12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Saturday September 28, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1 (view source)
Revision as of 03:44, 16 September 2010
, 03:44, 16 September 2010→1.3.4.13. Formalization of OF : Objective Levels: change white space formatting in TeX
One way to approach the formalization of an objective genre <math>G\!</math> is through an indexed collection of dyadic relations:
One way to approach the formalization of an objective genre <math>G\!</math> is through an indexed collection of dyadic relations:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq P_j \times Q_j ~ (\forall j \in J)</math>.
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq P_j \times Q_j ~ (\forall j \in J)</math>.
|}
|}
Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, <math>P_j\!</math> and <math>Q_j\!</math> for all <math>j\!</math> in <math>J\!</math>. Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts. Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'', can be defined as follows:
Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, <math>P_j\!</math> and <math>Q_j\!</math> for all <math>j\!</math> in <math>J\!</math>. Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts. Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'', can be defined as follows:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>X_j = P_j \cup Q_j</math>,
| <math>X_j = P_j \cup Q_j</math>,
| <math>P = \textstyle \bigcup_j P_j</math>,
| <math>P = \textstyle \bigcup_j P_j</math>,
The last and perhaps the best way to form an objective genre <math>G\!</math> is to present it as a triadic relation:
The last and perhaps the best way to form an objective genre <math>G\!</math> is to present it as a triadic relation:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q</math>,
| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q</math>,
|}
|}
or:
or:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X</math>.
| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X</math>.
|}
|}
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 genre, 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 genre, and it can be taken as an alternate way of defining the genre.
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \}</math>,
| <math>G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \}</math>,
|}
|}
or:
or:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}</math>.
| <math>G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}</math>.
|}
|}
The standing relation of a genre is denoted by the symbol <math>:\!\lessdot</math>, pronounced ''set-in'', with either of the following two type-markings:
The standing relation of a genre is denoted by the symbol <math>:\!\lessdot</math>, pronounced ''set-in'', with either of the following two type-markings:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>:\!\lessdot ~\subseteq~ J \times P \times Q</math>,
| <math>:\!\lessdot ~\subseteq~ J \times P \times Q</math>,
|-
|-
The propping relation of a genre is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:
The propping relation of a genre is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>:\!\gtrdot ~\subseteq~ J \times Q \times P</math>,
| <math>:\!\gtrdot ~\subseteq~ J \times Q \times P</math>,
|-
|-
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, <math>G = \{ G_j \}\!</math>, 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, <math>G = \{ G_j \}\!</math>, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}</math>.
| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}</math>.
|}
|}
When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X</math>.
When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X</math>.
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}</math>.
| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}</math>.
|}
|}
On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}</math> can be indicated by any one of the following equivalent expressions:
On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}</math> can be indicated by any one of the following equivalent expressions:
{| align="center" cellpadding="10" style="text-align:center; width:75%"
{| align="center" cellpadding="8" style="text-align:center; width:75%"
| <math>G : x \lessdot y</math>,
| <math>G : x \lessdot y</math>,
| <math>x \lessdot_G y</math>,
| <math>x \lessdot_G y</math>,
At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects. To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>(j, x, y) \in ~ :\!\lessdot</math>, or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>(j, y, x) \in ~ :\!\gtrdot</math>, all of the following notations are equivalent:
At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects. To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>(j, x, y) \in ~ :\!\lessdot</math>, or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>(j, y, x) \in ~ :\!\gtrdot</math>, all of the following notations are equivalent:
{| align="center" cellpadding="10" style="text-align:center; width:75%"
{| align="center" cellpadding="8" style="text-align:center; width:75%"
