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<div id="x6d28eafe58594dee933fc9f884e1194c" style="tab-interval:.5in;word-wrap:break-word">
<p class="MsoNormalCxSpFirst" style="line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><i>Yes, Joseph, I agree! The alteration of the sign indicates whether information (entropy) or redundancy is generated. </i></p><p class="MsoNormalCxSpFirst" style="line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">
</p><div><span style="font-family: Arial; font-size: 12pt;"><br /></span></div><div>
<span style="font-family: Arial; font-size: 12pt;">Leydesdorff, L., Petersen, A. M., & Ivanova, I. (2017). Self-organization of meaning and the reflexive communication of information. </span>
<i>
<span style="font-family: Arial; font-size: 12pt;">Social Science Information, 56</span>
</i>
<span style="font-family: Arial; font-size: 12pt;">(1), 4-27. doi: 10.1177/0539018416675074 . </span></div><div><font face="Arial"><span style="font-size: 16px;">We formulated on pp. , as follows: I owe this to my coauthors:</span></font></div><p class="MsoNormalCxSpFirst" style="line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><i><br /></i></p><p class="MsoNormalCxSpFirst" style="line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><i>Redundancy in three and four
dimensions<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></i></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">For the three-dimensional case and
using Figure 6, one can define, in addition to the two-dimensional values of <i>Υ</i> (in Eq. 3), a three-dimensional value including the redundancies
(in the respective overlaps) as follows:<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:0.5in;mso-add-space:auto;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:em8e097672-c165-404d-8fdd-52f2a58b3bf6@pc2014" width="307" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> (5) <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<table class="MsoTableGrid" border="0" cellspacing="0" cellpadding="0" style="border-width: initial; border-style: none; border-color: initial;" xmlns="http://www.w3.org/TR/REC-html40">
<tbody><tr>
<td width="319" valign="top" style="width:239.4pt;padding:0in 5.4pt 0in 5.4pt ;">
<p class="MsoNormalCxSpMiddle" align="center" style="text-align:center;line-height:115%;tab-stops:right 459pt ;"><img src="cid:emd05393f4-d071-4869-ac9e-abc0878d8ee3@pc2014" width="228" height="220" v:shapes="Picture_x0020_305" xmlns:v="urn:schemas-microsoft-com:vml" /><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
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<td width="319" valign="top" style="width:239.4pt;padding:0in 5.4pt 0in 5.4pt ;">
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;"><img src="cid:em69ed284a-670b-4f2d-9748-75263f723b58@pc2014" width="245" height="261" v:shapes="Picture_x0020_303" xmlns:v="urn:schemas-microsoft-com:vml" /><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
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<p class="MsoNormalCxSpMiddle" align="center" style="text-align:center;line-height:115%;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" align="center" style="text-align:center;line-height:115%;" xmlns="http://www.w3.org/TR/REC-html40"><b>Figure 6</b>: Overlapping uncertainties in three variables <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,
and <i>x</i><sub>3</sub>: <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" align="center" style="text-align:center;line-height:115%;" xmlns="http://www.w3.org/TR/REC-html40">two configurations with
opposite sign of <i>T</i><sub>123</sub>.<i><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></i></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">Using
information theory, however, one would make the following corrections for double
counting in the overlaps (by subtracting; see Figure 6):<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="text-indent:0.5in;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:em1ae5d112-70d9-44dc-8115-48faec57ea82@pc2014" width="311" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> (6)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">It
follows that the difference between the uncertainty in the historical system (<i>H</i><sub>123</sub> in Eq. 6) and the system
of expectations (<span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:ema7e0e571-9061-4177-a2ce-88fe44aee96d@pc2014" width="27" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> in Eq.
