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

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What do you know, it is the "Show Me" State, after all …  [[User:Jon Awbrey|Jon Awbrey]] 12:06, 13 October 2008 (PDT)
 
What do you know, it is the "Show Me" State, after all …  [[User:Jon Awbrey|Jon Awbrey]] 12:06, 13 October 2008 (PDT)
  
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 +
 
 +
==Propositions and Sentences==
 +
 
 +
<pre>
 +
The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
 +
whose arithmetic operations obey the rules of GF(2), the "galois field"
 +
of exactly two elements, whose addition and multiplication tables are
 +
tantamount to addition and multiplication of integers "modulo 2".
 +
 
 +
The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
 +
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
 +
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%
 +
and all of the structures that can be built on them become isomorphic
 +
at a high enough level of abstraction.  Consequently, the main reason
 +
for making this distinction in the immediate context appears to be more
 +
a matter of grammar than an issue of logical and mathematical substance,
 +
namely, so that the signs "%0%" and "%1%" can appear with a semblance of
 +
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)"
 +
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%.
 +
 
 +
Thus, negation is a monadic operation on boolean
 +
values, a function of the form (_) : %B% -> %B%.
 +
 
 +
It is convenient to transport the product and the sum operations of !B!
 +
into the logical setting of %B%, where they can be symbolized by signs
 +
of the same character, doubly underlined as necessary to avoid confusion.
 +
This yields the following definitions of a "product" and a "sum" in %B%
 +
and leads to the following forms of multiplication and addition tables.
 +
 
 +
The "product" of x and y, for values x, y in %B%, is given by Table 8.
 +
 
 +
Table 8.  Product Operation for the Boolean Domain
 +
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.
 +
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
 +
|  %+%  #  %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:
 +
 
 +
1.  "x =/= y", read "x is not equal to y", or "exactly one of x and y".
 +
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.
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
The utility of a suitable calculus would involve, among other things:
 +
 
 +
1.  Finding the values of given functions for given arguments.
 +
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.
 +
 
 +
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.  Considered in extensional form, f_Q is the subset of X x %B% that is given by the following formula:
 +
 
 +
    f_Q  =  {<x, b> in X x %B%  :  b = %1%  <=>  x in Q}.
 +
 
 +
2.  Considered in functional form, f_Q is the map from X to %B% that is given by the following condition:
 +
 
 +
    f_Q (x) = %1%  <=>  x in Q.
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
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:
 +
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:
 +
 
 +
Fiber of 1 under fQ  =  fQ-1(1)  =  {x in X  :  fQ(x) = 1}  =  Q.
 +
 
 +
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.
 +
 
 +
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:
 +
 
 +
If  f : X -> %B%,
 +
 
 +
then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x) = %1%}.
 +
 
 +
----
 +
 
 +
Some may recognize here fledgling efforts
 +
to reinforce flights of Fregean semantics
 +
with impish pitches of Peircean semiotics.
 +
Some may deem it Icarean, all too Icarean.
 +
 
 +
1.3.10.3  Propositions & Sentences (cont.)
 +
 
 +
The definition of a fiber, in either the general or the boolean case,
 +
is a purely nominal convenience for referring to the antecedent subset,
 +
the inverse image under a function, or the pre-image of a functional value.
 +
The definition of an operator on propositions, signified by framing the signs
 +
of propositions with fiber bars or ground signs, remains a purely notational
 +
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%,
 +
 
 +
then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x)}.
 +
 
 +
The set-builder frame "{x in X  :  ... }" requires a grammatical sentence or
 +
a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
 +
that serves to fill the frame in the initial definition of a logical fiber.
 +
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).
 +
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,
 +
    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
 +
    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,
 +
    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
 +
    as being its own logical interpretant, namely, the value that it signifies
 +
    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)"
 +
more directly names, that is, the value f(x).  In other words,
 +
the sentence "f(x) = %1%" has the same value to its interpretive
 +
context that the sign "f(x)" imparts to any comparable context,
 +
each by way of its respective evaluation for the same x in X.
 +
 
