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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
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, 18:32, 16 December 2008sub [\lessdot / <s><</s>], sub [\gtrdot / <s>></s>]
Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems. The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.
Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems. The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.
The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs "<font face="system"><s><</s></font>" and "<font face="system"><s>></s></font>". At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators "<font face="system"><s><</s></font>" and "<font face="system"><s>></s></font>" involve the application of dyadic relations analogous to class membership "∈" and its converse "∋", but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OM's the full power of triadic relations. Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context. In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.
The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs "<math>\lessdot</math>" and "<math>\gtrdot</math>". At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators "<math>\lessdot</math>" and "<math>\gtrdot</math>" involve the application of dyadic relations analogous to class membership "∈" and its converse "∋", but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OM's the full power of triadic relations. Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context. In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.
The staging relations divide into two orientations, "<font face="system"><s><</s></font>" versus "<font face="system"><s>></s></font>", indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:
The staging relations divide into two orientations, "<math>\lessdot</math>" versus "<math>\gtrdot</math>", indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:
# The ''standing relations'', indicated by "<font face="system"><s><</s></font>", are analogous to the ''element of'' or membership relation "∈". Another interpretation of "<font face="system"><s><</s></font>" is the ''instance of'' relation. At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.
# The ''standing relations'', indicated by "<math>\lessdot</math>", are analogous to the ''element of'' or membership relation "∈". Another interpretation of "<math>\lessdot</math>" is the ''instance of'' relation. At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.
# The ''propping relations'', indicated by "<font face="system"><s>></s></font>", are analogous to the ''class of'' relation or converse of the membership relation. An alternate meaning for "<font face="system"><s>></s></font>" is the ''property of'' relation. Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.
# The ''propping relations'', indicated by "<math>\gtrdot</math>", are analogous to the ''class of'' relation or converse of the membership relation. An alternate meaning for "<math>\gtrdot</math>" is the ''property of'' relation. Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.
Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, "<font face="system"><s><</s></font>" and "<font face="system"><s>></s></font>", and to maintain a formal calculus that treats analogous pairs of relations on an equal footing. Extra measures of convenience come into play when the relations are used as assignment operations or ''field promotions'', in other words, to create titles, define terms, and establish offices of objects in the active contexts of given relations. Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.
Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, "<math>\lessdot</math>" and "<math>\gtrdot</math>", and to maintain a formal calculus that treats analogous pairs of relations on an equal footing. Extra measures of convenience come into play when the relations are used as assignment operations or ''field promotions'', in other words, to create titles, define terms, and establish offices of objects in the active contexts of given relations. Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.
Next, I present several different ways of formalizing OG's and OM's. The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.
Next, I present several different ways of formalizing OG's and OM's. The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.
It is important to emphasize that the index set ''J'' and the particular attachments of indices to dyadic relations are part and parcel to ''G'', befitting the concrete character intended for the concept of an OG, which is expected to realistically embody in the character of each ''G''<sub>''j''</sub> both ''a local habitation and a name''. For this reason, among others, the ''G''<sub>''j''</sub> can safely be referred to as ''individual dyadic relations'' (IDR's). Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' (TIP's) that a concrete index ''j'' adds to its thematic object ''G''<sub>''j''</sub> , whether parenthetically or paraphatically.
It is important to emphasize that the index set ''J'' and the particular attachments of indices to dyadic relations are part and parcel to ''G'', befitting the concrete character intended for the concept of an OG, which is expected to realistically embody in the character of each ''G''<sub>''j''</sub> both ''a local habitation and a name''. For this reason, among others, the ''G''<sub>''j''</sub> can safely be referred to as ''individual dyadic relations'' (IDR's). Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' (TIP's) that a concrete index ''j'' adds to its thematic object ''G''<sub>''j''</sub> , whether parenthetically or paraphatically.
