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Inquiry Driven Systems : Part 8

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ContentsPart 1Part 2Part 3Part 4Part 5Part 6Part 7Part 8AppendicesReferencesDocument History


8. Overview of the Domain : Interpretive Inquiry

Interpretive Stance, Initial Theory, Concrete Examples

8.1. Interpretive Bearings : Conceptual and Descriptive Frameworks

In this section I review the conceptual and descriptive frameworks that I will deploy throughout this work. In passing, I explain my overall attitude toward the use of any theoretical outlook (scaffold or catwalk), namely, that it needs to be as flexible and as reflective as possible.

8.1.1. Catwalks : Flexible Frameworks and Peripatetic Categories

In order to have a term that expresses both the conceptual and the descriptive aspects of these perspective standpoints, I have chosen to call them "interpretive frameworks". When analyzed in depth and fully formalized they might be recognized as "theoretical frameworks". But not every manner of intuition (or slant on the world) can survive the reflective process and persist under examination as a viable style of interpretation. And I need a term to underscore the fact that these heuristic frameworks are already in operation, shunting attention and shifting selection on an automatic and informal basis, long before anyone thinks to articulate their axioms in theory or to criticize their biases in action.

The reason I refer to interpretive frameworks rather than "ontologies" is to emphasize that many of the categories listed in these systems are inclusive or overlapping in their scopes. Thus, the circumstance that the same object can be contemplated under several different headings of the framework is not of necessity intended to say anything substantive about the object itself.

The reason I refer to interpretive frameworks rather than "hierarchies", even though I will often settle on a standard sequence for considering the attributes of a contemplated object, is that there is in general no uniquely best order for taking up these properties.

This may seem like a trivial point, taken for granted by everyone as a part of understanding the use of language, but it serves to highlight an important issue, one still lacking in universal agreement.

I will say that a logical distinction is "interpretive" to mean that it depends on the choice of an interpreter to determine how anything is classified with respect to it. This does not mean that every option of consideration will always be found equally fitting, but only that it is possible to contemplate the alternatives in a form of mental experiment.

As much as possible I will try to exploit the available degrees of interpretive freedom to view all conceptual and descriptive distinctions as being in relation to a framework of interpretation. For ease of discussion, if not for any more substantive reason, interpretive frameworks are often depicted as enacted by interpretive agents or embodied by interpretive communities, all of which conditions of practice can be summed up in a parametric reference to a single "interpreter".

8.1.1.1. Eponymous Ancestors : The Precursors of Abstraction?

As one application of the flexible attitude just proposed, consider the following issue.

An important problem in the evolution or development of intelligence is the question how genuine concepts (categorical abstractions and hypothetical constructions) can be derived from particular percepts. The gap between individual acquaintance and comprehensive description always seems too vast to explain how incipient minds can vault it with any sense of security.

In formal language theory, this distinction corresponds to the difference between "terminal" and "non terminal" symbols in a formal grammar. That is, it signifies the contrast between lexical items with narrowly defined extensions and atomic instances as opposed to grammatical categories with infinite extensions and complex constituencies. Asking the question in this setting: How does a burgeoning language facility make the transition from finite state grammars, where terminals yield handles on non terminal symbols that obviate the need for a parser to backtrack, to higher level grammars, where a strategy of hypothetical trial and error is inevitable?

One way of visualizing a continuity in this transformation is by supposing that the potential to serve as an abstract sign is already available to interpreters in the flexible use of concrete signs. This suggests that generative categories and genuine hypotheses may arise by degrees in a gradual turning of phrases from fixed meanings to functional roles. Thus, authentic concepts can be derived from the interpretive recycling of individual names and nominal idioms into paradigmatic and schematic senses.

Peripatetically speaking, this illustrates a way that fledgling interpreters might pace themselves itself through the steps of this jump (the leap of abstraction) and trace a smooth progress over the intervening space: first, let them reposition discrete names in paradigmatic and schematic senses; then, allow them enough sense to recapture terminal and formulaic stereotypes as newly productive archetypes.

8.1.1.2. Reticles : Interpretive Flexibility as a Design Issue

As separate objects and independent constructs in and of themselves, interpretive frameworks are like the templates that observers impose on the scenes viewed through a reticle. Outside the slim chance of a pre-established harmony among them, there is no guarantee that the forms of intuition permitted by these instruments are essentially designed to fit the objects surveyed. Any notion of the world is a compromise between the specious and the factitious and yet supplies the mind with its only available grasp on reality.

