wolframscience.com

A New Kind of Science: The NKS Forum : Powered by vBulletin version 2.3.0 A New Kind of Science: The NKS Forum > NKS Way of Thinking > Language Of Cacti
Pages (3): « 1 2 [3]   Last Thread   Next Thread
Author
Thread Post New Thread    Post A Reply
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 30

1.3.10.12. The Cactus Language: Semantics (cont.)

In formal settings, it is common to speak of "interpretation" as if it
created a direct connection between the signs of a formal language and
the objects of the intended domain, in other words, as if it determined
the denotative component of a sign relation. But a closer attention to
what goes on reveals that the process of interpretation is more indirect,
that what it does is to provide each sign of a prospectively meaningful
source language with a translation into an already established target
language, where "already established" means that its relationship to
pragmatic objects is taken for granted at the moment in question.

With this in mind, it is clear that interpretation is an affair of signs
that at best respects the objects of all of the signs that enter into it,
and so it is the connotative aspect of semiotics that is at stake here.
There is nothing wrong with my saying that I interpret a sentence of a
formal language as a sign that refers to a function or to a proposition,
so long as you understand that this reference is likely to be achieved
by way of more familiar and perhaps less formal signs that you already
take to denote those objects.

On entering a context where a logical interpretation is intended for the
sentences of a formal language there are a few conventions that make it
easier to make the translation from abstract syntactic forms to their
intended semantic senses. Although these conventions are expressed in
unnecessarily colorful terms, from a purely abstract point of view, they
do provide a useful array of connotations that help to negotiate what is
otherwise a difficult transition. This terminology is introduced as the
need for it arises in the process of interpreting the cactus language.

The task of this Subsection is to specify a "semantic function" for
the sentences of the cactus language !L! = !C!(!P!), in other words,
to define a mapping that "interprets" each sentence of !C!(!P!) as
a sentence that says something, as a sentence that bears a meaning,
in short, as a sentence that denotes a proposition, and thus as a
sign of an indicator function. When the syntactic sentences of a
formal language are given a referent significance in logical terms,
for example, as denoting propositions or indicator functions, then
each form of syntactic combination takes on a corresponding form
of logical significance.

By way of providing a logical interpretation for the cactus language,
I introduce a family of operators on indicator functions that are
called "propositional connectives", and I distinguish these from
the associated family of syntactic combinations that are called
"sentential connectives", where the relationship between these
two realms of connection is exactly that between objects and
their signs. A propositional connective, as an entity of a
well-defined functional and operational type, can be treated
in every way as a logical or a mathematical object, and thus
as the type of object that can be denoted by the corresponding
form of syntactic entity, namely, the sentential connective that
is appropriate to the case in question.

There are two basic types of connectives, called the "blank connectives"
and the "bound connectives", respectively, with one connective of each
type for each natural number k = 0, 1, 2, 3, ... .


  • The "blank connective" of k places is signified by the
    concatenation of the k sentences that fill those places.

    For the special case of k = 0, the "blank connective" is taken to
    be an empty string or a blank symbol -- it does not matter which,
    since both are assigned the same denotation among propositions.
    For the generic case of k > 0, the "blank connective" takes
    the form "S_1 · ... · S_k". In the type of data that is
    called a "text", the raised dots "·" are usually omitted,
    supplanted by whatever number of spaces and line breaks
    serve to improve the readability of the resulting text.

  • The "bound connective" of k places is signified by the
    surcatenation of the k sentences that fill those places.

    For the special case of k = 0, the "bound connective" is taken to
    be an expression of the form "-()-", "-( )-", "-( )-", and so on,
    with any number of blank symbols between the parentheses, all of
    which are assigned the same logical denotation among propositions.
    For the generic case of k > 0, the "bound connective" takes the
    form "-(S_1, ..., S_k)-".

Jon Awbrey

-----------------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 12-16-2004 07:33 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 31

1.3.10.12. The Cactus Language: Semantics (cont.)

At this point, there are actually two different "dialects", "scripts",
or "modes" of presentation for the cactus language that need to be
interpreted, in other words, that need to have a semantic function
defined on their domains.


