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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 15:28, 2 May 2012
, 15:28, 2 May 2012→6.10. Higher Order Sign Relations : Examples: renumber tables
The intent of this succession, as interpreted in FL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> denotes or refers to <math>{}^{\langle} x {}^{\rangle},</math> which denotes or refers to <math>x.\!</math> Moreover, its computational realization, as implemented in CL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> addresses or evaluates to <math>{}^{\langle} x {}^{\rangle},</math> which addresses or evaluates to <math>x.\!</math>
The intent of this succession, as interpreted in FL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> denotes or refers to <math>{}^{\langle} x {}^{\rangle},</math> which denotes or refers to <math>x.\!</math> Moreover, its computational realization, as implemented in CL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> addresses or evaluates to <math>{}^{\langle} x {}^{\rangle},</math> which addresses or evaluates to <math>x.\!</math>
The designations ''higher order'' and ''lower order'' are attributed to signs in a casual, local, and transitory way. At this point they signify nothing beyond the occurrence in a sign relation of a pair of triples having the form shown in Table 38.
The designations ''higher order'' and ''lower order'' are attributed to signs in a casual, local, and transitory way. At this point they signify nothing beyond the occurrence in a sign relation of a pair of triples having the form shown in Table 37.
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Sign Relation Containing a Higher Order Sign}\!</math>
|+ style="height:30px" | <math>\text{Table 37.} ~~ \text{Sign Relation Containing a Higher Order Sign}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
This is all it takes to make <math>\text{s}\!</math> a lower order sign and <math>\text{t}\!</math> a higher order sign in relation to each other at the moments in question. Whether a global ordering of a more generally justifiable sort can be constructed from an arbitrary series of such purely local impressions is another matter altogether.
This is all it takes to make <math>\text{s}\!</math> a lower order sign and <math>\text{t}\!</math> a higher order sign in relation to each other at the moments in question. Whether a global ordering of a more generally justifiable sort can be constructed from an arbitrary series of such purely local impressions is another matter altogether.
Nevertheless, the preceding observations do show a way to give a definition of higher order signs that does not depend on the peculiarities of quotational devices. For example, consider the previously described sequence of increasingly higher order signs stemming from the object <math>x.\!</math> Table 39.1 shows how this succession can be transcribed into the form of a sign relation. But this is formally no different from the sign relation suggested in Table 39.2, one whose individual signs are not constructed in any special way. Both of these representations of sign relations, if continued in a consistent manner, would have the same abstract structure. If one of them is higher order then so is the other, at least, if the attributes of order are meant to have any formally invariant meaning.
Nevertheless, the preceding observations do show a way to give a definition of higher order signs that does not depend on the peculiarities of quotational devices. For example, consider the previously described sequence of increasingly higher order signs stemming from the object <math>x.\!</math> Table 38 shows how this succession can be transcribed into the form of a sign relation. But this is formally no different from the sign relation suggested in Table 39, one whose individual signs are not constructed in any special way. Both of these representations of sign relations, if continued in a consistent manner, would have the same abstract structure. If one of them is higher order then so is the other, at least, if the attributes of order are meant to have any formally invariant meaning.
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<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 39.1} ~~ \text{Sign Relation for a Succession of Higher Order Signs (1)}\!</math>
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (1)}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 39.2} ~~ \text{Sign Relation for a Succession of Higher Order Signs (2)}\!</math>
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Sign Relation for a Succession of Higher Order Signs (2)}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>