Dynamics And Logic
Note 1
I am going to excerpt some of my previous explorations on differential logic and dynamic systems and bring them to bear on the sorts of discrete dynamical themes that we find of interest in the NKS Forum. This adaptation draws on the "Cactus Rules", "Propositional Equation Reasoning Systems", and "Reductions Among Relations" threads, and will in time be applied to the "Differential Analytic Turing Automata" thread:
One of the first things that you can do, once you have a moderately functional calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.
Let us start with a proposition of the form \(p ~\operatorname{and}~ q\) that is graphed as two labels attached to a root node:
o-------------------------------------------------o | | | p q | | @ | | | o-------------------------------------------------o | p and q | o-------------------------------------------------o |
Written as a string, this is just the concatenation "\(p~q\)".
The proposition \(pq\!\) may be taken as a boolean function \(f(p, q)\!\) having the abstract type \(f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\) where \(\mathbb{B} = \{ 0, 1 \}\) is read in such a way that \(0\!\) means \(\operatorname{false}\) and \(1\!\) means \(\operatorname{true}.\)
In this style of graphical representation, the value \(\operatorname{true}\) looks like a blank label and the value \(\operatorname{false}\) looks like an edge.
o-------------------------------------------------o | | | | | @ | | | o-------------------------------------------------o | true | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | | | | @ | | | o-------------------------------------------------o | false | o-------------------------------------------------o |
Back to the proposition \(pq.\!\) Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition \(pq\!\) is true, as shown here:
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Now ask yourself: What is the value of the proposition \(pq\!\) at a distance of \(\operatorname{d}p\) and \(\operatorname{d}q\) from the cell \(pq\!\) where you are standing?
Don't think about it — just compute:
o-------------------------------------------------o | | | dp o o dq | | / \ / \ | | p o---@---o q | | | o-------------------------------------------------o | (p, dp) (q, dq) | o-------------------------------------------------o |
To make future graphs easier to draw in ASCII, I will use devices like @=@=@ and o=o=o to identify several nodes into one, as in this next redrawing:
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | (p, dp) (q, dq) | o-------------------------------------------------o |
However you draw it, these expressions follow because the expression \(p + \operatorname{d}p,\) where the plus sign indicates addition in \(\mathbb{B},\) that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
o-------------------------------------------------o | | | p dp | | o---o | | \ / | | @ | | | o-------------------------------------------------o | (p, dp) | o-------------------------------------------------o |
Next question: What is the difference between the value of the proposition \(pq\!\) "over there" and the value of the proposition \(pq\!\) where you are, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ p q | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | ((p, dp)(q, dq), p q) | o-------------------------------------------------o |
There is one thing that I ought to mention at this point: Computed over \(\mathbb{B},\) plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where \(pq\!\) is true? Well, substituting \(1\!\) for \(p\!\) and \(1\!\) for \(q\!\) in the graph amounts to erasing the labels \(p\!\) and \(q\!,\) as shown here:
o-------------------------------------------------o | | | dp dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | (( , dp)( , dq), ) | o-------------------------------------------------o |
And this is equivalent to the following graph:
o-------------------------------------------------o | | | dp dq | | o o | | \ / | | o | | | | | @ | | | o-------------------------------------------------o | ((dp) (dq)) | o-------------------------------------------------o |
Note 2
We have just met with the fact that the differential of the and is the or of the differentials.
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\(p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q\) |
o-------------------------------------------------o | | | dp dq | | o o | | \ / | | o | | p q | | | @ --Diff--> @ | | | o-------------------------------------------------o | p q --Diff--> ((dp) (dq)) | o-------------------------------------------------o |
It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.
If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the Boole–De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax that was adequate to handle the complexity of expressions that evolve.
Let us run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way.
We begin with a proposition or a boolean function \(f(p, q) = pq.\!\)
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A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like \(f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\) or \(f : \mathbb{B}^2 \to \mathbb{B}.\) The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
| Let \(P\!\) be the set of values \(\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.\) |
| Let \(Q\!\) be the set of values \(\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \operatorname{not}~ q,~ q \} ~\cong~ \mathbb{B}.\) |
Then interpret the usual propositions about \(p, q\!\) as functions of the concrete type \(f : P \times Q \to \mathbb{B}.\)
We are going to consider various operators on these functions. Here, an operator \(\operatorname{F}\) is a function that takes one function \(f\!\) into another function \(\operatorname{F}f.\)
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
| The difference operator \(\Delta,\!\) written here as \(\operatorname{D}.\) |
| The enlargement" operator \(\Epsilon,\!\) written here as \(\operatorname{E}.\) |
These days, \(\operatorname{E}\) is more often called the shift operator.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space \(X = P \times Q,\) its (first order) differential extension \(\operatorname{E}X\) is constructed according to the following specifications:
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\(\begin{array}{rcc} \operatorname{E}X & = & X \times \operatorname{d}X \end{array}\) |
where:
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\(\begin{array}{rcc} X & = & P \times Q \\[4pt] \operatorname{d}X & = & \operatorname{d}P \times \operatorname{d}Q \\[4pt] \operatorname{d}P & = & \{ \texttt{(} \operatorname{d}p \texttt{)},~ \operatorname{d}p \} \\[4pt] \operatorname{d}Q & = & \{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \} \end{array}\) |
The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that \(\operatorname{d}p\) means "change \(p\!\)" and \(\operatorname{d}q\) means "change \(q\!\)".
Drawing a venn diagram for the differential extension \(\operatorname{E}X = X \times \operatorname{d}X\) requires four logical dimensions, \(P, Q, \operatorname{d}P, \operatorname{d}Q,\) but it is possible to project a suggestion of what the differential features \(\operatorname{d}p\) and \(\operatorname{d}q\) are about on the 2-dimensional base space \(X = P \times Q\) by drawing arrows that cross the boundaries of the basic circles in the venn diagram for \(X\!,\) reading an arrow as \(\operatorname{d}p\) if it crosses the boundary between \(p\!\) and \(\texttt{(} p \texttt{)}\) in either direction and reading an arrow as \(\operatorname{d}q\) if it crosses the boundary between \(q\!\) and \(\texttt{(} q \texttt{)}\) in either direction.
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Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone. For example, the proposition \(\texttt{(} \operatorname{d}p \texttt{(} \operatorname{d}q \texttt{))}\) says the same thing as \(\operatorname{d}p \Rightarrow \operatorname{d}q,\) in other words, that there is no change in \(p\!\) without a change in \(q.\!\)
Note 3
Given the proposition \(f(p, q)\!\) over the space \(X = P \times Q,\) the (first order) enlargement of \(f\!\) is the proposition \(\operatorname{E}f\) over the differential extension \(\operatorname{E}X\) that is defined by the following formula:
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\(\begin{matrix} \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & f(p + \operatorname{d}p,~ q + \operatorname{d}q) & = & f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} ) \end{matrix}\) |
In the example \(f(p, q) = pq,\!\) the enlargement \(\operatorname{E}f\) is computed as follows:
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\(\begin{matrix} \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & (p + \operatorname{d}p)(q + \operatorname{d}q) & = & \texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)} \end{matrix}\) |
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef = (p, dp) (q, dq) | o-------------------------------------------------o |
Given the proposition \(f(p, q)\!\) over \(X = P \times Q,\) the (first order) difference of \(f\!\) is the proposition \(\operatorname{D}f\) over \(\operatorname{E}X\) that is defined by the formula \(\operatorname{D}f = \operatorname{E}f - f,\) or, written out in full:
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\(\begin{matrix} \operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & f(p + \operatorname{d}p,~ q + \operatorname{d}q) - f(p, q) & = & \texttt{(} f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} ),~ f(p, q) \texttt{)} \end{matrix}\) |
In the example \(f(p, q) = pq,\!\) the difference \(\operatorname{D}f\) is computed as follows:
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\(\begin{matrix} \operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & (p + \operatorname{d}p)(q + \operatorname{d}q) - pq & = & \texttt{((} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}, pq \texttt{)} \end{matrix}\) |
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ p q | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df = ((p, dp)(q, dq), pq) | o-------------------------------------------------o |
We did not yet go through the trouble to interpret this (first order) difference of conjunction fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition \(pq,\!\) that is, at the place where \(p = 1\!\) and \(q = 1.\!\) This evaluation is written in the form \(\operatorname{D}f|_{pq}\) or \(\operatorname{D}f|_{(1, 1)},\) and we arrived at the locally applicable law that is stated and illustrated as follows:
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\(f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q\) |
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The picture shows the analysis of the inclusive disjunction \(\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}\) into the following exclusive disjunction:
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\(\begin{matrix} \operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)} & + & \texttt{(} \operatorname{d}p \texttt{)}~ \operatorname{d}q & + & \operatorname{d}p ~\operatorname{d}q \end{matrix}\) |
The differential proposition that results may be interpreted to say "change \(p\!\) or change \(q\!\) or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.
