Vielbein - Relativity and other

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Some notes on Relativity and other arguments

Esempio di informazione semantica - II

In un report del 1952 Carnap e Bar-Hillel presentarono la loro teoria dell'informazione semantica, proponendo numerosi esempi tratti da un linguaggio basato su predicati monadici con un numero finito di costanti individuali.
In questa nota continuiamo i calcoli legati all'informazione semantica delle proposizioni del linguaggio di esempio fornito da Carnap, partendo dai risultati e informazioni della nota precedente. Per comodit&agrave; ricordiamo che stiamo trattando il caso di un linguaggio&nbsp;&nbsp;<img title="L_3^2" src="https://latex.codecogs.com/gif.latex?L_3^2" />&nbsp;composto da 3 costanti individuali {a,b,c} e 2 predicati {M,Y} che stanno ad indicare la propriet&agrave; di essere&nbsp; Maschio e Giovane. Inoltre riproponiamo qui di seguito lo spazio logico in cui sono indicati i vari state-description e le relative probabilit&agrave; logiche:
<table style="height: 609px; width: 550px;">
<tbody>
<tr style="height: 13px;">
<td style="width: 38px; background-color: yellow; text-align: center; height: 13px;">State</td>
<td style="width: 43px; background-color: yellow; text-align: center; height: 13px;">M,Y</td>
<td style="width: 41px; background-color: yellow; text-align: center; height: 13px;">M,!Y</td>
<td style="width: 41px; background-color: yellow; text-align: center; height: 13px;">!M,Y</td>
<td style="width: 41px; background-color: yellow; text-align: center; height: 13px;">!M,!Y</td>
<td style="width: 39px; background-color: yellow; text-align: center; height: 13px;">m*</td>
<td style="width: 38px; background-color: yellow; text-align: center; height: 13px;">State</td>
<td style="width: 40px; background-color: yellow; text-align: center; height: 13px;">M,Y</td>
<td style="width: 39px; background-color: yellow; text-align: center; height: 13px;">M,!Y</td>
<td style="width: 39.8882px; background-color: yellow; text-align: center; height: 13px;">!M,Y</td>
<td style="width: 39.1118px; background-color: yellow; text-align: center; height: 13px;">!M,!Y</td>
<td style="width: 42px; background-color: yellow; text-align: center; height: 13px;">m*</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z1</td>
<td style="width: 43px; height: 13px;">a,b,c&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">1/20</td>
<td style="width: 38px; height: 13px;">Z33</td>
<td style="width: 40px; height: 13px;">b&nbsp;</td>
<td style="width: 39px; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">-</td>
<td style="width: 39.1118px; height: 13px;">a,c&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13.313px;">
<td style="width: 38px; height: 13.313px;">Z2</td>
<td style="width: 43px; height: 13.313px;">&nbsp;-</td>
<td style="width: 41px; height: 13.313px;">a,b,c&nbsp;</td>
<td style="width: 41px; height: 13.313px;">-</td>
<td style="width: 41px; height: 13.313px;">-&nbsp;</td>
<td style="width: 39px; height: 13.313px;">1/20</td>
<td style="width: 38px; height: 13.313px;">Z34</td>
<td style="width: 40px; height: 13.313px;">a&nbsp;</td>
<td style="width: 39px; height: 13.313px;">-&nbsp;</td>
<td style="width: 39.8882px; height: 13.313px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13.313px;">b,c&nbsp;</td>
<td style="width: 42px; height: 13.313px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z3</td>
<td style="width: 43px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">a,b,c&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">1/20</td>
<td style="width: 38px; height: 13px;">Z35</td>
<td style="width: 40px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">a,b&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z4</td>
<td style="width: 43px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">a,b,c&nbsp;</td>
<td style="width: 39px; height: 13px;">1/20</td>
<td style="width: 38px; height: 13px;">Z36</td>
<td style="width: 40px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">a,c&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z5</td>
<td style="width: 43px; height: 13px;">a,b</td>
<td style="width: 41px; height: 13px;">c</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">-</td>
<td style="width: 39px; height: 13px;">1/60</td>
<td style="width: 38px; height: 13px;">Z37</td>
<td style="width: 40px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">b,c&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z6</td>
<td style="width: 43px; height: 13px;">a,c&nbsp;</td>
<td style="width: 41px; height: 13px;">b&nbsp;</td>
<td style="width: 41px; height: 13px;">-</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">1/60</td>