  • There is the literal formal language of strings in PARCE(!P!),
    the "painted and rooted cactus expressions" that constitute
    the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*.

  • There is the figurative formal language of graphs in PARC(!P!),
    the "painted and rooted cacti" themselves, a parametric family
    of graphs or a species of computational data structures that
    is graphically analogous to the language of literal strings.

Of course, these two modalities of formal language, like written and
spoken natural languages, are meant to have compatible interpretations,
and so it is usually sufficient to give just the meanings of either one.
All that remains is to provide a "codomain" or a "target space" for the
intended semantic function, in other words, to supply a suitable range
of logical meanings for the memberships of these languages to map into.
Out of the many interpretations that are formally possible to arrange,
one way of doing this proceeds by making the following definitions:

  • The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},
    is a proposition that is true if and only if each one of the Q_j is true.

    Conj^J_j Q_j is true <=> Q_j is true for every j in J.

  • The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
    is a proposition that is true if and only if just one of the Q_j is untrue.

    Surj^J_j Q_j is true <=> Q_j is untrue for unique j in J.

If the number of propositions that are being joined together is finite,
then the conjunction and the surjunction can be represented by means of
sentential connectives, incorporating the sentences that represent these
propositions into finite strings of symbols.

If J is finite, for instance, if J constitutes the interval j = 1 to k,
and if each proposition Q_j is represented by a sentence S_j, then the
following strategies of expression are open:

  • The conjunction Conj^J_j Q_j can be represented by a sentence that
    is constructed by concatenating the S_j in the following fashion:

    Conj^J_j Q_j <-< S_1 S_2 ... S_k.

  • The surjunction Surj^J_j Q_j can be represented by a sentence that
    is constructed by surcatenating the S_j in the following fashion:

    Surj^J_j Q_j <-< -(S_1, S_2, ..., S_k)-.

If one opts for a mode of interpretation that moves more directly from
the parse graph of a sentence to the potential logical meaning of both
the PARC and the PARCE, then the following specifications are in order:

A cactus rooted at a particular node is taken to represent what that
node denotes, its logical denotation or its logical interpretation.

  • The logical denotation of a node is the logical conjunction of that node's
    "arguments", which are defined as the logical denotations of that node's
    attachments. The logical denotation of either a blank symbol or an empty
    node is the boolean value %1% = "true". The logical denotation of the
    paint p_j is the proposition P_j, a proposition that is regarded as
    "primitive", at least, with respect to the level of analysis that
    is represented in the current instance of !C!(!P!).

  • The logical denotation of a lobe is the logical surjunction of that lobe's
    "arguments", which are defined as the logical denotations of that lobe's
    accoutrements. As a corollary, the logical denotation of the parse graph
    of "-()-", otherwise called a "needle", is the boolean value %0% = "false".

If one takes the point of view that PARC's and PARCE's amount to a
pair of intertranslatable languages for the same domain of objects,
then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",
can be used on either domain of signs to indicate the logical
denotation of a cactus C_j or the logical denotation of
a sentence S_j, respectively.

Jon Awbrey

-----------------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 12-16-2004 08:32 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 32

1.3.10.12. The Cactus Language: Semantics (cont.)

Tables 13.1 and 13.2 summarize the relations that serve to connect the
formal language of sentences with the logical language of propositions.
Between these two realms of expression there is a family of graphical
data structures that arise in parsing the sentences and that serve to
facilitate the performance of computations on the indicator functions.
The graphical language supplies an intermediate form of representation
between the formal sentences and the indicator functions, and the form
of mediation that it provides is very useful in rendering the possible
connections between the other two languages conceivable in fact, not to
mention in carrying out the necessary translations on a practical basis.
These Tables include this intermediate domain in their Central Columns.
Between their First and Middle Columns they illustrate the mechanics of
parsing the abstract sentences of the cactus language into the graphical
data structures of the corresponding species. Between their Middle and
Final Columns they summarize the semantics of interpreting the graphical
forms of representation for the purposes of reasoning with propositions.