| <math>j : x \lessdot y</math>,
| <math>j : x \lessdot y</math>,
| <math>x \lessdot_j y</math>,
| <math>x \lessdot_j y</math>,
<br>
<br>
{| align="center" border="1" cellpadding="4" cellspacing="2" style="text-align:left; width:100%"
{| align="center" border="1" cellpadding="4" cellspacing="2" style="text-align:left; width:100%"
|
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:ghostwhite; text-align:left; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:#f0f0ff; text-align:left; width:100%"
|-
|-
| width="50%" | <math>j : x \lessdot y</math>
| width="50%" | <math>j : x \lessdot y</math>
|
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:left; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:left; width:100%"
| width="50%" | <math>j\ \text{sets}\ x\ \text{in}\ y.</math>
| width="50%" | <math>j ~\text{sets}~ x ~\text{in}~ y.</math>
| width="50%" | <math>j\ \text{sets}\ y\ \text{on}\ x.</math>
| width="50%" | <math>j ~\text{sets}~ y ~\text{on}~ x.</math>
|-
|-
| <math>j\ \text{makes}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{makes}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{makes}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{makes}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{thinks}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{thinks}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{thinks}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{thinks}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{attests}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{attests}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{attests}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{attests}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{appoints}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{appoints}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{appoints}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{appoints}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{witnesses}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{witnesses}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{witnesses}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{witnesses}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{interprets}\ x\ \text{an instance of}\ y.</math>
| <math>j ~\text{interprets}~ x ~\text{an instance of}~ y.</math>
| <math>j\ \text{interprets}\ y\ \text{a property of}\ x.</math>
| <math>j ~\text{interprets}~ y ~\text{a property of}~ x.</math>
|-
|-
| <math>j\ \text{contributes}\ x\ \text{to}\ y.</math>
| <math>j ~\text{contributes}~ x ~\text{to}~ y.</math>
| <math>j\ \text{attributes}\ y\ \text{to}\ x.</math>
| <math>j ~\text{attributes}~ y ~\text{to}~ x.</math>
|-
|-
| <math>j\ \text{determines}\ x\ \text{an example of}\ y.</math>
| <math>j ~\text{determines}~ x ~\text{an example of}~ y.</math>
| <math>j\ \text{determines}\ y\ \text{a quality of}\ x.</math>
| <math>j ~\text{determines}~ y ~\text{a quality of}~ x.</math>
|-
|-
| <math>j\ \text{evaluates}\ x\ \text{an example of}\ y.</math>
| <math>j ~\text{evaluates}~ x ~\text{an example of}~ y.</math>
| <math>j\ \text{evaluates}\ y\ \text{a quality of}\ x.</math>
| <math>j ~\text{evaluates}~ y ~\text{a quality of}~ x.</math>
|-
|-
| <math>j\ \text{proposes}\ x\ \text{an example of}\ y.</math>
| <math>j ~\text{proposes}~ x ~\text{an example of}~ y.</math>
| <math>j\ \text{proposes}\ y\ \text{a quality of}\ x.</math>
| <math>j ~\text{proposes}~ y ~\text{a quality of}~ x.</math>
|-
|-
| <math>j\ \text{musters}\ x\ \text{under}\ y.</math>
| <math>j ~\text{musters}~ x ~\text{under}~ y.</math>
| <math>j\ \text{marshals}\ y\ \text{over}\ x.</math>
| <math>j ~\text{marshals}~ y ~\text{over}~ x.</math>
|-
|-
| <math>j\ \text{indites}\ x\ \text{among}\ y.</math>
| <math>j ~\text{indites}~ x ~\text{among}~ y.</math>
| <math>j\ \text{ascribes}\ y\ \text{about}\ x.</math>
| <math>j ~\text{ascribes}~ y ~\text{about}~ x.</math>
|-
|-
| <math>j\ \text{imputes}\ x\ \text{among}\ y.</math>
| <math>j ~\text{imputes}~ x ~\text{among}~ y.</math>
| <math>j\ \text{imputes}\ y\ \text{about}\ x.</math>
| <math>j ~\text{imputes}~ y ~\text{about}~ x.</math>
|-
|-
| <math>j\ \text{judges}\ x\ \text{beneath}\ y.</math>
| <math>j ~\text{judges}~ x ~\text{beneath}~ y.</math>
| <math>j\ \text{judges}\ y\ \text{beyond}\ x.</math>
| <math>j ~\text{judges}~ y ~\text{beyond}~ x.</math>
|-
|-
| <math>j\ \text{finds}\ x\ \text{preceding}\ y.</math>
| <math>j ~\text{finds}~ x ~\text{preceding}~ y.</math>
| <math>j\ \text{finds}\ y\ \text{succeeding}\ x.</math>
| <math>j ~\text{finds}~ y ~\text{succeeding}~ x.</math>
|-
|-
| <math>j\ \text{poses}\ x\ \text{before}\ y.</math>
| <math>j ~\text{poses}~ x ~\text{before}~ y.</math>
| <math>j\ \text{poses}\ y\ \text{after}\ x.</math>
| <math>j ~\text{poses}~ y ~\text{after}~ x.</math>
|-
|-
| <math>j\ \text{forms}\ x\ \text{below}\ y.</math>
| <math>j ~\text{forms}~ x ~\text{below}~ y.</math>
| <math>j\ \text{forms}\ y\ \text{above}\ x.</math>
| <math>j ~\text{forms}~ y ~\text{above}~ x.</math>
|}
|}
|}
|}
<br>
<br>