The artifactual nature of the mind's handle on things is a commonplace observation of most philosophies, but there is a job here that remains to be carried out. Descriptively, the task that falls to this project is to consider how computational models of interpretive systems can be designed to take this factor into explicit account. Instrumentally, it is a design goal of this project to reflect this aspect of interpretive frameworks in the implementation of their supporting software, and thus to recognize and incorporate a feature in the artifact that seems unavoidable in the natural case.

Later, in making use of formal calculi, I will propose that the distinction between constants and variables can be treated as interpretive and need not be a fixture of the syntactic specification. This means it will be part of the meaning that is left up to the interpreter which symbols have fixed interpretations and which are taken as surrogates for a variety of substitutions.

8.1.2. Heuristic Inclinations and Regulative Principles

This discussion involves itself in a relationship with objective systems, linguistic and mathematical signs and descriptions, and a broad span of mental bearings that range through the following list: sensations and impressions, percepts and intensions, concepts and ideas, affects and irritations, actions and impulses, purposes and intentions.

8.2. Features of Inquiry Driven Systems

I have described inquiry as a process of determination that takes an agent from a state of uncertainty to a state of relative certainty or increased information, of a kind and to a degree sufficient for action, ...

I am operating on the assumption that / If inquiry is a process of determination that leads from uncertainty to the kind of certainty, sufficient for action, that an agent experiences as a state of belief or knowledge, then I need to say something about / articulate / examine the underlying epistemology, the implicit theory of knowledge and belief, that I employ / is employed in this project.

... then I need to say something about the kind of certainty that can be the goal of inquiry, and how I intend to use words like "belief" and "knowledge" in this discussion / understand concepts like belief and knowledge.

I do not believe I know of any difference in my immediate experience between belief and knowledge. To be more precise, I do not think I can tell a difference/ I detect no difference, from any quality present in the moment of experience itself, between an experience of believing something to be true and an experience of knowing something to be true. I do not think that, by itself, any agent can tell a difference between what it believes and what it knows.

By myself, I do not see how I can draw a distinction / tell a difference between what I believe and what I know. The distinction posed between them is not essential, but serves rhetorical and statistical functions, as a measure of intensity and commonality.

To say I "know" something is true is to mean that I really believe it. To say that someone else "knows" something is to say that the other believes the same as oneself.

Within the moment / I believe momentary experience has no quality in itself that distinguishes/ In the moment of experience /

The distinction made between belief and knowledge serves a largely / is partly rhetorical and partly statistical. The word "knowledge" operates as an intensifier, to say that one "really" believes something and to measure / as a modifier to indicate the intensity of belief or the measure of commonality / shared belief across a community/ to say that one has checked a belief by various means, verified it with others, including one's recollective former and preconceivable future selves, and found it to be a widely shared belief across this group.

There is nothing about the experience itself that distinguishes a state of belief from a state of knowledge. The distinction of knowledge serves as an intensifier, to say that one "really" believes something, or as a statistical function, to say that one has checked this belief by various means, with others, including one's past future selves.

For my purposes, I can see no difference present in the quality of the state itself between an agent "believing" a sign (expression or indicator) to be true and the same agent "knowing" the sign to be true.

If there is a difference between belief and knowledge, then it must have something to do with the way that one state of experience can refer to other states of experience outside of itself. In other words, it has to do with global and relational properties of the manifold of experience, and with the possibility that information about these constraints can be reflected and articulated within the individual moments of the manifold itself.

Thus, the distinction of "knowledge" is not essential or phenomenal, but incidental and epiphenomenal. That is, it has to do with the way that relations between basic levels of phenomena can reflect themselves within/ The way I use these words is not perjorative, but taxonomic. It neither diminishes the reality of epiphenomenal features and accidental attributes nor excuses me from the task of analyzing the geometry of their incidence/ but merely classifies / and does not diminish the importance of epiphenomena or the reality of accidental events ...

For my purposes, I can see no difference in the state itself between an agent believing a sign (statement or indicator) to be true and that agent knowing that sign to be true.

The intention of this section is to discuss in some detail two examples of inquiry driven systems that have already been implemented in the form of computer programs.

The goal of this section is to present in concrete detail significant examples of two different kinds of inquiry driven system which have already been implemented in the form of computer programs.