5) is: <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:0.5in;mso-add-space:auto;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%;"><img src="cid:em00569763-83d9-4cd1-9e6f-b064d104b561@pc2014" width="242" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:0.5in;mso-add-space:auto;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:em3bb9a840-4220-47bf-9b9e-ffa1dbfe7a00@pc2014" width="226" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> (7)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">Furthermore,
the mutual information in three dimensions can be derived in information theory
using the Shannon formulas (e.g., Abramson, 1963, at p. 129; McGill, 1954; Yeung, 2008)
as: <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="text-indent:0.5in;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:em5e4250dc-9099-46ae-aa06-ba4c9862bbec@pc2014" width="320" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> (8)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">Using <i>Υ</i>-values
instead of <i>H</i>-values for the joint
entropies in Eq. 8, one obtains the mutual redundancy in three dimensions as
follows: <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:0.5in;mso-add-space:auto;line-height:115%;page-break-after:avoid;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%;"><img src="cid:em825aee26-160c-4c5b-b958-d7f6b15a9d42@pc2014" width="460" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:76.5pt;mso-add-space:auto;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%;"><img src="cid:embf318d98-d909-40d3-a6dc-7afbe55e7f50@pc2014" width="207" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:63pt;mso-add-space:auto;line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 4pt;"><img src="cid:eme5910d60-2cc6-488c-8285-e3ac55a82bbf@pc2014" width="44" height="20" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span> (9)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">In the
three-dimensional case, the mutual redundancy is thus equal to the mutual
information in three dimensions. Furthermore, Leydesdorff & Ivanova (2014,
at p. 392) show that in the case of four dimensions <i>R</i><sub>1234</sub> = – <i>T</i><sub>1234</sub>.
The sign of the mutual redundancy alternates with the number of dimensions.
This corrects for the otherwise inexplicable sign changes in the mutual
information with increasing dimensionality. This sign change of the mutual
information with dimensions is a well-known problem in information theory, but
beyond the scope of the present study.<a href="#_ftn1" name="_ftnref1" title=""><span class="MsoFootnoteReference"><span class="MsoFootnoteReference"><span style="font-size: 12pt; line-height: 107%; color: black;">[1]</span></span></span></a>
In other words, mutual redundancy is a consistent measure of negative entropy,
while mutual information <o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40">is
not, because of its sign changes with the dimensionality. We prove this claim
in the next section by generalizing the formulation.<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
<p class="MsoNormalCxSpMiddle" style="line-height:115%;tab-stops:right 459pt ;" xmlns="http://www.w3.org/TR/REC-html40"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></p>
<div xmlns="http://www.w3.org/TR/REC-html40"><br clear="all" />
<hr align="left" size="1" width="33%" />
<div id="ftn1">
<p class="MsoNormalCxSpFirst" style="line-height:normal;tab-stops:right 459pt ;"><a href="#_ftnref1" name="_ftn1" title=""><span class="MsoFootnoteReference"><span class="MsoFootnoteReference"><span style="font-size: 12pt; line-height: 107%;">[1]</span></span></span></a> <span style="font-size: 10pt;">Krippendorff (2009b, at p. 670) provided a general <i>notation</i> for this alteration with
changing dimensionality—but with the opposite sign (which further complicate the
issue; cf. Leydesdorff, 2010: 68)—as follows:<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="line-height:normal;tab-stops:right 459pt ;"><span style="font-size: 10pt;"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:0.5in;mso-add-space:auto;line-height:normal;tab-stops:right 459pt ;"><span style="font-size: 12pt; line-height: 107%; position: relative; top: 3pt;"><img src="cid:em0a8712dc-1c1b-42be-9817-f165fc412181@pc2014" width="178" height="18" v:shapes="_x0000_i1025" xmlns:v="urn:schemas-microsoft-com:vml" /></span><span style="font-size: 10pt;"> (9)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></span></p>
<p class="MsoNormalCxSpLast" style="line-height:normal;tab-stops:right 459pt ;"><span style="font-size: 10pt;"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"> </o:p></span></p>
<p class="MsoFootnoteText"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></p>
</div>
</div></div><div>
<span style="font-family: "Times New Roman", serif; font-size: 10pt;">In this equation, Γ is the set of variables
of which </span><i style="font-family: "Times New Roman", serif; font-size: 10pt;">X</i><span style="font-family: "Times New Roman", serif; font-size: 10pt;"> is a subset, and </span><i style="font-family: "Times New Roman", serif; font-size: 10pt;">H</i><span style="font-family: "Times New Roman", serif; font-size: 10pt;">(</span><i style="font-family: "Times New Roman", serif; font-size: 10pt;">X</i><span style="font-family: "Times New Roman", serif; font-size: 10pt;">)
is the uncertainty of the distribution; |Γ| is the cardinality of Γ, and |</span><i style="font-family: "Times New Roman", serif; font-size: 10pt;">X</i><span style="font-family: "Times New Roman", serif; font-size: 10pt;">| the cardinality of </span><i style="font-family: "Times New Roman", serif; font-size: 10pt;">X.</i><i><span style="font-family: "Times New Roman"; font-size: 12pt;">Generalization</span>
