Difference between revisions of "Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems"

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==Fragmata==
 
==Fragmata==
  
−
# [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/inquiry.htm Arisbe Site, 30 Jun 2000, Draft 8.2]
+
{| cellpadding="4"
−
# [http://stderr.org/pipermail/arisbe/2002-January/thread.html#1247 Arisbe List, 05 Jan 2002, Drafts 8.69–8.70 and Discussion]
+
| [http://www.cspeirce.com/menu/library/aboutcsp/awbrey/inquiry.htm Arisbe Site, "Inquiry Driven Systems", 04 Jul 2000]
−
# [http://forum.wolframscience.com/showthread.php?threadid=649 NKS Forum, 13 Dec 2004, Cf. Draft 8.70]
+
| IDS 1 – 1.3.4.19, 30 Jun 2000, Draft 8.2
−
# [http://forum.wolframscience.com/printthread.php?threadid=649 NKS Document, 13 Dec 2004, Cf. Draft 8.70]
+
|-
 +
| [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==
Line 63: Line 126:
 
: 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! — [[User:MyWikiBiz|MyWikiBiz]] 06:29, 23 October 2008 (PDT)
  
−
==Propositions and Sentences==
+
==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>
 
<pre>
−
Introducing the realm of "values" is a stopgap measure that temporarily
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
permits the discussion to avoid certain singularities in the embedding
 
−
sign relation, and allowing the process of "evaluation" as a compromise
 
−
mode of signification between connotation and denotation only manages to
 
−
steer around a topic that eventually has to be mapped in full, but these
 
−
strategies do allow the discussion to proceed a little further without
 
−
having to answer questions that are too difficult to be settled fully
 
−
or even tackled directly at this point.  As far as the relations among
 
−
connoting, denoting, and evaluing are concerned, it is possible that
 
−
all of these constitute independent dimensions of significance that
 
−
a sign might be able to enjoy, but since the notion of connotation
 
−
is already generic enough to contain multitudes of subspecies, I am
 
−
going to subsume, on a tentative basis, all of the conceivable modes
 
−
of "evaluing" within the broader concept of connotation.
 
  
−
With this degree of flexibility in mind, one can say that the sentence
+
IDS -- RT
−
"f(x) = %1%" latently connotes what the sign "f(x)" patently connotes.
 
−
Taken in abstraction, both syntactic entities fall into an equivalence
 
−
class of signs that constitutes an abstract object, a thing of value
 
−
that is "identified by" the sign "f(x)", and thus an object that might
 
−
as well be "identified with" the value f(x).
 
  
−
The upshot of this whole discussion of evaluation is that it allows one to
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
rewrite the definitions of indicator functions and their fibers as follows:
 
  
−
The "indicator function" or the "characteristic function" of a set Q c X,
+
RT.  Recurring Themes
−
written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
 
−
that is defined in the following ways:
 
  
−
1.  Considered in its extensional form, f_Q is the subset of X x %B%
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
    that is given by the following formula:
 
  
−
    f_Q = {<x, b> in X x %B% :  b <=>  x in Q}.
+
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
  
−
2.  Considered in its functional form, f_Q is the map from X to %B%
+
Outline of Excerpt
−
    that is given by the following condition:
 
  
−
    f_Q (x) <=> x in Q.
+
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
  
−
The "fibers" of truth and falsity under a proposition f : X -> %B%
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
are subsets of X that are variously described as follows:
 
  
−
1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)
+
RT.  Note 8
  
−
                              =  {x in X  :  f(x) = %1%}
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
                              = {x in X  :  f(x) }.
+
1.3.10.4. Empirical Types and Rational Types
  
−
2. The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)
+
I make a brief detour to explain what are likely to be
 +
the unfamiliar features of my definition of a sentence.
  
−
                              =  {x in X  :  f(x) = %0%}
+
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.
  
−
                              = {x in X :  (f(x)) }.
+
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.
  
