Changes

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<pre>

The proposition <math>(u, v, w)\!</math> evaluates to true if and only if just one of <math>u, v, w\!</math> is false. In the same way, the proposition <math>(x,(y),(z))\!</math> evaluates to true if and only if exactly one of <math>x, (y), (z)\!</math> is false.  Taking it by cases, let us first suppose that <math>x\!</math> is true.  Then it has to be that just one of <math>(y)\!</math> or <math>(z)\!</math> is false, which is tantamount to the proposition <math>((y),(z)),\!</math> which is equivalent to the proposition <math>(y, z).\!</math>  On the other hand, let us suppose that <math>x\!</math> is the false one.  Then both <math>(y)\!</math> and <math>(z)\!</math> must be true, which is to say that <math>y\!</math> is false and <math>z\!</math> is false.

The proposition (u, v, w) evaluates to true

if and only if just one of u, v, w is false.

In the same way, the proposition (x,(y),(z))

evaluates to true if and only if exactly one

of x, (y), (z) is false.  Taking it by cases,

let us first suppose that x is true.  Then it

has to be that just one of (y) or (z) is false,

which is tantamount to the proposition ((y),(z)),

which is equivalent to the proposition ( y , z ).

On the other hand, let us suppose that x is the

false one.  Then both (y) and (z) must be true,

which is to say that y is false and z is false.

What we have just said here is that the region

What we have just said here is that the region where <math>x\!</math> is true is partitioned into the regions where <math>y\!</math> and <math>z\!</math> are true, respectively, while the region where <math>x\!</math> is false has both <math>y\!</math> and <math>z\!</math> false. In other words, we have a ''pie-chart'' structure, where the genus <math>X\!</math> is divided into the disjoint and <math>X\!</math>-haustive couple of species <math>Y\!</math> and <math>Z.\!</math>

where x is true is partitioned into the regions

where y and z are true, respectively, while the

region where x is false has both y and z false.

In other words, we have a "pie-chart" structure,

where the genus X is divided into the disjoint

and X-haustive couple of species Y and Z.

The same analysis applies to the generic form

The same analysis applies to the generic form <math>(x, (x_1), \ldots, (x_k)),\!</math> specifying a pie-chart with a genus <math>X\!</math> and the <math>k\!</math> species <math>X_1, \ldots, X_k.\!</math>

(x, (x_1), ..., (x_k)), specifying a pie-chart

with a genus X and the k species X_1, ..., X_k.

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==Differential Logic : Series C==

==Differential Logic : Series C==