Changes

I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.

I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.

In the present examples, S = I = Syntactic Domain.

In the present examples, S = I = Syntactic Domain.

Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting them in the form of relational databases.  Thus, the rows of each Table list the ordered triples of the form <o, s, i> that make up the corresponding sign relations:  A, B  ?  O?S?I.  The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated.

Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting them in the form of relational databases.  Thus, the rows of each Table list the ordered triples of the form ‹o, s, i› that make up the corresponding sign relations:  A, B  ?  O?S?I.  The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated.

These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <o, s, i> that is called an "elementary relation", that is, one element of the relation's set-theoretic extension.

These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form ‹o, s, i› that is called an "elementary relation", that is, one element of the relation's set-theoretic extension.

Already in this elementary context, there are several different meanings that might attach to the project of a "formal semantics".  In the process of discussing these alternatives, I will introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

Already in this elementary context, there are several different meanings that might attach to the project of a "formal semantics".  In the process of discussing these alternatives, I will introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its "denotation".  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its "denotation".  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

The dyadic relation that constitutes the "denotative component" of a sign relation L is denoted by "Den(L)".  Information about the denotative component of semantics can be derived from L by taking its "dyadic projection" on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, "ProjOS(L)", "LOS", or "L12", and defined as follows:

The dyadic relation that constitutes the "denotative component" of a sign relation L is denoted by "Den(L)".  Information about the denotative component of semantics can be derived from L by taking its "dyadic projection" on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, "ProjOS(L)", "LOS", or "L12", and defined as follows:

Den(L)  =  ProjOS(L)  =  LOS  =  {<o, s> ? O?S : <o, s, i> ? L for some i ? I}.

Den(L)  =  ProjOS(L)  =  LOS  =  {‹o, s› ? O?S : ‹o, s, i› ? L for some i ? I}.

Looking to the denotative aspects of the present example, various rows of the Tables specify that A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

Looking to the denotative aspects of the present example, various rows of the Tables specify that A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

Formally, these new aspects of semantics present no additional problem:

Formally, these new aspects of semantics present no additional problem:

The connotative component of a sign relation L can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

The connotative component of a sign relation L can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

Con(L)  =  ProjSI(L)  =  LSI  =  {<s, i> ? S?I : <o, s, i> ? L for some o ? O}.

Con(L)  =  ProjSI(L)  =  LSI  =  {‹s, i› ? S?I : ‹o, s, i› ? L for some o ? O}.

The intentional component of semantics for a sign relation L, or its "second moment of denotation", is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

The intentional component of semantics for a sign relation L, or its "second moment of denotation", is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

Int(L)  =  ProjOI(L)  =  LOI  =  {<o, i> ? O?I : <o, s, i> ? L for some s ? S}.

Int(L)  =  ProjOI(L)  =  LOI  =  {‹o, i› ? O?I : ‹o, s, i› ? L for some s ? S}.

As it happens, the sign relations A and B in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of AOS and BOS is merely echoed in AOI and BOI, respectively.

As it happens, the sign relations A and B in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of AOS and BOS is merely echoed in AOI and BOI, respectively.

The principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes.  In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary.  Instead, this relationship must be naturally constrained or deliberately designed in such a way that it:

The principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes.  In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary.  Instead, this relationship must be naturally constrained or deliberately designed in such a way that it:

To discuss these types of situations further, I introduce the square bracket notation "[x]E" for "the equivalence class of the element x under the equivalence relation E".  A statement that the elements x and y are equivalent under E is called an "equation", and can be written in either one of two ways, as  "[x]E = [y]E"  or as  "x =E y".

To discuss these types of situations further, I introduce the square bracket notation "[x]E" for "the equivalence class of the element x under the equivalence relation E".  A statement that the elements x and y are equivalent under E is called an "equation", and can be written in either one of two ways, as  "[x]E = [y]E"  or as  "x =E y".

In the application to sign relations I extend this notation in the following ways.  When L is a sign relation whose "syntactic projection" or connotative component LSI is an equivalence relation on S, I write "[s]L" for "the equivalence class of s under LSI".  A statement that the signs x and y are synonymous under a semiotic equivalence relation LSI is called a "semiotic equation" (SEQ), and can be written in either of the forms:  "[x]L = [y]L"  or  "x =L y".

In the application to sign relations I extend this notation in the following ways.  When L is a sign relation whose "syntactic projection" or connotative component LSI is an equivalence relation on S, I write "[s]L" for "the equivalence class of s under LSI".  A statement that the signs x and y are synonymous under a semiotic equivalence relation LSI is called a "semiotic equation" (SEQ), and can be written in either of the forms:  "[x]L = [y]L"  or  "x =L y".

In many situations there is one further adaptation of the square bracket notation that can be useful.  Namely, when there is known to exist a particular triple <o, s, i> ? L, it is permissible to use "[o]L" to mean the same thing as "[s]L".  These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

In many situations there is one further adaptation of the square bracket notation that can be useful.  Namely, when there is known to exist a particular triple ‹o, s, i› ? L, it is permissible to use "[o]L" to mean the same thing as "[s]L".  These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

In these terms, the SER for interpreter A yields the semiotic equations:

In these terms, the SER for interpreter A yields the semiotic equations:

["A"]A  =  ["i"]A, ["B"]A  =  ["u"]A,

["A"]A  =  ["i"]A, ["B"]A  =  ["u"]A,

1.3.4.4  Graphical Representations

1.3.4.4  Graphical Representations

The dyadic components of sign relations can be given graph-theoretic representations, as "digraphs" (or "directed graphs"), that provide concise pictures of their structural and potential dynamic prop

The dyadic components of sign relations can be given graph-theoretic representations, as "digraphs" (or "directed graphs"), that provide concise pictures of their structural and potential dynamic prop

erties.  By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self-loop <x, x> is called a "sling" at x.

erties.  By way of terminology, a directed edge ‹x, y› is called an "arc" from point x to point y, and a self-loop ‹x, x› is called a "sling" at x.

The denotative components Den(A) and Den(B) can be represented as digraphs on the six points of their common world set W = O ? S ? I = { A, B, "A", "B", "i", "u"}.  The arcs are given as follows:

The denotative components Den(A) and Den(B) can be represented as digraphs on the six points of their common world set W = O ? S ? I = { A, B, "A", "B", "i", "u"}.  The arcs are given as follows:

1. Den(A) has an arc from each point of {"A", "i"} to A  

1. Den(A) has an arc from each point of {"A", "i"} to A  

2. The various OM's of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM.  If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them "relements", in this way jostling the mind to ask:  "Relement to what?"

2. The various OM's of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM.  If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them "relements", in this way jostling the mind to ask:  "Relement to what?"

The last and likely the best way one can choose to follow in order to form an objective genre G is to present it as a triadic relation:

The last and likely the best way one can choose to follow in order to form an objective genre G is to present it as a triadic relation:

G  =  {<j, p, q>}  ?  JxPxQ, or

G  =  {‹j, p, q›}  ?  JxPxQ, or

G  =  {<j, x, y>}  ?  JxXxX.

G  =  {‹j, x, y›}  ?  JxXxX.

For some reason the ultimately obvious method seldom presents itself exactly in this wise without diligent work on the part of the inquirer, or one who would arrogate the roles of both its former and its follower.  Perhaps this has to do with the problematic role of "synthetic a priori" truths in constructive mathematics.  Perhaps the mystery lies encrypted, no doubt buried in some obscure dead letter office, due to the obliterate indicia on the letters "P", "Q", and "X" inscribed above.  No matter - at the moment there are far more pressing rounds to make.

For some reason the ultimately obvious method seldom presents itself exactly in this wise without diligent work on the part of the inquirer, or one who would arrogate the roles of both its former and its follower.  Perhaps this has to do with the problematic role of "synthetic a priori" truths in constructive mathematics.  Perhaps the mystery lies encrypted, no doubt buried in some obscure dead letter office, due to the obliterate indicia on the letters "P", "Q", and "X" inscribed above.  No matter - at the moment there are far more pressing rounds to make.

Given a genre G whose OM's are indexed by a set J and whose objects form a set X, there is a triadic relation among an OM and a pair of objects that exists when the first object belongs to the second object according to that OM.  This is called the "standing relation" of the OG, and it can be taken as one way of defining and establishing the genre.  In the way that triadic relations usually give rise to dyadic operations, the associated "standing operation" of the OG can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated OM.