−
Perhaps this looks like a lot of work for the sake of what seems to be
+
Notice the general character of this development.  I start by
−
such a trivial form of syntactic transformation, but it is an important
+
defining a type of sign according to the type of object that it
−
step in loosening up the syntactic privileges that are held by the sign
+
happens to denote, ignoring at first the structural potential that
−
of logical equivalence "<=>", as written between logical sentences, and
+
it brings to the task.  According to this mode of definition, a type
−
by the sign of equality "=", as written between their logical values, or
+
of sign is singled out from other signs in terms of the type of object
−
else between propositions and their boolean values.  Doing this removes
+
that it actually denotes and not according to the type of object that it
−
a longstanding but wholly unnecessary conceptual confound between the
+
is designed or destined to denote, nor in terms of the type of structure
−
idea of an "assertion" and notion of an "equation", and it allows one
+
that it possesses in itself.  This puts the empirical categories, the
−
to treat logical equality on a par with the other logical operations.
+
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
  
−
Where are we? We just defined the concept of a functional fiber in several
+
RT.  Note 9
−
of the most excruciating ways possible, but that's just because this method
+
 
−
of refining functional fibers is intended partly for machine consumputation,
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
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:
+
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".
  
−
|  X-[| f |] , [| f |]  c  X
+
  3. Let the references that an indicator function f has to the elements
−
|  o      o  o  o  o      |
+
       on which it evaluates to %1% be called its "positive references"
−
|    \    /    \  |  /      |
+
      or its "indications".
−
|    \  /      \ | /       | f
 
−
|      \ /        \|/        |
 
−
|      o          o          v
 
−
|  {  %0%    ,    %1% }  =  %B%
 
  
−
For the sake of current reference:
+
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.
  
−
| The "fibers" of truth and falsity in a proposition f : X -> %B%
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
| are the subsets [| f |] and X - [| f |] of X that are variously
 
−
| described as follows:
 
−
|
 
−
| The fiber of %1% under f
 
−
|
 
−
| =  [| f |]  =  f^(-1)(%1%)
 
−
|
 
−
| =  {x in X  :  f(x) = %1%}
 
−
|
 
−
| =  {x in X  :  f(x) }.
 
−
|
 
−
| The fiber of %0% under f
 
−
|
 
−
| =  ~[| f |]  =  f^(-1)(%0%)
 
−
|
 
−
| =  {x in X  :  f(x) = %0%}
 
−
|
 
−
| =  {x in X  :  (f(x)) }.
 
  
−
Oh, by the way, the outer parentheses in "(f(g))" signify negation.
+
RT. Note 10
−
I did not have here the "stricken parentheses" that I normally use.
 
  
−
Why are we doing this?  The immediate reason -- whose critique I defer --
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
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
+
1.3.10.5.  Articulate Sentences (concl.)
−
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.
 
  
−
----
+
I conclude my pragmatic semiotic treatment of the relation between
 +
a sentence (a logical sign) and a proposition (a logical object).
  
−
In this episode, I compile a collection of definitions,
+
The universe of discourse for a sentence, the set whose elements the
−
leading up to the particular conception of a "sentence"
+
sentence is interpreted to be about, is not a property of the sentence
−
that I'll be using throughout the rest of this inquiry.
+
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.
  
−
1.3.10.3  Propositions & Sentences (cont.)
+
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.
  
−
As a purely informal aid to interpretation, I frequently use the letters
+
To assist the reading of informal examples, I frequently use the letters
−
"p", "q" to denote propositions.  This can serve to tip off the reader
+
"t", "u", "v", "z" to denote sentences.  Thus, it is conceivable to have
−
that a function is intended as the indicator function of a set, and
+
a situation where z = "q" and where q : X -> %B%.  Altogether, this means
−
it saves us the trouble of declaring the type f : X -> %B% each
+
that the sign "z" denotes the sentence z, that the sentence z is the same
−
time that a function is introduced as a proposition.
+
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.
  
−
Another convention of use in this context is to let boldface letters
+
A "sentential connective" is a sign, a coordinated sequence of signs,
−
stand for k-tuples, lists, or sequences of objects.  Typically, the
+
a syntactic pattern of contextual arrangement, or any other syntactic
−
elements of the k-tuple, list, or sequence are all of one type, and
+
device that can be used to connect a number of sentences together in
−
typically the boldface letter is of the same basic character as the
+
order to form a single sentence.  If k is the number of sentences that
−
indexed or subscripted letters that are used denote the components
+
are thereby connected, then the connective is said to be of "order k".
−
of the k-tuple, list, or sequence.  When the dimension of elements
+
If the sentences acquire a logical relationship through this mechanism,
−
and functions is clear from the context, we may elect to drop the
+
and are not just strung together by this device, then the connective
−
bolding of characters that name k-tuples, lists, and sequences.
+
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".
  
