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
+
{| 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==
Line 62: 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>
 
<pre>
−
The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
+
|  X-[| f |] ,  [| f |]  c  X
−
whose arithmetic operations obey the rules of GF(2), the "galois field"
+
|  o      o  o  o  o      |
−
of exactly two elements, whose addition and multiplication tables are
+
|    \    /    \  |  /      |
−
tantamount to addition and multiplication of integers "modulo 2".
+
|    \  /      \ | /        | f
 +
|      \ /        \|/        |
 +
|      o          o          v
 +
|  { %0%    ,   %1%  } =  %B%
 +
</pre>
  
−
The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
+
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.
−
whose elements are read as "false" and "true", or as "falsity" and "truth",
 
−
respectively.
 
  
−
At this point, I cannot tell whether the distinction between these two
+
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.
−
domains is slight or significant, and so this question must evolve its
 
−
own answer, while I pursue a larger inquiry by means of its hypothesis.
 
−
The weight of the matter appears to increase as the investigation moves
 
−
from abstract, algebraic, and formal settings to contexts where logical
 
−
semantics, natural language syntax, and concrete categories of grammar
 
−
are compelling considerations.  Speaking abstractly and roughly enough,
 
−
it is often acceptable to identify these two domains, and up until this
 
−
point there has rarely appeared to be a sufficient reason to keep their
 
−
concepts separately in mind.  The boolean domain %B% comes with at least
 
−
two operations, though often under different names and always included
 
−
in a number of others, that are analogous to the field operations of the
 
−
binary domain !B!, and operations that are isomorphic to the rest of the
 
−
boolean operations in %B% can always be built on the binary basis of !B!.
 
  
−
Of course, as sets of the same cardinality, the domains !B! and %B%
+
<pre>
−
and all of the structures that can be built on them become isomorphic
+
Where are we?  We just defined the concept of a functional fiber in several
−
at a high enough level of abstraction.  Consequently, the main reason
+
of the most excruciating ways possible, but that's just because this method
−
for making this distinction in the immediate context appears to be more
+
of refining functional fibers is intended partly for machine consumputation,
−
a matter of grammar than an issue of logical and mathematical substance,
+
so its schemata must be rendered free of all admixture of animate intuition.
−
namely, so that the signs "%0%" and "%1%" can appear with a semblance of
+
However, just between us, a single picture may suffice to sum up the notion:
−
syntactic legitimacy in linguistic contexts that call for a grammatical
 
−
sentence or a sentence surrogate to represent the classes of sentences
 
−
that are "always false" and "always true", respectively. The signs
 
−
"0" and "1", customarily read as nouns but not as sentences, fail
 
−
to be suitable for this purpose.  Whether these scruples, that are
 
−
needed to conform to a particular choice of natural language context,
 
−
are ultimately important, is another thing I do not know at this point.
 
  
−
The "negation" of x, for x in %B%, written as "(x)"
+
o-------------------------------------------------o
−
and read as "not x", is the boolean value (x) in %B%
+
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
−
that is %1% when x is %0%, and %0% when x is %1%.
+
| ` ` ` ` X-[| f |] , `[| f |]` ` `c` ` X ` ` ` ` |
 +
| ` ` ` ` o ` ` ` o ` o ` o ` o ` ` ` ` | ` ` ` ` |
 +
| ` ` ` ` `\` ` `/` ` `\` | `/` ` ` ` ` | ` ` ` ` |
 +
| ` ` ` ` ` \ ` / ` ` ` \ | / ` ` ` ` ` | f ` ` ` |
 +
| ` ` ` ` ` `\`/` ` ` ` `\|/` ` ` ` ` ` | ` ` ` ` |
 +
| ` ` ` ` ` ` o ` ` ` ` ` o ` ` ` ` ` ` v ` ` ` ` |
 +
| ` ` ` ` { `%0%` ` , ` `%1%` } ` `=` `%B%` ` ` ` |
 +
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
 +
o-------------------------------------------------o
  
−
Thus, negation is a monadic operation on boolean
+
Why are we doing this?  The immediate reason -- whose critique I defer --
−
values, a function of the form (_) : %B% -> %B%.
+
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.
  
