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Esempio di informazione semantica - III
<p>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.</p> <p style="text-align: justify;">In questa nota continuiamo i calcoli legati all'informazione semantica delle proposizioni del linguaggio di esempio fornito da Carnap, partendo dai risultati e informazioni delle note precedenti. Per comodità ricordiamo che stiamo trattando il caso di un linguaggio <img title="L_3^2" src="https://latex.codecogs.com/gif.latex?L_3^2" /> composto da 3 costanti individuali {a,b,c} e 2 predicati {M,Y} che stanno ad indicare la proprietà di essere Maschio e Giovane.</p> <p style="text-align: justify;">Inoltre riproponiamo qui di seguito lo spazio logico in cui sono indicati i vari state-description e le relative probabilità logiche:</p> <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 </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 41px; height: 13px;">- </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 </td> <td style="width: 39px; height: 13px;">- </td> <td style="width: 39.8882px; height: 13px;">-</td> <td style="width: 39.1118px; height: 13px;">a,c </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;"> -</td> <td style="width: 41px; height: 13.313px;">a,b,c </td> <td style="width: 41px; height: 13.313px;">-</td> <td style="width: 41px; height: 13.313px;">- </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 </td> <td style="width: 39px; height: 13.313px;">- </td> <td style="width: 39.8882px; height: 13.313px;">- </td> <td style="width: 39.1118px; height: 13.313px;">b,c </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;">- </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 41px; height: 13px;">a,b,c </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">1/20</td> <td style="width: 38px; height: 13px;">Z35</td> <td style="width: 40px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">c </td> <td style="width: 39.8882px; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px;">a,b </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;">- </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 41px; height: 13px;">a,b,c </td> <td style="width: 39px; height: 13px;">1/20</td> <td style="width: 38px; height: 13px;">Z36</td> <td style="width: 40px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">b </td> <td style="width: 39.8882px; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px;">a,c </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;">- </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;">- </td> <td style="width: 39px; height: 13px;">a </td> <td style="width: 39.8882px; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px;">b,c </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 </td> <td style="width: 41px; height: 13px;">b </td> <td style="width: 41px; height: 13px;">-</td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">1/60</td> <td style="width: 38px; height: 13px;">Z38</td> <td style="width: 40px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">- </td> <td style="width: 39.8882px; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px;">a,b </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 </td> <td style="width: 41px; height: 14px;">- </td> <td style="width: 41px; height: 14px;">- </td> <td style="width: 39px; height: 14px;">1/60</td> <td style="width: 38px; height: 14px;">Z39</td> <td style="width: 40px; height: 14px;">- </td> <td style="width: 39px; height: 14px;">- </td> <td style="width: 39.8882px; height: 14px;">b </td> <td style="width: 39.1118px; height: 14px;">a,c </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;">- </td> <td style="width: 41px; height: 13px;">c </td> <td style="width: 41px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">1/60</td> <td style="width: 38px; height: 13px;">Z40</td> <td style="width: 40px; height: 13px;">- </td> <td style="width: 39px; height: 13px;">- </td> <td style="width: 39.8882px; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px;">b,c </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;">- </td> <td style="width: 41px; height: 13px;">b</td> <td style="width: 41px; height: 13px;">- </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 </td> <td style="width: 39px; height: 13px;">b </td> <td style="width: 39.8882px; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px;">- </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;"> -</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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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;">Z43</td> <td style="width: 40px; text-align: center; height: 13px;">b </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px; text-align: center;">- </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">b </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 </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">- </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a </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 </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;">- </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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">b </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">- </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">b </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;"> 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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;">b </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,b</td> <td style="width: 41px; text-align: center; height: 13px;">c </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;">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;">Z18</td> <td style="width: 43px; text-align: center; height: 13px;">- </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;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">b,c </td> <td style="width: 41px; text-align: center; height: 13px;">a </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;">b </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,b </td> <td style="width: 41px; text-align: center; height: 13px;">- </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;">Z52</td> <td style="width: 40px; text-align: center; height: 13px;">c </td> <td style="width: 39px; text-align: center; height: 13px;">b </td> <td style="width: 39.8882px; text-align: center; height: 13px;">- </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,c </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">b </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 </td> <td style="width: 39px; text-align: center; height: 13px;">- </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;">c </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;">- </td> <td style="width: 41px; text-align: center; height: 15px;">b,c</td> <td style="width: 41px; text-align: center; height: 15px;">- </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 </td> <td style="width: 39px; text-align: center; height: 15px;">- </td> <td style="width: 39.8882px; text-align: center; height: 15px;">c </td> <td style="width: 39.1118px; height: 15px; text-align: center;">b </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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,b </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">- </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">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;">Z24</td> <td style="width: 43px; text-align: center; height: 13px;">b</td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,c </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">- </td> <td style="width: 39.8882px; text-align: center; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">b,c </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">- </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">b </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">c </td> <td style="width: 41px; text-align: center; height: 13px;">a,b </td> <td style="width: 41px; text-align: center; height: 13px;">- </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 </td> <td style="width: 39px; text-align: center; height: 13px;">- </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </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;">- </td> <td style="width: 41px; 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;">- </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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;">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;">Z28</td> <td style="width: 43px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a </td> <td style="width: 41px; text-align: center; height: 13px;">b,c </td> <td style="width: 41px; text-align: center; height: 13px;">- </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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">a </td> <td style="width: 39.8882px; text-align: center; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px; text-align: center;">b </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">a,b </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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">b </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">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;">Z30</td> <td style="width: 43px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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: 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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">b </td> <td style="width: 39.8882px; text-align: center; height: 13px;">c </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </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;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">b,c </td> <td style="width: 41px; text-align: center; height: 13px;">a </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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">a </td> <td style="width: 39.1118px; height: 13px; text-align: center;">b </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 </td> <td style="width: 41px; text-align: center; height: 13px;">- </td> <td style="width: 41px; text-align: center; height: 13px;">- </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;">- </td> <td style="width: 39px; text-align: center; height: 13px;">c </td> <td style="width: 39.8882px; text-align: center; height: 13px;">b </td> <td style="width: 39.1118px; height: 