In this section I describe two examples of inquiry driven systems that have already been implemented as computer programs. The basic terms of description are taken from the pragmatic theory of signs, which I introduce as briefly as possible.

In this section I describe two examples of inquiry driven systems that have already been implemented as computer programs. The description is cast within the pragmatic theory of signs, which I review briefly and only to the extent necessary for discussing the examples.

8.2.1. The Pragmatic Theory of Signs

The treatment of inquiry to be developed in this project makes constant use of a philosophical perspective on thinking and communication known as the "pragmatic theory of signs". The subject matter of the theory of signs is a class of three place relations called "sign relations". These relations can be understood as set theoretic objects, as sets of ordered triples, and it is often useful in building concrete intuitions to consider elementary examples of this sort. But the sign relations of ultimate interest have infinite extensions and extremely complex internal structures.

Thus it develops that significant examples of sign relations are typically described and analyzed indirectly, by referring to a postulated agent that enacts or embodies the three place transactions involved in a particular case. The agent of sign relation is commonly called an "interpreter", who is variously said to partake in, embody, enact, compute, implement, execute, or carry out the sign relation in question. This brand of personification and its idioms serve a narrative function, supplying the observer in theory with an identifiable character and a point of view that reflects incidental variations in attitude toward the subject, but their purpose at heart remains one and the the same, which is merely to indicate or convey a particular sign relation.

Because inquiry systems will be described as special types of sign relations, inquiry driven systems and inquiry agents will be treated respectively as sign relations and interpreters that enjoy certain types of additional features.

Viewed within the pragmatic theory of signs, inquiry driven systems come at the end of a chain of incremental specification and increasing specialization. Starting from the bare conception of a triadic relation, notions of determination and correspondence are added to obtain the definition of a sign relation. To the static form of a sign relation, a notion of dynamic change is added to reach the idea of a sign process. To the aimless flow of a sign process, a notion of value is added that gives the succession of signs a motive, a direction, and a goal. Altogether, a system with non trivial values specified for each of these attributes constitutes an inquiry driven system.

What I just gave is a convenient order for taking up the attributes of an inquiry driven system. There is nothing unique about this approach, in particular, it is often useful to consider the dimension of value before discussing the dynamics of change. The most important thing about this list of properties is that it makes it possible to discuss the extent to which the changes of the system are in accord with the values of the system. A major part of the work remaining in this project is concerned with analyzing these global attributes into more detailed features, examining their relationships to one another, and ultimately translating their potential qualities into operational terms.

In this description inquiry driven systems are viewed as special cases of sign systems, those to which a notion of value has been added, by which a particular interpreter distinguishes what it considers to be better and worse signs of a given object.

This is a comparative dimension along which a particular interpreter distinguishes / recognizes better and worse signs of a given object, and differentially measures the quality of messages that otherwise have exactly the same meaning / along which a interpreting agent assesses a measure of quality among expressions, a comparative dimension of better and worse representations / signs of an object to a particular interpreter has been added, a comparative dimension of better and worse representation.

8.2.1.1. Sign Relations

Conceived in logical terms, a "sign relation" R is a certain kind of three place relationship that exists among the elements of three domains: the object domain O, the sign domain S, and the interpretant domain I. To qualify as a sign relation in this setting, a three place relation R is required to satisfy a few additional properties to be named later.

Expressed in terms of its set theoretic extension, a sign relation R is associated with a set of ordered triples <o, s, i> that forms a subset of the cartesian product OxSxI. The notation R = Set(R) c OxSxI can be used to single out this interpretation.

Expressed in terms of its computational intension, a sign relation R is associated with a predicate or program that values ordered triples <o, s, i> according to their fitness for the logical functions of the intended sign relationship. The notation R = Fun(R) : OxSxI > B, where B = {0, 1}, can be used to single out this interpretation.

Ways of Knowing a Relation

Knowledge by acquaintance:              extension or enumeration;
Knowledge by description:               intension or comprehension;
Knowledge by regulation:                intention or competence.

When the extension of a concept is infinite, or for any reason inconvenient/ or inconvenient for pragmatic reasons to enumerate in detail, then knowledge of its objects must be achieved by means of description rather than acquaintance.

When the extension of a concept is infinite, or inconvenient for pragmatic reasons to enumerate in final detail, then a finite agent's comprehension of it is required to be knowledge by indirect description rather than knowledge by direct acquaintance (Russell). That is, the objects of the concept are known by means of the concept's intensions, the common properties of its objects as expressed in symbolic reminders. But the intensions of a concept can be still more numerous than the objects of its extension, and even when a finite selection of these intensions is enough to specify the extension uniquely, there can always be many different collections which do so, and many different ways of approaching the concept by proceeding through a sequence of features in the subset chosen.