 +
What is the relation among connoting, denoting, and "evaluing", where
 +
the last term is coined to describe all the ways of bearing, conveying,
 +
developing, or evolving a value in, to, or into an interpretive context?
 +
In other words, when a sign is evaluated to a particular value, one can
 +
say that the sign "evalues" that value, using the verb in a way that is
 +
categorically analogous or grammatically conjugate to the times when one
 +
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
 +
formal discussions, especially by the fact that an equivalence class of signs
 +
is tantamount to a formal object.  This has the effect of allowing an abstract
 +
connotation to work as a formal denotation.  In other words, if the purpose of
 +
a sign is merely to lead its interpreter up to a sign in an equivalence class
 +
of signs, then it follows that this equivalence class is the object of the
 +
sign, that connotation can achieve denotation, at least, to some degree,
 +
and that the interpretant domain collapses with the object domain,
 +
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
 +
permits the discussion to avoid certain singularities in the embedding
 +
sign relation, and allowing the process of "evaluation" as a compromise
 +
mode of signification between connotation and denotation only manages to
 +
steer around a topic that eventually has to be mapped in full, but these
 +
strategies do allow the discussion to proceed a little further without
 +
having to answer questions that are too difficult to be settled fully
 +
or even tackled directly at this point.  As far as the relations among
 +
connoting, denoting, and evaluing are concerned, it is possible that
 +
all of these constitute independent dimensions of significance that
 +
a sign might be able to enjoy, but since the notion of connotation
 +
is already generic enough to contain multitudes of subspecies, I am
 +
going to subsume, on a tentative basis, all of the conceivable modes
 +
of "evaluing" within the broader concept of connotation.
 +
 
 +
With this degree of flexibility in mind, one can say that the sentence
 +
"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
 +
rewrite the definitions of indicator functions and their fibers as follows:
 +
 
 +
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.  Considered in its extensional form, f_Q is the subset of X x %B%
 +
    that is given by the following formula:
 +
 
 +
    f_Q  =  {<x, b> in X x %B%  :  b  <=>  x in Q}.
 +
 
 +
2.  Considered in its functional form, f_Q is the map from X to %B%
 +
    that is given by the following condition:
 +
 
 +
    f_Q (x)  <=>  x in Q.
 +
 
 +
The "fibers" of truth and falsity under a proposition f : X -> %B%
 +
are subsets of X that are variously described as follows:
 +
 
 +
1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)
 +
 
 +
                              =  {x in X  :  f(x) = %1%}
 +
 
 +
                              =  {x in X  :  f(x) }.
 +
 
 +
2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)
 +
 
 +
                              =  {x in X  :  f(x) = %0%}
 +
 
 +
                              =  {x in X  :  (f(x)) }.
 +
 
 +
Perhaps this looks like a lot of work for the sake of what seems to be
 +
such a trivial form of syntactic transformation, but it is an important
 +
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.
 +
 
 +
----
 +
 
 +
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%
 +
 
 +
For the sake of current reference:
 +
 
 +
| The "fibers" of truth and falsity in a proposition f : X -> %B%
 +
| are the subsets [| f |] and X - [| f |] of X that are variously
 +
| described as follows:
 +
|
 +
| The fiber of %1% under f
 +
|
 +
| =  [| f |]  =  f^(-1)(%1%)
 +
|
 +
| =  {x in X  :  f(x) = %1%}
 +
|
 +
| =  {x in X  :  f(x) }.
 +
|
 +
| The fiber of %0% under f
 +
|
 +
| =  ~[| f |]  =  f^(-1)(%0%)
 +
|
 +
| =  {x in X  :  f(x) = %0%}
 +
|
 +
| =  {x in X  :  (f(x)) }.
 +
 
 +
Oh, by the way, the outer parentheses in "(f(g))" signify negation.
 +
I did not have here the "stricken parentheses" that I normally use.
 +
 
 +
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.
 +
 
 +
----
 +
 
 +
In this episode, I compile a collection of definitions,
 +
leading up to the particular conception of a "sentence"
 +
that I'll be using throughout the rest of this inquiry.
 +
 
 +
1.3.10.3  Propositions & Sentences (cont.)
 +
 
 +
As a purely informal aid to interpretation, I frequently use the letters
 +
"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
 +
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.  If x_1, ..., x_k in X,      then #x# = <x_1, ..., x_k> in X' = X^k.
 +
 
 +
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 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>.
 +
 
 +
A "sentence about things in the universe", for short, a "sentence",
 +
is a sign that denotes a proposition.  In other words, a sentence is
 +
any sign that denotes an indicator function, any sign whose object is
 +
a function of the form f : X -> B.
 +
 