<td style="width: 38px; height: 13px;">Z38</td>
<td style="width: 40px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">a,b&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 14px;">
<td style="width: 38px; height: 14px;">Z7</td>
<td style="width: 43px; height: 14px;">b,c</td>
<td style="width: 41px; height: 14px;">a&nbsp;</td>
<td style="width: 41px; height: 14px;">-&nbsp;</td>
<td style="width: 41px; height: 14px;">-&nbsp;</td>
<td style="width: 39px; height: 14px;">1/60</td>
<td style="width: 38px; height: 14px;">Z39</td>
<td style="width: 40px; height: 14px;">-&nbsp;</td>
<td style="width: 39px; height: 14px;">-&nbsp;</td>
<td style="width: 39.8882px; height: 14px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 14px;">a,c&nbsp;</td>
<td style="width: 42px; height: 14px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z8</td>
<td style="width: 43px; height: 13px;">a,b</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">c&nbsp;</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">1/60</td>
<td style="width: 38px; height: 13px;">Z40</td>
<td style="width: 40px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">b,c&nbsp;</td>
<td style="width: 42px; height: 13px;">1/60</td>
</tr>
<tr style="text-align: center; height: 13px;">
<td style="width: 38px; height: 13px;">Z9</td>
<td style="width: 43px; height: 13px;">a,c</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 41px; height: 13px;">b</td>
<td style="width: 41px; height: 13px;">-&nbsp;</td>
<td style="width: 39px; height: 13px;">1/60</td>
<td style="width: 38px; height: 13px;">Z41</td>
<td style="width: 40px; height: 13px;">a&nbsp;</td>
<td style="width: 39px; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px;">-&nbsp;</td>
<td style="width: 42px; height: 13px;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z10</td>
<td style="width: 43px; text-align: center; height: 13px;">b,c</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z42</td>
<td style="width: 40px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">&nbsp;-</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z11</td>
<td style="width: 43px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z43</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">-&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z12</td>
<td style="width: 43px; text-align: center; height: 13px;">a,c</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z44</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">-&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z13</td>
<td style="width: 43px; text-align: center; height: 13px;">b,c</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z45</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">-&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z14</td>
<td style="width: 43px; text-align: center; height: 13px;">c</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z46</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">-&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z15</td>
<td style="width: 43px; text-align: center; height: 13px;">b</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z47</td>
<td style="width: 40px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">&nbsp;c</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z16</td>
<td style="width: 43px; text-align: center; height: 13px;">a</td>
<td style="width: 41px; text-align: center; height: 13px;">b,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z48</td>
<td style="width: 40px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z17</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b</td>
<td style="width: 41px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z49</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">c&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z18</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c</td>
<td style="width: 41px; text-align: center; height: 13px;">b</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z50</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z19</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z51</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z20</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z52</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z21</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z53</td>
<td style="width: 40px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">c&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 15px;">
<td style="width: 38px; text-align: center; height: 15px;">Z22</td>
<td style="width: 43px; text-align: center; height: 15px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 15px;">b,c</td>