Note 4
Last time we computed what is variously called the difference map, the difference proposition, or the local proposition \(\operatorname{D}f_x\) of the proposition \(f(p, q) = pq\!\) at the point \(x\!\) where \(p = 1\!\) and \(q = 1.\!\)
In the universe \(X = P \times Q,\) the four propositions \(pq,~ p \texttt{(} q \texttt{)},~ \texttt{(} p \texttt{)} q,~ \texttt{(} p \texttt{)(} q \texttt{)}\) that indicate the "cells", or the smallest regions of the venn diagram, are called singular propositions. These serve as an alternative notation for naming the points \((1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!\) respectively.
Thus we can write \(\operatorname{D}f_x = \operatorname{D}f|x = \operatorname{D}f|(1, 1) = \operatorname{D}f|pq,\) so long as we know the frame of reference in force.
In the example \(f(p, q) = pq,\!\) the value of the difference proposition \(\operatorname{D}f_x\) at each of the four points in \(x \in X\!\) may be computed in graphical fashion as shown below:
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ p q | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df = ((p, dp)(q, dq), pq) | o-------------------------------------------------o |
o-------------------------------------------------o | | | dp dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df|pq = ((dp) (dq)) | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | dp | dq | | o---o o---o | | \ | | / | | \ | | / o | | \| |/ | | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df|p(q) = (dp) dq | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | | dp dq | | o---o o---o | | \ | | / | | \ | | / o | | \| |/ | | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df|(p)q = dp (dq) | o-------------------------------------------------o |
o-------------------------------------------------o | | | o o | | | dp | dq | | o---o o---o | | \ | | / | | \ | | / o o | | \| |/ \ / | | o=o-----------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ | | | o-------------------------------------------------o | Df|(p)(q) = dp dq | o-------------------------------------------------o |
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
o-------------------------------------------------o | | | e | | o-o-o-...-o-o-o | | \ / | | \ / | | \ / | | \ / e | | \ / o | | \ / | | | @ = @ | | | o-------------------------------------------------o | (e, , ... , , ) = (e) | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | e_1 e_2 e_k | | | o---o-...-o---o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / e_1 ... e_k | | @ = @ | | | o-------------------------------------------------o | (e_1, ..., e_k, ()) = e_1 ... e_k | o-------------------------------------------------o |
Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.
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The Figure shows the points of the extended universe \(\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q\) that are indicated by the difference map \(\operatorname{D}f : \operatorname{E}X \to \mathbb{B},\) namely, the following six points or singular propositions::
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\(\begin{array}{rcccc} 1. & p & q & \operatorname{d}p & \operatorname{d}q \\ 2. & p & q & \operatorname{d}p & (\operatorname{d}q) \\ 3. & p & q & (\operatorname{d}p) & \operatorname{d}q \\ 4. & p & (q) & (\operatorname{d}p) & \operatorname{d}q \\ 5. & (p) & q & \operatorname{d}p & (\operatorname{d}q) \\ 6. & (p) & (q) & \operatorname{d}p & \operatorname{d}q \end{array}\) |
The information borne by \(\operatorname{D}f\) should be clear enough from a survey of these six points — they tell you what you have to do from each point of \(X\!\) in order to change the value borne by \(f(p, q),\!\) that is, the move you have to make in order to reach a point where the value of the proposition \(f(p, q)\!\) is different from what it is where you started.
Note 5
We have been studying the action of the difference operator \(\operatorname{D}\) on propositions of the form \(f : P \times Q \to \mathbb{B},\) as illustrated by the example \(f(p, q) = pq\!\) that is known in logic as the conjunction of \(p\!\) and \(q.\!\) The resulting difference map \(\operatorname{D}f\) is a (first order) differential proposition, that is, a proposition of the form \(\operatorname{D}f : P \times Q \times \operatorname{d}P \times \operatorname{d}Q \to \mathbb{B}.\)
Abstracting from the augmented venn diagram that shows how the models or satisfying interpretations of \(\operatorname{D}f\) distribute over the extended universe of discourse \(\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,\) the difference map \(\operatorname{D}f\) can be represented in the form of a digraph or directed graph, one whose points are labeled with the elements of \(X = P \times Q\) and whose arrows are labeled with the elements of \(\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,\) as shown in the following Figure.
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\(\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \operatorname{D}f & = & p & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] & + & p & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] & + & (p) & \cdot & q & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] & + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~ \end{array}\) |
Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning. We will encounter more and more of these alternative readings as we go.
Note 6
The enlargement or shift operator \(\operatorname{E}\) exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, \(f(p, q) = pq.\!\)
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
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\(\begin{array}{lccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \operatorname{d}X & = & \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k. \\[6pt] \text{Then} & \operatorname{E}X & = & X \times \operatorname{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k \end{array}\) |
For a proposition of the form \(f : X_1 \times \ldots \times X_k \to \mathbb{B},\) the (first order) enlargement of \(f\!\) is the proposition \(\operatorname{E}f : \operatorname{E}X \to \mathbb{B}\) that is defined by the following equation:
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\(\begin{array}{l} \operatorname{E}f(x_1, \ldots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) \\[6pt] = \quad f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k) \\[6pt] = \quad f( \texttt{(} x_1, \operatorname{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \operatorname{d}x_k \texttt{)} ) \end{array}\) |
The differential variables \(\operatorname{d}x_j\) are boolean variables of the same basic type as the ordinary variables \(x_j.\!\) Although it is conventional to distinguish the (first order) differential variables with the operative prefix "\(\operatorname{d}\)" this way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
In the example of logical conjunction, \(f(p, q) = pq,\!\) the enlargement \(\operatorname{E}f\) is formulated as follows:
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\(\begin{array}{l} \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) \\[6pt] = \quad (p + \operatorname{d}p)(q + \operatorname{d}q) \\[6pt] = \quad \texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)} \end{array}\) |
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
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\(\begin{matrix} \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & p~q & + & p~\operatorname{d}q & + & q~\operatorname{d}p & + & \operatorname{d}p~\operatorname{d}q \end{matrix}\) |
To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for \(\operatorname{E}f\) in the same way that we did for \(\operatorname{D}f.\) Toward that end, the value of \(\operatorname{E}f_x\) at each \(x \in X\) may be computed in graphical fashion as shown below:
o-------------------------------------------------o | | | p dp q dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef = (p, dp) (q, dq) | o-------------------------------------------------o |
o-------------------------------------------------o | | | dp dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef|pq = (dp) (dq) | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | dp | dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef|p(q) = (dp) dq | o-------------------------------------------------o |
o-------------------------------------------------o | | | o | | | dp dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef|(p)q = dp (dq) | o-------------------------------------------------o |
o-------------------------------------------------o | | | o o | | | dp | dq | | o---o o---o | | \ | | / | | \ | | / | | \| |/ | | @=@ | | | o-------------------------------------------------o | Ef|(p)(q) = dp dq | o-------------------------------------------------o |
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition \(\operatorname{E}f.\)
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\(\begin{matrix} \operatorname{E}f & = & pq \cdot \operatorname{E}f_{pq} & + & p(q) \cdot \operatorname{E}f_{p(q)} & + & (p)q \cdot \operatorname{E}f_{(p)q} & + & (p)(q) \cdot \operatorname{E}f_{(p)(q)} \end{matrix}\) |
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element \((\operatorname{d}p)(\operatorname{d}q)\) is drawn as a loop at the point \(p~q.\)
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\(\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \operatorname{E}f & = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q) \\[4pt] & + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] & + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}\) |
We may understand the enlarged proposition \(\operatorname{E}f\) as telling us all the different ways to reach a model of the proposition \(f\!\) from each point of the universe \(X.\!\)
Note 7
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type \(P \times Q \to \mathbb{B}\) and abstract type \(\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\) A few Tables are set here that detail the actions of \(\operatorname{E}\) and \(\operatorname{D}\) on each of these functions, allowing us to view the results in several different ways.
Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
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\(\mathcal{L}_1\) \(\text{Decimal}\) |
\(\mathcal{L}_2\) \(\text{Binary}\) |
\(\mathcal{L}_3\) \(\text{Vector}\) |
\(\mathcal{L}_4\) \(\text{Cactus}\) |
\(\mathcal{L}_5\) \(\text{English}\) |
\(\mathcal{L}_6\) \(\text{Ordinary}\) |
| \(p\colon\!\) | \(1~1~0~0\!\) | ||||
| \(q\colon\!\) | \(1~0~1~0\!\) | ||||
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\(\begin{matrix} f_0 \\[4pt] f_1 \\[4pt] f_2 \\[4pt] f_3 \\[4pt] f_4 \\[4pt] f_5 \\[4pt] f_6 \\[4pt] f_7 \end{matrix}\) |
\(\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}\) |
\(\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (p)(q) \\[4pt] (p)~q~ \\[4pt] (p)~~~ \\[4pt] ~p~(q) \\[4pt] ~~~(q) \\[4pt] (p,~q) \\[4pt] (p~~q) \end{matrix}\) |
\(\begin{matrix} \text{false} \\[4pt] \text{neither}~ p ~\text{nor}~ q \\[4pt] q ~\text{without}~ p \\[4pt] \text{not}~ p \\[4pt] p ~\text{without}~ q \\[4pt] \text{not}~ q \\[4pt] p ~\text{not equal to}~ q \\[4pt] \text{not both}~ p ~\text{and}~ q \end{matrix}\) |
\(\begin{matrix} 0 \\[4pt] \lnot p \land \lnot q \\[4pt] \lnot p \land q \\[4pt] \lnot p \\[4pt] p \land \lnot q \\[4pt] \lnot q \\[4pt] p \ne q \\[4pt] \lnot p \lor \lnot q \end{matrix}\) |
|
\(\begin{matrix} f_8 \\[4pt] f_9 \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}\) |
\(\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}\) |
\(\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}\) |
\(\begin{matrix} ~~p~~q~~ \\[4pt] ((p,~q)) \\[4pt] ~~~~~q~~ \\[4pt] ~(p~(q)) \\[4pt] ~~p~~~~~ \\[4pt] ((p)~q)~ \\[4pt] ((p)(q)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} p ~\text{and}~ q \\[4pt] p ~\text{equal to}~ q \\[4pt] q \\[4pt] \text{not}~ p ~\text{without}~ q \\[4pt] p \\[4pt] \text{not}~ q ~\text{without}~ p \\[4pt] p ~\text{or}~ q \\[4pt] \text{true} \end{matrix}\) |
\(\begin{matrix} p \land q \\[4pt] p = q \\[4pt] q \\[4pt] p \Rightarrow q \\[4pt] p \\[4pt] p \Leftarrow q \\[4pt] p \lor q \\[4pt] 1 \end{matrix}\) |
|
\(\mathcal{L}_1\) \(\text{Decimal}\) |
\(\mathcal{L}_2\) \(\text{Binary}\) |
\(\mathcal{L}_3\) \(\text{Vector}\) |
\(\mathcal{L}_4\) \(\text{Cactus}\) |
\(\mathcal{L}_5\) \(\text{English}\) |
\(\mathcal{L}_6\) \(\text{Ordinary}\) |
| \(p\colon\!\) | \(1~1~0~0\!\) | ||||
| \(q\colon\!\) | \(1~0~1~0\!\) | ||||
| \(f_0\!\) | \(f_{0000}\!\) | \(0~0~0~0\) | \((~)\) | \(\text{false}\!\) | \(0\!\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0100} \\[4pt] f_{1000} \end{matrix}\) |
\(\begin{matrix} 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~1~0~0 \\[4pt] 1~0~0~0 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} \text{neither}~ p ~\text{nor}~ q \\[4pt] q ~\text{without}~ p \\[4pt] p ~\text{without}~ q \\[4pt] p ~\text{and}~ q \end{matrix}\) |
\(\begin{matrix} \lnot p \land \lnot q \\[4pt] \lnot p \land q \\[4pt] p \land \lnot q \\[4pt] p \land q \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} f_{0011} \\[4pt] f_{1100} \end{matrix}\) |
\(\begin{matrix} 0~0~1~1 \\[4pt] 1~1~0~0 \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} \text{not}~ p \\[4pt] p \end{matrix}\) |
\(\begin{matrix} \lnot p \\[4pt] p \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} f_{0110} \\[4pt] f_{1001} \end{matrix}\) |
\(\begin{matrix} 0~1~1~0 \\[4pt] 1~0~0~1 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} p ~\text{not equal to}~ q \\[4pt] p ~\text{equal to}~ q \end{matrix}\) |
\(\begin{matrix} p \ne q \\[4pt] p = q \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} f_{0101} \\[4pt] f_{1010} \end{matrix}\) |
\(\begin{matrix} 0~1~0~1 \\[4pt] 1~0~1~0 \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} \text{not}~ q \\[4pt] q \end{matrix}\) |
\(\begin{matrix} \lnot q \\[4pt] q \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} f_{0111} \\[4pt] f_{1011} \\[4pt] f_{1101} \\[4pt] f_{1110} \end{matrix}\) |
\(\begin{matrix} 0~1~1~1 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \end{matrix}\) |
\(\begin{matrix} ~(p~~q)~ \\[4pt] ~(p~(q)) \\[4pt] ((p)~q)~ \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} \text{not both}~ p ~\text{and}~ q \\[4pt] \text{not}~ p ~\text{without}~ q \\[4pt] \text{not}~ q ~\text{without}~ p \\[4pt] p ~\text{or}~ q \end{matrix}\) |
\(\begin{matrix} \lnot p \lor \lnot q \\[4pt] p \Rightarrow q \\[4pt] p \Leftarrow q \\[4pt] p \lor q \end{matrix}\) |
| \(f_{15}\!\) | \(f_{1111}\!\) | \(1~1~1~1\) | \(((~))\) | \(\text{true}\!\) | \(1\!\) |
Note 8
The next four Tables expand the expressions of \(\operatorname{E}f\) and \(\operatorname{D}f\) in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.