Table 13.1. Semantic Translations: Functional Form
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | Par | ` ` ` ` ` ` ` ` ` | Den | ` ` ` ` ` ` ` ` ` |
| Sentence` ` ` ` ` | --> | Graph ` ` ` ` ` ` | --> | Proposition ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| S_j ` ` ` ` ` ` ` | --> | C_j ` ` ` ` ` ` ` | --> | Q_j ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| Conc^0` ` ` ` ` ` | --> | Node^0` ` ` ` ` ` | --> | %1% ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| Conc^k_j` S_j ` ` | --> | Node^k_j` C_j ` ` | --> | Conj^k_j` Q_j ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| Surc^0` ` ` ` ` ` | --> | Lobe^0` ` ` ` ` ` | --> | %0% ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| Surc^k_j` S_j ` ` | --> | Lobe^k_j` C_j ` ` | --> | Surj^k_j` Q_j ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o


Table 13.2. Semantic Translations: Equational Form
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | Par | ` ` ` ` ` ` ` ` ` | Den | ` ` ` ` ` ` ` ` ` |
| -[Sentence]-` ` ` | `=` | -[Graph]- ` ` ` ` | `=` | Proposition ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| -[S_j]- ` ` ` ` ` | `=` | -[C_j]- ` ` ` ` ` | `=` | Q_j ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| -[Conc^0]-` ` ` ` | `=` | -[Node^0]-` ` ` ` | `=` | %1% ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| -[Conc^k_j` S_j]- | `=` | -[Node^k_j` C_j]- | `=` | Conj^k_j` Q_j ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| -[Surc^0]-` ` ` ` | `=` | -[Lobe^0]-` ` ` ` | `=` | %0% ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| -[Surc^k_j` S_j]- | `=` | -[Lobe^k_j` C_j]- | `=` | Surj^k_j` Q_j ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-----o-------------------o-----o-------------------o

Aside from their common topic, the two Tables present slightly different
ways of conceptualizing the operations that go to establish their maps.
Table 13.1 records the functional associations that connect each domain
with the next, taking the triplings of a sentence S_j, a cactus C_j, and
a proposition Q_j as basic data, and fixing the rest by recursion on these.
Table 13.2 records these associations in the form of equations, treating
sentences and graphs as alternative kinds of signs, and generalizing the
spiny bracket operator to indicate the proposition that either denotes.
It should be clear at this point that either scheme of translation puts
the sentences, the graphs, and the propositions that it associates with
each other roughly in the roles of the signs, the interpretants, and the
objects, respectively, whose triples define an appropriate sign relation.
Indeed, the "roughly" can be made "exactly" as soon as the domains of
a suitable sign relation are specified precisely.

Jon Awbrey

-----------------------------------------------------------------------------------------

Last edited by Jon Awbrey on 12-16-2004 at 09:32 PM

Report this post to a moderator | IP: Logged

Old Post 12-16-2004 09:22 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 33

1.3.10.12. The Cactus Language: Semantics (cont.)

A good way to illustrate the action of the conjunction and surjunction
operators is to demonstate how they can be used to construct all of the
boolean functions on k variables, just now, let us say, for k = 0, 1, 2.

A boolean function on 0 variables is just a boolean constant F^0 in the
boolean domain %B% = {%0%, %1%}. Table 14 shows several different ways
of referring to these elements, just for the sake of consistency using
the same format that will be used in subsequent Tables, no matter how
degenerate it tends to appears in the immediate case:


  • Column 1 lists each boolean element or boolean function under its
    ordinary constant name or under a succinct nickname, respectively.

  • Column 2 lists each boolean function in a style of function name "F^i_j"
    that is constructed as follows: The superscript "i" gives the dimension
    of the functional domain, that is, the number of its functional variables,
    and the subscript "j" is a binary string that recapitulates the functional
    values, using the obvious translation of boolean values into binary values.

  • Column 3 lists the functional values for each boolean function, or possibly
    a boolean element appearing in the guise of a function, for each combination
    of its domain values.

