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In the next few installments, I will be working toward the definition of an operation called the ''stretch''. This is related to the concept from category theory that is called a ''pullback''. As a few will know the uses of that already, maybe there's hope of stretching the number.
In the next few installments, I will be working toward the definition of an operation called the ''stretch''. This is related to the concept from category theory that is called a ''pullback''. As a few will know the uses of that already, maybe there's hope of stretching the number.
<pre>
Where are we? We just defined the concept of a functional fiber in several
of the most excruciating ways possible, but that's just because this method
of refining functional fibers is intended partly for machine consumputation,
so its schemata must be rendered free of all admixture of animate intuition.
However, just between us, a single picture may suffice to sum up the notion:
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` X-[| f |] , `[| f |]` ` `c` ` X ` ` ` ` |
| ` ` ` ` o ` ` ` o ` o ` o ` o ` ` ` ` | ` ` ` ` |
| ` ` ` ` `\` ` `/` ` `\` | `/` ` ` ` ` | ` ` ` ` |
| ` ` ` ` ` \ ` / ` ` ` \ | / ` ` ` ` ` | f ` ` ` |
| ` ` ` ` ` `\`/` ` ` ` `\|/` ` ` ` ` ` | ` ` ` ` |
| ` ` ` ` ` ` o ` ` ` ` ` o ` ` ` ` ` ` v ` ` ` ` |
| ` ` ` ` { `%0%` ` , ` `%1%` } ` `=` `%B%` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Why are we doing this? The immediate reason -- whose critique I defer --
has to do with finding a modus vivendi, whether a working compromise or
a genuine integration, between the assertive-declarative languages and
the functional-procedural languages that we have available for the sake
of conceptual-logical-ontological analysis, clarification, description,
inference, problem-solving, programming, representation, or whatever.
In the next few installments, I will be working toward the definition
of an operation called the "stretch". This is related to the concept
from category theory that is called a "pullback". As a few will know
the uses of that already, maybe there's hope of stretching the number.
</pre>
==Empirical Types and Rational Types==
<pre>
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
IDS -- RT
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Recurring Themes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Resource: Inquiry Driven Systems: An Inquiry Into Inquiry
Creation: 23 Jun 1996
Revision: 16 Dec 2001
Location: http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
Outline of Excerpt
1.3.10.3. Propositions and Sentences
1.3.10.4. Empirical Types and Rational Types
1.3.10.5. Articulate Sentences
1.3.10.6. Stretching Principles
1.3.10.7. Stretching Operations
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 8
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.4. Empirical Types and Rational Types
I make a brief detour to explain what are likely to be
the unfamiliar features of my definition of a sentence.
In this Subsection, I want to examine the style of definition that I used
to define a sentence as a type of sign, to adapt its application to other
problems of defining types, and to draw a lesson of general significance.
Notice that I am defining a sentence in terms of what it denotes, and not
in terms of its structure as a sign. In this way of reckoning, a sign is
not a sentence on account of any property that it has in itself, but only
due to the sign relation that actually works to interpret it. This makes
the property of being a sentence a question of actualities and contingent
relations, not merely a question of potentialities and absolute categories.
This does nothing to alter the level of interest that one is bound to have
in the structures of signs, it merely shifts the axis of the question from
the logical plane of definition to the pragmatic plane of effective action.
As a practical matter, of course, some signs are better for a given purpose
than others, more conducive to a particular result than others, and turn out
to be more effective in achieving an assigned objective than others, and the
reasons for this are at least partly explained by the relationships that can
be found to exist among a sign's structure, its object, and the sign relation
that fits the sign and its object to each other.
