html#2602

BU: Jon Awbrey wrote:

JA: But symbols of logical operations can be given real denotations
    in the form of particular formal or mathemtical objects.  These
    may be understood as hypostatic abstractions, but they are none
    the less solid for all that.  Symbols of logical operations can
    be used to construct propositions that are fictitious, but that
    is not their sin, whether by original or acquired imputation.

BU: I don't see how a first-order logical operator like "and" can be given an
    abstraction as its denotation.  "Andness" or "bothness" can be given such
    an abstraction as the denotation.  "And" doesn't denote an abstraction any
    more than "blue" denotes blueness.  "Blue" denotes blue things.  If the term
    "and" denotes at all (I think that it does), it denotes all xy in the sense
    of "Jack and Jill", its denotation is everything-and-everything, or, if the
    term "and" links two predicates, then it denotes anything x such that x is Y
    and Z or such-and-such or even T and T;  in that case "and" denotes everything
    too.  "Not" denotes nothing, and has zero denotation, or one might say that it
    denotes objects in the universe of "non-existent objects".  I can't think of
    a way to express the denotation of the term "and" to cover all the ways which
    "and" can be used to link, but I'm unsure whether this would be best solved by
    treating the term "and" as a term with various senses dependent on whether it
    links subjects, predicates, or other things (its use to mean "plus" should be
    distinguished).  In first-order logic the conjunctive appears to be always
    a link between predicates.  Anyway, it's true that the term "and," with
    a denotation of everything, would be denotationally indistinguishable
    from many other words.  But it's inconsistent to have "and" denote an
    abstraction yet "blue" denote blue things instead of blueness.  It seems
    to me that instead one can have one's cake and eat it too, in this case --
    one doesn't need to give up the abstraction.  "Andness" or "conjunction"
    or some such word can denote the abstraction.

Ben,

Life is too short to keep pretending that mathematics was born yesterday,
even if this tends to be the received view for many philosophies of math,
so let me take up one of the more established perspectives that actually
gets used to some effect in math and computer science, what is sometimes
called "denotational semantics", however much that might sound like some
kind of tautology to our minds' ears.  In this view, very roughly stated,
signoid entities like mathematical expressions or program codes are said
to denote abstract objects, like spaces, functions, relations, and so on.

Pick a mathematical object at random, let's say, a function
f : N -> Z from the non-negative integers N = {0, 1, 2, ...}
to the integers Z = {..., -2, -1, 0, 1, 2, ...}.  By way of
a few suggestions, here's a hit parade of favorite numbers:

| N.J.A. Sloane, "Integer Sequence WebCam"
|'On-Line Encyclopedia of Integer Sequences'
| http://www.research.att.com/~njas/sequences/WebCam.html

In general, such a function is a countable set of ordered pairs,
{0:f(0), 1:f(1), 2:f(2), ...}, and if the sequence is countably
infinite we can hardly list all the pairs, so we tend to prefer
having some type of formulaic expression or at least a computer
program code that describes the set.  At other times, of course,
we have a formula or a program and wish to produce the function,
but that is usually the less problematic case.

In this sort of enterprise, we observe a more general form of the
fundamental distinction between numbers, the objects of denotation,
and numerals, the signs that denote them.  Only now we have complex
objects like functions and complex signs like formulas and programs.

Have to break for now ...

Jon Awbrey

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