  • Column 4 shows the usual expressions of these elements in the cactus language,
    conforming to the practice of omitting the strike-throughs in display formats.
    Here I illustrate also the useful convention of sending the expression "(())"
    as a visible stand-in for the expression of a constantly "true" truth value,
    one that would otherwise be represented by a blank expression, and tend to
    elude our giving it much notice in the context of more demonstrative texts.

Table 14. Boolean Functions on Zero Variables
o----------o----------o-------------------------------------------o----------o
| Constant | Function | ` ` ` ` ` ` ` ` ` `F()` ` ` ` ` ` ` ` ` ` | Function |
o----------o----------o-------------------------------------------o----------o
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| %0% ` ` `| F^0_0 ` `| ` ` ` ` ` ` ` ` ` `%0%` ` ` ` ` ` ` ` ` ` | ` `() ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| %1% ` ` `| F^0_1 ` `| ` ` ` ` ` ` ` ` ` `%1%` ` ` ` ` ` ` ` ` ` | ` (())` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o----------o----------o-------------------------------------------o----------o

Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
of which there are precisely four. Here, Column 1 codes the contents of
Column 2 in a more concise form, compressing the lists of boolean values,
recorded as bits in the subscript string, into their decimal equivalents.
Naturally, the boolean constants reprise themselves in this new setting
as constant functions on one variable. Thus, one has the synonymous
expressions for constant functions that are expressed in the next
two chains of equations:

  • F^1_0 = F^1_00 = %0% : %B% -> %B%

  • F^1_3 = F^1_11 = %1% : %B% -> %B%

As for the rest, the other two functions are easily recognized as corresponding
to the one-place logical connectives, or the monadic operators on %B%. Thus,
the function F^1_1 = F^1_01 is recognizable as the negation operation, and
the function F^1_2 = F^1_10 is obviously the identity operation.

Table 15. Boolean Functions on One Variable
o----------o----------o-------------------------------------------o----------o
| Function | Function | ` ` ` ` ` ` ` ` ` F(x)` ` ` ` ` ` ` ` ` ` | Function |
o----------o----------o---------------------o---------------------o----------o
| ` ` ` ` `| ` ` ` ` `| ` ` ` F(%0%)` ` ` ` | ` ` ` F(%1%)` ` ` ` | ` ` ` ` `|
o----------o----------o---------------------o---------------------o----------o
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| F^1_0 ` `| F^1_00` `| ` ` ` ` %0% ` ` ` ` | ` ` ` ` %0% ` ` ` ` | ` ( ) ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| F^1_1 ` `| F^1_01` `| ` ` ` ` %0% ` ` ` ` | ` ` ` ` %1% ` ` ` ` | ` (x) ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| F^1_2 ` `| F^1_10` `| ` ` ` ` %1% ` ` ` ` | ` ` ` ` %0% ` ` ` ` | ` `x` ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| F^1_3 ` `| F^1_11` `| ` ` ` ` %1% ` ` ` ` | ` ` ` ` %1% ` ` ` ` | `(( ))` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o----------o----------o---------------------o---------------------o----------o

Jon Awbrey

-----------------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 12-16-2004 11:48 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 34

1.3.10.12. The Cactus Language: Semantics (concl.)

Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,
of which there are precisely sixteen in number. As before, all of the boolean
functions of fewer variables are subsumed in this Table, though under a set of
alternative names and possibly different interpretations. Just to acknowledge
a few of the more notable pseudonyms:


  • The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.

  • The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.

  • The negation and identity of the first variable are F^2_03 and F^2_12, resp.

  • The negation and identity of the other variable are F^2_05 and F^2_10, resp.

  • The logical conjunction is given by the function F^2_08 (x, y) = x · y.

  • The logical disjunction is given by the function F^2_14 (x, y) = ((x)(y)).

Functions expressing the "conditionals", "implications",
or "if-then" statements are given in the following ways:

  • [x => y] = F^2_11 (x, y) = (x (y)) = [not x without y].

  • [x <= y] = F^2_13 (x, y) = ((x) y) = [not y without x].

