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==Fragmata==
 
==Fragmata==
   −
# http://www.cspeirce.com/menu/library/aboutcsp/awbrey/inquiry.htm
+
{| cellpadding="4"
−
# http://forum.wolframscience.com/showthread.php?threadid=649
+
| [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/inquiry.htm Arisbe Site, "Inquiry Driven Systems", 04 Jul 2000]
−
# http://forum.wolframscience.com/printthread.php?threadid=649
+
| IDS 1 – 1.3.4.19, 30 Jun 2000, Draft 8.2
 +
|-
 +
| [http://suo.ieee.org/email/thrd125.html#07409 SUO List, "Critique Of Non-Functional Reason", 27 Nov 2001]
 +
| IDS 1.3.10.3, 27 Nov 2001, Draft 8.63
 +
|-
 +
| [http://suo.ieee.org/email/thrd125.html#07455 SUO List, "Critique Of Non-Functional Reason", 29 Nov 2001]
 +
| IDS 1.3.10.4, 28 Nov 2001, Draft 8.64
 +
|-
 +
| [http://suo.ieee.org/ontology/thrd39.html#03473 Ontology List, "Critique Of Non-Functional Reason", 05 Dec 2001]
 +
| IDS 1.3.10, 01 Dec 2001, Draft 8.65
 +
|-
 +
| [http://stderr.org/pipermail/arisbe/2002-January/thread.html#1247 Arisbe List, "Inquiry Driven Systems", 05 Jan 2002]
 +
| IDS, Drafts 8.69 – 8.70
 +
|-
 +
| [http://suo.ieee.org/ontology/thrd25.html#04226 Ontology List, "Pragmatic Maxim", 10 Jun 2002]
 +
| IDS 3.3, 24 Apr 2002, Draft 8.73
 +
|-
 +
| [http://suo.ieee.org/ontology/thrd25.html#04242 Ontology List, "All Ways Lead to Inquiry", 13 Jun 2002]
 +
| IDS 1.4, 10 Jun 2002, Draft 8.75
 +
|-
 +
| [http://suo.ieee.org/ontology/thrd25.html#04264 Ontology List, "Priorisms of Normative Sciences", 20 Jun 2002]
 +
| IDS 3.2.8, 10 Jun 2002, Draft 8.75
 +
|-
 +
| [http://suo.ieee.org/ontology/thrd25.html#04266 Ontology List, "Principle of Rational Action", 20 Jun 2002]
 +
| IDS 3.2.9, 10 Jun 2002, Draft 8.75
 +
|-
 +
| [http://stderr.org/pipermail/inquiry/2004-April/thread.html#1328 Inquiry List, "Reflective Inquiry", 13 Apr 2004]
 +
| IDS 3.2
 +
|-
 +
| [http://stderr.org/pipermail/inquiry/2004-November/thread.html#1996 Inquiry List, "Higher Order Signs", 24 Nov 2004]
 +
| IDS 3.4.9 – 3.4.10
 +
|-
 +
| [http://forum.wolframscience.com/showthread.php?threadid=629 NKS Forum, "Higher Order Signs", 24 Nov 2004]
 +
| IDS 3.4.9 – 3.4.10
 +
|-
 +
| [http://forum.wolframscience.com/archive/topic/629-1.html NKS Archive, "Higher Order Signs", 24 Nov 2004]
 +
| IDS 3.4.9 – 3.4.10
 +
|-
 +
| [http://forum.wolframscience.com/printthread.php?threadid=629 NKS Printable, "Higher Order Signs", 24 Nov 2004]
 +
| IDS 3.4.9 – 3.4.10
 +
|-
 +
| [http://stderr.org/pipermail/inquiry/2004-December/thread.html#2171 Inquiry List, "Recurring Themes", 17 Dec 2004]
 +
| IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
 +
|-
 +
| [http://forum.wolframscience.com/showthread.php?threadid=654 NKS Forum, "Recurring Themes", 17 Dec 2004]
 +
| IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
 +
|-
 +
| [http://forum.wolframscience.com/archive/topic/654-1.html NKS Archive, "Recurring Themes", 17 Dec 2004]
 +
| IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
 +
|-
 +
| [http://forum.wolframscience.com/printthread.php?threadid=654 NKS Printable, "Recurring Themes", 17 Dec 2004]
 +
| IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
 +
|-
 +
| [http://stderr.org/pipermail/inquiry/2004-December/thread.html#2135 Inquiry List, "Language Of Cacti", 13 Dec 2004]
 +
| IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
 +
|-
 +
| [http://forum.wolframscience.com/showthread.php?threadid=649 NKS Forum, "Language Of Cacti", 13 Dec 2004]
 +
| IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
 +
|-
 +
| [http://forum.wolframscience.com/archive/topic/649-1.html NKS Archive, "Language Of Cacti", 13 Dec 2004]
 +
| IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
 +
|-
 +
| [http://forum.wolframscience.com/printthread.php?threadid=649 NKS Printable, "Language Of Cacti", 13 Dec 2004]
 +
| IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
 +
|}
    