Depending on the prevailing direction of interest in the genre ''G'', "<font face="system"><s><</s></font>" or "<font face="system"><s>></s></font>", the same symbol is used equivocally for all the relations ''G''<sub>''j''</sub> . The ''G''<sub>''j''</sub> can be regarded as formalizing the OM's that make up the genre ''G'', provided it is understood that the information corresponding to the parameter ''j'' constitutes an integral part of the ''motive'' or ''motif'' of ''G''<sub>''j''</sub> .
Depending on the prevailing direction of interest in the genre ''G'', "<math>\lessdot</math>" or "<math>\gtrdot</math>", the same symbol is used equivocally for all the relations ''G''<sub>''j''</sub> . The ''G''<sub>''j''</sub> can be regarded as formalizing the OM's that make up the genre ''G'', provided it is understood that the information corresponding to the parameter ''j'' constitutes an integral part of the ''motive'' or ''motif'' of ''G''<sub>''j''</sub> .
In this formulation, ''G'' constitutes an ''ontological hierarchy'' (OH) of a plenary and potentiating type, one that determines the complete array of objects and relationships that are conceivably available and describably ''effable'' within a given discussion. Operating with reference to the global field of possibilities presented by ''G'', each ''G''<sub>''j''</sub> corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.
In this formulation, ''G'' constitutes an ''ontological hierarchy'' (OH) of a plenary and potentiating type, one that determines the complete array of objects and relationships that are conceivably available and describably ''effable'' within a given discussion. Operating with reference to the global field of possibilities presented by ''G'', each ''G''<sub>''j''</sub> corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.
:{| style="text-align:left; width:90%"
:{| style="text-align:left; width:90%"
| ''G'' : ''x'' <font face="system"><s><</s></font> ''y'' ,
| ''G'' : ''x'' <math>\lessdot</math> ''y'' ,
| ''x'' <font face="system"><s><</s></font><sub>''G''</sub> ''y'' ,
| ''x'' <math>\lessdot</math><sub>''G''</sub> ''y'' ,
| ''x'' <font face="system"><s><</s></font> ''y'' : ''G'' ,
| ''x'' <math>\lessdot</math> ''y'' : ''G'' ,
|-
|-
| ''G'' : ''y'' <font face="system"><s>></s></font> ''x'' ,
| ''G'' : ''y'' <math>\gtrdot</math> ''x'' ,
| ''y'' <font face="system"><s>></s></font><sub>''G''</sub> ''x'' ,
| ''y'' <math>\gtrdot</math><sub>''G''</sub> ''x'' ,
| ''y'' <font face="system"><s>></s></font> ''x'' : ''G'' .
| ''y'' <math>\gtrdot</math> ''x'' : ''G'' .
|}
|}
:{| style="text-align:left; width:90%"
:{| style="text-align:left; width:90%"
| ''j'' : ''x'' <font face="system"><s><</s></font> ''y'' ,
| ''j'' : ''x'' <math>\lessdot</math> ''y'' ,
| ''x'' <font face="system"><s><</s></font><sub>''j''</sub> ''y'' ,
| ''x'' <math>\lessdot</math><sub>''j''</sub> ''y'' ,
| ''x'' <font face="system"><s><</s></font> ''y'' : ''j'' ,
| ''x'' <math>\lessdot</math> ''y'' : ''j'' ,
|-
|-
| ''j'' : ''y'' <font face="system"><s>></s></font> ''x'' ,
| ''j'' : ''y'' <math>\gtrdot</math> ''x'' ,
| ''y'' <font face="system"><s>></s></font><sub>''j''</sub> ''x'' ,
| ''y'' <math>\gtrdot</math><sub>''j''</sub> ''x'' ,
| ''y'' <font face="system"><s>></s></font> ''x'' : ''j'' .
| ''y'' <math>\gtrdot</math> ''x'' : ''j'' .
