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

[Jon Awbrey]

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QUIPS.  Discussion Note 52

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BU = Ben Udell
JA = Jon Awbrey

Re: QUIPS-DIS 49.  http://stderr.org/pipermail/inquiry/2005-May/002743.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.

Ben,

Given that preamble on the relation between formal objects and formal signs,
the cut-to-the-chase answer is just to let the truth table for "and" be its
canonical denotation, its sign-theoretic object.  The truth table for "and",
taking up its relational domains in the order of the expression "x = y & z",
is a 3-adic relation over the boolean domain B = {falsity, truth} = {0, 1},
a subset of B^3 'of the form' given here:  {0:0:0, 0:0:1, 0:1:0, 1:1:1}.

BU: "Andness" or "bothness" can be given such an abstraction as the denotation.
    "And" doesn't denote an abstraction any more than "blue" denotes blueness.

The "of the form" clause in the above statement is critically important.
It says that we are dealing with formal structures, that is, hypostatic
abstractions, only up to isomorphism.  When we are engaged in a "formal"
inquiry, that is, mainly concerned with or primarily having to do with
the forms of the matters under view, then we are dealing with abstract
objects or hypostatic abstractions.

As far as what "blue" and "true" denote, we always have
the options of letting them plurally denote many things,
or singly denote their corresponding abstract qualities.

I will elaborate these themes a little further
on the "Factorization and Reification" thread.

BU: "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.

BU: "Not" denotes nothing, and has zero denotation, or one might say
    that it denotes objects in the universe of "non-existent objects".

"Not" may be taken to denote (the truth table of) a function ~ : B -> B.

BU: 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).

BU: 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.

One of the reasons why we form abstractions, prescissive or hypostatic,
is to bring into relief and to highlight for future recognition patterns
that are shared by and transferable across many different real situations.

Jon Awbrey

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