Thus, knowledge by description is approximate knowledge, knowledge whose quality and character can depend on the current stage and overall manner of approach. This sort of knowledge is contingent on and biased by each agent's particular way of approaching the objective, or the objects of the concept in question.

Due to the inclusion of these secondary facets in the character of the knowledge cut out, not every aspect of it is invariant over changes in the means of approach. The artifacts of the resulting knowledge that are not indifferent to the path of approach are called intensional features of the method, procedure, or computation. For example, programs that effectively compute the same function, the same set of ordered pairs in extension, but do it with non identical profiles of efficiency are said to differ in their intensional properties. Here, it is not the intensions of the functions as objects which differ, but the intensions of the programs as objects which do. The intensions of a program are related to the intensions of a function in the complex way that information about functional domain elements is traded for information about functional range elements throughout the progress of a computation.

Sometimes our grasp of the objects coming under a concept is even more tentative and tenuous.

Sometimes the mind's reach toward an objective is still more tentative and tenuous, exceeding the grasp of any present concept or familiar description, but represented only in the hope that certain rules of procedure or regulative principles are bound to converge on it in time. Thus, the object of knowledge is the object of an intention, and so is the hopeful knowledge itself. This kind of epistemological stance or orientation toward knowledge can be called "intentional knowledge" or "knowledge by intention", but is really more like an "intention to know".

To the extent that the computational intentions of this project are successful, more and more of the theoretical concepts employed in the unformalized parts of the inquiry will be operationalized as computable functions, serving to accomplish the actions or recognize the objects intended by each concept. In practical terms, this means that the functional interpretation of relational concepts, including the notion of a sign relation that founds the whole enterprise, will become paramount to the approach I have chosen. However, ...

The letters "o", "s", and "i" are examples of identifier names (variables or constants) that are used in discussing sign relations. They denote elements of the relational domains that fill the object, the sign, and the interpretant roles of the sign relation in question.

When the object is a formal system then its elements are regarded as signs (words or phrases, terms or formulas, pixels or pictures).

When the object is a dynamic system then its elements are regarded as states (points, moments, positions, vectors, configurations).

In general, the only constraint placed on a sign relation is the following definition.

(Peirce, NEM)

To complete this definition, it would be necessary to say what is meant by the notions of "determination" and "correspondence" that it invokes. This I defer to a later discussion. For now, I can limit discussion to the kinds of sign relations that are useful in systems theory and that occur in the computational representation of formal systems.

For the purposes of systems theory, and staying within the frame of computable representations, a number of additional restrictions and simplifying assumptions can be attached to the generic specifications of a sign system.

Because this discussion will stay within the framework of systems theory and limit its scope to the computational representation of formal systems, a number of restrictions and simplifications can be imposed on the general definition of a sign relation.

The roles of the sign relation are filled by systems or states of systems.

The object system o is a member of the object domain O.

In the cases of interest here, the object name (variable or identifier) "o" refers to a system or a state of system.

8.2.1.2. Types of Signs

8.2.2. The Pragmatic Theory of Inquiry

8.2.2.1. Abduction
8.2.2.2. Deduction
8.2.2.3. Induction

8.3. Examples of Inquiry Driven Systems

8.3.1. “Index” : A Program for Learning Formal Languages

The program "Index" actualizes an inquiry driven system that learns formal languages, operating under a restricted notion of learning that is explained next.

To specify an inquiry driven system I can first describe it in static terms as a sign relation, and then elaborate the more dynamic aspects of its sign process. In this example the role of the object is played by a special kind of mathematical object, a formal system known as a "two level formal language".

The object o is a two level formal language over a finite alphabet A. The object domain O is the set of all such languages over the same alphabet.

A two level formal languge o is specified by giving its words and its phrases, o = <W, P> = <o.lex, o.lit>, two sets that comprise its "lexical" and "literal" levels, respectively.

The words of o are finite sequences of letters from the alphabet A, collectively forming a set W called the "lexicon" of o. In symbols, W(o) = o.lex c A*.

The phrases of o are finite sequences of words from the lexicon W, collectively forming a set P called the "liturgy" of o. In symbols, P(o) = o.lit c W*.