| \(f\!\) |
\(\operatorname{T}_{11} f\) \(\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}\) |
\(\operatorname{T}_{10} f\) \(\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}\) |
\(\operatorname{T}_{01} f\) \(\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}\) |
\(\operatorname{T}_{00} f\) \(\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}\) | |
| \(f_0\!\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} ~p~~q~ \\[4pt] ~p~(q) \\[4pt] (p)~q~ \\[4pt] (p)(q) \end{matrix}\) |
\(\begin{matrix} ~p~(q) \\[4pt] ~p~~q~ \\[4pt] (p)(q) \\[4pt] (p)~q~ \end{matrix}\) |
\(\begin{matrix} (p)~q~ \\[4pt] (p)(q) \\[4pt] ~p~~q~ \\[4pt] ~p~(q) \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} ~p~ \\[4pt] (p) \end{matrix}\) |
\(\begin{matrix} ~p~ \\[4pt] (p) \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ~q~ \\[4pt] (q) \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ~q~ \\[4pt] (q) \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} ((p)(q)) \\[4pt] ((p)~q~) \\[4pt] (~p~(q)) \\[4pt] (~p~~q~) \end{matrix}\) |
\(\begin{matrix} ((p)~q~) \\[4pt] ((p)(q)) \\[4pt] (~p~~q~) \\[4pt] (~p~(q)) \end{matrix}\) |
\(\begin{matrix} (~p~(q)) \\[4pt] (~p~~q~) \\[4pt] ((p)(q)) \\[4pt] ((p)~q~) \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
| \(f_{15}\!\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) |
| \(\text{Fixed Point Total}\!\) | \(4\!\) | \(4\!\) | \(4\!\) | \(16\!\) | |
| \(f\!\) |
\(\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}\) |
\(\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}\) |
\(\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}\) |
\(\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}\) | |
| \(f_0\!\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \\[4pt] ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \\[4pt] (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] (p) \\[4pt] ~p~ \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \\[4pt] (~) \\[4pt] (~) \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
\(\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} ~(p~~q)~ \\[4pt] ~(p~(q)) \\[4pt] ((p)~q)~ \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \\[4pt] ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ~q~ \\[4pt] (q) \\[4pt] ~q~ \\[4pt] (q) \end{matrix}\) |
\(\begin{matrix} ~p~ \\[4pt] ~p~ \\[4pt] (p) \\[4pt] (p) \end{matrix}\) |
\(\begin{matrix} (~) \\[4pt] (~) \\[4pt] (~) \\[4pt] (~) \end{matrix}\) |
| \(f_{15}\!\) | \(((~))\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
Note 9
| \(f\!\) | \(\operatorname{E}f|_{xy}\) | \(\operatorname{E}f|_{p(q)}\) | \(\operatorname{E}f|_{(p)q}\) | \(\operatorname{E}f|_{(p)(q)}\) | |
| \(f_0\!\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}p~~\operatorname{d}q~ \\[4pt] ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] (\operatorname{d}p)(\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] ~\operatorname{d}p~~\operatorname{d}q~ \\[4pt] (\operatorname{d}p)(\operatorname{d}q) \\[4pt] (\operatorname{d}p)~\operatorname{d}q~ \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] (\operatorname{d}p)(\operatorname{d}q) \\[4pt] ~\operatorname{d}p~~\operatorname{d}q~ \\[4pt] ~\operatorname{d}p~(\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p)(\operatorname{d}q) \\[4pt] (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] ~\operatorname{d}p~~\operatorname{d}q~ \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}p~ \\[4pt] (\operatorname{d}p) \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}p~ \\[4pt] (\operatorname{d}p) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p) \\[4pt] ~\operatorname{d}p~ \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p) \\[4pt] ~\operatorname{d}p~ \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ~(\operatorname{d}p,~\operatorname{d}q)~ \\[4pt] ((\operatorname{d}p,~\operatorname{d}q)) \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p,~\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p,~\operatorname{d}q)~ \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p,~\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p,~\operatorname{d}q)~ \end{matrix}\) |
\(\begin{matrix} ~(\operatorname{d}p,~\operatorname{d}q)~ \\[4pt] ((\operatorname{d}p,~\operatorname{d}q)) \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}q~ \\[4pt] (\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}q) \\[4pt] ~\operatorname{d}q~ \end{matrix}\) |
\(\begin{matrix} ~\operatorname{d}q~ \\[4pt] (\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}q) \\[4pt] ~\operatorname{d}q~ \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ((\operatorname{d}p)~\operatorname{d}q~) \\[4pt] (~\operatorname{d}p~(\operatorname{d}q)) \\[4pt] (~\operatorname{d}p~~\operatorname{d}q~) \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p)~\operatorname{d}q~) \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] (~\operatorname{d}p~~\operatorname{d}q~) \\[4pt] (~\operatorname{d}p~(\operatorname{d}q)) \end{matrix}\) |
\(\begin{matrix} (~\operatorname{d}p~(\operatorname{d}q)) \\[4pt] (~\operatorname{d}p~~\operatorname{d}q~) \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ((\operatorname{d}p)~\operatorname{d}q~) \end{matrix}\) |
\(\begin{matrix} (~\operatorname{d}p~~\operatorname{d}q~) \\[4pt] (~\operatorname{d}p~(\operatorname{d}q)) \\[4pt] ((\operatorname{d}p)~\operatorname{d}q~) \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \end{matrix}\) |
| \(f_{15}\!\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) |
| \(f\!\) | \(\operatorname{D}f|_{xy}\) | \(\operatorname{D}f|_{p(q)}\) | \(\operatorname{D}f|_{(p)q}\) | \(\operatorname{D}f|_{(p)(q)}\) | |
| \(f_0\!\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \end{matrix}\) |
\(\begin{matrix} ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \end{matrix}\) |
\(\begin{matrix} ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} \operatorname{d}p \\[4pt] \operatorname{d}p \end{matrix}\) |
\(\begin{matrix} \operatorname{d}p \\[4pt] \operatorname{d}p \end{matrix}\) |
\(\begin{matrix} \operatorname{d}p \\[4pt] \operatorname{d}p \end{matrix}\) |
\(\begin{matrix} \operatorname{d}p \\[4pt] \operatorname{d}p \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p,~\operatorname{d}q) \\[4pt] (\operatorname{d}p,~\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p,~\operatorname{d}q) \\[4pt] (\operatorname{d}p,~\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p,~\operatorname{d}q) \\[4pt] (\operatorname{d}p,~\operatorname{d}q) \end{matrix}\) |
\(\begin{matrix} (\operatorname{d}p,~\operatorname{d}q) \\[4pt] (\operatorname{d}p,~\operatorname{d}q) \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} \operatorname{d}q \\[4pt] \operatorname{d}q \end{matrix}\) |
\(\begin{matrix} \operatorname{d}q \\[4pt] \operatorname{d}q \end{matrix}\) |
\(\begin{matrix} \operatorname{d}q \\[4pt] \operatorname{d}q \end{matrix}\) |
\(\begin{matrix} \operatorname{d}q \\[4pt] \operatorname{d}q \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \end{matrix}\) |
\(\begin{matrix} ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \end{matrix}\) |
\(\begin{matrix} ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \end{matrix}\) |
\(\begin{matrix} ~~\operatorname{d}p~~\operatorname{d}q~~ \\[4pt] ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] ((\operatorname{d}p)(\operatorname{d}q)) \end{matrix}\) |
| \(f_{15}\!\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) |
Note 10
If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of \(\operatorname{E}\) and \(\operatorname{D}\) at once whelm over its discrete and finite powers to grasp them. But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.
So let us do just that.
I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table A3.