−
For example:
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
1.  If x_1, ..., x_k in X,      then #x# = <x_1, ..., x_k> in X' = X^k.
+
RT.  Note 11
  
−
2.  If x_1, ..., x_k  : X,      then #x# = <x_1, ..., x_k>  : X' = X^k.
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
3. If f_1, ..., f_k : X -> Y,  then #f# = <f_1, ..., f_k>  : (X -> Y)^k.
+
1.3.10.6.  Stretching Principles
  
−
There is usually felt to be a slight but significant distinction between
+
We are in the home stretch of what I promised to bring home this time around.
−
the "membership statement" that uses the sign "in" as in Example (1) and
+
Let me set up the play by bringing back to mind a deuce of basic definitions
−
the "type statement" that uses the sign ":" as in examples (2) and (3).
+
from earlier in the game:
−
The difference that appears to be perceived in categorical statements,
 
−
when those of the form "x in X" and those of the form "x : X" are set
 
−
in side by side comparisons with each other, is that a multitude of
 
−
objects can be said to have the same type without having to posit
 
−
the existence of a set to which they all belong.  Without trying
 
−
to decide whether I share this feeling or even fully understand
 
−
the distinction in question, I can only try to maintain a style
 
−
of notation that respects it to some degree. It is conceivable
 
−
that the question of belonging to a set is rightly sensed to be
 
−
the more serious matter, one that has to do with the reality of
 
−
an object and the substance of a predicate, than the question of
 
−
falling under a type, that may have more to do with the way that
 
−
a sign is interpreted and the way that information about an object
 
−
is organized.  When it comes to the kinds of hypothetical statements
 
−
that appear in these Examples, those of the form "x in X => #x# in X'"
 
−
and "x : X => #x# : X'", these are usually read as implying some order
 
−
of synthetic construction, one whose contingent consequences involve the
 
−
constitution of a new space to contain the elements being compounded and
 
−
the recognition of a new type to characterize the elements being moulded,
 
−
respectively.  In these applications, the statement about types is again
 
−
taken to be less presumptive than the corresponding statement about sets,
 
−
since the apodosis is intended to do nothing more than to abbreviate and
 
−
to summarize what is already stated in the protasis.
 
  
 
A "boolean connection" of degree k, also known as a "boolean function"
 
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,
 
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
 
a boolean connection of degree k is a proposition about things in the
−
universe X = %B%^k.
+
universe of discourse X = %B%^k.
  
 
An "imagination" of degree k on X is a k-tuple of propositions about things
 
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 kinds of notation that
+
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
 
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
 
as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k.  All of
Line 255: Line 394:
 
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
 
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
 
the former.
 
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 images" that is determined by #f# and x, more specifically,
Line 265: Line 504:
 
defined as follows:
 
defined as follows:
  
−
If        #b#  =      <b_1, ..., b_k>          in  %B%^k,
+
  If        #b#  =      <b_1, ..., b_k>          in  %B%^k,
  
−
then  p_j (#b#)  =  p_j (<b_1, ..., b_k>)  =  b_j  in  %B%.
+
  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>.
 
The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.
  
−
A "sentence about things in the universe", for short, a "sentence",
+
Just as a sentence is a sign that denotes a proposition,
−
is a sign that denotes a proposition.  In other words, a sentence is
+
which thereby serves to indicate a set, a propositional
−
any sign that denotes an indicator function, any sign whose object is
+
connective is a provision of syntax whose mediate effect
−
a function of the form f : X -> B.
+
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%,
  
−
To emphasize the empirical contingency of this definition, one can say
+
      then  -[z]-  =  q.
−
that a sentence is any sign that is interpreted as naming a proposition,
 
−
any sign that is taken to denote an indicator function, or any sign whose
 
−
object happens to be a function of the form f : X -> B.
 
  
−
----
+
  2.  If the sentence z denotes the proposition q : X -> %B%,
  
−
I finish out the Subsection on "Propositions & Sentences" with
+
      hence  [|q|]  =  q^(-1)(%1%)  =  Q  c  X,
−
an account of how I use concepts like "assertion" and "denial".
 
  
−
1.3.10.3 Propositions & Sentences (cont.)
+
      then  -[z]-  =  q  =  f_Q  = -{Q}-.
  
−
An "expression" is a type of sign, for instance, a term or a sentence,
+
  3.   Q    = {x in X  : x in Q}
−
that has a value.  In forming this conception of an expression, I am
 
−
deliberately leaving a number of options open, for example, whether
 
−
the expression amounts to a term or to a sentence and whether it
 
−
ought to be accounted as denoting a value or as connoting a value.
 