As mathematical objects, not to mention objects that are potentially infinite in their extension and presumably unknown to the agent at the beginning of inquiry, o and O have the status of "external objects". This means they do not inhabit the minds or computers, the original or extended media, of inquiry agents. External objects never have their being in the locus of representation but become known to the agent only by means of the signs that gradually come to inhere in its being.

Because the agent of inquiry has a limited capacity for taking up information, the process of becoming informed about an external object cannot be any form of direct instruction on the part of the object or perfect intuition on the part of the agent. It is always a matter of stepwise approximation to better representations of the object. The progress of inquiry accumulates the tokens of the object's features that successively impress themselves on the agent's medium of attention and integrates them into the ongoing process that constitutes a particular agent's total activity.

8.3.2. “Study” : A Program for Reasoning with Propositions

The "Study" module implements an inquiry driven system that helps the user reason with expressions in propositional calculus.

The "Model" function within this program is a generic routine that implements a type of interpreter for propositional calculus. It takes in a proposition expressed in a particular syntax for propositional calculus and generates a data structure that is tantamount to the Disjunctive Normal Form (DNF) of the proposition. I will use the function notation "DNF(P)" and "Model(P)" to indicate the output of this routine for the proposition P. The DNF of a proposition P, as expressed in the data structure Model(P), is in a sense the clearest expression of the proposition P relative to the particular class of purposes that are embodied in a given interpreter.

The Study module contains several functions which compute different kinds of normal forms for propositions. These procedures constitute "modelers" or "interpreters" of propositional syntax in the sense that they generate the logical "models" or satisfying "interpretations" of propositions.

Any procedure that computes a normal form exemplifies an important kind of inquiry driven system. The dimension of value, or motivation, associated with the process can be regarded as a measure of "clarity". In computing a normal form the interpreter passes from an arbitrary representation of an indicated objective, one that can be as obscure as possible within the bounds of acceptable syntax, to a standardized formulation, one that manifests a patently clear and readily readable expression of the same objective. Thus, the operation is one that preserves meaning while maximizing clarity and ease of application.

In computing a normal form the system passes from an arbitrarily obscure representation of a propositional objective to a maximally clear expression of the same objective.

Need to clarify that a normal form is defined relative to a particular class of purposes or questions. For example, a sorted list is a normal form for questions about the multiplicity of items on the list, that is, about the existence and the number of occurrences of given items on the list.

It needs to be understood that the cpncept of a canonical form is defined relative to a particular purpose, a purpose which is embodied in a particular interpreter, or which a particular interpreter is intended to realize. Often this purpose can be expressed as a task of answering a particular class of questions about the object domain.

For the purposes of this discussion, I will draw a distinction between "canonical forms" and "normal forms". Distinquish canonical form in a semantic equivalence class, the intentional concept, from normal form of a transformation, the operational concept. A canonical form is an expression that is especially well suited to represent its equivalence class. A normal form is a fixed point of a grammatical transformation, that is, a stable point of a rewrite procedure that acts on the space of expressions. When the intentional canon/ canonical intention is rendered operational/ put into operation by a particular interpreter, then the two notions coincide, but only then.

To illustrate how the Model program actualizes an inquiry process, I will treat two examples in detail, …

Example 1:  P(x, y)  =
"x implies y".

Table 101.1  Standard Truth Table for P(x, y)
        x       y       P
        1       1       1
        1       0       0
        0       1       1
        0       0       1

Table 101.2  Variant Truth Table for P(x, y)
                        P
        x       y       1
        x       (y)     0
        (x)     y       1
        (x)     (y)     1

Table 101.3  Model Tree for P(x, y)

___.____ x ___ y     *
   |     |            
   |     |____(y)    -
   |                  
   |____(x)___ y     *
         |            
         |____(y)    *

Table 102.1  Logical Input for P(x, y)
        File:  "P.log"        Translation
 ( x ( y ))      Not x without y.

Table 102.2  Model Output for P(x, y)
        File:  "P.mod"        Model Value
 x
  y *   *
  (y ) -        -
 (x ) * *

Table 102.3  Tenor Output for P(x, y)
        File:  "P.ten"        Model Count
 x
  y *   1
 (x ) * 2

Table 102.4  Disjunctive Normal Form for P(x, y)
        DNF     Translation
 ((    x    y    Either x and y
 )(  ( x )       or not x
 ))      .