−
It is convenient to transport the product and the sum operations of !B!
+
In the next few installments, I will be working toward the definition
−
into the logical setting of %B%, where they can be symbolized by signs
+
of an operation called the "stretch". This is related to the concept
−
of the same character, doubly underlined as necessary to avoid confusion.
+
from category theory that is called a "pullback".  As a few will know
−
This yields the following definitions of a "product" and a "sum" in %B%
+
the uses of that already, maybe there's hope of stretching the number.
−
and leads to the following forms of multiplication and addition tables.
+
</pre>
  
−
The "product" of x and y, for values x, y in %B%, is given by Table 8.
+
==Empirical Types and Rational Types==
  
−
Table 8.  Product Operation for the Boolean Domain
+
<pre>
−
o---------o---------o---------o
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
|  %.%  #  %0%  |  %1%  |
 
−
o=========o=========o=========o
 
−
|  %0%  #  %0%  |  %0%  |
 
−
o---------o---------o---------o
 
−
|  %1%  #  %0%  |  %1%  |
 
−
o---------o---------o---------o
 
  
−
Viewed as a function on logical values, %.% : %B% x %B% -> %B%, the product corresponds to the logical operation that is commonly called "conjunction" and that is otherwise expressed as "x and y".  In accord with common practice, the raised dot ".", doubly underlined or otherwise, is frequently omitted from written expressions of the product.
+
IDS -- RT
−
The "sum" of x and y, for values x, y in %B%, is given by Table 9.
 
  
−
Table 9.  Sum Operation for the Boolean Domain
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
o---------o---------o---------o
 
−
|  %+%  #  %0%  |  %1%  |
 
−
o=========o=========o=========o
 
−
|  %0%  #  %0%  |  %1%  |
 
−
o---------o---------o---------o
 
−
|  %1%  #  %1%  |  %0%  |
 
−
o---------o---------o---------o
 
  
−
Viewed as a function on logical values, %+% : %B% x %B% -> %B%, the sum corresponds to the logical operation that is generally called "exclusive disjunction" and that is otherwise expressed as "x or y, but not both".  Depending on the context, a couple of other signs and readings that can invoke this operation are:
+
RT.  Recurring Themes
  
−
1.  "x =/= y", read "x is not equal to y", or "exactly one of x and y".
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
2.  "x <=/=> y", read "x is not equivalent to y", or "x opposes y".
 
  
−
For sentences, the signs of equality ("=") and inequality ("=/=") are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence ("<=>") and inequivalence ("<=/=>") refer to the logical values, if any, of these strings, and serve to signify the equality and inequality, respectively, of their conceivable boolean values. For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.
+
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
  
−
In logical studies, one tends to be interested in all of the operations or all of the functions of a given type, at least, to the extent that their totalities and their individualities can be comprehended, and not just the specialized collections that define particular algebraic structures.
+
Outline of Excerpt
  
−
Although the rest of the conceivably possible dyadic operations on boolean values, in other words, the remainder of the sixteen functions f : %B% x %B% -> %B%, could be presented in the same way as the multiplication and addition tables, it is better to look for a more efficient style of representation, one that is able to express all of the boolean functions on the same number of variables on a roughly equal basis, and with a bit of luck, affords us with a calculus for computing with these functions.
+
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 utility of a suitable calculus would involve, among other things:
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
1.  Finding the values of given functions for given arguments.
+
RT.  Note 8
−
2.  Inverting boolean functions, that is, "finding the fibers" of boolean functions, or solving logical equations that are expressed in terms of boolean functions.
 
−
3.  Facilitating the recognition of invariant forms that take boolean functions as their functional components.
 
  
−
The whole point of formal logic, the reason for doing logic formally and the measure that determines how far it is possible to reason abstractly, is to discover functions that do not vary as much as their variables do, in other words, to identify forms of logical functions that, though they express a dependence on the values of their constituent arguments, do not vary as much as possible, but approach the way of being a function that constant functions enjoy.  Thus, the recognition of a logical law amounts to identifying a logical function, that, though it ostensibly depends on the values of its putative arguments, is not as variable in its values as the values of its variables are allowed to be.
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
The "indicator function" or the "characteristic function" of a set Q c X, written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%} that is defined in the following ways:
+
1.3.10.4.  Empirical Types and Rational Types
  
−
1.  Considered in extensional form, f_Q is the subset of X x %B% that is given by the following formula:
+
I make a brief detour to explain what are likely to be
 +
the unfamiliar features of my definition of a sentence.
  
