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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

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More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get: <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math> Amazing!

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==~~Note 11==~~

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==Operational Representation==

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We have been contemplating functions of the type <math>f : X \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.

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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''.

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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.

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For example, consider the proposition <math>f\!</math> of concrete type <math>f : P \times Q \times R \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} p, q, r \texttt{)}</math> in cactus syntax. Taken as an assertion in what Peirce called the ''existential interpretation'', the proposition <math>\texttt{(} p, q, r \texttt{)}</math> says that just one of <math>p, q, r\!</math> is false. It is instructive to consider this assertion in relation to the logical conjunction <math>pqr\!</math> of the same propositions. A venn diagram of <math>\texttt{(} p, q, r \texttt{)}</math> looks like this:

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~~{| align="center" cellpadding="10"~~

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~~| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]~~

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|}

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In relation to the center cell indicated by the conjunction <math>pqr,\!</math> the region indicated by <math>\texttt{(} p, q, r \texttt{)}</math> is comprised of the adjacent or bordering cells. Thus they are the cells that are just across the boundary of the center cell, reached as if by way of Leibniz's ''minimal changes'' from the point of origin, in this case, <math>pqr.\!</math>

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More generally speaking, in a <math>k\!</math>-dimensional universe of discourse that is based on the ''alphabet'' of features <math>\mathcal{X} = \{ x_1, \ldots, x_k \},</math> the same form of boundary relationship is manifested for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form <math>e_1 \cdot \ldots \cdot e_k,</math> where <math>e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},</math> for <math>j = 1 ~\text{to}~ k.</math> The proposition <math>\texttt{(} e_1, \ldots, e_k \texttt{)}</math> indicates the disjunctive region consisting of the cells that are just next door to <math>e_1 \cdot \ldots \cdot e_k.</math>

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~~==Note 12~~==

{| align="center" cellpadding="0" cellspacing="0" width="90%"

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In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.

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==~~Applications~~==

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==Least Action Operators==

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We have been contemplating functions of the type <math>f : X \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 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''.

+

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 [[zeroth order logic]].

+

For example, consider the proposition <math>f\!</math> of concrete type <math>f : P \times Q \times R \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} p, q, r \texttt{)}</math> in cactus syntax. Taken as an assertion in what Peirce called the ''existential interpretation'', the proposition <math>\texttt{(} p, q, r \texttt{)}</math> says that just one of <math>p, q, r\!</math> is false. It is instructive to consider this assertion in relation to the logical conjunction <math>pqr\!</math> of the same propositions. A venn diagram of <math>\texttt{(} p, q, r \texttt{)}</math> looks like this:

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{| align="center" cellpadding="10"

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| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]

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|}

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In relation to the center cell indicated by the conjunction <math>pqr,\!</math> the region indicated by <math>\texttt{(} p, q, r \texttt{)}</math> is comprised of the adjacent or bordering cells. Thus they are the cells that are just across the boundary of the center cell, reached as if by way of Leibniz's ''minimal changes'' from the point of origin, in this case, <math>pqr.\!</math>

+

More generally speaking, in a <math>k\!</math>-dimensional universe of discourse that is based on the ''alphabet'' of features <math>\mathcal{X} = \{ x_1, \ldots, x_k \},</math> the same form of boundary relationship is manifested for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form <math>e_1 \cdot \ldots \cdot e_k,</math> where <math>e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},</math> for <math>j = 1 ~\text{to}~ k.</math> The proposition <math>\texttt{(} e_1, \ldots, e_k \texttt{)}</math> indicates the disjunctive region consisting of the cells that are just next door to <math>e_1 \cdot \ldots \cdot e_k.</math>

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==Goal-Oriented Systems==

I want to continue developing the basic tools of differential logic, which arose from exploring the connections between dynamics and logic, but I also wanted to give some hint of the applications that have motivated this work all along. One of these applications is to cybernetic systems, whether we see these systems as agents or cultures, individuals or species, organisms or organizations.

Jon Awbrey

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