Given a genre G whose OM's are indexed by a set J and whose objects form a set X, there is a triadic relation among an OM and a pair of objects that exists when the first object belongs to the second object according to that OM.  This is called the "standing relation" of the OG, and it can be taken as one way of defining and establishing the genre.  In the way that triadic relations usually give rise to dyadic operations, the associated "standing operation" of the OG can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated OM.

There is a "partial converse" of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the "propping relation" of the OG, and it can be taken as an alternate way of defining the genre.

There is a "partial converse" of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the "propping relation" of the OG, and it can be taken as an alternate way of defining the genre.

G^  =  {<j, q, p> ? J?Q?P : <j, p, q> ? G}, or

G^  =  {‹j, q, p› ? J?Q?P : ‹j, p, q› ? G}, or

G^  =  {<j, y, x> ? J?X?X : <j, x, y> ? G}.

G^  =  {‹j, y, x› ? J?X?X : ‹j, x, y› ? G}.

The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an OG:

The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an OG:

1. The standing relation of an OG is denoted by the symbol ":<", pronounced "set-in", so that  :< ? JxPxQ  or  :< ? J?X?X.

1. The standing relation of an OG is denoted by the symbol ":<", pronounced "set-in", so that  :< ? JxPxQ  or  :< ? J?X?X.

2. The propping relation of an OG is denoted by the symbol ":>", pronounced "set-on", so that  :> ? J?Q?P  or  :> ? J?X?X.

2. The propping relation of an OG is denoted by the symbol ":>", pronounced "set-on", so that  :> ? J?Q?P  or  :> ? J?X?X.

Often one's level of interest in a genre is "purely generic".  When the relevant genre is regarded as an indexed family of dyadic relations, G = {Gj}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

Often one's level of interest in a genre is "purely generic".  When the relevant genre is regarded as an indexed family of dyadic relations, G = {Gj}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

UJG  =  Uj Gj  =  {<x, y> ? X?X : <x, y> ? Gj for some j ? J}.

UJG  =  Uj Gj  =  {‹x, y› ? X?X : ‹x, y› ? Gj for some j ? J}.

When the relevant genre is contemplated as a triadic relation, G ? J?X?X, then one is dealing with the projection of G on the object dyad XxX.

When the relevant genre is contemplated as a triadic relation, G ? J?X?X, then one is dealing with the projection of G on the object dyad XxX.

GXX  =  ProjXX(G)  =  {<x, y> ? X?X : <j, x, y> ? G for some j ? J}.

GXX  =  ProjXX(G)  =  {‹x, y› ? X?X : ‹j, x, y› ? G for some j ? J}.

On these occasions, the assertion that  <x, y> ?  UJG  =  GXX  can be indicated by any one of the following equivalent expressions:

On these occasions, the assertion that  ‹x, y› ?  UJG  =  GXX  can be indicated by any one of the following equivalent expressions:

G : x < y, x <G y, x < y : G,

G : x < y, x <G y, x < y : G,

G : y > x, y >G x, y > x : G.

G : y > x, y >G x, y > x : G.

At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation (IDR) that links two objects.  To indicate that a triple consisting of an OM j and two objects x and y belongs to the standing relation of the OG, <j, x, y> ? :<, or equally, to indicate that a triple consisting of an OM j and two objects y and x belongs to the propping relation of the OG, <j, y, x> ? :>, all of the following notations are equivalent:

At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation (IDR) that links two objects.  To indicate that a triple consisting of an OM j and two objects x and y belongs to the standing relation of the OG, ‹j, x, y› ? :<, or equally, to indicate that a triple consisting of an OM j and two objects y and x belongs to the propping relation of the OG, ‹j, y, x› ? :>, all of the following notations are equivalent:

j : x < y, x <j y, x < y : j,

j : x < y, x <j y, x < y : j,

j : y > x, y >j x, y > x : j.

j : y > x, y >j x, y > x : j.

1. In a cognitive context, if j is a considered opinion that S is true, and j is a considered opinion that T is true, then it does not have to automatically follow that j is a considered opinion that S and T are true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of S and T.

1. In a cognitive context, if j is a considered opinion that S is true, and j is a considered opinion that T is true, then it does not have to automatically follow that j is a considered opinion that S and T are true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of S and T.

