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In this and the four subsections that follow, I describe a calculus for representing propositions as sentences, in other words, as syntactically defined sequences of signs, and for manipulating these sentences chiefly in the light of their semantically defined contents, in other words, with respect to their logical values as propositions.  In their computational representation, the expressions of this calculus parse into a class of tree-like data structures called ''painted cacti''.  This is a family of graph-theoretic data structures that can be observed to have especially nice properties, turning out to be not only useful from a computational standpoint but also quite interesting from a theoretical point of view. The rest of this subsection serves to motivate the development of this calculus and treats a number of general issues that surround the topic.

In this and the four subsections that follow, I describe a calculus for

representing propositions as sentences, in other words, as syntactically

defined sequences of signs, and for manipulating these sentences chiefly

in the light of their semantically defined contents, in other words, with

respect to their logical values as propositions.  In their computational

representation, the expressions of this calculus parse into a class of

tree-like data structures called "painted cacti".  This is a family of

graph-theoretic data structures that can be observed to have especially

nice properties, turning out to be not only useful from a computational

standpoint but also quite interesting from a theoretical point of view.

The rest of this subsection serves to motivate the development of this

calculus and treats a number of general issues that surround the topic.

In order to facilitate the use of propositions as indicator functions

In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains. If none of the formalisms that are readily available or in common use are able to meet all of the design requirements that come to mind, then it is necessary to contemplate the design of a new language that is especially tailored to the purpose.  In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe that are indicated by them.

it helps to acquire a flexible notation for referring to propositions

in that light, for interpreting sentences in a corresponding role, and

for negotiating the requirements of mutual sense between the two domains.

If none of the formalisms that are readily available or in common use are

able to meet all of the design requirements that come to mind, then it is

necessary to contemplate the design of a new language that is especially

tailored to the purpose.  In the present application, there is a pressing

need to devise a general calculus for composing propositions, computing

their values on particular arguments, and inverting their indications to

arrive at the sets of things in the universe that are indicated by them.

For computational purposes, it is convenient to have a middle ground or

For computational purposes, it is convenient to have a middle ground or an intermediate language for negotiating between the ''koine'' of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if this renders it necessary to introduce an artificial medium of exchange between these two domains. If one envisions these computations to be carried out in any organized fashion, and ultimately or partially by means of the familiar sorts of machines, then the strings that express these logical propositions are likely to find themselves parsed into tree-like data structures at some stage of the game.  With regard to their abstract structures as graphs, there are several species of graph-theoretic data structures that can be used to accomplish this job in a reasonably effective and efficient way.

an intermediate language for negotiating between the koine of sentences

regarded as strings of literal characters and the realm of propositions

regarded as objects of logical value, even if this renders it necessary

to introduce an artificial medium of exchange between these two domains.

If one envisions these computations to be carried out in any organized

fashion, and ultimately or partially by means of the familiar sorts of

machines, then the strings that express these logical propositions are

likely to find themselves parsed into tree-like data structures at some

stage of the game.  With regard to their abstract structures as graphs,

there are several species of graph-theoretic data structures that can be

used to accomplish this job in a reasonably effective and efficient way.

Over the course of this project, I plan to use two species of graphs:

Over the course of this project, I plan to use two species of graphs:

1.  "Painted And Rooted Cacti" (PARCAI).

# "Painted And Rooted Cacti" (PARCAI).

# "Painted And Rooted Conifers" (PARCOI).

2.  "Painted And Rooted Conifers" (PARCOI).

For now, it is enough to discuss the former class of data structures, leaving the consideration of the latter class to a part of the project where their distinctive features are key to developments at that stage.  Accordingly, within the context of the current patch of discussion, or until it becomes necessary to attach further notice to the conceivable varieties of parse graphs, the acronym "PARC" is sufficient to indicate the pertinent genus of abstract graphs that are under consideration.

For now, it is enough to discuss the former class of data structures,

By way of making these tasks feasible to carry out on a regular basis, a prospective language designer is required not only to supply a fluent medium for the expression of propositions, but further to accompany the assertions of their sentences with a canonical mechanism for teasing out the fibers of their indicator functions.  Accordingly, with regard to a body of conceivable propositions, one needs to furnish a standard array of techniques for following the threads of their indications from their objective universe to their values for the mind and back again, that is, for tracing the clues that sentences provide from the universe of their objects to the signs of their values, and, in turn, from signs to objects.  Ultimately, one seeks to render propositions so functional as indicators of sets and so essential for examining the equality of sets that they can constitute a veritable criterion for the practical conceivability of sets.  Tackling this task requires me to introduce a number of new definitions and a collection of additional notational devices, to which I now turn.

leaving the consideration of the latter class to a part of the project

where their distinctive features are key to developments at that stage.

Accordingly, within the context of the current patch of discussion, or

until it becomes necessary to attach further notice to the conceivable

varieties of parse graphs, the acronym "PARC" is sufficient to indicate

the pertinent genus of abstract graphs that are under consideration.