| \(f\!\) |
\(\operatorname{T}_{11} f\) \(\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}\) |
\(\operatorname{T}_{10} f\) \(\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}\) |
\(\operatorname{T}_{01} f\) \(\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}\) |
\(\operatorname{T}_{00} f\) \(\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}\) | |
| \(f_0\!\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) | \((~)\) |
|
\(\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
\(\begin{matrix} ~p~~q~ \\[4pt] ~p~(q) \\[4pt] (p)~q~ \\[4pt] (p)(q) \end{matrix}\) |
\(\begin{matrix} ~p~(q) \\[4pt] ~p~~q~ \\[4pt] (p)(q) \\[4pt] (p)~q~ \end{matrix}\) |
\(\begin{matrix} (p)~q~ \\[4pt] (p)(q) \\[4pt] ~p~~q~ \\[4pt] ~p~(q) \end{matrix}\) |
\(\begin{matrix} (p)(q) \\[4pt] (p)~q~ \\[4pt] ~p~(q) \\[4pt] ~p~~q~ \end{matrix}\) |
|
\(\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} ~p~ \\[4pt] (p) \end{matrix}\) |
\(\begin{matrix} ~p~ \\[4pt] (p) \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
\(\begin{matrix} (p) \\[4pt] ~p~ \end{matrix}\) |
|
\(\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \end{matrix}\) |
\(\begin{matrix} ((p,~q)) \\[4pt] ~(p,~q)~ \end{matrix}\) |
\(\begin{matrix} ~(p,~q)~ \\[4pt] ((p,~q)) \end{matrix}\) |
|
\(\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ~q~ \\[4pt] (q) \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
\(\begin{matrix} ~q~ \\[4pt] (q) \end{matrix}\) |
\(\begin{matrix} (q) \\[4pt] ~q~ \end{matrix}\) |
|
\(\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
\(\begin{matrix} ((p)(q)) \\[4pt] ((p)~q~) \\[4pt] (~p~(q)) \\[4pt] (~p~~q~) \end{matrix}\) |
\(\begin{matrix} ((p)~q~) \\[4pt] ((p)(q)) \\[4pt] (~p~~q~) \\[4pt] (~p~(q)) \end{matrix}\) |
\(\begin{matrix} (~p~(q)) \\[4pt] (~p~~q~) \\[4pt] ((p)(q)) \\[4pt] ((p)~q~) \end{matrix}\) |
\(\begin{matrix} (~p~~q~) \\[4pt] (~p~(q)) \\[4pt] ((p)~q~) \\[4pt] ((p)(q)) \end{matrix}\) |
| \(f_{15}\!\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) | \(((~))\) |
| \(\text{Fixed Point Total}\!\) | \(4\!\) | \(4\!\) | \(4\!\) | \(16\!\) | |
The shift operator \(\operatorname{E}\) can be understood as enacting a substitution operation on the propositional form \(f(p, q)\!\) that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:
|
\(\begin{array}{lcl} \operatorname{E} ~:~ (X \to \mathbb{B}) & \to & (\operatorname{E}X \to \mathbb{B}) \\[6pt] \operatorname{E} ~:~ f(p, q) & \mapsto & \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) \\[6pt] \operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q) & = & f(p + \operatorname{d}p, q + \operatorname{d}q) \\[6pt] & = & f( \texttt{(} p, \operatorname{d}p \texttt{)}, \texttt{(} q, \operatorname{d}q \texttt{)} ) \end{array}\) |
Therefore, if we evaluate Ef at particular values of dp and dq,
for example, dp = i and dq = j, where i, j are in B, we obtain:
E_ij : (X -> B) -> (X -> B)
E_ij : f -> E_ij f
E_ij f
= Ef | <dp = i, dq = j>
= f<p + i, q + j>
= f<(p, i), (q, j)>
The notation is a little bit awkward, but the data of the Table should
make the sense clear. The important thing to observe is that E_ij has
the effect of transforming each proposition f : X -> B into some other
proposition f' : X -> B. As it happens, the action is one-to-one and
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
is an example of what is called a "transformation group" on the set
of sixteen propositions. Bowing to a longstanding linear and local
tradition, I will therefore redub the four elements of this group
as T_00, T_01, T_10, T_11, to bear in mind their transformative
character, or nature, as the case may be. Abstractly viewed,
this group of order four has the following operation table:
o----------o----------o----------o----------o----------o
| % | | | |
| * % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o==========o==========o==========o==========o==========o
| % | | | |
| T_00 % T_00 | T_01 | T_10 | T_11 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_01 % T_01 | T_00 | T_11 | T_10 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_10 % T_10 | T_11 | T_00 | T_01 |
| % | | | |
o----------o----------o----------o----------o----------o
| % | | | |
| T_11 % T_11 | T_10 | T_01 | T_00 |
| % | | | |
o----------o----------o----------o----------o----------o
It happens that there are just two possible groups of 4 elements.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
The other is Klein's four-group V_4 (German "Vier"), which it is.
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, T_00 operates as the
group identity, fixing all 16 propositions, while the other
three group elements fix 4 propositions each, and so we get:
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. -- Amazing!
Note 11
We have been contemplating functions of the type f : X -> 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
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 another sort of "differential aspect" that concerns
the "boundary operator" or the "marked connective" that serves as
one of a pair of basic connectives in the cactus language for ZOL.
Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
cactus syntax. Taken as an assertion in what C.S. Peirce called the
"existential interpretation", the so-called boundary form "(p, q, r)"
asserts that one and only one of the propositions p, q, r is false.
It is instructive to consider this assertion in relation to the
conjunction "p q r" of the same propositions. A venn diagram
for the boundary form (p, q, r) is shown in Figure 11.
o-----------------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | | |
| | P | |
| | | |
| | | |
| | | |
| o--o----------o o----------o--o |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| / \%%%%%%%%%%o%%%%%%%%%%/ \ |
| / \%%%%%%%%/ \%%%%%%%%/ \ |
| / \%%%%%%/ \%%%%%%/ \ |
| / \%%%%/ \%%%%/ \ |
| o o--o-------o--o o |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | Q |%%%%%%%| R | |
| | |%%%%%%%| | |
| o o%%%%%%%o o |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 11. Boundary Form (p, q, r)
In relation to the center cell indicated by the conjunction pqr
the region indicated by (p, q, r) is comprised of the "adjacent"
or the "bordering" cells. Thus they are the cells that are just
across the boundary of the center cell, as if reached by way of
Leibniz's "minimal changes" from the point of origin, here, pqr.
More generally speaking, in a k-dimensional universe of discourse
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
the same form of boundary relationship is manifested for any cell
of origin that one might choose to indicate, say, by means of the
conjunction of positive and negative basis features "u_1 ... u_k",
where u_j = x_j or u_j = (x_j), for j = 1 to k. The proposition
(u_1, ..., u_k) indicates the disjunctive region consisting of
the cells that are "just next door" to the cell u_1 ... u_k.
Note 12
| Consider what effects that might conceivably have | practical bearings you conceive the objects of your | conception to have. Then, your conception of those | effects is the whole of your conception of the object. | | C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438 One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as "representation principles". As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a "closure principle". We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations. Let us return to the example of the so-called "four-group" V_4. We encountered this group in one of its concrete representations, namely, as a "transformation group" that acts on a set of objects, in this particular case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, say, in the form of the group operation table copied here: o-------o-------o-------o-------o-------o | % | | | | | * % e | f | g | h | | % | | | | o=======o=======o=======o=======o=======o | % | | | | | e % e | f | g | h | | % | | | | o-------o-------o-------o-------o-------o | % | | | | | f % f | e | h | g | | % | | | | o-------o-------o-------o-------o-------o | % | | | | | g % g | h | e | f | | % | | | | o-------o-------o-------o-------o-------o | % | | | | | h % h | g | f | e | | % | | | | o-------o-------o-------o-------o-------o This table is abstractly the same as, or isomorphic to, the versions with the E_ij operators and the T_ij transformations that we took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called "regular representations", that are always readily available, as they can be generated from the mere data of the abstract operation table itself. For example, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a logical "aggregate" of elementary dyadic relatives, that is to say, a disjunction or a logical sum whose terms represent the ordered pairs of <input : output> transactions that are produced by each group element in turn. This yields what is usually known as one of the "regular representations" of the group, specifically, the "first", the "post-", or the "right" regular representation. It has long been conventional to organize the terms in the form of a matrix: Reading "+" as a logical disjunction: G = e + f + g + h, And so, by expanding effects, we get: G = e:e + f:f + g:g + h:h + e:f + f:e + g:h + h:g + e:g + f:h + g:e + h:f + e:h + f:g + g:f + h:e More on the pragmatic maxim as a representation principle later.
Note 13
The above-mentioned fact about the regular representations
of a group is universally known as "Cayley's Theorem". It
is usually stated in the form: "Every group is isomorphic
to a subgroup of Aut(X), where X is a suitably chosen set
and Aut(X) is the group of its automorphisms". There is
in Peirce's early papers a considerable generalization
of the concept of regular representations to a broad
class of relational algebraic systems. The crux of
the whole idea can be summed up as follows:
Contemplate the effects of the symbol
whose meaning you wish to investigate
as they play out on all the stages of
conduct on which you have the ability
to imagine that symbol playing a role.
This idea of definition by way of context transforming operators
is basically the same as Jeremy Bentham's notion of "paraphrasis",
a "method of accounting for fictions by explaining various purported
terms away" (Quine, in Van Heijenoort, 'From Frege to Gödel', p. 216).
Today we'd call these constructions "term models". This, again, is
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
of lambda calculus, and I reckon you all know where that leads.
Note 14
The next few excursions in this series will provide
a scenic tour of various ideas in group theory that
will turn out to be of constant guidance in several
of the settings that are associated with our topic.
Let me return to Peirce's early papers on the algebra of relatives
to pick up the conventions that he used there, and then rewrite my
account of regular representations in a way that conforms to those.
Peirce expresses the action of an "elementary dual relative" like so:
| [Let] A:B be taken to denote
| the elementary relative which
| multiplied into B gives A.
|
| Peirce, 'Collected Papers', CP 3.123.
Peirce is well aware that it is not at all necessary to arrange the
elementary relatives of a relation into arrays, matrices, or tables,
but when he does so he tends to prefer organizing 2-adic relations
in the following manner:
a:b + a:b + a:c +
b:a + b:b + b:c +
c:a + c:b + c:c
For example, given the set X = {a, b, c}, suppose that
we have the 2-adic relative term m = "marker for" and
the associated 2-adic relation M c X x X, the general
pattern of whose common structure is represented by
the following matrix:
M =
M_aa a:a + M_ab a:b + M_ac a:c +
M_ba b:a + M_bb b:b + M_bc b:c +
M_ca c:a + M_cb c:b + M_cc c:c
It has long been customary to omit the implicit plus signs
in these matrical displays, but I have restored them here
simply as a way of separating terms in this blancophage
web format.
For at least a little while, I will make explicit
the distinction between a "relative term" like m
and a "relation" like M c X x X, but it is best
to think of both of these entities as involving
different applications of the same information,
and so we could just as easily write this form:
m =
m_aa a:a + m_ab a:b + m_ac a:c +
m_ba b:a + m_bb b:b + m_bc b:c +
m_ca c:a + m_cb c:b + m_cc c:c
By way of making up a concrete example,
let us say that M is given as follows:
a is a marker for a
a is a marker for b
b is a marker for b
b is a marker for c
c is a marker for c
c is a marker for a
In sum, we have this matrix:
M =
1 a:a + 1 a:b + 0 a:c +
0 b:a + 1 b:b + 1 b:c +
1 c:a + 0 c:b + 1 c:c
I think that will serve to fix notation
and set up the remainder of the account.
Note 15
In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients". When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:
M =
1 1 0
0 1 1
1 0 1
However the specification may come to be written, this
is all just convenient schematics for stipulating that:
M = a:a + b:b + c:c + a:b + b:c + c:a
Recognizing !1! = a:a + b:b + c:c as the identity transformation,
the 2-adic relative term m = "marker for" can be represented as an
element !1! + a:b + b:c + c:a of the so-called "group ring", all of
which makes this element just a special sort of linear transformation.
Up to this point, we are still reading the elementary relatives
of the form i:j in the way that Peirce customarily read them in
logical contexts: i is the relate, j is the correlate, and in
our current example we reading i:j, or more exactly, m_ij = 1,
to say that i is a marker for j. This is the mode of reading
that we call "multiplying on the left".
In the algebraic, permutational, or transformational contexts of
application, however, Peirce converts to the alternative mode of
reading, although still calling i the relate and j the correlate,
the elementary relative i:j now means that i gets changed into j.
In this scheme of reading, the transformation a:b + b:c + c:a is
a permutation of the aggregate $1$ = a + b + c, or what we would
now call the set {a, b, c}, in particular, it is the permutation
that is otherwise notated as:
( a b c )
< >
( b c a )
This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
Note 16
We've been exploring the applications of a certain technique
for clarifying abstruse concepts, a rough-cut version of the
pragmatic maxim that I've been accustomed to refer to as the
"operationalization" of ideas. The basic idea is to replace
the question of "What it is", which modest people comprehend
is far beyond their powers to answer any time soon, with the
question of "What it does", which most people know at least
a modicum about.
In the case of regular representations of groups we found
a non-plussing surplus of answers to sort our way through.
So let us track back one more time to see if we can learn
any lessons that might carry over to more realistic cases.
Here is is the operation table of V_4 once again:
o-------o-------o-------o-------o-------o
| % | | | |
| * % e | f | g | h |
| % | | | |
o=======o=======o=======o=======o=======o
| % | | | |
| e % e | f | g | h |
| % | | | |
o-------o-------o-------o-------o-------o
| % | | | |
| f % f | e | h | g |
| % | | | |
o-------o-------o-------o-------o-------o
| % | | | |
| g % g | h | e | f |
| % | | | |
o-------o-------o-------o-------o-------o
| % | | | |
| h % h | g | f | e |
| % | | | |
o-------o-------o-------o-------o-------o
A group operation table is really just a device for recording
a certain 3-adic relation, specifically, the set of 3-tuples
of the form <x, y, z> that satisfy the equation x * y = z,
where the sign '*' that indicates the group operation is
frequently omitted in contexts where it is understood.
In the case of V_4 = (G, *), where G is the "underlying set"
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
whose triples are listed below:
e:e:e
e:f:f
e:g:g
e:h:h
f:e:f
f:f:e
f:g:h
f:h:g
g:e:g
g:f:h
g:g:e
g:h:f
h:e:h
h:f:g
h:g:f
h:h:e
It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:
1. Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to xy.
2. Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, _>, with
the effect of producing the saturated
rheme <y, x> that evaluates to yx.
In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {<y : xy> : y in G}.
The pairs <y : xy> can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run across the top margin. This aspect of
pragmatic definition we recognize as the regular
ante-representation:
e = e:e + f:f + g:g + h:h
f = e:f + f:e + g:h + h:g
g = e:g + f:h + g:e + h:f
h = e:h + f:g + g:f + h:e
In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {<y : yx> : y in G}.
The pairs <y : yx> can be found by picking an x
from the top margin of the group operation table
and considering its effects on each y in turn as
these run down the left margin. This aspect of
pragmatic definition we recognize as the regular
post-representation:
e = e:e + f:f + g:g + h:h
f = e:f + f:e + g:h + h:g
g = e:g + f:h + g:e + h:f
h = e:h + f:g + g:f + h:e
If the ante-rep looks the same as the post-rep,
now that I'm writing them in the same dialect,
that is because V_4 is abelian (commutative),
and so the two representations have the very
same effects on each point of their bearing.
Note 17
So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group. This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {a, b, c}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):
Table 17-a. Permutations or Substitutions in Sym_{a, b, c}
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| a b c | a b c | a b c | a b c | a b c | a b c |
| | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | |
| v v v | v v v | v v v | v v v | v v v | v v v |
| | | | | | |
| a b c | c a b | b c a | a c b | c b a | b a c |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
Here is the operation table for S_3, given in abstract fashion:
Table 17-b. Symmetric Group S_3
o-------------------------------------------------o
| |
| o |
| e / \ e |
| / \ |
| / e \ |
| f / \ / \ f |
| / \ / \ |
| / f \ f \ |
| g / \ / \ / \ g |
| / \ / \ / \ |
| / g \ g \ g \ |
| h / \ / \ / \ / \ h |
| / \ / \ / \ / \ |
| / h \ e \ e \ h \ |
| i / \ / \ / \ / \ / \ i |
| / \ / \ / \ / \ / \ |
| / i \ i \ f \ j \ i \ |
| j / \ / \ / \ / \ / \ / \ j |
| / \ / \ / \ / \ / \ / \ |
| o j \ j \ j \ i \ h \ j o |
| \ / \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / \ / |
| \ h \ h \ e \ j \ i / |
| \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / |
| \ i \ g \ f \ h / |
| \ / \ / \ / \ / |
| \ / \ / \ / \ / |
| \ f \ e \ g / |
| \ / \ / \ / |
| \ / \ / \ / |
| \ g \ f / |
| \ / \ / |
| \ / \ / |
| \ e / |
| \ / |
| \ / |
| o |
| |
o-------------------------------------------------o
I think that the NKS reader can guess how we might apply
this group to the space of propositions of type B^3 -> B.
By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
Note 18
By way of collecting a short-term pay-off for all the work that we
did on the regular representations of the Klein 4-group V_4, let us
write out as quickly as possible in "relative form" a minimal budget
of representations for the symmetric group on three letters, Sym(3).
After doing the usual bit of compare and contrast among the various
representations, we will have enough concrete material beneath our
abstract belts to tackle a few of the presently obscured details
of Peirce's early "Algebra + Logic" papers.
Writing the permutations or substitutions of Sym {a, b, c}
in relative form generates what is generally thought of as
a "natural representation" of S_3.
e = a:a + b:b + c:c
f = a:c + b:a + c:b
g = a:b + b:c + c:a
h = a:a + b:c + c:b
i = a:c + b:b + c:a
j = a:b + b:a + c:c
I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as x:y
constitutes the turning of x into y. It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.