−
    f_Q  =  {<x, b> in X x %B%  :  b = %1%  <=>  x in Q}.
+
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.
  
−
2.  Considered in functional form, f_Q is the map from X to %B% that is given by the following condition:
+
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.
  
−
    f_Q (x) = %1% <=> x in Q.
+
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.
  
−
A "proposition about things in the universe", for short, a "proposition", is the same thing as an indicator function, that is, a function of the form f : X -> %B%.  The convenience of this seemingly redundant usage is that it allows one to refer to an indicator function without having to specify right away, as a part of its designated subscript, exactly what set it indicates, even though a proposition always indicates some subset of its designated universe, and even though one will probably or eventually want to know exactly what subset that is.
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some of which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.
+
RT.  Note 9
  
−
The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.  In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f.  All of these equivalent concepts are defined as follows:
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
Fiber of y under f  =  f^(-1)(y)  =  {x in X  :  f(x) = y}.
 
  
−
In the special case where f is the indicator function f_Q of the set Q c X, the fiber of 1 under fQ is just the set Q back again:
+
1.3.10.5.  Articulate Sentences
  
−
Fiber of 1 under fQ  =  fQ-1(1)  =  {x in X  : fQ(x) = 1}  =  Q.
+
A sentence is called "articulate" if:
  
−
In this specifically boolean setting, as in the more generally logical context, where "truth" under any name is especially valued, it is worth devoting a specialized notation to the "fiber of truth" in a proposition, to mark the set that it indicates with a particular ease and explicitness.
+
  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.
  
−
For this purpose, I introduce the use of "fiber bars" or "ground signs", written as "[| ... |]" around a sentence or the sign of a proposition, and whose application is defined as follows:
+
  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.
  
−
If f : X -> %B%,
+
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.
  
−
then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x) = %1%}.
+
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".
  
−
Some may recognize here fledgling efforts
+
Since the value of a sentence or a proposition depends on the universe of discourse
−
to reinforce flights of Fregean semantics
+
to which it is "referred", and since it also depends on the element of the universe
−
with impish pitches of Peircean semiotics.
+
with regard to which it is evaluated, it is conventional to say that a sentence or
−
Some may deem it Icarean, all too Icarean.
+
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.
  
−
1.3.10.3  Propositions & Sentences (cont.)
+
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:
  
−
The definition of a fiber, in either the general or the boolean case,
+
  1.  Let the reference of a sentence or a proposition to a universe of discourse,
−
is a purely nominal convenience for referring to the antecedent subset,
+
      the one that it acquires by way of taking on any interpretation at all, be
−
the inverse image under a function, or the pre-image of a functional value.
+
      taken as its "general reference", the kind of reference that one can safely
−
The definition of an operator on propositions, signified by framing the signs
+
      ignore as irrelevant, at least, so long as one stays immersed in only one
−
of propositions with fiber bars or ground signs, remains a purely notational
+
      context of discourse or only one moment of discussion.
−
device, and yet the notion of a fiber in a logical context serves to raise
 
−
an interesting point. By way of illustration, it is legitimate to rewrite
 
−
the above definition in the following form:
 
  
−
If f : X -> %B%,
+
  2. Let the references that an indicator function f has to the elements
 +
      on which it evaluates to %0% be called its "negative references".
  
−
then [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x)}.
+
  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".
  
−
The set-builder frame "{x in X  :  ... }" requires a grammatical sentence or
+
Finally, unspecified references to the "references" of a sentence,
−
a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
+
a proposition, or an indicator function can be taken by default
−
that serves to fill the frame in the initial definition of a logical fiber.
+
as references to their specific, positive references.
−
And what is a sentence but the expression of a proposition, in other words,
 
−
the name of an indicator function?  As it happens, the sign "f(x)" and the
 
−
sentence "f(x) = %1%" represent the very same value to this context, for
 
−
all x in X, that is, they will appear equal in their truth or falsity
 
−
to any reasonable interpreter of signs or sentences in this context,
 
−
and so either one of them can be tendered for the other, in effect,
 
−
exchanged for the other, within this context, frame, and reception.
 
  
−
The sign "f(x)" manifestly names the value f(x).
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
This is a value that can be seen in many lights.
 
−
It is, at turns:
 
  
−
1.  The value that the proposition f has at the point x,
+
RT.  Note 10
−
    in other words, the value that f bears at the point x
 
−
    where f is being evaluated, the value that f takes on
 
−
    with respect to the argument or the object x that the
 
−
    whole proposition is taken to be about.
 
  
−
2.  The value that the proposition f not only takes up at
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
    the point x, but that it carries, conveys, transfers,
 
−
    or transports into the setting "{x in X  :  ... }" or
 
−
    into any other context of discourse where f is meant
 
−
    to be evaluated.
 
  
−
3.  The value that the sign "f(x)" has in the context where it is placed,
+
1.3.10.5.  Articulate Sentences (concl.)
−
    that it stands for in the context where it stands, and that it continues
 
−
    to stand for in this context just so long as the same proposition f and the
 
−
    same object x are borne in mind.
 
  
−
4.  The value that the sign "f(x)" represents to its full interpretive context
+
I conclude my pragmatic semiotic treatment of the relation between
−
    as being its own logical interpretant, namely, the value that it signifies
+
a sentence (a logical sign) and a proposition (a logical object).
−
    as its canonical connotation to any interpreter of the sign that is cognizant
 
−
    of the context in which it appears.
 
  
−
The sentence "f(x) = %1%" indirectly names what the sign "f(x)"
+
The universe of discourse for a sentence, the set whose elements the
−
more directly names, that is, the value f(x).  In other words,
+
sentence is interpreted to be about, is not a property of the sentence
−
the sentence "f(x) = %1%" has the same value to its interpretive
+
by itself, but of the sentence in the presence of its interpretation.
−
context that the sign "f(x)" imparts to any comparable context,
+
Independently of how many explicit variables a sentence contains, its
−
each by way of its respective evaluation for the same x in X.
+
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.
  
−
What is the relation among connoting, denoting, and "evaluing", where
+
Notice that the letters "p" and "q", interpreted as signs that denote
−
the last term is coined to describe all the ways of bearing, conveying,
+
the indicator functions p, q : X -> %B%, have the character of sentences
−
developing, or evolving a value in, to, or into an interpretive context?
+
in relation to propositions, at least, they have the same status in this
−
In other words, when a sign is evaluated to a particular value, one can
+
abstract discussion as genuine sentences have in concrete applications.
−
say that the sign "evalues" that value, using the verb in a way that is
+
This illustrates the relation between sentences and propositions as
−
categorically analogous or grammatically conjugate to the times when one
+
a special case of the relation between signs and objects.
−
says that a sign "connotes" an idea or that a sign "denotes" an object.
 
−
This does little more than provide the discussion with a "weasel word",
 
−
a term that is designed to avoid the main issue, to put off deciding the
 
−
exact relation between formal signs and formal values, and ultimately to
 
−
finesse the question about the nature of formal values, whether they are
 
−
more akin to conceptual signs and figurative ideas or to the kinds of
 
−
literal objects and platonic ideas that are independent of the mind.
 
  
−
These questions are confounded by the presence of certain peculiarities in
+
To assist the reading of informal examples, I frequently use the letters
−
formal discussions, especially by the fact that an equivalence class of signs
+
"t", "u", "v", "z" to denote sentences.  Thus, it is conceivable to have
−
is tantamount to a formal object.  This has the effect of allowing an abstract
+
a situation where z = "q" and where q : X -> %B%.  Altogether, this means
−
connotation to work as a formal denotation.  In other words, if the purpose of
+
that the sign "z" denotes the sentence z, that the sentence z is the same
−
a sign is merely to lead its interpreter up to a sign in an equivalence class
+
thing as the sentence "q", and that the sentence "q" denotes the proposition,
−
of signs, then it follows that this equivalence class is the object of the
+
characteristic function, or indicator function q : X -> %B%.  In settings where
−
sign, that connotation can achieve denotation, at least, to some degree,
+
it is necessary to keep track of a large number of sentences, I use subscripted
−
and that the interpretant domain collapses with the object domain,
+
letters like "e_1", ..., "e_n" to refer to the various expressions in question.
−
at least, in some respect, all things being relative to the
 
−
sign relation that embeds the discussion.
 