13px; text-align: center;">a </td> <td style="width: 42px; height: 13px; text-align: center;">1/120</td> </tr> </tbody> </table> <p style="text-align: justify;">In generale la quantità est(in, H, e) - dove 'in' rappresenta sia la funzione 'cont' che 'inf' - sarà diversa dalla quantità est(in, H, e.k) dove si è aggiunta la proposizione 'k' all'evidenza 'e'. Per gestire questi casi Carnap ha introdotto la '<strong>ammount of specification of H through k</strong>' definita come:</p> <p style="text-align: center;"><img title="sp(in, H, e, k)= est(in, H, e) - est(in, H, e.k)" src="https://latex.codecogs.com/gif.latex?sp(in,&space;H,&space;e,&space;k)=&space;est(in,&space;H,&space;e)&space;-&space;est(in,&space;H,&space;e.k)" /></p> <p style="text-align: justify;">Supponiamo ora di riprendere il nostro esempio dove l'evidenza e = Ma.Mb.Ya.Yc, H = {Mc, !Mc} e k = Yc; in questo modo la nuova evidenza e.k = Ma.Mb.Ya.Yc.Yc e avevamo già calcolato est*(inf, H, e.k) = 0.81, così come avevamo est*(inf, H, e) = 0.91. Applicando ora la formula precedente si ottiene</p> <p style="text-align: justify;">sp(inf, H, e, k) = 0.91 - 0.81 = 0.10</p> <p style="text-align: justify;">E' interessante anche affrontare il caso in cui un'insieme di ipotesi H = {h1 .. hn} sia in qualche modo determinato da un'altro insieme di ipotesi K = {k1 .. km}; questo ci porta alla definizione di '<strong>posterior estimate of the amount of information carried by H on e and K</strong>', che definita come:</p> <p style="text-align: center;"><img title="est(in, H |K, e)=\sum_{q=1}^{m} c(k_q , e) * est(in, H, e.k_q)" src="https://latex.codecogs.com/gif.latex?est(in,&space;H&space;|K,&space;e)=\sum_{q=1}^{m}&space;c(k_q&space;,&space;e)&space;*&space;est(in,&space;H,&space;e.k_q)" /></p> <p style="text-align: justify;">Riprendiamo il nostro esempio, dove l'evidenza e = 'Ma.Mb.Ya.Yb', H = {Mc, !Mc}, K = {Yc, !Yc}. Per applicare le formule precedenti utilizziamo la misura di probabilità m*. Valgono inoltre i calcoli seguenti: </p> <p style="text-align: justify;">m*(e) = m*(Ma.Mb.Ya.Yb) = 1/10</p> <p style="text-align: justify;">m*(k1.e) = m*(Yc.Ma.Mb.Ya.Yb) = 1*1/20 + 1*1/60 + 0*1/120 = 4/60 = 1/15</p> <p style="text-align: justify;">m*(k2.e) = m*(!Yc.Ma.Mb.Ya.Yb) = 0*1/20 + 1/60 + 0*1/120 = 1/60</p> <p style="text-align: justify;">c*(k1, e) = c*(Yc, Ma.Mb.Ya.Yb) = m*(Yc.Ma.Mb.Ya.Yb) / m*(Ma.Mb.Ya.Yb) = 1/60/1/10 = 1/6</p> <p style="text-align: justify;">c*(k2, e) = c*(!Yc, Ma.Mb.Ya.Yb) = m*(!Yc.Ma.Mb.Ya.Yb) / m*(Ma.Mb.Ya.Yb) = 1/60/1/10 = 1/6</p> <p style="text-align: justify;">m*(e.k1.h1) = m*(Ma.Mb.Ya.Yb.Yc.Mc) = 1*1/20 + 0*1/60 + 0*1/120 = 1/20</p> <p style="text-align: justify;">m*(e.k2.h1) = m*(Ma.Mb.Ya.Yb.!Yc.Mc) = 0*1/20 + 1*1/60 + 0*1/120 = 1/60</p> <p style="text-align: justify;">m*(e.k1.h2) = m*(Ma.Mb.Ya.Yb.Yc.!Mc) = 0*1/20 + 1*1/60 + 0*1/120 = 1/60</p> <p style="text-align: justify;">m*(e.k2.h2) = m*(Ma.Mb.Ya.Yb.!Yc.!Mc) = 0*1/20 + 1*1/60 + 0*1/120 = 1/60</p> <p style="text-align: justify;">c*(h1, e.k1) = m*(e.k1.h1)/m*(e.k1) = 1/20 / 1/15 = 3/4</p> <p style="text-align: justify;">c*(h2, e.k1) = m*(e.k1.h2)/m*(e.k1) = 1/60 / 1/15 = 1/4</p> <p style="text-align: justify;">c*(h1, e.k2) = m*(e.k2.h1)/m*(e.k2) = 1/60 / 1/60 = 1</p> <p style="text-align: justify;">c*(h2, e.k2) = m*(e.k2.h2)/m*(e.k2) = 1/60 / 1/60 = 1</p> <p style="text-align: justify;">inf*(h1 | e.k1) = Log(m*(e.k1)) - Log(m*(e.k1.h1) = Log(1/15) - Log(1/20) = -4.05 + 4.32 = 0.27</p> <p style="text-align: justify;">inf*(h1 | e.k2) = Log(m*(e.k2)) - Log(m*(e.k2.h1) = Log(1/60) - Log(1/60) = 0</p> <p style="text-align: justify;">inf*(h2 | e.k1) = Log(m*(e.k1)) - Log(m*(e.k1.h2) = Log(1/15) - Log(1/60) = -4.05 + 5.96 = 1.89</p> <p style="text-align: justify;">inf*(h2 | e.k2) = Log(m*(e.k2)) - Log(m*(e.k2.h2) = Log(1/60) - Log(1/60) = 0</p> <p style="text-align: justify;">est(inf, H, e.k1) = c*(h1, e.k1) * inf*(h1 | e.k1) + c*(h2, e.k1) * inf*(h2 | e.k1) = 3/4 * 0.27 + 1/4 * 1.89 = 0.20 + 0.47 = 0.67</p> <p style="text-align: justify;">est(inf, H, e.k2) = c*(h1, e.k2) * inf*(h1 | e.k2) + c*(h2, e.k2) * inf*(h2 | e.k2) = 0 + 0 = 0</p> <p style="text-align: justify;">Da queste relazioni si ottiene:</p> <p style="text-align: justify;">est(inf, H|K, e) = c*(k1,e) * est(inf, H, e.k1) + c*(k2,e) * est(inf, H, e.k2) = 1/6 * 0.67 + 0 = 0.111 </p> <p style="text-align: justify;">Vale anche l'importante teorema:</p> <p style="text-align: center;"><img title="est(in, H | K, e) = est(in, H.K, e) - est(in, K, e)" src="https://latex.codecogs.com/gif.latex?est(in,&space;H&space;|&space;K,&space;e)&space;=&space;est(in,&space;H.K,&space;e)&space;-&space;est(in,&space;K,&space;e)" /></p> <p style="text-align: justify;">che riscorda la regola fondamentale del calcolo probabilistico. Questa formula si può anche invertire per ottenere:</p> <p style="text-align: center;"><img title="est(in, H.K, e) = est(in, H |K, e) + est(in, K, e)" src="https://latex.codecogs.com/gif.latex?est(in,&space;H.K,&space;e)&space;=&space;est(in,&space;H&space;|K,&space;e)&space;+&space;est(in,&space;K,&space;e)" /></p> <p style="text-align: justify;">che può essere utile nel caso dei dynamic frames.</p> <p style="text-align: justify;"> </p>
Some notes on Relativity and other arguments