Notice the general character of this development. I start by
defining a type of sign according to the type of object that it
happens to denote, ignoring at first the structural potential that
it brings to the task. According to this mode of definition, a type
of sign is singled out from other signs in terms of the type of object
that it actually denotes and not according to the type of object that it
is designed or destined to denote, nor in terms of the type of structure
that it possesses in itself. This puts the empirical categories, the
classes based on actualities, at odds with the rational categories,
the classes based on intentionalities. In hopes that this much
explanation is enough to rationalize the account of types that
I am using, I break off the digression at this point and
return to the main discussion.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 9
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.5. Articulate Sentences
A sentence is called "articulate" if:
1. It has a significant form, a compound construction,
a multi-part constitution, a well-developed composition,
or a non-trivial structure as a sign.
2. There is an informative relationship that exists
between its structure as a sign and the content
of the proposition that it happens to denote.
A sentence of the articulate kind is typically given in the form of
a "description", an "expression", or a "formula", in other words, as
an articulated sign or a well-structured element of a formal language.
As a general rule, the category of sentences that one will be willing to
contemplate is compiled from a particular selection of complex signs and
syntactic strings, those that are assembled from the basic building blocks
of a formal language and held in especial esteem for the roles that they
play within its grammar. Still, even if the typical sentence is a sign
that is generated by a formal regimen, having its form, its meaning,
and its use governed by the principles of a comprehensive grammar,
the class of sentences that one has a mind to contemplate can also
include among its number many other signs of an arbitrary nature.
Frequently this "formula" has a "variable" in it that "ranges over" the
universe X. A "variable" is an ambiguous or equivocal sign that can be
interpreted as denoting any element of the set that it "ranges over".
If a sentence denotes a proposition f : X -> %B%, then the "value" of the
sentence with regard to x in X is the value f(x) of the proposition at x,
where "%0%" is interpreted as "false" and "%1%" is interpreted as "true".
Since the value of a sentence or a proposition depends on the universe of discourse
to which it is "referred", and since it also depends on the element of the universe
with regard to which it is evaluated, it is conventional to say that a sentence or
a proposition "refers" to a universe of discourse and to its elements, though often
in a variety of different senses. Furthermore, a proposition, acting in the guise
of an indicator function, "refers" to the elements that it "indicates", namely, the
elements on which it takes a positive value. In order to sort out the potential
confusions that are capable of arising here, I need to examine how these various
notions of reference are related to the notion of denotation that is used in the
pragmatic theory of sign relations.
One way to resolve the various and sundry senses of "reference" that arise
in this setting is to make the following brands of distinctions among them:
1. Let the reference of a sentence or a proposition to a universe of discourse,
the one that it acquires by way of taking on any interpretation at all, be
taken as its "general reference", the kind of reference that one can safely
ignore as irrelevant, at least, so long as one stays immersed in only one
context of discourse or only one moment of discussion.
2. Let the references that an indicator function f has to the elements
on which it evaluates to %0% be called its "negative references".
3. Let the references that an indicator function f has to the elements
on which it evaluates to %1% be called its "positive references"
or its "indications".
Finally, unspecified references to the "references" of a sentence,
a proposition, or an indicator function can be taken by default
as references to their specific, positive references.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 10
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.5. Articulate Sentences (concl.)
I conclude my pragmatic semiotic treatment of the relation between
a sentence (a logical sign) and a proposition (a logical object).
The universe of discourse for a sentence, the set whose elements the
sentence is interpreted to be about, is not a property of the sentence
by itself, but of the sentence in the presence of its interpretation.
Independently of how many explicit variables a sentence contains, its
value can always be interpreted as depending on any number of implicit
variables. For instance, even a sentence with no explicit variable,
a constant expression like "%0%" or "%1%", can be taken to denote
a constant proposition of the form c : X -> %B%. Whether or not it
has an explicit variable, I always take a sentence as referring to
a proposition, one whose values refer to elements of a universe X.
Notice that the letters "p" and "q", interpreted as signs that denote
the indicator functions p, q : X -> %B%, have the character of sentences
in relation to propositions, at least, they have the same status in this
abstract discussion as genuine sentences have in concrete applications.
This illustrates the relation between sentences and propositions as
a special case of the relation between signs and objects.