|}
|}
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| ''j'' : ''x'' <font face="system"><s><</s></font> ''y''
| ''j'' : ''x'' <math>\lessdot</math> ''y''
| ''j'' : ''y'' <font face="system"><s>></s></font> ''x''
| ''j'' : ''y'' <math>\gtrdot</math> ''x''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| ''x'' <font face="system"><s><</s></font><sub>''j''</sub> ''y''
| ''x'' <math>\lessdot</math><sub>''j''</sub> ''y''
| ''y'' <font face="system"><s>></s></font><sub>''j''</sub> ''x''
| ''y'' <math>\gtrdot</math><sub>''j''</sub> ''x''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| ''x'' <font face="system"><s><</s></font> ''y'' : ''j''
| ''x'' <math>\lessdot</math> ''y'' : ''j''
| ''y'' <font face="system"><s>></s></font> ''x'' : ''j''
| ''y'' <math>\gtrdot</math> ''x'' : ''j''
|-
|-
| ''j'' sets ''x'' in ''y''.
| ''j'' sets ''x'' in ''y''.
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 { <font face="system"><s><</s></font> , <font face="system"><s>></s></font> }.
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 { <math>\lessdot</math> , <math>\gtrdot</math> }.
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.
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 re-introduce the symbols "<font face="system"><s><</s></font>" and "<font face="system"><s>></s></font>" 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) = 〈 <font face="system"><s><</s></font> , <font face="system"><s>></s></font> 〉, where:
I recall the OG of ''properties and instances'' and re-introduce the symbols "<math>\lessdot</math>" and "<math>\gtrdot</math>" 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) = ? <math>\lessdot</math> , <math>\gtrdot</math> ?, where:
:{|
:{|
| ''x'' <font face="system"><s><</s></font>
| ''x'' <math>\lessdot</math>
| =
| =
| ''x''’s Property
| ''x''’s Property
| Object above ''x'' ,
| Object above ''x'' ,
|-
|-
| ''x'' <font face="system"><s>></s></font>
| ''x'' <math>\gtrdot</math>
| =
| =
| ''x''’s Instance
| ''x''’s Instance
|}
|}
A symbol like "''x'' <font face="system"><s><</s></font>" or "''x'' <font face="system"><s>></s></font>", with extra spaces or dots being optional, is called a ''catenation'', where "''x''" is the ''catenand'' and "<font face="system"><s><</s></font>" or "<font face="system"><s>></s></font>" is the ''catenator''. Due to the fact that "<font face="system"><s><</s></font>" and "<font face="system"><s>></s></font>" indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:
A symbol like "''x'' <math>\lessdot</math>" or "''x'' <math>\gtrdot</math>", with extra spaces or dots being optional, is called a ''catenation'', where "''x''" is the ''catenand'' and "<math>\lessdot</math>" or "<math>\gtrdot</math>" is the ''catenator''. Due to the fact that "<math>\lessdot</math>" and "<math>\gtrdot</math>" indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:
:{|
:{|
| ''x'' <font face="system"><s><</s></font>
| ''x'' <math>\lessdot</math>
| =
| =
| ''x'' is the Instance of what?
| ''x'' is the Instance of what?
| ''x''’s Property ,
| ''x''’s Property ,
|-
|-
| ''x'' <font face="system"><s>></s></font>
| ''x'' <math>\gtrdot</math>
| =
| =
| ''x'' is the Property of what?
| ''x'' is the Property of what?
| ''x''’s Property's Instance
| ''x''’s Property's Instance
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s><</s></font><font face="system"><s>></s></font> ,
| ''x'' <math>\cdot</math> <math>\lessdot</math><math>\gtrdot</math> ,
|-
|-
| ''x''’s Index
| ''x''’s Index
| ''x''’s Instance's Property
| ''x''’s Instance's Property
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s>></s></font><font face="system"><s><</s></font> .
| ''x'' <math>\cdot</math> <math>\gtrdot</math><math>\lessdot</math> .