  
−
Introducing the realm of "values" is a stopgap measure that temporarily
+
A "sentential connective" is a sign, a coordinated sequence of signs,
−
permits the discussion to avoid certain singularities in the embedding
+
a syntactic pattern of contextual arrangement, or any other syntactic
−
sign relation, and allowing the process of "evaluation" as a compromise
+
device that can be used to connect a number of sentences together in
−
mode of signification between connotation and denotation only manages to
+
order to form a single sentence.  If k is the number of sentences that
−
steer around a topic that eventually has to be mapped in full, but these
+
are thereby connected, then the connective is said to be of "order k".
−
strategies do allow the discussion to proceed a little further without
+
If the sentences acquire a logical relationship through this mechanism,
−
having to answer questions that are too difficult to be settled fully
+
and are not just strung together by this device, then the connective
−
or even tackled directly at this point.  As far as the relations among
+
is called a "logical connective".  If the value of the constructed
−
connoting, denoting, and evaluing are concerned, it is possible that
+
sentence depends on the values of the component sentences in such
−
all of these constitute independent dimensions of significance that
+
a way that the value of the whole is a boolean function of the
−
a sign might be able to enjoy, but since the notion of connotation
+
values of the parts, then the connective earns the title of
−
is already generic enough to contain multitudes of subspecies, I am
+
a "propositional connective".
−
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
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
"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
+
RT.  Note 11
−
rewrite the definitions of indicator functions and their fibers as follows:
 
  
−
The "indicator function" or the "characteristic function" of a set Q c X,
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
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%
+
1.3.10.6.  Stretching Principles
−
    that is given by the following formula:
 
  
−
    f_Q  =  {<x, b> in X x %B%  : b  <=>  x in Q}.
+
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:
  
−
2.  Considered in its functional form, f_Q is the map from X to %B%
+
A "boolean connection" of degree k, also known as a "boolean function"
−
    that is given by the following condition:
+
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.
  
−
    f_Q (x) <=> x in Q.
+
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 "fibers" of truth and falsity under a proposition f : X -> %B%
+
The purpose of this exercise is to illuminate how a sentence,
−
are subsets of X that are variously described as follows:
+
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.
  
−
1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
                              = {x in X  :  f(x) = %1%}
+
RT. Note 12
  
−
                              =  {x in X  :  f(x) }.
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)
+
1.3.10.6.  Stretching Principles (cont.)
  
−
                              =  {x in X  : f(x) = %0%}
+
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:
  
−
                              = {x in X  :  (f(x)) }.
+
  1. Any relation of values extends to a relation of what is valued.
  
−
Perhaps this looks like a lot of work for the sake of what seems to be
+
  2.  Any statement about values says something about the things
−
such a trivial form of syntactic transformation, but it is an important
+
      that are given these values.
−
step in loosening up the syntactic privileges that are held by the sign
 
−
of logical equivalence "<=>", as written between logical sentences, and
 
−
by the sign of equality "=", as written between their logical values, or
 
−
else between propositions and their boolean values.  Doing this removes
 
−
a longstanding but wholly unnecessary conceptual confound between the
 
−
idea of an "assertion" and notion of an "equation", and it allows one
 
−
to treat logical equality on a par with the other logical operations.
 
  
−
----
+
  3.  Any association among a range of values establishes
 +
      an association among the domains of things
 +
      that these values are the values of.
  
−
Where are we? We just defined the concept of a functional fiber in several
+
  4. Any connection between two values can be stretched to create a connection,
−
of the most excruciating ways possible, but that's just because this method
+
      of analogous form, between the objects, persons, qualities, or relationships
−
of refining functional fibers is intended partly for machine consumputation,
+
      that are valued in these connections.
−
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
+
   5. For every operation on values, there is a corresponding operation on the actions,
−
|  o      o  o  o  o      |
+
       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
−
|    \  /      \ | /       | f
+
      various proceedings.
−
|      \ /        \|/        |
 
−
|      o          o          v
 
−
|  {  %0%    ,    %1%  }  =  %B%
 
  
−
For the sake of current reference:
+
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.
  