To assist the reading of informal examples, I frequently use the letters
"t", "u", "v", "z" to denote sentences. Thus, it is conceivable to have
a situation where z = "q" and where q : X -> %B%. Altogether, this means
that the sign "z" denotes the sentence z, that the sentence z is the same
thing as the sentence "q", and that the sentence "q" denotes the proposition,
characteristic function, or indicator function q : X -> %B%. In settings where
it is necessary to keep track of a large number of sentences, I use subscripted
letters like "e_1", ..., "e_n" to refer to the various expressions in question.
A "sentential connective" is a sign, a coordinated sequence of signs,
a syntactic pattern of contextual arrangement, or any other syntactic
device that can be used to connect a number of sentences together in
order to form a single sentence. If k is the number of sentences that
are thereby connected, then the connective is said to be of "order k".
If the sentences acquire a logical relationship through this mechanism,
and are not just strung together by this device, then the connective
is called a "logical connective". If the value of the constructed
sentence depends on the values of the component sentences in such
a way that the value of the whole is a boolean function of the
values of the parts, then the connective earns the title of
a "propositional connective".
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 11
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.6. Stretching Principles
We are in the home stretch of what I promised to bring home this time around.
Let me set up the play by bringing back to mind a deuce of basic definitions
from earlier in the game:
A "boolean connection" of degree k, also known as a "boolean function"
on k variables, is a map of the form F : %B%^k -> %B%. In other words,
a boolean connection of degree k is a proposition about things in the
universe of discourse X = %B%^k.
An "imagination" of degree k on X is a k-tuple of propositions about things
in the universe X. By way of displaying the various brands of notation that
are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of
these features of the typical imagination #f# can be summed up in either one
of two ways: either in the form of a membership statement, to the effect that
#f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
the former.
The purpose of this exercise is to illuminate how a sentence,
a sign constituted as a string of characters, can be enfused
with a proposition, an object of no slight abstraction, in a
way that can speak about an external universe of discourse X.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 12
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.6. Stretching Principles (cont.)
There is a principle, of constant use in this work, that needs to be made explicit.
In order to give it a name, I refer to this idea as the "stretching principle".
Expressed in different ways, it says that:
1. Any relation of values extends to a relation of what is valued.
2. Any statement about values says something about the things
that are given these values.
3. Any association among a range of values establishes
an association among the domains of things
that these values are the values of.
4. Any connection between two values can be stretched to create a connection,
of analogous form, between the objects, persons, qualities, or relationships
that are valued in these connections.
5. For every operation on values, there is a corresponding operation on the actions,
conducts, functions, procedures, or processes that lead to these values, as well
as there being analogous operations on the objects that instigate all of these
various proceedings.
Nothing about the application of the stretching principle guarantees that
the analogues it generates will be as useful as the material it works on.
It is another question entirely whether the links that are forged in this
fashion are equal in their strength and apposite in their bearing to the
tried and true utilities of the original ties, but in principle they
are always there.
In particular, a connection F : %B%^k -> %B% can be understood to
indicate a relation among boolean values, namely, the k-ary relation
L = F^(-1)(%1%) c %B%^k. If these k values are values of things in a
universe X, that is, if one imagines each value in a k-tuple of values
to be the functional image that results from evaluating an element of X
under one of its possible aspects of value, then one has in mind the
k propositions f_j : X -> %B%, for j = 1 to k, in sum, one embodies
the imagination #f# = <f_1, ..., f_k>. Together, the imagination
#f# in (X -> %B%)^k and the connection F : %B%^k -> %B% stretch
each other to cover the universe X, yielding a new proposition
q : X -> %B%.
