12,080
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 02:24, 28 April 2008
, 02:24, 28 April 2008→Note 2: markup
The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of stimulus patterns that can be described in terms of the three features "round" <math>u\!</math>, "doubly outlined" <math>v\!</math>, and "centrally dark" <math>w\!</math>. We may regard these simple features as logical propositions <math>u, v, w : X \to \mathbb{B}.</math> The target concept <math>\mathcal{Q}</math> is one whose extension is a polymorphous set <math>Q\!</math>, the subset <math>Q\!</math> of the universe <math>X\!</math> where the complex feature <math>q : X \to \mathbb{B}</math> holds true. The <math>Q\!</math> in question is defined by the requirement: "Having at least 2 of the 3 features in the set <math>\{ u, v, w \}\!</math>".
The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of stimulus patterns that can be described in terms of the three features "round" <math>u\!</math>, "doubly outlined" <math>v\!</math>, and "centrally dark" <math>w\!</math>. We may regard these simple features as logical propositions <math>u, v, w : X \to \mathbb{B}.</math> The target concept <math>\mathcal{Q}</math> is one whose extension is a polymorphous set <math>Q\!</math>, the subset <math>Q\!</math> of the universe <math>X\!</math> where the complex feature <math>q : X \to \mathbb{B}</math> holds true. The <math>Q\!</math> in question is defined by the requirement: "Having at least 2 of the 3 features in the set <math>\{ u, v, w \}\!</math>".
Taking the symbols <math>u\!</math> = "round", <math>v\!</math> = "doubly outlined", <math>w\!</math> = "centrally dark", and using the corresponding capital letters to label the circles of a venn diagram, we get a picture of the target set <math>Q\!</math> as the shaded region in Figure 1. Using these symbols as "sentence letters" in a truth table, let the truth function <math>q\!</math> mean the very same thing as the expression "(<math>u\!</math> and <math>v\!</math>) or (<math>u\!</math> and <math>w\!</math>) or (<math>v\!</math> and <math>w\!</math>)".
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| X |
| X |