<!DOCTYPE html><HTML lang="en"> <head> <meta charset="utf-8"> <link href="../../TheosStyle.css" rel="stylesheet" type="text/css"><link rel="shortcut icon" href="../../TT_ICO.png" /> <title>Note: Christian Tractatus (Theo Todman's Web Page)</title> </head><body> <a name="Top"></a> <h1>Theo Todman's Web Page - Notes Pages</h1><hr><h2>Christian Tractatus</h2><p class = "Centered">(Text as at 12/08/2007 10:17:46)<hr> <P><FONT COLOR = "0000FF"><B>Therefore, scaling factors are required to ensure that no individual is left out. </B><ol type="1"><li>Let us call the scaling factor the utilitarian metric, Mij and let us assume that the population (including future generations) capable of being affected by the potential actions of a particular individual <I>i</I> is <FONT FACE="Symbol">W</FONT><FONT FACE="Arial">, that the quantity or value of a quantifiable good G available to <I>i</I> is X, and that the utilitarian metric applicable to <I>i</I> is M<FONT FACE="Arial" SIZE=2>i</FONT><FONT FACE="Arial">. Then, the total utility, U(X, <FONT FACE="Symbol">W</FONT><FONT FACE="Arial">), resulting from i's apportionment of X is:-<BR>U(X, <FONT FACE="Symbol">W</FONT><FONT FACE="Arial">) = </FONT><FONT FACE="Symbol">S</FONT><FONT FACE="Symbol" SIZE=2>W</FONT><FONT FACE="Arial">X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial">M</FONT><FONT FACE="Arial" SIZE=2>ij</FONT><FONT FACE="Arial">, where </FONT><FONT FACE="Symbol">S</FONT><FONT FACE="Arial">X</FONT><FONT FACE="Arial" SIZE=2>j </FONT><FONT FACE="Arial">= X and </FONT><FONT FACE="Symbol">S</FONT><FONT FACE="Symbol" SIZE=2>W</FONT><FONT FACE="Arial"> sums over all j </FONT><FONT FACE="Symbol">e W</FONT><FONT FACE="Arial">.</li><li>What form does the utilitarian metric take? We can set some bounds as follows:- <ul type="disc"><li></FONT><FONT FACE="Symbol">- &lt;</FONT><FONT FACE="Arial"> </FONT><FONT FACE="Symbol">S</FONT><FONT FACE="Symbol" SIZE=2>W</FONT><FONT FACE="Arial">X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial">M</FONT><FONT FACE="Arial" SIZE=2>ij</FONT><FONT FACE="Arial"> </FONT><FONT FACE="Symbol">&lt; + </FONT><FONT FACE="Arial"> , whenever </FONT><FONT FACE="Symbol"> - &lt;</FONT><FONT FACE="Arial"> X </FONT><FONT FACE="Symbol">&lt; + .</FONT><FONT FACE="Arial"> </li><li>M will vary with the good X and the population </FONT><FONT FACE="Symbol">W</FONT><FONT FACE="Arial"> under consideration. Some goods are optional luxuries while others are essentials. Also, the quantity of good G already possessed by the individuals j </FONT><FONT FACE="Symbol">e W</FONT><FONT FACE="Arial"> will affect the marginal utility of a further X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial"> of G to j.</li><li>Because of the decreasing marginal utility of any good, in all cases, U(X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial">) / X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial"> </FONT><FONT FACE="Symbol">&reg;</FONT><FONT FACE="Arial"> 0 as X</FONT><FONT FACE="Arial" SIZE=2>j </FONT><FONT FACE="Symbol">&reg;</FONT><FONT FACE="Arial"> </FONT><FONT FACE="Symbol"></FONT><FONT FACE="Arial">.</li><li>In the case of any essential good G of utility X, of which the X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial"> represent the j's total holdings, U(X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial">) </FONT><FONT FACE="Symbol">&reg; - </FONT><FONT FACE="Arial">as X</FONT><FONT FACE="Arial" SIZE=2>j </FONT><FONT FACE="Symbol">&reg;</FONT><FONT FACE="Arial"> 0.</li><li>A correction will be needed to prevent U(X</FONT><FONT FACE="Arial" SIZE=2>j</FONT><FONT FACE="Arial">=0) </FONT><FONT FACE="Symbol">= -</FONT><FONT FACE="Arial"> for even an essential G. This is because on occasions, such as in time of war or where X is insufficient to satisfy all of </FONT><FONT FACE="Symbol">W</FONT><FONT FACE="Arial">, we will have no option but to sacrifice some individuals.