To encapsulate the form of this general result, I define a scheme of composition
that takes an imagination #f# = <f_1, ..., f_k> in (X -> %B%)^k and a boolean
connection F : %B%^k -> %B% and gives a proposition q : X -> %B%. Depending
on the situation, specifically, according to whether many F and many #f#,
a single F and many #f#, or many F and a single #f# are being considered,
I refer to the resultant q under one of three descriptions, respectively:
1. In a general setting, where the connection F and the imagination #f#
are both permitted to take up a variety of concrete possibilities,
call q the "stretch of F and #f# from X to %B%", and write it in
the style of a composition as "F $ #f#". This is meant to suggest
that the symbol "$", here read as "stretch", denotes an operator
of the form $ : (%B%^k -> %B%) x (X -> %B%)^k -> (X -> %B%).
2. In a setting where the connection F is fixed but the imagination #f#
is allowed to vary over a wide range of possibilities, call q the
"stretch of F to #f# on X", and write it in the style "F^$ #f#",
as if "F^$" denotes an operator F^$ : (X -> %B%)^k -> (X -> %B%)
that is derived from F and applied to #f#, ultimately yielding
a proposition F^$ #f# : X -> %B%.
3. In a setting where the imagination #f# is fixed but the connection F
is allowed to range over a wide variety of possibilities, call q the
"stretch of #f# by F to %B%", and write it in the fashion "#f#^$ F",
as if "#f#^$" denotes an operator #f#^$ : (%B%^k -> %B%) -> (X -> %B%)
that is derived from #f# and applied to F, ultimately yielding
a proposition #f#^$ F : X -> %B%.
Because the stretch notation is used only in settings
where the imagination #f# : (X -> %B%)^k and the
connection F : %B%^k -> %B% are distinguished
by their types, it does not really matter
whether one writes "F $ #f#" or "#f# $ F"
for the initial form of composition.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 13
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.6. Stretching Principles (concl.)
To complete the general discussion of stretching principles,
we will need to call back to mind the following definitions:
The "play of images" that is determined by #f# and x, more specifically,
the play of the imagination #f# = <f_1, ..., f_k> that has to with the
element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
that satisfies the equations b_j = f_j (x), for all j = 1 to k.
A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
defined as follows:
If #b# = <b_1, ..., b_k> in %B%^k,
then p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%.
The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.
Just as a sentence is a sign that denotes a proposition,
which thereby serves to indicate a set, a propositional
connective is a provision of syntax whose mediate effect
is to denote an operation on propositions, which thereby
manages to indicate the result of an operation on sets.
In order to see how these compound forms of indication
can be defined, it is useful to go through the steps
that are needed to construct them. In general terms,
the ingredients of the construction are as follows:
1. An imagination of degree k on X, in other words, a k-tuple
of propositions f_j : X -> %B%, for j = 1 to k, or an object
of the form #f# = <f_1, ..., f_k> : (X -> %B%)^k.
2. A connection of degree k, in other words, a proposition
about things in %B%^k, or a boolean function of the form
F : %B%^k -> %B%.
From this 2-ply of material, it is required to construct a proposition
q : X -> %B% such that q(x) = F(f_1(x), ..., f_k(x)), for all x in X.
The desired construction is determined as follows:
The cartesian power %B%^k, as a cartesian product, is characterized
by the possession of a "projective imagination" #p# = <p_1, ..., p_k>
of degree k on %B%^k, along with the property that any imagination
#f# = <f_1, ..., f_k> of degree k on an arbitrary set W determines
a unique map !f! : W -> %B%^k, the play of whose projective images
<p_1(!f!(w)), ..., p_k(!f!(w))> on the functional image !f!(w) matches
the play of images <f_1(w), ..., f_k(w)> under #f#, term for term and
at every element w in W.
Just to be on the safe side, I state this again in more standard terms.
The cartesian power %B%^k, as a cartesian product, is characterized by
the possession of k projection maps p_j : %B%^k -> %B%, for j = 1 to k,
along with the property that any k maps f_j : W -> %B%, from an arbitrary
set W to %B%, determine a unique map !f! : W -> %B%^k satisfying the system
of equations p_j(!f!(w)) = f_j(w), for all j = 1 to k, and for all w in W.