−
| The "fibers" of truth and falsity in a proposition f : X -> %B%
+
In particular, a connection F : %B%^k -> %B% can be understood to
−
| are the subsets [| f |] and X - [| f |] of X that are variously
+
indicate a relation among boolean values, namely, the k-ary relation
−
| described as follows:
+
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
−
| The fiber of %1% under f
+
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
−
| = [| f |]  =  f^(-1)(%1%)
+
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
−
| =  {x in X  :  f(x) = %1%}
+
#f# in (X -> %B%)^k and the connection F : %B%^k -> %B% stretch
−
|
+
each other to cover the universe X, yielding a new proposition
−
| = {x in X :  f(x) }.
+
q : X -> %B%.
−
|
 
−
| 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.
+
To encapsulate the form of this general result, I define a scheme of composition
−
I did not have here the "stricken parentheses" that I normally use.
+
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:
  
−
Why are we doing this? The immediate reason -- whose critique I defer --
+
  1. In a general setting, where the connection F and the imagination #f#
−
has to do with finding a modus vivendi, whether a working compromise or
+
      are both permitted to take up a variety of concrete possibilities,
−
a genuine integration, between the assertive-declarative languages and
+
      call q the "stretch of F and #f# from X to %B%", and write it in
−
the functional-procedural languages that we have available for the sake
+
      the style of a composition as "F $ #f#".  This is meant to suggest
−
of conceptual-logical-ontological analysis, clarification, description,
+
      that the symbol "$", here read as "stretch", denotes an operator
−
inference, problem-solving, programming, representation, or whatever.
+
      of the form $ : (%B%^k -> %B%) x (X -> %B%)^k -> (X -> %B%).
  
−
In the next few installments, I will be working toward the definition
+
  2.  In a setting where the connection F is fixed but the imagination #f#
−
of an operation called the "stretch".  This is related to the concept
+
      is allowed to vary over a wide range of possibilities, call q the
−
from category theory that is called a "pullback".  As a few will know
+
      "stretch of F to #f# on X", and write it in the style "F^$ #f#",
−
the uses of that already, maybe there's hope of stretching the number.
+
      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%.
  
−
In this episode, I compile a collection of definitions,
+
Because the stretch notation is used only in settings
−
leading up to the particular conception of a "sentence"
+
where the imagination #f# : (X -> %B%)^k and the
−
that I'll be using throughout the rest of this inquiry.
+
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.
  
−
1.3.10.3  Propositions & Sentences (cont.)
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
  
−
As a purely informal aid to interpretation, I frequently use the letters
+
RT.  Note 13
−
"p", "q" to denote propositions.  This can serve to tip off the reader
 
−
that a function is intended as the indicator function of a set, and
 
−
it saves us the trouble of declaring the type f : X -> %B% each
 
−
time that a function is introduced as a proposition.
 
  
−
Another convention of use in this context is to let boldface letters
+
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
−
stand for k-tuples, lists, or sequences of objects.  Typically, the
 
−
elements of the k-tuple, list, or sequence are all of one type, and
 
−
typically the boldface letter is of the same basic character as the
 
−
indexed or subscripted letters that are used denote the components
 
−
of the k-tuple, list, or sequence.  When the dimension of elements
 
−
and functions is clear from the context, we may elect to drop the
 
−
bolding of characters that name k-tuples, lists, and sequences.
 
  
−
For example:
+
1.3.10.6.  Stretching Principles (concl.)
  
−
1.  If x_1, ..., x_k in X,      then #x# = <x_1, ..., x_k> in X' = X^k.
+
To complete the general discussion of stretching principles,
−
 
+
we will need to call back to mind the following definitions:
−
2.  If x_1, ..., x_k  : X,      then #x# = <x_1, ..., x_k>  : X' = X^k.
 
−
 
 
−
3.  If f_1, ..., f_k  : X -> Y,  then #f# = <f_1, ..., f_k>  : (X -> Y)^k.
 
−
 
 
−
There is usually felt to be a slight but significant distinction between
 
−
the "membership statement" that uses the sign "in" as in Example (1) and
 
−
the "type statement" that uses the sign ":" as in examples (2) and (3).
 
−
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"
 
−
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 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 kinds 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 "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 476: 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

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IDS -- RT

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RT.  Recurring Themes

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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

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RT.  Note 8

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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.

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RT.  Note 9

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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.

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RT.  Note 10

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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".

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RT.  Note 11

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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.

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RT.  Note 12

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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.

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RT.  Note 15

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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.

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