Note 19
To construct the regular representations of S_3,
we pick up from the data of its operation table,
DAL 17, Table 17-b, at either one of these sites:
http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
Just by way of staying clear about what we are doing,
let's return to the recipe that we worked out before:
It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:
1. Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to xy.
2. Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, _>, with
the effect of producing the saturated
rheme <y, x> that evaluates to yx.
In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {<y : xy> : y in G}.
The pairs <y : xy> can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin. This produces
the regular ante-representation of S_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:j + i:h + j:i
g = e:g + f:e + g:f + h:i + i:j + j:h
h = e:h + f:i + g:j + h:e + i:f + j:g
i = e:i + f:j + g:h + h:g + i:e + j:f
j = e:j + f:h + g:i + h:f + i:g + j:e
In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {<y : yx> : y in G}.
The pairs <y : yx> can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin. This generates
the regular post-representation of S_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:i + i:j + j:h
g = e:g + f:e + g:f + h:j + i:h + j:i
h = e:h + f:j + g:i + h:e + i:g + j:f
i = e:i + f:h + g:j + h:f + i:e + j:g
j = e:j + f:i + g:h + h:g + i:f + j:e
If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.
Note 20
You may be wondering what happened to the announced subject
of "Dynamics And Logic". What occurred was a bit like this:
We happened to make the observation that the shift operators {E_ij}
form a transformation group that acts on the set of propositions of
the form f : B x B -> B. Group theory is a very attractive subject,
but it did not draw us so far from our intended course as one might
initially think. For one thing, groups, especially the groups that
are named after the Norwegian mathematician Marius Sophus Lie, turn
out to be of critical importance in solving differential equations.
For another thing, group operations provide us with an ample supply
of triadic relations that have been extremely well-studied over the
years, and thus they give us no small measure of useful guidance in
the study of sign relations, another brand of 3-adic relations that
have significance for logical studies, and in our acquaintance with
which we have scarcely begun to break the ice. Finally, I couldn't
resist taking up the links between group representations, amounting
to the very archetypes of logical models, and the pragmatic maxim.
Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
Note 21
We have seen a couple of groups, V_4 and S_3, represented in
several different ways, and we have seen each of these types
of representation presented in several different fashions.
Let us look at one other stylistic variant for presenting
a group representation that is often used, the so-called
"matrix representation" of a group.
Returning to the example of Sym(3), we first encountered
this group in concrete form as a set of permutations or
substitutions acting on a set of letters X = {a, b, c}.
This set of permutations was displayed in Table 17-a,
copies of which can be found here:
http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
These permutations were then converted to "relative form":
e = a:a + b:b + c:c
f = a:c + b:a + c:b
g = a:b + b:c + c:a
h = a:a + b:c + c:b
i = a:c + b:b + c:a
j = a:b + b:a + c:c
From this relational representation of Sym {a, b, c} ~=~ S_3,
one easily derives a "linear representation", regarding each
permutation as a linear transformation that maps the elements
of a suitable vector space into each other, and representing
each of these linear transformations by means of a matrix,
resulting in the following set of matrices for the group:
Table 21. Matrix Representations of the Permutations in S_3
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 |
| 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 |
| 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
The key to the mysteries of these matrices is revealed by
observing that their coefficient entries are arrayed and
overlayed on a place mat that's marked like so:
o-----o-----o-----o
| a:a | a:b | a:c |
o-----o-----o-----o
| b:a | b:b | b:c |
o-----o-----o-----o
| c:a | c:b | c:c |
o-----o-----o-----o
Note 22
It would be good to summarize, in rough but intuitive terms,
the outlook on differential logic that we have reached so far.
We've been considering a class of operators on universes
of discourse, each of which takes us from considering one
universe of discourse, X%, to considering a larger universe
of discourse, EX%.
Each of these operators, in broad terms having the form
W : X% -> EX%, acts on each proposition f : X -> B of the
source universe X% to produce a proposition Wf : EX -> B
of the target universe EX%.
The two main operators that we have worked with up to this
point are the enlargement or shift operator E : X% -> EX%
and the difference operator D : X% -> EX%.
E and D take a proposition in X%, that is, a proposition f : X -> B
that is said to be "about" the subject matter of X, and produce the
extended propositions Ef, Df : EX -> B, which may be interpreted as
being about specified collections of changes that might occur in X.
Here we have need of visual representations,
some array of concrete pictures to anchor our
more earthy intuitions and to help us keep our
wits about us before we try to climb any higher
into the ever more rarefied air of abstractions.
One good picture comes to us by way of the "field" concept.
Given a space X, a "field" of a specified type Y over X is
formed by assigning to each point of X an object of type Y.
If that sounds like the same thing as a function from X to
the space of things of type Y -- it is -- but it does seem
helpful to vary the mental images and to take advantage of
the figures of speech that spring to mind under the emblem
of this field idea.
In the field picture, a proposition f : X -> B becomes
a "scalar" field, that is, a field of values in B, or
a "field of model indications" (FOMI).
Let us take a moment to view an old proposition
in this new light, for example, the conjunction
pq : X -> B that is depicted in Figure 22-a.
o-------------------------------------------------o
| |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| / /%%%\ \ |
| o o%%%%%o o |
| | |%%%%%| | |
| | P |%%%%%| Q | |
| | |%%%%%| | |
| o o%%%%%o o |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
Figure 22-a. Conjunction pq : X -> B
Each of the operators E, D : X% -> EX% takes us from considering
propositions f : X -> B, here viewed as "scalar fields" over X,
to considering the corresponding "differential fields" over X,
analogous to what are usually called "vector fields" over X.
The structure of these differential fields can be described this way.
To each point of X there is attached an object of the following type:
a proposition about changes in X, that is, a proposition g : dX -> B.
In this frame, if X% is the universe that is generated by the set of
coordinate propositions {p, q}, then dX% is the differential universe
that is generated by the set of differential propositions {dp, dq}.
These differential propositions may be interpreted as indicating
"change in p" and "change in q", respectively.
A differential operator W, of the first order sort that we have
been considering, takes a proposition f : X -> B and gives back
a differential proposition Wf: EX -> B.
In the field view, we see the proposition f : X -> B as a scalar field
and we see the differential proposition Wf: EX -> B as a vector field,
specifically, a field of propositions about contemplated changes in X.
The field of changes produced by E on pq is shown in Figure 22-b.
o-------------------------------------------------o
| |
| |
| o-------------o o-------------o |
| / \ / \ |
| / P o Q \ |
| / /%\ \ |
| / /%%%\ \ |
| o o.->-.o o |
| | p(q)(dp)dq |%\%/%| (p)q dp(dq) | |
| | o---------------|->o<-|---------------o | |
| | |%%^%%| | |
| o o%%|%%o o |
| \ \%|%/ / |
| \ \|/ / |
| \ o / |
| \ /|\ / |
| o-------------o | o-------------o |
| | |
| | |
| | |
| o |
| (p)(q) dp dq |
| |
o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
| |
| Ef = p q (dp)(dq) |
| |
| + p (q) (dp) dq |
| |
| + (p) q dp (dq) |
| |
| + (p)(q) dp dq |
| |
o-------------------------------------------------o
Figure 22-b. Enlargement E[pq] : EX -> B
The differential field E[pq] specifies the changes
that need to be made from each point of X in order
to reach one of the models of the proposition pq,
that is, in order to satisfy the proposition pq.
The field of changes produced by D on pq is shown in Figure 22-c.
o-------------------------------------------------o
| |
| |
| o-------------o o-------------o |
| / \ / \ |
| / P o Q \ |
| / /%\ \ |
| / /%%%\ \ |
| o o%%%%%o o |
| | (dp)dq |%%%%%| dp(dq) | |
| | o<--------------|->o<-|-------------->o | |
| | |%%^%%| | |
| o o%%|%%o o |
| \ \%|%/ / |
| \ \|/ / |
| \ o / |
| \ /|\ / |
| o-------------o | o-------------o |
| | |
| | |
| v |
| o |
| dp dq |
| |
o-------------------------------------------------o
| f = p q |
o-------------------------------------------------o
| |
| Df = p q ((dp)(dq)) |
| |
| + p (q) (dp) dq |
| |
| + (p) q dp (dq) |
| |
| + (p)(q) dp dq |
| |
o-------------------------------------------------o
Figure 22-c. Difference D[pq] : EX -> B
The differential field D[pq] specifies the changes
that need to be made from each point of X in order
to feel a change in the felt value of the field pq.