 +
To emphasize the empirical contingency of this definition, one can say
 +
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.
 +
 
 +
----
 +
 
 +
I finish out the Subsection on "Propositions & Sentences" with
 +
an account of how I use concepts like "assertion" and "denial".
 +
 
 +
1.3.10.3  Propositions & Sentences (cont.)
 +
 
 +
An "expression" is a type of sign, for instance, a term or a sentence,
 +
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
 +
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,
 +
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,
 +
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
 +
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)-.
 +
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
 +
a proposition, any sign that is taken as denoting an indicator function,
 +
by that very fact alone successfully qualifies as a sentence.  That is,
 +
a sentence is any sign that actually succeeds in denoting a proposition,
 +
any sign that one way or another brings to mind, as its actual object,
 +
a function of the form f : X -> B.
 +
 
 +
There are many features of this definition that need to be understood.
 +
Indeed, there are problems involved in this whole style of definition
 +
that need to be discussed, and doing this requires a slight excursion.
 +
</pre>

Revision as of 18:56, 14 January 2009

Fragmata

  1. http://www.cspeirce.com/menu/library/aboutcsp/awbrey/inquiry.htm
  2. http://forum.wolframscience.com/showthread.php?threadid=649
  3. http://forum.wolframscience.com/printthread.php?threadid=649

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

The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
whose arithmetic operations obey the rules of GF(2), the "galois field"
of exactly two elements, whose addition and multiplication tables are
tantamount to addition and multiplication of integers "modulo 2".

The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
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
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%
and all of the structures that can be built on them become isomorphic
at a high enough level of abstraction.  Consequently, the main reason
for making this distinction in the immediate context appears to be more
a matter of grammar than an issue of logical and mathematical substance,
namely, so that the signs "%0%" and "%1%" can appear with a semblance of
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)"
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%.

Thus, negation is a monadic operation on boolean
values, a function of the form (_) : %B% -> %B%.

It is convenient to transport the product and the sum operations of !B!
into the logical setting of %B%, where they can be symbolized by signs
of the same character, doubly underlined as necessary to avoid confusion.
This yields the following definitions of a "product" and a "sum" in %B%
and leads to the following forms of multiplication and addition tables.

The "product" of x and y, for values x, y in %B%, is given by Table 8.

Table 8.  Product Operation for the Boolean Domain
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.
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
|   %+%   #   %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:

1.  "x =/= y", read "x is not equal to y", or "exactly one of x and y".
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.

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.

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.

The utility of a suitable calculus would involve, among other things:

1.  Finding the values of given functions for given arguments.
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.

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.  Considered in extensional form, f_Q is the subset of X x %B% that is given by the following formula:

    f_Q  =  {<x, b> in X x %B%  :  b = %1%  <=>  x in Q}.

2.  Considered in functional form, f_Q is the map from X to %B% that is given by the following condition:

    f_Q (x) = %1%  <=>  x in Q.

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.

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.

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:
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:

Fiber of 1 under fQ  =  fQ-1(1)  =  {x in X  :  fQ(x) = 1}  =  Q.

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.

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:

If  f : X -> %B%,

then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x) = %1%}.

----

Some may recognize here fledgling efforts
to reinforce flights of Fregean semantics
with impish pitches of Peircean semiotics.
Some may deem it Icarean, all too Icarean.

1.3.10.3  Propositions & Sentences (cont.)

The definition of a fiber, in either the general or the boolean case,
is a purely nominal convenience for referring to the antecedent subset,
the inverse image under a function, or the pre-image of a functional value.
The definition of an operator on propositions, signified by framing the signs
of propositions with fiber bars or ground signs, remains a purely notational
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%,

then  [| f |]  =  f^(-1)(%1%)  =  {x in X  :  f(x)}.

The set-builder frame "{x in X  :  ... }" requires a grammatical sentence or
a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
that serves to fill the frame in the initial definition of a logical fiber.
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).
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,
    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
    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,
    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
    as being its own logical interpretant, namely, the value that it signifies
    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)"
more directly names, that is, the value f(x).  In other words,
the sentence "f(x) = %1%" has the same value to its interpretive
context that the sign "f(x)" imparts to any comparable context,
each by way of its respective evaluation for the same x in X.