−
Perhaps the expression has different values under different lights,
 
−
and perhaps it relates to them differently in different respects.
 
−
In the end, what one calls an expression matters less than where
 
−
its value lies. Of course, no matter whether one chooses to call
 
−
an expression a "term" or a "sentence", if the value is an element
 
−
of %B%, then the expression affords the option of being treated as
 
−
a sentence, meaning that it is subject to assertion and composition
 
−
in the same way that any sentence is, having its value figure into
 
−
the values of larger expressions through the linkages of sentential
 
−
connectives, and affording us the consideration of what things in
 
−
what universe the corresponding proposition happens to indicate.
 
  
−
Expressions with this degree of flexibility in the types under
+
              = [| -{Q}- |]  =  -{Q}-^(-1)(%1%)
−
which they can be interpreted are difficult to translate from
 
−
their formal settings into more natural contexts. Indeed,
 
−
the whole issue can be difficult to talk about, or even
 
−
to think about, since the grammatical categories of
 
−
sentential clauses and noun phrases are rarely so
 
−
fluid in natural language settings are they can
 
−
be rendered in artificially formal arenas.
 
  
−
To finesse the issue of whether an expression denotes or connotes its value,
+
              =  [|  f_Q  |]  =  (f_Q)^(-1)(%1%).
−
or else to create a general term that covers what both possibilities have
 
−
in common, one can say that an expression "evalues" its value.
 
  
−
An "assertion" is just a sentence that is being used in a certain way,
+
  4.  -{Q}-  =  -{ {x in X  :  x in Q} }-
−
namely, to indicate the indication of the indicator function that the
 
−
sentence is usually used to denote.  In other words, an assertion is
 
−
a sentence that is being converted to a certain use or that is being
 
−
interpreted in a certain role, and one whose immediate denotation is
 
−
being pursued to its substantive indication, specifically, the fiber
 
−
of truth of the proposition that the sentence potentially denotes.
 
−
Thus, an assertion is a sentence that is held to denote the set of
 
−
things in the universe for which the sentence is held to be true.
 
  
−
Taken in a context of communication, an assertion is basically a request
+
              =  -[x in Q]-
−
that the interpreter consider the things for which the sentence is true,
 
−
in other words, to find the fiber of truth in the associated proposition,
 
−
or to invert the indicator function that is denoted by the sentence with
 
−
respect to its possible value of truth.
 
  
−
A "denial" of a sentence z is an assertion of its negation -(z)-.
+
              =  f_Q.
−
The denial acts as a request to think about the things for which the
 
−
sentence is false, in other words, to find the fiber of falsity in the
 
−
indicted proposition, or to invert the indicator function that is being
 
−
denoted by the sentence with respect to its possible value of falsity.
 
  
−
According to this manner of definition, any sign that happens to denote
+
Now if a sentence z really denotes a proposition q,
−
a proposition, any sign that is taken as denoting an indicator function,
+
and if the notation "-[z]-" is meant to supply merely
−
by that very fact alone successfully qualifies as a sentence.  That is,
+
another name for the proposition that z already denotes,
−
a sentence is any sign that actually succeeds in denoting a proposition,
+
then why is there any need for this additional notation?
−
any sign that one way or another brings to mind, as its actual object,
+
It is because the interpretive mind habitually races from
−
a function of the form f : X -> B.
+
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.
  
−
There are many features of this definition that need to be understood.
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
Indeed, there are problems involved in this whole style of definition
 
−
that need to be discussed, and doing this requires a slight excursion.
 