</li><li>A further correction will usually need to be added to protect the interests of the person <I>i</I> making the ethical decision. That is, M</FONT><FONT FACE="Arial" SIZE=2>ij</FONT><FONT FACE="Arial"> increases as j </FONT><FONT FACE="Symbol">&reg;</FONT><FONT FACE="Arial"> i. This protective zone may also apply to the interests of others of his choice (such as his family), though not to the exclusion of all others. Both moderate selfishness &amp; directed altruism are to be considered ethical.</li><li>If j </FONT><FONT FACE="Symbol">e W</FONT><FONT FACE="Arial"> is a member of a future generation, M</FONT><FONT FACE="Arial" SIZE=2>ij</FONT><FONT FACE="Arial"> </FONT><FONT FACE="Symbol">&reg;</FONT><FONT FACE="Arial"> 0 as generations progress.</li><li>There is no reason why any mathematical function should satisfy these criteria. However, it is to be noted that a logarithmic function satisfies c & d above, and that a negative exponential satisfies f & g, these being functions of quantity of good (X) and distance of relationship respectively. </li></ul></li><li>The only problem we have addressed above is one of partition. We have not yet discussed the issue of the choice of production of goods or of the utilitarian difference between goods.</li><li>A critical issue is that, with resource R at our disposal, we have the ability to produce goods G, H or I . . . (or a combination of goods). How do we decide what to produce? Our calculation so far shows how the effective quantity (utility) of a good may vary with its distribution, but not how to arbitrate between goods: how to weigh G versus H, for instance.</li><li>This is a complex question because, as we have noted, the marginal utility of Xj to j depends on j as well as (quantitatively & qualitatively) on X & the underlying good G. </li></ol></P> <BR><HR><BR><CENTER><TABLE class = "Bridge" WIDTH=950><TR> <TH WIDTH="25%">Note last updated</TH> <TH WIDTH="50%">Reference for this Topic</TH> <TH WIDTH="25%">Parent Topic</TH></TR> <TR><TD WIDTH="25%">12/08/2007 10:17:46</TD> <TD WIDTH="50%">511 (Consequentialism - Algorithms)</TD> <TD WIDTH="25%"><A href ="../../Notes/Notes_4/Notes_436.htm">Non-theistic Ethics - Consequentialism</A></TD></TR> </TABLE></center> <BR><HR><BR><h3>Summary of Note Links to this Page</h3> <CENTER> <TABLE Class = "Bridge" WIDTH=950> <TR> <td bgcolor="#ffff4d" WIDTH="20%"><A href = "../../Notes/Notes_4/Notes_436.htm#2"><span title="Medium Quality">Non-theistic Ethics - Consequentialism</span></A></TD> <TD WIDTH="20%">&nbsp;</TD> <TD WIDTH="20%">&nbsp;</TD> <TD WIDTH="20%">&nbsp;</TD> <TD WIDTH="20%">&nbsp;</TD> </TR> </TABLE> </CENTER> <P class = "Centered">To access information, click on one of the links in the table above.</P> <a name="ColourConventions"></a><br><hr><br><h3 class = "Left">Text Colour Conventions</h3><OL TYPE="1"><li><FONT COLOR = "0000FF">Blue</FONT>: Text by me; &copy; Theo Todman, 2018</li></OL><BR> <center><BR><HR><BR><TABLE class = "Bridge" WIDTH=950><TR><TD WIDTH="30%">&copy; Theo Todman, June 2007 - August 2018.</TD><TD WIDTH="40%">Please address any comments on this page to <A HREF="mailto:theo@theotodman.com">theo@theotodman.com</A>.</TD><TD WIDTH="30%">File output: <time datetime="2018-08-01T03:27" pubdate>01/08/2018 03:27:09</time> <br><A HREF="../../Notes/Notes_10/Notes_1010.htm">Website Maintenance Dashboard</A></TD></TR><TD WIDTH="30%"><A HREF="#Top">Return to Top of this Page</A></TD><TD WIDTH="40%"><A HREF="../../Notes/Notes_11/Notes_1140.htm">Return to Theo Todman's Philosophy Page</A></TD><TD WIDTH="30%"><A HREF="../../index.htm">Return to Theo Todman's Home Page</A></TD></TR></TABLE></CENTER><HR></BODY></HTML>