The function that corresponds to the "biconditional",
the "equivalence", or the "if and only" statement is
exhibited in the following fashion:

  • [x <=> y] = [x = y] = F^2_09 (x, y) = ((x , y)).

Finally, there is a boolean function that is logically associated with
the "exclusive disjunction", "inequivalence", or "not equals" statement,
algebraically associated with the "binary sum" or "bitsum" operation,
and geometrically associated with the "symmetric difference" of sets.
This function is given by:

  • [x =/= y] = [x + y] = F^2_06 (x, y) = (x , y).

Table 16. Boolean Functions on Two Variables
o----------o----------o-------------------------------------------o----------o
| Function | Function | ` ` ` ` ` ` ` ` `F(x, y)` ` ` ` ` ` ` ` ` | Function |
o----------o----------o----------o----------o----------o----------o----------o
| ` ` ` ` `| ` ` ` ` `| %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% | ` ` ` ` `|
o----------o----------o----------o----------o----------o----------o----------o
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_00` `| F^2_0000 | ` %0% ` `| ` %0% ` `| ` %0% ` `| ` %0% ` `| ` `() ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_01` `| F^2_0001 | ` %0% ` `| ` %0% ` `| ` %0% ` `| ` %1% ` `| `(x)(y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_02` `| F^2_0010 | ` %0% ` `| ` %0% ` `| ` %1% ` `| ` %0% ` `| `(x) y` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_03` `| F^2_0011 | ` %0% ` `| ` %0% ` `| ` %1% ` `| ` %1% ` `| `(x)` ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_04` `| F^2_0100 | ` %0% ` `| ` %1% ` `| ` %0% ` `| ` %0% ` `| ` x (y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_05` `| F^2_0101 | ` %0% ` `| ` %1% ` `| ` %0% ` `| ` %1% ` `| ` ` (y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_06` `| F^2_0110 | ` %0% ` `| ` %1% ` `| ` %1% ` `| ` %0% ` `| `(x, y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_07` `| F^2_0111 | ` %0% ` `| ` %1% ` `| ` %1% ` `| ` %1% ` `| `(x `y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_08` `| F^2_1000 | ` %1% ` `| ` %0% ` `| ` %0% ` `| ` %0% ` `| ` x `y` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_09` `| F^2_1001 | ` %1% ` `| ` %0% ` `| ` %0% ` `| ` %1% ` `| ((x, y)) |
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_10` `| F^2_1010 | ` %1% ` `| ` %0% ` `| ` %1% ` `| ` %0% ` `| ` ` `y` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_11` `| F^2_1011 | ` %1% ` `| ` %0% ` `| ` %1% ` `| ` %1% ` `| `(x (y)) |
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_12` `| F^2_1100 | ` %1% ` `| ` %1% ` `| ` %0% ` `| ` %0% ` `| ` x ` ` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_13` `| F^2_1101 | ` %1% ` `| ` %1% ` `| ` %0% ` `| ` %1% ` `| ((x) y) `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_14` `| F^2_1110 | ` %1% ` `| ` %1% ` `| ` %1% ` `| ` %0% ` `| ((x)(y)) |
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
| F^2_15` `| F^2_1111 | ` %1% ` `| ` %1% ` `| ` %1% ` `| ` %1% ` `| ` (())` `|
| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` `|
o----------o----------o----------o----------o----------o----------o----------o

Let me now address one last question that may have occurred to some.
What has happened, in this suggested scheme of functional reasoning,
to the distinction that is quite pointedly made by careful logicians
between (1) the connectives called "conditionals" and symbolized by
the signs "->" and "<-", and (2) the assertions called "implications"
and symbolized by the signs "=>" and "<=", and, in a related question:
What has happened to the distinction that is equally insistently made
between (3) the connective called the "biconditional" and signified by
the sign "<->" and (4) the assertion that is called an "equivalence"
and signified by the sign "<=>"? My answer is this: For my part,
I am deliberately avoiding making these distinctions at the level
of syntax, preferring to treat them instead as distinctions in
the use of boolean functions, turning on whether the function
is mentioned directly and used to compute values on arguments,
or whether its inverse is being invoked to indicate the fibers
of truth or untruth under the propositional function in question.