<td style="width: 41px; text-align: center; height: 15px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 15px;">a</td>
<td style="width: 39px; text-align: center; height: 15px;">1/60</td>
<td style="width: 38px; text-align: center; height: 15px;">Z54</td>
<td style="width: 40px; text-align: center; height: 15px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 15px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 15px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 15px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 15px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z23</td>
<td style="width: 43px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z55</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">c&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z24</td>
<td style="width: 43px; text-align: center; height: 13px;">b</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z56</td>
<td style="width: 40px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z25</td>
<td style="width: 43px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z57</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z26</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z58</td>
<td style="width: 40px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z27</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z59</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">c&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z28</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z60</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z29</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">c</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z61</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">c&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z30</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z62</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z31</td>
<td style="width: 43px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">b,c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z63</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">a&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">b&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
<tr style="height: 13px;">
<td style="width: 38px; text-align: center; height: 13px;">Z32</td>
<td style="width: 43px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 41px; text-align: center; height: 13px;">a,b</td>
<td style="width: 39px; text-align: center; height: 13px;">1/60</td>
<td style="width: 38px; text-align: center; height: 13px;">Z64</td>
<td style="width: 40px; text-align: center; height: 13px;">-&nbsp;</td>
<td style="width: 39px; text-align: center; height: 13px;">c&nbsp;</td>
<td style="width: 39.8882px; text-align: center; height: 13px;">b&nbsp;</td>
<td style="width: 39.1118px; height: 13px; text-align: center;">a&nbsp;</td>
<td style="width: 42px; height: 13px; text-align: center;">1/120</td>
</tr>
</tbody>
</table>
Ricordiamo infine che il nostro esempio &egrave; legato all'evidenza e=Ma.Mb e stiamo&nbsp;ragionando&nbsp;sull'ipotesi h=Mc; su questo esempio avevamo ottenuto che m*(Ma.Mb) = 0.29 e che m*(Ma.Mb.Mc) = 0.20.
Veniamo ora a considerare le definizioni di informazione semantica proposte da Carnap (ci limiteremo nel considerare la misura di probabilit&agrave; m* e non altre); egli introduce il concetto di 'contenuto di informazione' come:
<img title="Cont^*(i)=1-m^*(i)" src="https://latex.codecogs.com/gif.latex?Cont^*(i)=1-m^*(i)" />
Se&nbsp;consideriamo il nostro esempio con i = 'Ma.Mb' abbiamo che Cont*(Ma.Mb)=1 - 0.29 = 0.71. Vale inoltre la relazione:
<img title="Cont^*(h|e)=m^*(e)-m^*(e.h)" src="https://latex.codecogs.com/gif.latex?Cont^*(h|e)=m^*(e)-m^*(e.h)" />
che sta ad indicare il contenuto di informazione dell'ipotesi 'h' in base all'evidenza 'e'. Nel nostro esempio Cont*(Mc | Ma.Mb) = m*(Ma.Mb) - m*(Ma.Mb.Mc) = 0.29 - 0.20 = 0.09.
Carnap introduce una seconda definizione di informazione - che chiama 'quantit&agrave; di informazione' con la seguente formula:
<img title="inf^*(i)=-Log(m^*(i))" src="https://latex.codecogs.com/gif.latex?inf^*(i)=-Log(m^*(i))" />
dove il logaritmo &egrave; scelto in base 2. Se prendiamo in considerazione il nostro esempio ed imponiamo i = 'Ma.Mb' otteniamo Inf*(Ma.Mb)=-Log(0.29) = 1.78. Anche in questo caso vale la relazione:
<img title="Inf^*(h|e)=Log(m^*(e))-Log(m^*(e.h))" src="https://latex.codecogs.com/gif.latex?Inf^*(h|e)=Log(m^*(e))-Log(m^*(e.h))" />
che sta ad indicare la quantit&agrave; di informazione associata all'ipotesi 'h',&nbsp;nota l'evidenza empirica 'e'. Nel nostro esempio abbiamo Inf*(Mc, Ma.Mb)=Log(Ma.Mb) - Log(Ma.Mb.Mc) = Log(0.29) - Log(0.20) = -1.78 + 2.32 = 0.54.