By way of making these tasks feasible to carry out on a regular basis,

Depending on whether a formal language is called by the type of sign that makes it up or whether it is named after the type of object that its signs are intended to denote, one may refer to this cactus language as a ''sentential calculus'' or as a ''propositional calculus'', respectively.

a prospective language designer is required not only to supply a fluent

medium for the expression of propositions, but further to accompany the

assertions of their sentences with a canonical mechanism for teasing out

objective universe to their values for the mind and back again, that is,

for tracing the clues that sentences provide from the universe of their

objects to the signs of their values, and, in turn, from signs to objects.

Ultimately, one seeks to render propositions so functional as indicators

constitute a veritable criterion for the practical conceivability of sets.

and a collection of additional notational devices, to which I now turn.

Depending on whether a formal language is called by the type of sign

When the syntactic definition of the language is well enough understood, then the language can begin to acquire a semantic function.  In natural circumstances, the syntax and the semantics are likely to be engaged in a process of co-evolution, whether in ontogeny or in phylogeny, that is, the two developments probably form parallel sides of a single bootstrap.  But this is not always the easiest way, at least, at first, to formally comprehend the nature of their action or the power of their interaction.

that makes it up or whether it is named after the type of object that

its signs are intended to denote, one may refer to this cactus language

as a "sentential calculus" or as a "propositional calculus", respectively.

When the syntactic definition of the language is well enough understood,

According to the customary mode of formal reconstruction, the language is first presented in terms of its syntax, in other words, as a formal language of strings called ''sentences'', amounting to a particular subset of the possible strings that can be formed on a finite alphabet of signs. A syntactic definition of the ''cactus language'', one that proceeds along purely formal lines, is carried out in the next Subsection. After that, the development of the language's more concrete aspects can be seen as a matter of defining two functions:

then the language can begin to acquire a semantic function.  In natural

circumstances, the syntax and the semantics are likely to be engaged in

a process of co-evolution, whether in ontogeny or in phylogeny, that is,

the two developments probably form parallel sides of a single bootstrap.

But this is not always the easiest way, at least, at first, to formally

comprehend the nature of their action or the power of their interaction.

According to the customary mode of formal reconstruction, the language

# The first is a function that takes each sentence of the language into a computational data structure, to be exact, a tree-like parse graph called a ''painted cactus''.

is first presented in terms of its syntax, in other words, as a formal

# The second is a function that takes each sentence of the language, or its interpolated parse graph, into a logical proposition, in effect, ending up with an indicator function as the object denoted by the sentence.

language of strings called "sentences", amounting to a particular subset

of the possible strings that can be formed on a finite alphabet of signs.

A syntactic definition of the "cactus language", one that proceeds along

purely formal lines, is carried out in the next Subsection.  After that,

the development of the language's more concrete aspects can be seen as

1.  The first is a function that takes each sentence of the language

The discussion of syntax brings up a number of associated issues that have to be clarified before going on.  These are questions of ''style'', that is, the sort of description, ''grammar'', or theory that one finds available or chooses as preferable for a given language.  These issues are discussed in the Subsection after next (Subsection 1.3.10.10).

    into a computational data structure, to be exact, a tree-like

    parse graph called a "painted cactus".

2.  The second is a function that takes each sentence of the language,

There is an aspect of syntax that is so schematic in its basic character that it can be conveyed by computational data structures, so algorithmic in its uses that it can be automated by routine mechanisms, and so fixed in its nature that its practical exploitation can be served by the usual devices of computation.  Because it involves the transformation of signs, it can be recognized as an aspect of semiotics.  Since it can be carried out in abstraction from meaning, it is not up to the level of semantics, much less a complete pragmatics, though it does incline to the pragmatic aspects of computation that are auxiliary to and incidental to the human use of language.  Therefore, I refer to this aspect of formal language use as the ''algorithmics'' or the ''mechanics'' of language processing.  A mechanical conversion of the cactus language into its associated data structures is discussed in Subsection 1.3.10.11.

    or its interpolated parse graph, into a logical proposition, in effect,

    ending up with an indicator function as the object denoted by the sentence.

In the usual way of proceeding on formal grounds, meaning is added by giving each grammatical sentence, or each syntactically distinguished string, an interpretation as a logically meaningful sentence, in effect, equipping or providing each abstractly well-formed sentence with a logical proposition for it to denote.  A semantic interpretation of the cactus language is carried out in Subsection 1.3.10.12.

 

 

In the usual way of proceeding on formal grounds, meaning is added by giving

each "grammatical sentence", or each syntactically distinguished string, an

interpretation as a logically meaningful sentence, in effect, equipping or

providing each abstractly well-formed sentence with a logical proposition

for it to denote.  A semantic interpretation of the "cactus language" is

carried out in Subsection 1.3.10.12.

==The Cactus Language : Syntax==

==The Cactus Language : Syntax==