Table 103.1  Structure of the Sign Relation Rel(P)
        Object  Sign    Interpretant
        o1      s1      s2
        o2      s1      s2
        o3      s1      s2
        o1      s2      s3
        o2      s2      s3
        o3      s2      s3
        o1      s3      s3
        o2      s3      s3
        o3      s3      s3

Table 103.2  Contents of the Sign Relation Rel(P)
        Element  Description
        o1       Point " x  y "   =   <1, 1>
        o2       Point "(x) y "   =   <0, 1>
        o3       Point "(x)(y)"   =   <0, 0>
        s1       Parse "(x (y))"
        s2       Parse "(x y, (x))"
        s3       Parse "(x (y, ()(y)), (x))"

Example 2:  Q(x, y, z)  =
"just one false of x, y, z".

Table 104.1  Standard Truth Table for Q(x, y, z)
        x       y       z       Q
        1       1       1       0
        1       1       0       1
        1       0       1       1
        1       0       0       0
        0       1       1       1
        0       1       0       0
        0       0       1       0
        0       0       0       0

Table 104.2  Variant Truth Table for Q(x, y, z)
                                Q
        x       y       z       0
        x       y       (z)     1
        x       (y)     z       1
        x       (y)     (z)     0
        (x)     y       z       1
        (x)     y       (z)     0
        (x)     (y)     z       0
        (x)     (y)     (z)     0

Table 104.3  Model Tree for Q(x, y, z)

___.____ x ___ y ___ z     -
   |     |     |            
   |     |     |____(z)    *
   |     |                  
   |     |____(y)___ z     *
   |           |            
   |           |____(z)    -
   |                        
   |____(x)___ y ___ z     *
         |     |            
         |     |____(z)    -
         |                  
         |____(y)___ z     -
               |            
               |____(z)    -

Table 105.1  Logical Input for Q(x, y, z)
        File:  "Q.log"        Translation
 ( x , y , z )   Just one false
         among x, y, z.

Table 105.2  Model Output for Q(x, y, z)
        File:  "Q.mod"        Model Value
 x
  y
   z -  -
   (z ) *       *
  (y )
   z *  *
   (z ) -       -
 (x )
  y
   z *  *
   (z ) -       -
  (y ) -        -

Table 105.3  Tenor Output for Q(x, y, z)
        File:  "Q.ten"        Model Count
 x
  y
   (z ) *       1
  (y )
   z *  2
 (x )
  y
   z *  3

Table 105.4  Disjunctive Normal Form for Q(x, y, z)
        DNF     Translation
 ((    x    y  ( z )     Either  x  &  y  & -z
 )(    x  ( y )  z         or    x  & -y  &  z
 )(  ( x )  y    z         or   -x  &  y  &  z
 ))      .

Table 106.1  Structure of the Sign Relation Rel(Q)
        Object  Sign    Interpretant
        o1      s1      s2
        o2      s1      s2
        o3      s1      s2
        o1      s2      s3
        o2      s2      s3
        o3      s2      s3
        o1      s3      s4
        o2      s3      s4
        o3      s3      s4
        o1      s4      s4
        o2      s4      s4
        o3      s4      s4

Table 106.2  Contents of Rel(Q):  Objects
        Element  Description
        o1       Point " x  y (z)"   =   <1, 1, 0>
        o2       Point " x (y) z "   =   <1, 0, 1>
        o3       Point "(x) y  z "   =   <0, 1, 1>

Table 106.3  Contents of Rel(Q):  Signs
        Element  Description
        s1       Parse "(x, y, z)"
        s2       Parse "( x (y, z)
                        ,(x) y  z
                        )"
        s3       Parse "( x ( y (z)
                            ,(y) z
                            )
                        ,(x)( y  z
                            ,(y)()
                            )
                        )"
        s4       Parse "( x ( y ( z ()
                                ,(z) *
                                )
                            ,(y)( z  *
                                ,(z)()
                                )
                            )
                        ,(x)( y ( z  *
                                ,(z)()
                                )
                            ,(y)()
                            )
                        )"