 
</pre>
 
</pre>

Latest revision as of 18:46, 16 January 2009

Fragmata

Arisbe Site, "Inquiry Driven Systems", 04 Jul 2000 IDS 1 – 1.3.4.19, 30 Jun 2000, Draft 8.2
SUO List, "Critique Of Non-Functional Reason", 27 Nov 2001 IDS 1.3.10.3, 27 Nov 2001, Draft 8.63
SUO List, "Critique Of Non-Functional Reason", 29 Nov 2001 IDS 1.3.10.4, 28 Nov 2001, Draft 8.64
Ontology List, "Critique Of Non-Functional Reason", 05 Dec 2001 IDS 1.3.10, 01 Dec 2001, Draft 8.65
Arisbe List, "Inquiry Driven Systems", 05 Jan 2002 IDS, Drafts 8.69 – 8.70
Ontology List, "Pragmatic Maxim", 10 Jun 2002 IDS 3.3, 24 Apr 2002, Draft 8.73
Ontology List, "All Ways Lead to Inquiry", 13 Jun 2002 IDS 1.4, 10 Jun 2002, Draft 8.75
Ontology List, "Priorisms of Normative Sciences", 20 Jun 2002 IDS 3.2.8, 10 Jun 2002, Draft 8.75
Ontology List, "Principle of Rational Action", 20 Jun 2002 IDS 3.2.9, 10 Jun 2002, Draft 8.75
Inquiry List, "Reflective Inquiry", 13 Apr 2004 IDS 3.2
Inquiry List, "Higher Order Signs", 24 Nov 2004 IDS 3.4.9 – 3.4.10
NKS Forum, "Higher Order Signs", 24 Nov 2004 IDS 3.4.9 – 3.4.10
NKS Archive, "Higher Order Signs", 24 Nov 2004 IDS 3.4.9 – 3.4.10
NKS Printable, "Higher Order Signs", 24 Nov 2004 IDS 3.4.9 – 3.4.10
Inquiry List, "Recurring Themes", 17 Dec 2004 IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
NKS Forum, "Recurring Themes", 17 Dec 2004 IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
NKS Archive, "Recurring Themes", 17 Dec 2004 IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
NKS Printable, "Recurring Themes", 17 Dec 2004 IDS 1.3.10.3 – 1.3.10.7, 16 Dec 2001
Inquiry List, "Language Of Cacti", 13 Dec 2004 IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
NKS Forum, "Language Of Cacti", 13 Dec 2004 IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
NKS Archive, "Language Of Cacti", 13 Dec 2004 IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002
NKS Printable, "Language Of Cacti", 13 Dec 2004 IDS 1.3.10.8 – 1.3.10.13, 06 Jan 2002

Symbol Sandbox

  • Default : < > < > < > < >
  • Courier : < > < > < > < >
  • Fixedsys : < > < > < > < >
  • Pmingliu : < > < > < > < >
  • System : < > < > < > < >
  • Terminal : < > < > < > < >
  • LaTeX \[< >\] \(< >\!\) \(\lessdot \gtrdot\)


\[\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}\]


Xj = Pj ∪ Qj ,

P = ∪j Pj ,

Q = ∪j Qj .


\[\begin{matrix} X_j = P_j \cup Q_j , & P = \bigcup_j P_j , & Q = \bigcup_j Q_j . \end{matrix}\]

Notes & Queries

JA: I'm in the process of merging and reconciling two slightly different versions of this paper, but it may be the end of the summer before I can finish doing that. Jon Awbrey 09:48, 29 May 2007 (PDT)

Jon, your content soars way over my head, but I am nonetheless delighted that you're using Centiare so effectively (if at least to get #1 Google search results for inquiry driven systems — even though that's currently not happening … Google's a bit quirky as it digests our site and "learns" where to put us in the rankings). I hope that you can keep up the effort, and that we can help you from an operational standpoint. MyWikiBiz 13:26, 29 May 2007 (PDT)

JA: Thanks for the interest, and I've been "pleased as punch" with the environment so far, mostly for reasons independent of the SEO factor — the quality of the working environment is more important to me than any need to corner the market in a given subject area. As far as I know, I coined the term "inquiry driven system" back in the (19)80's — though I know as soon as I say that, it will turn out that C.S. Peirce scooped me by a century or so — anyway, it's already the case that 90% of the stuff on the web about inquiry driven systems was written by yours truly. On the other hand, when my Centiare user and directory pages depose my Wikipedia user and discussion pages from the top of the Google heap, that will be the test case for me! Jon Awbrey 14:36, 29 May 2007 (PDT)

Congratulations!

Congratulations! Someone from Missouri visited this page today as a result of this search. — MyWikiBiz 11:57, 13 October 2008 (PDT)

What do you know, it is the "Show Me" State, after all … 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 system inquiry examples. Congratulations, again! — 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: 

|   X-[| f |] ,  [| f |]   c   X
|   o       o   o   o   o      |
|    \     /     \  |  /       |
|     \   /       \ | /        | f
|      \ /         \|/         |
|       o           o          v
|   {  %0%    ,    %1%  }  =  %B%

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. 

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.

Empirical Types and Rational Types

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.

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RT.  Note 13

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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.

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RT.  Note 14

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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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