What is the relation among connoting, denoting, and "evaluing", where
the last term is coined to describe all the ways of bearing, conveying,
developing, or evolving a value in, to, or into an interpretive context?
In other words, when a sign is evaluated to a particular value, one can
say that the sign "evalues" that value, using the verb in a way that is
categorically analogous or grammatically conjugate to the times when one
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
formal discussions, especially by the fact that an equivalence class of signs
is tantamount to a formal object.  This has the effect of allowing an abstract
connotation to work as a formal denotation.  In other words, if the purpose of
a sign is merely to lead its interpreter up to a sign in an equivalence class
of signs, then it follows that this equivalence class is the object of the
sign, that connotation can achieve denotation, at least, to some degree,
and that the interpretant domain collapses with the object domain,
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
permits the discussion to avoid certain singularities in the embedding
sign relation, and allowing the process of "evaluation" as a compromise
mode of signification between connotation and denotation only manages to
steer around a topic that eventually has to be mapped in full, but these
strategies do allow the discussion to proceed a little further without
having to answer questions that are too difficult to be settled fully
or even tackled directly at this point.  As far as the relations among
connoting, denoting, and evaluing are concerned, it is possible that
all of these constitute independent dimensions of significance that
a sign might be able to enjoy, but since the notion of connotation
is already generic enough to contain multitudes of subspecies, I am
going to subsume, on a tentative basis, all of the conceivable modes
of "evaluing" within the broader concept of connotation.

With this degree of flexibility in mind, one can say that the sentence
"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
rewrite the definitions of indicator functions and their fibers as follows:

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.  Considered in its extensional form, f_Q is the subset of X x %B%
    that is given by the following formula:

    f_Q  =  {<x, b> in X x %B%  :  b  <=>  x in Q}.

2.  Considered in its functional form, f_Q is the map from X to %B%
    that is given by the following condition:

    f_Q (x)  <=>  x in Q.

The "fibers" of truth and falsity under a proposition f : X -> %B%
are subsets of X that are variously described as follows:

1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)

                              =  {x in X  :  f(x) = %1%}

                              =  {x in X  :  f(x) }.

2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)

                              =  {x in X  :  f(x) = %0%}

                              =  {x in X  :  (f(x)) }.

Perhaps this looks like a lot of work for the sake of what seems to be
such a trivial form of syntactic transformation, but it is an important
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.

----

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%

For the sake of current reference:

| The "fibers" of truth and falsity in a proposition f : X -> %B%
| are the subsets [| f |] and X - [| f |] of X that are variously
| described as follows:
|
| The fiber of %1% under f
|
| =  [| f |]  =  f^(-1)(%1%)
|
| =  {x in X  :  f(x) = %1%}
|
| =  {x in X  :  f(x) }.
|
| The fiber of %0% under f
|
| =  ~[| f |]  =  f^(-1)(%0%)
|
| =   {x in X  :  f(x) = %0%}
|
| =   {x in X  :  (f(x)) }.

Oh, by the way, the outer parentheses in "(f(g))" signify negation.
I did not have here the "stricken parentheses" that I normally use.

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.

----

In this episode, I compile a collection of definitions,
leading up to the particular conception of a "sentence"
that I'll be using throughout the rest of this inquiry.

1.3.10.3  Propositions & Sentences (cont.)

As a purely informal aid to interpretation, I frequently use the letters
"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
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.  If x_1, ..., x_k in X,       then #x# = <x_1, ..., x_k> in X' = X^k.

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 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>.

A "sentence about things in the universe", for short, a "sentence",
is a sign that denotes a proposition.  In other words, a sentence is
any sign that denotes an indicator function, any sign whose object is
a function of the form f : X -> B.

To emphasize the empirical contingency of this definition, one can say
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.

----

I finish out the Subsection on "Propositions & Sentences" with
an account of how I use concepts like "assertion" and "denial".

1.3.10.3  Propositions & Sentences (cont.)

An "expression" is a type of sign, for instance, a term or a sentence,
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
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,
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,
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
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)-.
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
a proposition, any sign that is taken as denoting an indicator function,
by that very fact alone successfully qualifies as a sentence.  That is,
a sentence is any sign that actually succeeds in denoting a proposition,
any sign that one way or another brings to mind, as its actual object,
a function of the form f : X -> B.

There are many features of this definition that need to be understood.
Indeed, there are problems involved in this whole style of definition
that need to be discussed, and doing this requires a slight excursion.
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