Jon Awbrey

-----------------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 12-17-2004 12:16 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 35

1.3.10.13. Stretching Exercises

In this Subsection I bring together a number of what
may have seemed almost wholly divergent developments.
For ease of reference, I repeat here a number of the
definitions that are needed again in this discussion.

A "boolean connection" of degree k, also known as a "boolean function"
of 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 "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>.

The definition of the "stretch" operation and the uses
of the various brands of denotational operators can be
reviewed on thread that was titled "Recurring Themes":

RT. http://forum.wolframscience.com/sho...hp?threadid=654
RT. http://stderr.org/pipermail/inquiry...hread.html#2171

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:

  • Any relation of values extends to a relation of what is valued.

  • Any statement about values says something about the things
    that are given these values.

  • Any association among a range of values establishes
    an association among the domains of things
    that these values are the values of.

  • 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.

  • 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:

  • 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%).

  • 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%.

  • 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.

Jon Awbrey

-----------------------------------------------------------------------------------------

Report this post to a moderator | IP: Logged

Old Post 12-19-2004 03:46 AM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Jon Awbrey


Registered: Feb 2004
Posts: 557

Language Of Cacti

LOC. Note 36

1.3.10.13. Stretching Exercises (concl.)

Taking up the preceding arrays of particular connections, namely,
the boolean functions on two or less variables, it is possible to
illustrate the use of the stretch operation in a variety of
concrete cases.

For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
that is, any one of the sixteen possibilities in Table 16, while p and q
are propositions of the form p, q : X -> %B%, that is, propositions about
things in the universe X, or else the indicators of sets contained in X.

Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,
and the stretch of the connection F to #f# on X amounts to a proposition
F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
the "stretch of F to p and q". If one is concerned with many different
propositions about things in X, or if one is abstractly indifferent to
the particular choices for p and q, then one can detach the operator
F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",
and consider it in isolation from any concrete application.

When the "cactus notation" is used to represent boolean functions,
a single "$" sign at the end of the expression is enough to remind
a reader that the connections are meant to be stretched to several
propositions on a universe X.

For instance, take the connection F : %B%^2 -> %B% such that:


  • F(x, y) = F^2_06 (x, y) = -(x, y)-

This connection is the boolean function on a couple of variables x, y
that yields a value of %1% if and only if just one of x, y is not %1%,
that is, if and only if just one of x, y is %1%. There is clearly an
isomorphism between this connection, viewed as an operation on the
boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary
values x, y in !B! = GF(2) that is otherwise known as "x + y".

The same connection F : %B%^2 -> %B% can also be read as a proposition
about things in the universe X = %B%^2. If S is a sentence that denotes
the proposition F, then the corresponding assertion says exactly what one
otherwise states by uttering "x is not equal to y". In such a case, one
has -[S]- = F, and all of the following expressions are ordinarily taken
as equivalent descriptions of the same set:

  • [| -[S]- |]

    = [| F |]

    = F^(-1)(%1%)

    = {<x, y> in %B%^2 : S}

    = {<x, y> in %B%^2 : F(x, y) = %1%}

    = {<x, y> in %B%^2 : F(x, y)}

    = {<x, y> in %B%^2 : -(x, y)- = %1%}

    = {<x, y> in %B%^2 : -(x, y)- }

    = {<x, y> in %B%^2 : x exclusive-or y}

    = {<x, y> in %B%^2 : just one true of x, y}

    = {<x, y> in %B%^2 : x not equal to y}

    = {<x, y> in %B%^2 : x <=/=> y}

    = {<x, y> in %B%^2 : x =/= y}

    = {<x, y> in %B%^2 : x + y}.