==Symbol Sandbox==
 
==Symbol Sandbox==
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* Terminal : <font face="terminal">&lt; &gt; '''&lt;''' '''&gt;''' <s>&lt;</s> <s>&gt;</s> <s>'''&lt;'''</s> <s>'''&gt;'''</s></font>
 
* Terminal : <font face="terminal">&lt; &gt; '''&lt;''' '''&gt;''' <s>&lt;</s> <s>&gt;</s> <s>'''&lt;'''</s> <s>'''&gt;'''</s></font>
   −
* LaTeX : <math>< ></math> <math>< >\!</math>
+
* LaTeX : <math>< ></math> <math>< >\!</math> <math>\lessdot \gtrdot</math>
    
<br>
 
<br>
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==Congratulations!==
 
==Congratulations!==
   −
'''Congratulations!'''  Someone from Missouri visited this page today as a result of [http://search.yahoo.com/search;_ylt=A0geu.Sdf_NI1nUANQNXNyoA?fr2=sg-gac&sado=1&p=system%20of%20inquiry%20papers&fr=fptb-hptb8-s&ei=UTF-8 this search]. -- [[User:MyWikiBiz|MyWikiBiz]] 11:57, 13 October 2008 (PDT)
+
'''Congratulations!'''  Someone from Missouri visited this page today as a result of [http://search.yahoo.com/search;_ylt=A0geu.Sdf_NI1nUANQNXNyoA?fr2=sg-gac&sado=1&p=system%20of%20inquiry%20papers&fr=fptb-hptb8-s&ei=UTF-8 this search]. &mdash; [[User:MyWikiBiz|MyWikiBiz]] 11:57, 13 October 2008 (PDT)
    
What do you know, it is the "Show Me" State, after all &hellip;  [[User:Jon Awbrey|Jon Awbrey]] 12:06, 13 October 2008 (PDT)
 
What do you know, it is the "Show Me" State, after all &hellip;  [[User:Jon Awbrey|Jon Awbrey]] 12:06, 13 October 2008 (PDT)
   −
:Furthermore, someone from New York City visited the page today, via a #1 search result on Yahoo! for [http://search.yahoo.com/search?p=system%20inquiry%20examples&fr=yfp-t-501&toggle=1&cop=mss&ei=UTF-8 system inquiry examples].  Congratulations, again! -- [[User:MyWikiBiz|MyWikiBiz]] 06:29, 23 October 2008 (PDT)
+
: Furthermore, someone from New York City visited the page today, via a #1 search result on Yahoo! for [http://search.yahoo.com/search?p=system%20inquiry%20examples&fr=yfp-t-501&toggle=1&cop=mss&ei=UTF-8 system inquiry examples].  Congratulations, again! &mdash; [[User:MyWikiBiz|MyWikiBiz]] 06:29, 23 October 2008 (PDT)
 +
 
 +
==Propositions And Sentences : Residual Remarks==
 +
 
 +
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:
 +
 
 +
<pre>
 +
|  X-[| f |] ,  [| f |]  c  X
 +
|  o      o  o  o  o      |
 +
|    \    /    \  |  /      |
 +
|    \  /      \ | /        | f
 +
|      \ /        \|/        |
 +
|      o          o          v
 +
|  {  %0%    ,    %1%  }  =  %B%
 +
</pre>
 +
 
 +
Why are we doing this?  The immediate reason &mdash; whose critique I defer &mdash; 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>
 +
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
 +
</pre>
Jon Awbrey
12,080

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