Now suppose that the arbitrary set W in this construction is just
the relevant universe X. Given that the function !f! : X -> %B%^k
is uniquely determined by the imagination #f# : (X -> %B%)^k, or what
is the same thing, by the k-tuple of propositions #f# = <f_1, ..., f_k>,
it is safe to identify !f! and #f# as being a single function, and this
makes it convenient on many occasions to refer to the identified function
by means of its explicitly descriptive name "<f_1, ..., f_k>". This facility
of address is especially appropriate whenever a concrete term or a constructive
precision is demanded by the context of discussion.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 14
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.7. Stretching Operations
The preceding discussion of stretch operations is slightly more general
than is called for in the present context, and so it is probably a good
idea to draw out the particular implications that are needed right away.
If F : %B%^k -> %B% is a boolean function on k variables, then it is possible
to define a mapping F^$ : (X -> %B%)^k -> (X -> %B%), in effect, an operation
that takes k propositions into a single proposition, where F^$ satisfies the
following conditions:
F^$ (f_1, ..., f_k) : X -> %B%
such that:
F^$ (f_1, ..., f_k)(x)
= F(#f#(x))
= F(<f_1, ..., f_k>(x))
= F(f_1(x), ..., f_k(x)).
Thus, F^$ is just the sort of entity that a propositional connective denotes,
a particular way of connecting the propositions that are denoted by a number
of sentences into a proposition that is denoted by a single sentence.
Now "f_X" is sign that denotes the proposition f_X,
and it certainly seems like a sufficient sign for it.
Why would we need to recognize any other signs of it?
If one takes a sentence as a type of sign that denotes a proposition and
a proposition as a type of function whose values serve to indicate a set,
then one needs a way to grasp the overall relation between the sentence
and the set as taking place within a "higher order" (HO) sign relation.
Sketched very roughly, the relationships of denotation and indication that exist
among sets, propositions, sentences, and values can be diagrammed as in Table 10.
Table 10. Levels of Indication
o-------------------o-------------------o-------------------o
| Object` ` ` ` ` ` | Sign` ` ` ` ` ` ` | Higher Order Sign |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Set ` ` ` ` ` ` ` | Proposition ` ` ` | Sentence` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| f^(-1)(b) ` ` ` ` | f ` ` ` ` ` ` ` ` | "f" ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Q ` ` ` ` ` ` ` ` | %1% ` ` ` ` ` ` ` | "%1%" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| X-Q ` ` ` ` ` ` ` | %0% ` ` ` ` ` ` ` | "%0%" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
Strictly speaking, propositions are too abstract to be signs, hence the
contents of Table 10 have to be taken with the indicated grains of salt.
Propositions, as indicator functions, are abstract mathematical objects,
not any kinds of syntactic elements, thus propositions cannot literally
constitute the orders of concrete signs that remain of ultimate interest
in the pragmatic theory of signs, or in any theory of effective meaning.
Therefore, it needs to be understood that a proposition f can be said
to "indicate" the set Q only insofar as the values of %1% and %0% that
it assigns to the elements of the universe X are positive and negative
indications, respectively, of the elements in Q, and thus indications
of the set Q and of its complement ~X = X - Q, respectively. It is
these logical values, when rendered by a concrete implementation of
the indicator function f, that are the actual signs of the objects
inside the set Q and the objects outside the set Q, respectively.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RT. Note 15
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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:
1. 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".
2. 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:
1. 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.
2. 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:
1. If the sentence z denotes the proposition q : X -> %B%,
then -[z]- = q.
2. If the sentence z denotes the proposition q : X -> %B%,
hence [|q|] = q^(-1)(%1%) = Q c X,
then -[z]- = q = f_Q = -{Q}-.
3. Q = {x in X : x in Q}
= [| -{Q}- |] = -{Q}-^(-1)(%1%)
= [| f_Q |] = (f_Q)^(-1)(%1%).
4. -{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.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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