Notice the slight distinction, that I continue to maintain at this point,
between the logical values {false, true} and the algebraic values {0, 1}.
This makes it legitimate to write a sentence directly into the right side
of the set-builder expression, for instance, weaving the sentence S or the
sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }",
thereby obtaining the corresponding expressions listed above, while the
proposition F(x, y) can also be asserted more directly without equating
it to %1%, since it already has a value in {false, true}, and thus can
be taken as tantamount to an actual sentence.

If the appropriate safeguards can be kept in mind, avoiding all danger of
confusing propositions with sentences and sentences with assertions, then
the marks of these distinctions need not be forced to clutter the account
of the more substantive indications, that is, the ones that really matter.
If this level of understanding can be achieved, then it may be possible
to relax these restrictions, along with the absolute dichotomy between
algebraic and logical values, which tends to inhibit the flexibility
of interpretation.

This covers the properties of the connection F(x, y) = -(x, y)-,
treated as a proposition about things in the universe X = %B%^2.
Staying with this same connection, it is time to demonstrate how
it can be "stretched" into an operator on arbitrary propositions.

To continue the exercise, let p and q be arbitrary propositions about
things in the universe X, that is, maps of the form p, q : X -> %B%,
and suppose that p, q are indicator functions of the sets P, Q c X,
respectively. In other words, one has the following set of data:

  • p = -{P}- : X -> %B%

  • q = -{Q}- : X -> %B%

  • <p, q> = < -{P}- , -{Q}- > : (X -> %B%)^2

Then one has an operator F^$, the stretch of the connection F over X,
and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
the following properties:

  • F^$ = -( , )-^$ : (X -> %B%)^2 -> (X -> %B%)

  • F^$ (p, q) = -(p, q)-^$ : X -> %B%

As a result, the application of the proposition F^$ (p, q) to each x in X
yields a logical value in %B%, all in accord with the following equations:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` F^$ (p, q)(x) ` = ` -(p, q)-^$ (x) `in `%B% ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ^ ` ` ` ` ` ` ` ` ` ^ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` = ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` v ` ` ` ` ` ` ` ` ` v ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` F(p(x), q(x)) ` = ` -(p(x), q(x))-` in` %B% ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

For each choice of propositions p and q about things in X, the stretch of
F to p and q on X is just another proposition about things in X, a simple
proposition in its own right, no matter how complex its current expression
or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in
relation to p and q. Like any other proposition about things in X, it
indicates a subset of X, namely, the fiber that is variously described
in the following ways:

  • [| F^$ (p, q) |]

    = [| -(p, q)-^$ |]

    = (F^$ (p, q))^(-1)(%1%)

    = {x in X : F^$ (p, q)(x)}

    = {x in X : -(p, q)-^$ (x)}

    = {x in X : -(p(x), q(x))- }

    = {x in X : p(x) ± q(x)}

    = {x in X : p(x) =/= q(x)}

    = {x in X : -{P}- (x) =/= -{Q}- (x)}

    = {x in X : x in P <=/=> x in Q}

    = {x in X : x in P-Q or x in Q-P}

    = {x in X : x in P-Q |_| Q-P}

    = {x in X : x in P ± Q}

    = P ± Q c X

    = [|p|] ± [|q|] c X.

Which was to be shown.

Jon Awbrey

-----------------------------------------------------------------------------------------

Last edited by Jon Awbrey on 12-20-2004 at 02:09 AM

Report this post to a moderator | IP: Logged

Old Post 12-19-2004 03:06 PM
Jon Awbrey is offline Click Here to See the Profile for Jon Awbrey Click here to Send Jon Awbrey a Private Message Visit Jon Awbrey's homepage! Edit/Delete Message Reply w/Quote
Post New Thread    Post A Reply
Pages (3): « 1 2 [3]   Last Thread   Next Thread
Show Printable Version | Email this Page | Subscribe to this Thread


 

wolframscience.com  |  wolfram atlas  |  NKS online  |  Wolfram|Alpha  |  Wolfram Science Summer School  |  web resources  |  contact us

Forum Sponsored by Wolfram Research

© 2004-14 Wolfram Research, Inc. | Powered by vBulletin 2.3.0 © 2000-2002 Jelsoft Enterprises, Ltd. | Disclaimer | Archives