|}
|}
| ''x'' <math>\cdot</math> ''M''<sub>''OS''</sub>
| ''x'' <math>\cdot</math> ''M''<sub>''OS''</sub>
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s><</s></font><font face="system"><s>></s></font> ,
| ''x'' <math>\cdot</math> <math>\lessdot</math><math>\gtrdot</math> ,
|-
|-
| ''x''’s Index
| ''x''’s Index
| ''x'' <math>\cdot</math> ''N''<sub>''OS''</sub>
| ''x'' <math>\cdot</math> ''N''<sub>''OS''</sub>
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s>></s></font><font face="system"><s><</s></font> .
| ''x'' <math>\cdot</math> <math>\gtrdot</math><math>\lessdot</math> .
|}
|}
| ''x'' <math>\cdot</math> ''M''<sub>''SO''</sub>
| ''x'' <math>\cdot</math> ''M''<sub>''SO''</sub>
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s><</s></font><font face="system"><s>></s></font> ,
| ''x'' <math>\cdot</math> <math>\lessdot</math><math>\gtrdot</math> ,
|-
|-
| For Indices:
| For Indices:
| ''x'' <math>\cdot</math> ''N''<sub>''SO''</sub>
| ''x'' <math>\cdot</math> ''N''<sub>''SO''</sub>
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s>></s></font><font face="system"><s><</s></font> .
| ''x'' <math>\cdot</math> <math>\gtrdot</math><math>\lessdot</math> .
|}
|}
| ''M''<sub>''SO''</sub>
| ''M''<sub>''SO''</sub>
| =
| =
| <font face="system"><s><</s></font><font face="system"><s>></s></font> ,
| <math>\lessdot</math><math>\gtrdot</math> ,
|-
|-
| For Indices:
| For Indices:
| ''N''<sub>''SO''</sub>
| ''N''<sub>''SO''</sub>
| =
| =
| <font face="system"><s>></s></font><font face="system"><s><</s></font> .
| <math>\gtrdot</math><math>\lessdot</math> .
|}
|}
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 ''H''<sub>''OS''</sub> and ''H''<sub>''OI''</sub> can be analyzed as compound relations over the basis supplied by the ''G''<sub>''j''</sub> 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 ''H''<sub>''OS''</sub> and ''H''<sub>''OI''</sub> can be analyzed as compound relations over the basis supplied by the ''G''<sub>''j''</sub> 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 〈 <font face="system"><s><</s></font> , <font face="system"><s>></s></font> 〉, 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 ? <math>\lessdot</math> , <math>\gtrdot</math> ?, 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'' ∪ ''S'' ∪ ''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'' ∪ ''S'' ∪ ''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 ''X''<sub>0</sub> is the initial collection of objects and signs, ''X''<sub>0</sub> = ''W''. The top layer ''Q'' is the relevant class of object qualities, ''Q'' = ''X''<sub>0</sub><font face="system"><s><</s></font> = ''W'' <math>\cdot</math> <font face="system"><s><</s></font>. The bottom layer ''E'' is a suitable set of object exemplars, ''E'' = ''X''<sub>0</sub><font face="system"><s>></s></font> = ''W'' <math>\cdot</math> <font face="system"><s>></s></font>.
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 ''X''<sub>0</sub> is the initial collection of objects and signs, ''X''<sub>0</sub> = ''W''. The top layer ''Q'' is the relevant class of object qualities, ''Q'' = ''X''<sub>0</sub><math>\lessdot</math> = ''W'' <math>\cdot</math> <math>\lessdot</math>. The bottom layer ''E'' is a suitable set of object exemplars, ''E'' = ''X''<sub>0</sub><math>\gtrdot</math> = ''W'' <math>\cdot</math> <math>\gtrdot</math>.