Table 107.1  Normal Form Expansion of Q(x, y, z):  Version 1
Sign     Expression      Translation
 s1      (x, y, z)       Just one false of x, y, z
 s2      ( x (y, z)      Either x & (y, z)
         ,(x) y  z         or  -x &  y  z
         )
 s3      ( x ( y (z)     Either x & either y & (z)
             ,(y) z                   or  -y &  z
             )
         ,(x)( y  z        or  -x & either y &  z
             ,(y)()                   or  -y & false
             )
         )
 s4      ( x ( y ( z ()  Either x & either y & either z & 0
                 ,(z) *                          or  -z & 1
                 )
             ,(y)( z  *               or  -y & either z & 1
                 ,(z)()                          or  -z & 0
                 )
             )
         ,(x)( y ( z  *    or  -x & either y & either z & 1
                 ,(z)()                          or  -z & 0
                 )
             ,(y)()                   or  -y & false
             )
         )

Table 107.2  Normal Form Expansion of Q(x, y, z):  Version 2
Sign     Expression      Translation
 s1      (x, y, z)       Just one false of x, y, z.
 s2      ( x (y, z)      Either x & (y, z)
         ,(x) y  z       or not x &  y  z
         )       .
 s3      ( x     Either x &
          ( y (z)          either y & (z)
          ,(y) z           or not y &  z
          )        ;
         ,(x)    or not x &
          ( y  z           either y &  z
          ,(y)()           or not y & false
          )        ;
         )       .
 s4      ( x     Either x &
          ( y      either y &
           ( z ()            either z & false
           ,(z) *            or not z & true
           )         ;
          ,(y)     or not y &
           ( z *             either z & true
           ,(z)()            or not z & false
           )         ;
          )        ;
         ,(x)    or not x &
          ( y      either y &
           ( z  *            either z & true
           ,(z)()            or not z & false
           )         ;
          ,(y)()           or not y & false
          )        ;
         )       .

8.4. Discussion and Development of Objectives

8.4.1. Objective 1a : Propositions as Types

In this component I investigate an important relationship that exists between two kinds of formal systems, called applicational calculi (AC's) and propositional calculi (PC's).

Expressions in an applicational calculus are called "terms" and are built up from a supply of primitive symbols called "basic terms" through the use of a binary operation called "application".

Expressions in a propositional calculus are called "propositions" and are built up from a supply of primitive symbols called "basic propositions" through the use of a stock of k ary operations called "connectives".

Exact formalization of symbolic calculi is absolutely necessary, especially if we want computers to serve as interpreters, or at least to lighten the burdens of interpretation by taking up the bit parts of formal routines. Because the problem environments where calculi develop show no signs of stopping in their growing complexity, it becomes inevitable that we turn to computers as auxiliary interpreters of formal systems, to secure their operational significance and manage the increasing complications of their practical use. But a formalized syntax, however necessary, is not enough to demonstrate utility.

A calculus is never a finished object, complete in itself, but a tool shaped to the end of interpretation.

I am using the word "interpretation" to denote the whole complex of activities through which signs acquire practical meaning. For human beings the process of interpretation is a largely automatic and usually unanalyzed affair, but it can also be highly flexible and unusually adaptive. Human language users or symbol manipulators have powers they hardly reflect on and barely understand, to entertain novel associations between signs and whatever it is that signs convey, to examine habitual assumptions about the meaning of symbols in practice.

One person can simply invite another to entertain new associations between signs and ideas or to examine old assumptions about their meaning in practice, and the other is somehow able to comply, if only for the sake of experiment and argument. But the need to supply a formal system with a computational basis, involving exact syntax, effective semantics, and computerized "pragmatics" (that is, a computer supported operational basis) requires us to analyze the process of interpretation in minute detail and on new grounds.

In use, AC's and PC's take on meaning according to a variety of interpretive rules. I am adopting an interpretation on either side that highlights the relation of interest between these two kinds of calculi and that brings the intended correspondence into sharper relief.

In both kinds of calculus the primitive symbols are distinguished as "constants" and "variables". A constant is a name for a definite object of thought, and is intended to maintain a fixed meaning throughout a given discussion. A variable is a symbol of indefinite reference, or a token without a pre assigned meaning, but is used as a site for the substitution of other expressions or as a placeholder for a multitude of conceptual objects and logical values.

Semantics. Meanings are provided or attached to symbols by means of:

1. Contexts of occurrence: the facts or rules of distribution.

2. Action induced on other symbols by means of rewrite rules:

generators and relations, or paraphrastic definitions.

3. Synonymy: membership in semantic equivalence classes.

4. Arbitrary fiat, external designation, or special convention.

8.4.2. Objective 1b : Proof Styles and Developments

8.4.3. Objective 1c : Interpretation and Authority


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