</i>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">Eq. 8 can be rewritten as follows: </span>
</div>
<div> </div>
<div>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> = </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>1</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>2</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>3</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>12</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>13</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>23</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">(8)</span>
</div>
<div>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> = [(</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>1</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>2</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>12</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">) + (</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>1</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>3</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>13</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">) + (</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>2</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>3</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>23</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">)] + [</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>1</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>2</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>3</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">]</span>
</div>
<div>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> = [</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>12</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>13</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> + </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>23</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">] + [</span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>1</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>2</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> - </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">H</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>3</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">]</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;">(10)</span>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">The second bracket in Eq. 10 makes a negative contribution, because of the subadditivity of the entropy: </span>
<img src="cid:emdb5fc684-e273-4914-80ec-441c546804ed@pc2014" width="169" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">, which holds for any dimension </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">n≥ 2</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">. The terms in the first bracket of Eq. 10 are strictly positive. As noted, the sign of the resulting value of </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">T</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 8pt;">
<sub>123</sub>
</span>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> depends on the empirical configuration (as indicated by the two configurations in Figure 6).</span>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">It follows (inductively) that for any given dimension </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">n</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">, one can formulate combinations of mutual informations corresponding to </span>
<img src="cid:em371126a3-b81c-4bd3-a4e7-7da2537ddc77@pc2014" width="163" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> that are by definition positive (or zero in the null case of complete independence). For example (up to four dimensions) as follows:</span>
</div>
<div> </div>
<div>
<img src="cid:em830200e7-597c-4147-a9a3-57cb31aa02dc@pc2014" width="232" height="22" />
</div>
<div>
<img src="cid:em647c60fb-bd51-4d9e-b9dc-bca15664333b@pc2014" width="321" height="23" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">(11)</span>
</div>
<div>
<img src="cid:em09300107-5e00-4c1a-9e8e-1326b973709e@pc2014" width="426" height="23" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">where the sums on the right-hand side are over the </span>
<img src="cid:emad0b1ea3-9771-48d5-b172-70fbdbeaa287@pc2014" width="22" height="24" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> permutations of the indices. This relation can be extended for general </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">n</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> as,</span>
</div>
<div> </div>
<div>
<img src="cid:emb798e0fd-b681-4571-a7be-22730b89d70e@pc2014" width="208" height="21" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
</div>
<div>
<img src="cid:em5f6f61a0-967e-4ef1-8f35-6963f77e9b34@pc2014" width="624" height="65" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">(12)</span>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">where the last term on the right-hand side is equal to </span>
<img src="cid:emd1ea5367-435d-4780-81a3-37909038815e@pc2014" width="86" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">. Returning to the relation between </span>
<img src="cid:embe3cbad6-5735-4a53-b7d4-66feb23ccf8f@pc2014" width="23" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> and </span>
<img src="cid:em0931ecd5-8563-4385-89fa-9c98d2e4f5e8@pc2014" width="22" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">, it now follows instructively that: </span>
