Changes

===Note 6===

===Note 6===

I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:

I think that we finally have enough of the preliminary

 

set-ups and warm-ups out of the way that we can begin

to tackle the differential analysis proper of our

(( u v )( u w )( v w )).

sample proposition q = (( u v )( u w )( v w )).

When X is the type of space that is generated by {u, v, w},

When <math>X\!</math> is the type of space that is generated by <math>\{ u, v, w \}\!</math>, let <math>\operatorname{d}X</math> be the type of space that is generated by <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>, and let <math>X \times \operatorname{d}X</math> be the type of space that is generated by the extended set of boolean basis elements <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. For convenience, define a notation "<math>\operatorname{E}X</math>" so that <math>\operatorname{E}X = X \times \operatorname{d}X</math>. Even though the differential variables are in some abstract sense no different than other boolean variables, it usually helps to mark their distinctive roles and their differential interpretation by means of the distinguishing domain name "<math>\operatorname{d}\mathbb{B}</math>". Using these designations of logical spaces, the propositions over them can be assigned both abstract and concrete types.

let dX be the type of space that is generated by (du, dv, dw},

and let X x dX be the type of space that is generated by the

extended set of boolean basis elements {u, v, w, du, dv, dw}.

For convenience, define a notation "EX" so that EX = X x dX.

Even though the differential variables are in some abstract

sense no different than other boolean variables, it usually

helps to mark their distinctive roles and their differential

interpretation by means of the distinguishing domain name "dB".

Using these designations of logical spaces, the propositions

over them can be assigned both abstract and concrete types.

For instance, consider the proposition q<u, v, w>, as before,

For instance, consider the proposition q<u, v, w>, as before,

and then consider its tacit extension q<u, v, w, du, dv, dw>,

and then consider its tacit extension q<u, v, w, du, dv, dw>,