Carnap introduce in seguito la 'estimate of the ammount of information carried by H with rispect to e', dove H &egrave; un'insieme di ipotesi {h1 ... hn}, con le fornule:
<img title="est(cont, H, e)=\sum_p c(h_p, e)*cont(h_p|e)" src="https://latex.codecogs.com/gif.latex?est(cont,&amp;space;H,&amp;space;e)=\sum_p&amp;space;c(h_p,&amp;space;e)*cont(h_p|e)" />
<img title="est(inf, H, e)=\sum_p c(h_p, e)*inf(h_p|e)" src="https://latex.codecogs.com/gif.latex?est(inf,&amp;space;H,&amp;space;e)=\sum_p&amp;space;c(h_p,&amp;space;e)*inf(h_p|e)" />
dove c(h,e) = m(h.e)/m(e).
Se ora consideriamo H = {M.c, !M.c} con evidenza e = Ma.Mb e la misura di probabilit&agrave; m* otteniamo che Cont*(M.c | Ma.Mb) = 0.09 - come abbiamo visto poco sopra - e
Cont*(!Mc | Ma.Mb) =&nbsp;m*(Ma.Mb) - m*(Ma.Mb.!Mc) = 0.29 - (0*1/20 + 4*1/60 + 0*1/120) = 0.29 - 0.06 = 0.23.
Ricordiamo inoltre che c*(Mc, Ma.Mb) = 0.2/0.29 = 0.68 e
c*(!Mc, Ma.Mb) = m*(Ma.Mb.!Mc) / m*(Ma.Mb) = 0.06 / 0.29 = 0.20.
Da ci&ograve; si deduce che:
est*(cont,H,e) = c*(Mc, Ma.Mb)*cont*(Mc | Ma.Mb) +&nbsp;c*(!Mc, Ma.Mb)*cont*(!Mc | Ma.Mb) = 0.68 * 0.09 + 0.20 * 0.23 = 0.061 + 0.046 = 0.107.
In modo analogo se consideriamo l'informazione inf* abbiamo che inf*(Mc | Ma.Mb) = 0.54 - come abbiamo visto poco sopra - e
inf*(!Mc | Ma.Mb) = Log(m*(Ma.Mb)) - log(m*(Ma.Mb.!Mc) = Log(0.29) - Log(0.06) = -1.78 + 4.05 = 2.27.
Da ci&ograve; si deduce che:
est*(inf, H, e) =&nbsp;c*(Mc, Ma.Mb)*inf*(Mc | Ma.Mb) +&nbsp;c*(!Mc, Ma.Mb)*inf*(!Mc | Ma.Mb) = 0.68*0.54 + 0.20*2.27 = 0.36 + 0.45 = 0.81.
Veniamo ora a considerare il caso in cui si abbiano due insiemi di ipotesi H={h1 .. hn} e K={k1 .. km}, che portano a definire la congiunzione H.K = {h1.k1, h1.k2 ... h1.km ... hn.k1 ... hn.km} e quindi l'estimate ammount of information di due gruppi di ipotesi H e K su un'evidenza 'e' come:
<img title="est(cont,H.K, e) =\sum_{q=1}^{m} \sum_{p=1}^{n}c(h_p.k_q,e)cont(h_p.k_q | e)" src="https://latex.codecogs.com/gif.latex?est(cont,H.K,&amp;space;e)&amp;space;=\sum_{q=1}^{m}&amp;space;\sum_{p=1}^{n}c(h_p.k_q,e)cont(h_p.k_q&amp;space;|&amp;space;e)" /><img title="est(inf,H.K, e) =\sum_{q=1}^{m} \sum_{p=1}^{n}c(h_p.k_q,e) inf(h_p.k_q | e)" src="https://latex.codecogs.com/gif.latex?est(inf,H.K,&amp;space;e)&amp;space;=\sum_{q=1}^{m}&amp;space;\sum_{p=1}^{n}c(h_p.k_q,e)&amp;space;inf(h_p.k_q&amp;space;|&amp;space;e)" />
Consideriamo ora&nbsp;l'evidenza e = Ma.Mb.Ya.Yb e le ipotesi H={Mc, !Mc} e K={Yc, !Yc}.&nbsp;Per applicare le formule precedenti utilizzando la misura di probabilit&agrave; m*.
