Changes

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>&nbsp;and&nbsp;<math>v\!</math>) or (<math>u\!</math>&nbsp;and&nbsp;<math>w\!</math>) or (<math>v\!</math>&nbsp;and&nbsp;<math>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>&nbsp;and&nbsp;<math>v\!</math>) or (<math>u\!</math>&nbsp;and&nbsp;<math>w\!</math>) or (<math>v\!</math>&nbsp;and&nbsp;<math>w\!</math>)".

{| align="center" cellpadding="10" style="text-align:center"

|

<p>[[Image:Polymorphous_Set.jpg|500px]]</p>

| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |

<p><math>\text{Figure 1.  Polymorphous Set}~ Q</math></p>

| . . . . . . . . . . .o-------------o. . . . . . . . . . . |

| . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |

| . . . . . . . . . ./. . . . . . . . .\. . . . . . . . . . |

| . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |

| . . . . . . . . ./. . . . . . . . . . .\. . . . . . . . . |

Figure 1.  Polymorphous Set Q

</pre>

In other words, the proposition <math>q\!</math> is a truth-function of the 3 logical variables <math>u\!</math>, <math>v\!</math>, <math>w\!</math>, and it may be evaluated according to the "truth table" scheme that is shown in Table 2.  In this representation the polymorphous set <math>Q\!</math> appears in the guise of what some people call the "pre-image" or the "fiber of truth" under the function <math>q\!</math>.  More precisely, the 3-tuples for which <math>q\!</math> evaluates to true are in an obvious correspondence with the shaded cells of the venn diagram.  No matter how we get down to the level of actual information, it's all pretty much the same stuff.

In other words, the proposition <math>q\!</math> is a truth-function of the 3 logical variables <math>u\!</math>, <math>v\!</math>, <math>w\!</math>, and it may be evaluated according to the "truth table" scheme that is shown in Table 2.  In this representation the polymorphous set <math>Q\!</math> appears in the guise of what some people call the "pre-image" or the "fiber of truth" under the function <math>q\!</math>.  More precisely, the 3-tuples for which <math>q\!</math> evaluates to true are in an obvious correspondence with the shaded cells of the venn diagram.  No matter how we get down to the level of actual information, it's all pretty much the same stuff.

For concreteness, consider the polymorphous set <math>Q\!</math> of Example&nbsp;1 and focus on the central cell, specifically, the cell described by the conjunction of logical features in the expression "<math>u\ v\ w</math>".

For concreteness, consider the polymorphous set <math>Q\!</math> of Example&nbsp;1 and focus on the central cell, specifically, the cell described by the conjunction of logical features in the expression "<math>u\ v\ w</math>".

{| align="center" cellpadding="10" style="text-align:center"

|

<p>[[Image:Polymorphous_Set.jpg|500px]]</p>

| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |

<p><math>\text{Figure 1.  Polymorphous Set}~ Q</math></p>

| . . . . . . . . . . .o-------------o. . . . . . . . . . . |

| . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |

| . . . . . . . . . ./. . . . . . . . .\. . . . . . . . . . |

| . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |

| . . . . . . . . ./. . . . . . . . . . .\. . . . . . . . . |

Figure 1.  Polymorphous Set Q

</pre>

The proposition or the truth-function <math>q\!</math> that describes <math>Q\!</math> is:

The proposition or the truth-function <math>q\!</math> that describes <math>Q\!</math> is:

Consider the polymorphous set <math>Q\!</math> of Example&nbsp;1 and focus on the central cell, described by the conjunction of logical features in the expression "<math>u\ v\ w\!</math>".

Consider the polymorphous set <math>Q\!</math> of Example&nbsp;1 and focus on the central cell, described by the conjunction of logical features in the expression "<math>u\ v\ w\!</math>".

{| align="center" cellpadding="10" style="text-align:center"

|

| X . . . . . . . . . . . . . . . . . . . . . . . |

<p>[[Image:Polymorphous_Set.jpg|500px]]</p>

<p><math>\text{Figure 1.  Polymorphous Set}~ Q</math></p>

| . . . . . . . . o-------------o . . . . . . . . |

| . . . . . . . ./. . . . . . . .\. . . . . . . . |

| . . . . . . . / . . . . . . . . \ . . . . . . . |

| . . . . . . ./. . . . . . . . . .\. . . . . . . |

| . . \ . . . . . . . . \%/ . . . . . . . . / . . |

Figure 1.  Polymorphous Set Q

</pre>

The proposition or truth-function <math>q : X \to \mathbb{B}</math> that describes <math>Q\!</math> is represented by the following graph and text expressions:

The proposition or truth-function <math>q : X \to \mathbb{B}</math> that describes <math>Q\!</math> is represented by the following graph and text expressions: