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Revision as of 18:57, 16 December 2008
, 18:57, 16 December 2008sub [\lessdot / <font face="system">:<s><</s></font>], sub [\gtrdot / <font face="system">:<s>></s></font>]
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:
The standing relation of an OG is denoted by the symbol "<font face="system">:<s><</s></font>", pronounced ''set-in'', with either of the following two type-markings:
The standing relation of an OG is denoted by the symbol "<math>:\!\lessdot</math>", pronounced ''set-in'', with either of the following two type-markings:
: <font face="system">:<s><</s></font> ⊆ ''J'' × ''P'' × ''Q'',
: <math>:\!\lessdot</math> ⊆ ''J'' × ''P'' × ''Q'',
: <font face="system">:<s><</s></font> ⊆ ''J'' × ''X'' × ''X''.
: <math>:\!\lessdot</math> ⊆ ''J'' × ''X'' × ''X''.
The propping relation of an OG is denoted by the symbol "<font face="system">:<s>></s></font>", pronounced ''set-on'', with either of the following two type-markings:
The propping relation of an OG is denoted by the symbol "<math>:\!\gtrdot</math>", pronounced ''set-on'', with either of the following two type-markings:
: <font face="system">:<s>></s></font> ⊆ ''J'' × ''Q'' × ''P'',
: <math>:\!\gtrdot</math> ⊆ ''J'' × ''Q'' × ''P'',
: <font face="system">:<s>></s></font> ⊆ ''J'' × ''X'' × ''X''.
: <math>:\!\gtrdot</math> ⊆ ''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'' = {''G''<sub>''j''</sub>}, 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'' = {''G''<sub>''j''</sub>}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.
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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, in symbols, (''j'', ''x'', ''y'') ∈ <font face="system">:<s><</s></font>, 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, in symbols, (''j'', ''y'', ''x'') ∈ <font face="system">:<s>></s></font>, 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, in symbols, (''j'', ''x'', ''y'') ∈ <math>:\!\lessdot</math>, 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, in symbols, (''j'', ''y'', ''x'') ∈ <math>:\!\gtrdot</math>, all of the following notations are equivalent:
:{| style="text-align:left; width:90%"
:{| style="text-align:left; width:90%"
* 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 the conjunction "''S'' and ''T''" is 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.
* 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 the conjunction "''S'' and ''T''" is 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''(<font face="system">:<s><</s></font>).
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''(<math>:\!\lessdot</math>).
=====1.3.4.14. Application of OF : Generic Level=====
=====1.3.4.14. Application of OF : Generic Level=====