</div>
<div> </div>
<div>
<img src="cid:em0e867bb6-8644-4665-83f4-5b6eab93a473@pc2014" width="81" height="20" />
</div>
<div>
<img src="cid:emdc636ba9-9ac1-4d14-9cf2-63533bdb65dc@pc2014" width="194" height="21" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">(13)</span>
</div>
<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">and the analogous relations for </span>
<img src="cid:em694f0e4c-6b2b-45e4-a64e-735a9019bb71@pc2014" width="30" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> and </span>
<img src="cid:emb4ae3dc7-b817-454d-89af-d0d1a0e2b449@pc2014" width="37" height="20" />
<span style="font-family: "Times New Roman"; font-size: 12pt;"> follow in the same way from Eq. (12). More generally, in the case of more than two dimension, </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">n > 2</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">:</span>
</div>
<div> </div>
<div>
<img src="cid:emd6d13e0a-fa46-49de-a1d6-bf6c70df0f1f@pc2014" width="334" height="20" />
<b>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
</b>
</div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> </span>
<img src="cid:em49a88861-2fef-46e1-93de-1ed111cc7652@pc2014" width="482" height="33" />
<span style="font-family: "Times New Roman"; font-size: 12pt;">(14)</span>
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<div> </div>
<div>
<span style="font-family: "Times New Roman"; font-size: 12pt;">The left-bracketed term of Eq. 14 is necessarily negative entropy (because of the subadditivity of the entropy), while the configuration of the remaining mutual information relations contribute a second term on the right which is positive (see the set of Equations 11 above). In other words, we model here the generation of redundancy on the one side versus the historical process of uncertainty generation in relating on the other, as an empirical balance in a system that operates with more than two codes (e.g., alphabets; Abramson, 1963, pp. 127 ff.). When the resulting </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">R</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> is negative, self-organization prevails over organization in the configuration under study, whereas a positive </span>
<i>
<span style="font-family: "Times New Roman"; font-size: 12pt;">R</span>
</i>
<span style="font-family: "Times New Roman"; font-size: 12pt;"> indicates conversely a predominance of organization over self-organization as two different subdynamics. </span>
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<p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><b><font face="Times New Roman" style="font-size: 9pt;" size="1">_______________</font></b></p><p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><b><font face="Times New Roman" style="font-size: 9pt;" size="1"><a href="https://www.springer.com/gp/book/9783030599508"></a>Loet Leydesdorff</font></b></p><p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><b><font face="Times New Roman" style="font-size: 9pt;" size="1"><br /></font></b></p><div id="x37dfff32a36b4410ad8e891fea6bb1b8"><p class="MsoNormal" xmlns="http://www.w3.org/TR/REC-html40" style="line-height:normal;"><b style=""><font style="font-size: 9pt;" face="Times New Roman" size="1"><a href="https://link.springer.com/book/10.1007/978-3-030-59951-5" style="">"The Evolutionary Dynamics of Discusive Knowledge"</a>(Open Access)</font></b></p></div><div id="x37dfff32a36b4410ad8e891fea6bb1b8"></div><p class="MsoNormal" style="line-height: normal;" xmlns="http://www.w3.org/TR/REC-html40"><font face="Times New Roman" style="font-size: 8pt;" size="1">Professor emeritus, University of Amsterdam </font></p><p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><font style="font-size: 8pt;" face="Times New Roman" size="1">
Amsterdam School of Communication Research (ASCoR)<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></font></p>
<p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><font style="font-size: 8pt;" face="Times New Roman" size="1"><a href="mailto:loet@leydesdorff.net" title="mailto:loet@leydesdorff.net" style="">loet@leydesdorff.net </a>; <a href="http://www.leydesdorff.net/" title="http://www.leydesdorff.net/" style="">http://www.leydesdorff.net/</a>
<o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></font></p><p class="MsoNormal" style="line-height:normal;" xmlns="http://www.w3.org/TR/REC-html40"><a href="http://scholar.google.com/citations?user=ych9gNYAAAAJ&hl=en" style="background-color: rgba(0, 0, 0, 0);"><font style="font-size: 8pt;" face="Times New Roman" size="1">http://scholar.google.com/citations?user=ych9gNYAAAAJ&hl=en</font></a></p></div><div id="x37dfff32a36b4410ad8e891fea6bb1b8"><span><font style="font-size: 8pt;" face="Times New Roman" size="1">
</font></span><p class="MsoNormal" xmlns="http://www.w3.org/TR/REC-html40" style="line-height:normal;background-color:rgba(0,0,0,0);"><font style="font-size: 8pt;"><font style="font-size: 8pt;" face="Times New Roman" size="1">ORCID: <a href="http://orcid.org/0000-0002-7835-3098" style="">http://orcid.org/0000-0002-7835-3098</a>; </font><span style="font-size: 10pt;"><o:p xmlns:o="urn:schemas-microsoft-com:office:office"></o:p></span></font></p><span>
</span><p class="MsoNormal" xmlns="http://www.w3.org/TR/REC-html40" style="line-height:normal;background-color:rgba(0,0,0,0);"></p></div>
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<div>------ Original Message ------</div>
<div>From: "<a href="mailto:joe.brenner@bluewin.ch">joe.brenner@bluewin.ch</a>" <<a href="mailto:joe.brenner@bluewin.ch">joe.brenner@bluewin.ch</a>></div>