Recall the reading of the staging relations:
Recall the reading of the staging relations:
:{| cellpadding="2" style="text-align:center"
:{| cellpadding="2" style="text-align:center"
| ''h'' : ''x'' <font face="system"><s><</s></font> ''m''
| ''h'' : ''x'' <math>\lessdot</math> ''m''
| ⇔
| ⇔
| ''h'' regards ''x''
| ''h'' regards ''x''
| instance of ''m'' ,
| instance of ''m'' ,
|-
|-
| ''h'' : ''m'' <font face="system"><s>></s></font> ''y''
| ''h'' : ''m'' <math>\gtrdot</math> ''y''
| ⇔
| ⇔
| ''h'' regards ''m''
| ''h'' regards ''m''
| property of ''y'' .
| property of ''y'' .
|-
|-
| ''h'' : ''x'' <font face="system"><s>></s></font> ''n''
| ''h'' : ''x'' <math>\gtrdot</math> ''n''
| ⇔
| ⇔
| ''h'' regards ''x''
| ''h'' regards ''x''
| property of ''n'' ,
| property of ''n'' ,
|-
|-
| ''h'' : ''n'' <font face="system"><s><</s></font> ''y''
| ''h'' : ''n'' <math>\lessdot</math> ''y''
| ⇔
| ⇔
| ''h'' regards ''n''
| ''h'' regards ''n''
| ''M''<sub>''OS''</sub>
| ''M''<sub>''OS''</sub>
| :
| :
| ''x'' <font face="system"><s><</s></font><font face="system"><s>></s></font> ''x''’s Sign,
| ''x'' <math>\lessdot</math><math>\gtrdot</math> ''x''’s Sign,
|-
|-
| For Indices:
| For Indices:
| ''N''<sub>''OS''</sub>
| ''N''<sub>''OS''</sub>
| :
| :
| ''x'' <font face="system"><s>></s></font><font face="system"><s><</s></font> ''x''’s Sign.
| ''x'' <math>\gtrdot</math><math>\lessdot</math> ''x''’s Sign.
|}
|}
| ''x'' <math>\cdot</math> ''M''<sub>''OS''</sub>
| ''x'' <math>\cdot</math> ''M''<sub>''OS''</sub>
| =
| =
| ''x'' <math>\cdot</math> <font face="system"><s><</s></font><sub>''j''</sub> <font face="system"><s>></s></font><sub>''j''</sub> ,
| ''x'' <math>\cdot</math> <math>\lessdot</math><sub>''j''</sub> <math>\gtrdot</math><sub>''j''</sub> ,
|-
|-
| For Indices:
| For Indices:
| ''x'' <math>\cdot</math> ''N''<sub>''OS''</sub>
| ''x'' <math>\cdot</math> ''N''<sub>''OS''</sub>
| =
| =
| x <math>\cdot</math> <font face="system"><s>></s></font><sub>''k''</sub><font face="system"><s><</s></font><sub>''k''</sub> .
| x <math>\cdot</math> <math>\gtrdot</math><sub>''k''</sub><math>\lessdot</math><sub>''k''</sub> .
|}
|}
| ''j'' : ''x'' <font face="system"><s><</s> <s>></s></font> ''y''
| ''j'' : ''x'' <font face="system"><s><</s> <s>></s></font> ''y''
| ⇔
| ⇔
| ''j'' : ''x'' <font face="system"><s><</s></font> ''m''
| ''j'' : ''x'' <math>\lessdot</math> ''m''
| and
| and
| ''j'' : ''m'' <font face="system"><s>></s></font> ''y'',
| ''j'' : ''m'' <math>\gtrdot</math> ''y'',
| for some ''m''
| for some ''m''
| ∈ ''Q'' ,
| ∈ ''Q'' ,
| ''k'' : ''x'' <font face="system"><s>></s> <s><</s></font> ''y''
| ''k'' : ''x'' <font face="system"><s>></s> <s><</s></font> ''y''
| ⇔
| ⇔
| ''k'' : ''x'' <font face="system"><s>></s></font> ''n''
| ''k'' : ''x'' <math>\gtrdot</math> ''n''
| and
| and
| ''k'' : ''n'' <font face="system"><s><</s></font> ''y'',
| ''k'' : ''n'' <math>\lessdot</math> ''y'',
| for some ''n''
| for some ''n''
| ∈ ''E'' .
| ∈ ''E'' .