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Jon Awbrey


Registered: Feb 2004
Posts: 558

Recurring Themes

RT. Note 15

1.3.10.7. Stretching Operations (concl.)

In order to deal with the HO sign relations that are involved
in the present setting, I introduce a couple of new notations:


  • To mark the relation of denotation between a sentence z and the proposition
    that it denotes, let the "spiny bracket" notation "-[z]-" be used for
    "the indicator function denoted by the sentence z".

  • To mark the relation of denotation between a proposition q and the set
    that it indicates, let the "spiny brace" notation "-{Q}-" be used for
    "the indicator function of the set Q".

Notice that the spiny bracket operator "-[ ]-" takes one "downstream",
confluent with the direction of denotation, from a sign to its object,
whereas the spiny brace operator "-{ }-" takes one "upstream", against
the usual direction of denotation, and thus from an object to its sign.

In order to make these notations useful in practice, it is necessary to note
a couple of their finer points, points that might otherwise seem too fine to
take much trouble over. For the sake their ultimate utility, never the less,
I express their usage a bit more carefully as follows:

  • Let "spiny brackets", like "-[ ]-", be placed around a name of a sentence z,
    as in the expression "-[z]-", or else around a token appearance of the sentence
    itself, to serve as a name for the proposition that z denotes.

  • Let "spiny braces", like "-{ }-", be placed around a name of a set Q, as in
    the expression "-{Q}-", to serve as a name for the indicator function f_Q.

In passing, let us recall the use of the "fiber bars" or the "ground marker"
as an alternate notation for the fiber of truth in a proposition q, like so:

  • [| q |] = q^(-1)(%1%).

Table 11 illustrates the use of this notation, listing in each Column
several different but equivalent ways of referring to the same entity.

Table 11. Illustrations of Notation
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | Higher Order Sign |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `Set` ` ` ` | ` `Proposition` ` | ` ` Sentence` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` Q ` ` ` ` | ` ` ` ` q ` ` ` ` | ` ` ` ` z ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `[| -[z]- |]` ` | ` ` ` -[z]- ` ` ` | ` ` ` ` z ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `[| q |]` ` ` | ` ` ` ` q ` ` ` ` | ` ` ` `"q"` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` [| f_Q |] ` ` | ` ` ` `f_Q` ` ` ` | ` ` ` "f_Q" ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` Q ` ` ` ` | ` ` ` -{Q}- ` ` ` | ` ` `"-{Q}-"` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o

In effect, one can observe the following relations
and formulas, all of a purely notational character:

  • If the sentence z denotes the proposition q : X -> %B%,

    then -[z]- = q.

  • If the sentence z denotes the proposition q : X -> %B%,

    hence [|q|] = q^(-1)(%1%) = Q c X,

    then -[z]- = q = f_Q = -{Q}-.

  • Q = {x in X : x in Q}

    = [| -{Q}- |] = -{Q}-^(-1)(%1%)

    = [| f_Q |] = (f_Q)^(-1)(%1%).

  • -{Q}- = -{ {x in X : x in Q} }-

    = -[x in Q]-

    = f_Q.

Now if a sentence z really denotes a proposition q,
and if the notation "-[z]-" is meant to supply merely
another name for the proposition that z already denotes,
then why is there any need for this additional notation?
It is because the interpretive mind habitually races from
the sentence z, through the proposition q that it denotes,
and on to the set Q = [|q|] that the proposition indicates,
often jumping to the conclusion that the set Q is the only
thing that the sentence z is intended to denote. This HO
sign situation and the mind's inclination when placed
within its setting calls for a linguistic mechanism
or a notational device that is capable of analyzing
the compound action and controlling its articulate
performance, and this requires a way to interrupt
the flow of assertion that typically takes place
from z to q to Q.

Jon Awbrey

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