Iniziamo con il considerare l'evidenza 'e' e calcoliamo m*(e) = m*(Ma.Mb.Ya.Yb) = 1/20 + 3*1/60 + 0*1/120 = 1/20 + 1/20 = 1/10. Calcoliamo in seguito m*(h1.k1.e) = m*(Mc.Yc.Ma.Mb.Ya.Yb) = 1/20 + 0*1/60 + 0*1/120 = 1/20; m*(h1.k2.e) = m*(Mc.!Yc.Ma.Mb.Ya.Yb) = 0*1/20 + 1/60 + 0*1/120 = 1/60; m*(h2.k1.e) = m*(!Mc.Yc.Ma.Mb.Ya.Yb) = 0*1/20 + 1/60 + 0*1/120 = 1/60; m*(h2.k2.e) = m*(!Mc.!Yc.Ma.Mb.Ya.Yb) = 0*1/20 + 1/60 + 0*1/120 = 1/60.
Da quanto calcolato poco sopra, riusciamo a&nbsp;determinare&nbsp;c*(h1.k1, e) = m*(h1.k1.e)/m*(e) = 1/20 / 1/10 = 1/2; c*(h1.k2, e) = m*(h1.k2.e) / m*(e) = 1/60 / 1/10 = 1/6; abbiamo inoltre che c*(h1.k2, e) = c*h2.k1, e) = c*(h2.k2, e) = 1/6.
Calcoliamo&nbsp;ora cont*(h1.k1 | e) = m*(e) - m*(h1.k1.e) = 1/10 - 1/20 = 1/20; cont*(h1.k2 | e) = m*(e) - m*(h1.k2.e) = 1/10 - 1/60 = 1/12; abbiamo inoltre che cont*(h1.k2 |e) = cont*(h2.k1 |e) = cont*(h2.k2 | e) = 1/12.
Possiamo ora calcolare:
est(cont, H.K, e) = c*(h1.k1, e) * cont*(h1.k1 | e) +&nbsp;c*(h1.k2, e) * cont*(h1.k2 | e) +&nbsp;c*(h2.k1, e) * cont*(h2.k1 | e) +&nbsp;c*(h2.k2, e) * cont*(h2.k2 | e) = 1/2 * 1/20 + 3 * (1/6 * 1/12) = 1/40 + 1/24 = 3/120 + 5/120 = 8/120 = 1/15.
Determiniamo ora inf*(h1.k1 | e) = Log(m*(e)) - Log(m*(h1.k1.e)) = Log(1/10) - Log(1/20) = -3.32 + 4.32 = 1.0; inf*(h1.h2 | e) =&nbsp;Log(m*(e)) - Log(m*(h1.k2.e)) = Log(1/10) - Log(1/60) = -3.32 + 6.64 = 3.32;&nbsp;abbiamo inoltre che inf*(h1.k2 |e) = inf*(h2.k1 |e) = inf*(h2.k2 | e) = 3.32.
Possiamo ora calcolare:
est(inf, H.K, e)&nbsp;= c*(h1.k1, e) * inf*(h1.k1 | e) +&nbsp;c*(h1.k2, e) * inf*(h1.k2 | e) +&nbsp;c*(h2.k1, e) * inf*(h2.k1 | e) +&nbsp;c*(h2.k2, e) * inf*(h2.k2&nbsp;| e) = 1/2 * 1.0 + 3 ( 1/6 * 3.32) = 0.5 + 1.66 = 2.16.