Changes

====Note 8====

====Note 8====

We have been contemplating functions of the type <math>f : U \to \mathbb{B}</math> and studying the action of the operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on this family. These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of ''scalar potential fields''. These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes. The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two:  standing still on level ground or falling off a bluff.

We have been contemplating functions of the type f : U -> B

studying the action of the operators E and D on this family.

These functions, that we may identify for our present aims

with propositions, inasmuch as they capture their abstract

forms, are logical analogues of "scalar potential fields".

These are the sorts of fields that are so picturesquely

presented in elementary calculus and physics textbooks

by images of snow-covered hills and parties of skiers

who trek down their slopes like least action heroes.

The analogous scene in propositional logic presents

us with forms more reminiscent of plateaunic idylls,

being all plains at one of two levels, the mesas of

verity and falsity, as it were, with nary a niche

to inhabit between them, restricting our options

for a sporting gradient of downhill dynamics to

just one of two, standing still on level ground

or falling off a bluff.

We are still working well within the logical analogue of the

We are still working well within the logical analogue of the classical finite difference calculus, taking in the novelties that the logical transmutation of familiar elements is able to bring to light.  Soon we will take up several different notions of approximation relationships that may be seen to organize the space of propositions, and these will allow us to define several different forms of differential analysis applying to propositions. In time we will find reason to consider more general types of maps, having concrete types of the form <math>X_1 \times \ldots \times X_k \to Y_1 \times \ldots \times Y_n</math> and abstract types <math>\mathbb{B}^k \to \mathbb{B}^n.</math> We will think of these mappings as transforming universes of discourse into themselves or into others, in short, as ''transformations of discourse''.

classical finite difference calculus, taking in the novelties

that the logical transmutation of familiar elements is able to

bring to light.  Soon we will take up several different notions

of approximation relationships that may be seen to organize the

space of propositions, and these will allow us to define several

different forms of differential analysis applying to propositions.

In time we will find reason to consider more general types of maps,

having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n

and abstract types B^k -> B^n.  We will think of these mappings as

transforming universes of discourse into themselves or into others,

in short, as "transformations of discourse".

Before we continue with this intinerary, however, I would like to highlight

Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the ''boundary operator'' or the ''marked connective'' that serves as one of the two basic connectives in the cactus language for ZOL.

another sort of "differential aspect" that concerns the "boundary operator"

or the "marked connective" that serves as one of the two basic connectives

in the cactus language for ZOL.

For example, consider the proposition f of concrete type f : X x Y x Z -> B

For example, consider the proposition <math>f\!</math> of concrete type <math>f : X \times Y \times Z \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} x, y, z \texttt{)}</math> in cactus syntax. Taken as an assertion in what Peirce called the ''existential interpretation'', <math>\texttt{(} x, y, z \texttt{)}</math> says that just one of <math>x, y, z\!</math> is false.  It is useful to consider this assertion in relation to the conjunction <math>xyz\!</math> of the features that are engaged as its arguments.  A venn diagram of <math>\texttt{(} x, y, z \texttt{)}</math> looks like this:

and abstract type f : B^3 -> B that is written "(x, y, z)" in cactus syntax.

Taken as an assertion in what Peirce called the "existential interpretation",

(x, y, z) says that just one of x, y, z is false.  It is useful to consider

this assertion in relation to the conjunction xyz of the features that are

engaged as its arguments.  A venn diagram of (x, y, z) looks like this:

o-----------------------------------------------------------o

o-----------------------------------------------------------o

| U                                                        |

| U                                                        |

|                                                          |

|                                                          |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

In relation to the center cell indicated by the conjunction xyz,

In relation to the center cell indicated by the conjunction xyz,

the region indicated by (x, y, z) is comprised of the "adjacent"

the region indicated by (x, y, z) is comprised of the "adjacent"