Note 23
I want to continue developing the basic tools of differential logic, which arose out of many years of thinking about the connections between dynamics and logic -- those there are and those there ought to be -- 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. A cybernetic system has goals and actions for reaching them. It has a state space X, giving us all of the states that the system can be in, plus it has a goal space G c X, the set of states that the system "likes" to be in, in other words, the distinguished subset of possible states where the system is regarded as living, surviving, or thriving, depending on the type of goal that one has in mind for the system in question. As for actions, there is to begin with the full set !T! of all possible actions, each of which is a transformation of the form T : X -> X, but a given cybernetic system will most likely have but a subset of these actions available to it at any given time. And even if we begin by thinking of actions in very general and very global terms, as arbitrarily complex transformations acting on the whole state space X, we quickly find a need to analyze and approximate them in terms of simple transformations acting locally. The preferred measure of "simplicity" will of course vary from one paradigm of research to another. A generic enough picture at this stage of the game, and one that will remind us of these fundamental features of the cybernetic system even as things get far more complex, is afforded by Figure 23. o---------------------------------------------------------------------o | | | X | | o-------------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | | | | | | | | | | | | | G | | | | | | | | | | | | | | | o o | | \ / | | \ / | | \ T / | | \ o<------------/-------------o | | \ / | | \ / | | \ / | | o-------------------o | | | | | o---------------------------------------------------------------------o Figure 23. Elements of a Cybernetic System
Note 24
Now that we've introduced the field picture for thinking about
propositions and their analytic series, a very pleasing way of
picturing the relationship among a proposition f : X -> B, its
enlargement or shift map Ef : EX -> B, and its difference map
Df : EX -> B can now be drawn.
To illustrate this possibility, let's return to the differential
analysis of the conjunctive proposition f<p, q> = pq, giving the
development a slightly different twist at the appropriate point.
Figure 24-1 shows the proposition pq once again, which we now view
as a scalar field, in effect, a potential "plateau" of elevation 1
over the shaded region, with an elevation of 0 everywhere else.
o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| / /%%%\ \ |
| / /%%%%%\ \ |
| / /%%%%%%%\ \ |
| / /%%%%%%%%%\ \ |
| o o%%%%%%%%%%%o o |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | P |%%%%%%%%%%%| Q | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| | |%%%%%%%%%%%| | |
| o o%%%%%%%%%%%o o |
| \ \%%%%%%%%%/ / |
| \ \%%%%%%%/ / |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------------o o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 24-1. Proposition pq : X -> B
Given any proposition f : X -> B, the "tacit extension" of f to EX
is notated !e!f : EX -> B and defined by the equation !e!f = f, so
it's really just the same proposition living in a bigger universe.
Tacit extensions formalize the intuitive idea that a new function
is related to an old function in such a way that it obeys the same
constraints on the old variables, with a "don't care" condition on
the new variables.
Figure 24-2 illustrates the "tacit extension" of the proposition
or scalar field f = pq : X -> B to give the extended proposition
or differential field that we notate as !e!f = !e![pq] : EX -> B.
o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (dp) (dq) o o |
| | | o-->--o | | |
| | | \ / | | |
| | (dp) dq | \ / | dp (dq) | |
| | o<-----------------o----------------->o | |
| | | | | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| dp | dq |
| | |
| v |
| o |
| |
o---------------------------------------------------------------------o
Figure 24-2. Tacit Extension !e![pq] : EX -> B
Thus we have a pictorial way of visualizing the following data:
!e![pq]
=
p q . dp dq
+
p q . dp (dq)
+
p q . (dp) dq
+
p q . (dp)(dq)
Note 25
Staying with the example pq : X -> B, Figure 25-1 shows the enlargement or shift map E[pq] : EX -> B in the same style of differential field picture that we drew for the tacit extension !e![pq] : EX -> B. o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (dp) (dq) o o | | | | o-->--o | | | | | | \ / | | | | | (dp) dq | \ / | dp (dq) | | | | o----------------->o<-----------------o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | | | | o | | | o---------------------------------------------------------------------o Figure 25-1. Enlargement E[pq] : EX -> B A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields !e!f and Ef, both of the type EX -> B, is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras. In this case one notices that the tacit extension !e!f and the enlargement Ef are in a certain sense dual to each other, with !e!f indicating all of the arrows out of the region where f is true, and with Ef indicating all of the arrows into the region where f is true. The only arc that they have in common is the no-change loop (dp)(dq) at pq. If we add the two sets of arcs mod 2, then the common loop drops out, leaving the 6 arrows of D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2. o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | (dp) dq | | dp (dq) | | | | o<---------------->o<---------------->o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 25-2. Difference Map D[pq] : EX -> B The differential features of D[pq] may be collected cell by cell of the underlying universe X% = [p, q] to give the following expansion: D[pq] = p q . ((dp)(dq)) + p (q) . (dp) dq + (p) q . dp (dq) + (p)(q) . dp dq
Note 26
If we follow the classical line that singles out linear functions
as ideals of simplicity, then we may complete the analytic series
of the proposition f = pq : X -> B in the following way.
Figure 26-1 shows the differential proposition df = d[pq] : EX -> B
that we get by extracting the cell-wise linear approximation to the
difference map Df = D[pq] : EX -> B. This is the logical analogue
of what would ordinarily be called 'the' differential of pq, but
since I've been attaching the adjective "differential" to just
about everything in sight, the distinction tends to be lost.
For the time being, I'll resort to using the alternative
name "tangent map" for df.
o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / o \ \ |
| / / ^ ^ \ \ |
| o o / \ o o |
| | | / \ | | |
| | | / \ | | |
| | |/ \| | |
| | (dp)/ dq dp \(dq) | |
| | /| |\ | |
| | / | | \ | |
| | / | | \ | |
| o / o o \ o |
| \ v \ dp dq / v / |
| \ o<--------------------->o / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------------o o-------------------o |
| |
| |
o---------------------------------------------------------------------o
Figure 26-1. Differential or Tangent d[pq] : EX -> B
Just to be clear about what's being indicated here,
it's a visual way of specifying the following data:
d[pq]
=
p q . (dp, dq)
+
p (q) . dq
+
(p) q . dp
+
(p)(q) . 0
To understand the extended interpretations, that is,
the conjunctions of basic and differential features
that are being indicated here, it may help to note
the following equivalences:
(dp, dq) = dp + dq = dp(dq) + (dp)dq
dp = dp dq + dp(dq)
dq = dp dq + (dp)dq
Capping the series that analyzes the proposition pq
in terms of succeeding orders of linear propositions,
Figure 26-2 shows the remainder map r[pq] : EX -> B,
that happens to be linear in pairs of variables.
o---------------------------------------------------------------------o
| |
| X |
| o-------------------o o-------------------o |
| / \ / \ |
| / P o Q \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | | dp dq | | |
| | o<------------------------------->o | |
| | | | | |
| | | | | |
| | | o | | |
| o o ^ o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ dp | dq / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| | |
| | |
| v |
| o |
| |
o---------------------------------------------------------------------o
Figure 26-2. Remainder r[pq] : EX -> B
Reading the arrows off the map produces the following data:
r[pq]
=
p q . dp dq
+
p (q) . dp dq
+
(p) q . dp dq
+
(p)(q) . dp dq
In short, r[pq] is a constant field,
having the value dp dq at each cell.
A more detailed presentation of Differential Logic can be found here:
DLOG D. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
Document History
Ontology List (Apr–Jul 2002)
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- http://suo.ieee.org/ontology/msg04079.html
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- http://suo.ieee.org/ontology/msg04268.html
- http://suo.ieee.org/ontology/msg04269.html
- http://suo.ieee.org/ontology/msg04272.html
- http://suo.ieee.org/ontology/msg04273.html
- http://suo.ieee.org/ontology/msg04290.html
Inquiry List (May & Jul 2004)
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NKS Forum (May & Jul 2004)
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