<div>To: <a href="mailto:christophe.menant@hotmail.fr">christophe.menant@hotmail.fr</a></div>
<div>Cc: <a href="mailto:fis@listas.unizar.es">fis@listas.unizar.es</a>; <a href="mailto:deacon@berkeley.edu">deacon@berkeley.edu</a></div>
<div>Sent: 3/13/2022 11:51:55 AM</div>
<div>Subject: Re: [Fis] How Molecules Became Signs. One Way Out</div><div><br /></div>
<div id="x8d99a0f091c642e"><blockquote cite="418905604.6353.1647168715505@bluewin.ch" type="cite" class="cite2">
<font size="3">Dear Friends and Colleagues,</font><div><font size="3"><br /></font></div><div><font size="3">There is one way out of the dilemma which has not been directly refuted. When Christophe and others ask if can we talk of information in an a-biotic or pre-biotic world, if information means something actual, present, the answer is no. If one asks instead did </font><span style="font-size: medium;">the </span><i style="font-size: medium;">potential </i><span style="font-size: medium;">for information as meaning exist, the answer for me is yes. </span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">An additional statement must be added, otherwise the above is no more than a trivial tautology. It is that information is defined in the process of its emergence from some energetic ground. Since no process, nothing <i>in process</i> is complete, complete and <b>incomplete </b>parts of the process are present simultaneously and dynamically. The locus of this "nascent" information is the detailed physico-chemical structure of the living entities involved and their non-living constituents. These or parts of them move from actual to potential and <i>vice versa </i>and this movement is what ultimately defines their meaning.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">One reading of the above is that I have given a new interpretation of the nature of a sign. I would gladly accept this, provided it can be subsequently decided whether or not the concept of sign adds further information. It may not.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Thank you and best wishes,</span></div><div><span style="font-size: medium;">Joseph</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><blockquote style="margin-right: 0px; margin-left:15px;">----Message d'origine----<br />De : <a href="mailto:christophe.menant@hotmail.fr">christophe.menant@hotmail.fr</a><br />Date : 13/03/2022 - 00:02 (CEST)<br />À : <a href="mailto:deacon@berkeley.edu">deacon@berkeley.edu</a>, <a href="mailto:fis@listas.unizar.es">fis@listas.unizar.es</a><br />Objet : [Fis] How Molecules Became Signs<br /><br /><style type="text/css"><!--#x8d99a0f091c642e p
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<span style="font-family: "Times New Roman" , serif;font-size: 12.0pt;">Dear Friends,</span>
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<span style="font-family: Calibri , sans-serif;font-size: 14.6667px;background-color: rgb(255,255,255);display: inline;"><span lang="EN-US" style="font-size: 12.0pt;font-family: "Times New Roman" , serif;border: none windowtext 1.0pt;padding: 0.0cm;">that war is a horrible drama for innocent civilians. And I agree with your comments.<br /> Our FIS discussions cannot bring much help to that human drama, but our tentative analysis of what is “information” may somehow lead to a better understanding of human motivations guide behaviors. <br /> Let me add another comment to Terry’s work. <br /></span><b><span lang="EN-US" style="font-size: 12.0pt;font-family: "Times New Roman" , serif;color: rgb(42,42,42);"><br /></span></b><a><span lang="EN-US" style="font-size: 12.0pt;font-family: "Times New Roman" , serif;color: rgb(42,42,42);">Dear Terry</span></a><span lang="EN-US" style="font-size: 12.0pt;font-family: "Times New Roman" , serif;color: rgb(42,42,42);">, <br /> In addition to my Feb 23 post I would appreciate some information on parts of your paper I may not have understood that well. <br /> Regarding the concept of information, you consider that information in a pragmatic-functional sense can be understood in terms of molecular evolution. <br /> This brings to consider that “information” is present in an a-biotic or pre-biotic world (a purely molecular world). Such pre-biotic world has existed before the emergence of life in our universe. But can we talk of information and meaning, of signs, in such a world? How should they be understood in a purely material world devoid of living entities? <br /> As said,<span> </span>it would be nice if you could clarify these points by making available definitions for information, meaning and sign in such an a-biotic world. This would allow a better understanding of your starting point.<br /> Also, I do not see that well using the Peircean term of “Interpretant” for an inert world. We know that the Interpretant (the meaning) needs an Interpreter (the meaning generator). So introducing Interpretants in your paper also brings to introduce Interpreters in an inert world. Your sentence “In Peircean terms, this amounts to asking what sort of molecular system is competent to produce the Interpretants” is equivalent to: “what sort of molecular system is competent to generate meanings”. I’m not sure that meaning generation by a molecular system <span style="background-color: rgb(255,255,255);display: inline;"> in a purely material and inert world </span>can be clearly understood by today science or philosophy. And I do not remember Peirce theory of sign being about inert matter. Could you tel us more about your position on these subject? <br /> Thanks again for your time <br /> Christophe</span><br /></span>
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