2. In a logical context, if j is a piece of evidence that S is true, and j is a piece of evidence that T is true, then it follows by these very facts alone that j is a piece of evidence that S and T are true.  This is analogous to a situation where, if a person j draws a set of three lines AB, BC, and AC, then j has drawn a triangle ABC, whether j recognizes the fact on reflection and further consideration or not.

2. In a logical context, if j is a piece of evidence that S is true, and j is a piece of evidence that T is true, then it follows by these very facts alone that j is a piece of evidence that S and T are true.  This is analogous to a situation where, if a person j draws a set of three lines AB, BC, and AC, then j has drawn a triangle ABC, whether j recognizes the fact on reflection and further consideration or not.

Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory.  For example, the predicate P : J -> B = {0, 1}, defined by P(j) <=> "j proposes x an instance of y", is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in J.  And yet, all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple <j, x, y> in the genre G(:<).

Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory.  For example, the predicate P : J -> B = {0, 1}, defined by P(j) <=> "j proposes x an instance of y", is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in J.  And yet, all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple ‹j, x, y› in the genre G(:<).

1.3.4.14  Application of OF:  Generic Level

1.3.4.14  Application of OF:  Generic Level

Given an ontological framework that can provide multiple perspectives and moving platforms for dealing with object structure, in other words, that can organize diverse hierarchies and developing orders of objects, attention can now return to the discussion of sign relations as models of intellectual processes.

Given an ontological framework that can provide multiple perspectives and moving platforms for dealing with object structure, in other words, that can organize diverse hierarchies and developing orders of objects, attention can now return to the discussion of sign relations as models of intellectual processes.

The OG I apply here is called the genre of "properties and instances".  One moves through its space, higher and lower in a particular ontology, by means of two dyadic relations, upward by taking a "property of" and downward by taking an "instance of" whatever object initially enters one's focus of attention.  Each object of this OG is reckoned to be the unique common property of the set of objects that lie one step below it, objects that are in turn reckoned to be instances of the given object.

The OG I apply here is called the genre of "properties and instances".  One moves through its space, higher and lower in a particular ontology, by means of two dyadic relations, upward by taking a "property of" and downward by taking an "instance of" whatever object initially enters one's focus of attention.  Each object of this OG is reckoned to be the unique common property of the set of objects that lie one step below it, objects that are in turn reckoned to be instances of the given object.

Pretty much the same relational structures could be found in the genre or paradigm of "qualities and examples", but the use of "examples" here is polymorphous enough to include experiential, exegetic, and executable examples (EXE's).  This points the way to a series of related genres, for example, the OG's of "principles and illustrations", "laws and existents", "precedents and exercises", and on to "lessons and experiences".  All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.

Pretty much the same relational structures could be found in the genre or paradigm of "qualities and examples", but the use of "examples" here is polymorphous enough to include experiential, exegetic, and executable examples (EXE's).  This points the way to a series of related genres, for example, the OG's of "principles and illustrations", "laws and existents", "precedents and exercises", and on to "lessons and experiences".  All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.

Any object of these OG's can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an "object quality" (OQ) or an "object example" (OE) of something else.  In future references, abbreviated notations like "OG (Prop, Inst)" or "OG = <Prop, Inst>" will be used to specify particular genres, giving the intended interpretations of their generating relations { < , > }.

Any object of these OG's can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an "object quality" (OQ) or an "object example" (OE) of something else.  In future references, abbreviated notations like "OG (Prop, Inst)" or "OG = ‹Prop, Inst›" will be used to specify particular genres, giving the intended interpretations of their generating relations { < , > }.

With respect to this OG, I can now characterize icons and indices.  Icons are signs by virtue of being instances of properties of objects.  Indices are signs by virtue of being properties of instances of objects.

With respect to this OG, I can now characterize icons and indices.  Icons are signs by virtue of being instances of properties of objects.  Indices are signs by virtue of being properties of instances of objects.

Because the initial discussion seems to flow more smoothly if I apply dyadic relations on the left, I formulate these definitions as follows:

Because the initial discussion seems to flow more smoothly if I apply dyadic relations on the left, I formulate these definitions as follows:

Is this prospect a utopian vision?  Perhaps it is exactly that.  But it is the hope that inquiry discovers resting first and last within itself, quietly guiding every other aim and motive of inquiry.

Is this prospect a utopian vision?  Perhaps it is exactly that.  But it is the hope that inquiry discovers resting first and last within itself, quietly guiding every other aim and motive of inquiry.

Turning to the language of objective concerns, what can now be said about the compositional structures of the iconic sign relation M and the indexical sign relation N?  In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.

Turning to the language of objective concerns, what can now be said about the compositional structures of the iconic sign relation M and the indexical sign relation N?  In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.

I recall the OG of "properties and instances" and introduce the symbols "<" and ">" for the converse pair of dyadic relations that generate it.  Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of x" and "instance of x" by means of a case inflection on x, that is, as "x's property" and "x's instance", respectively.  Described in this way, OG (Prop, Inst) = < < , > >, where:

I recall the OG of "properties and instances" and introduce the symbols "<" and ">" for the converse pair of dyadic relations that generate it.  Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of x" and "instance of x" by means of a case inflection on x, that is, as "x's property" and "x's instance", respectively.  Described in this way, OG (Prop, Inst) = ‹ < , > ›, where:

"x <" = "x's Property" = "Property of x" = "Object above x",

"x <" = "x's Property" = "Property of x" = "Object above x",

"x >" = "x's Instance" = "Instance of x" = "Object below x".

"x >" = "x's Instance" = "Instance of x" = "Object below x".

"x >"  =  "x is the Property of what?"  =  "x's Instance".

"x >"  =  "x is the Property of what?"  =  "x's Instance".

In this fashion, the definitions of icons and indices can be reformulated:

In this fashion, the definitions of icons and indices can be reformulated:

x's Icon  =  x's Property's Instance  =  x.<>,

x's Icon  =  x's Property's Instance  =  x.‹›,

x's Index  =  x's Instance's Property  =  x.><.

x's Index  =  x's Instance's Property  =  x.›‹.

According to the definitions of the homogeneous sign relations M and N:

According to the definitions of the homogeneous sign relations M and N:

x's Icon  =  x.MOS,

x's Icon  =  x.MOS,

The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind.  As always, it is best to regard OG's and OM's as "filters" and "reticles", as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.

The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind.  As always, it is best to regard OG's and OM's as "filters" and "reticles", as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.

With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to "factor the facets" or "decompose the components" of sign relations in a more systematic fashion.  Given a homogeneous sign relation H of iconic or indexical type, the dyadic projections HOS and HOI can be analyzed as compound relations over the basis supplied by the Gj in G.  As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.

With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to "factor the facets" or "decompose the components" of sign relations in a more systematic fashion.  Given a homogeneous sign relation H of iconic or indexical type, the dyadic projections HOS and HOI can be analyzed as compound relations over the basis supplied by the Gj in G.  As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.

To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type < < , > >, and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.

To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type ‹ < , > ›, and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.

Let X collect the objects of thought that fall within a particular OM, and let X include the whole world of a sign relation plus everything needed to support and contain it.  That is, X collects all the types of things that go into a sign relation, O U S U I = W ? X, plus whatever else in the way of distinct object qualities (OQ's) and object exemplars (OE's) is discovered or established to be generated out of this basis by the relations of the OM.

Let X collect the objects of thought that fall within a particular OM, and let X include the whole world of a sign relation plus everything needed to support and contain it.  That is, X collects all the types of things that go into a sign relation, O U S U I = W ? X, plus whatever else in the way of distinct object qualities (OQ's) and object exemplars (OE's) is discovered or established to be generated out of this basis by the relations of the OM.

In order to keep this X simple enough to contemplate on a single pass, but still make it deep enough to cover the issues of interest at present, I limit X to having just three disjoint layers of things to worry about.  The middle layer X0 is the initial collection of objects and signs, X0 = W.  The top layer Q is the relevant class of object qualities, Q = X0< = W.<.  The bottom layer E is a suitable set of object exemplars, E = X0> = W.>.

In order to keep this X simple enough to contemplate on a single pass, but still make it deep enough to cover the issues of interest at present, I limit X to having just three disjoint layers of things to worry about.  The middle layer X0 is the initial collection of objects and signs, X0 = W.  The top layer Q is the relevant class of object qualities, Q = X0< = W.<.  The bottom layer E is a suitable set of object exemplars, E = X0> = W.>.