Inquiry Into Inquiry

Propositional Forms on Two Variables

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar shapes into seven subclasses.  Thereby hangs many a tale, to be told in time.

Table A2.  Propositional Forms on Two Variables

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Re: Structural Modeling • Joseph Simpson

Thanks, Joe, glad you liked the table, I’ve got a million of ’em!  I’ll be setting another mess of tables directly as we continue studying the effects of differential operators on families of propositional forms.

For anyone wondering, Where’s the Peirce? — the efficiency of Peirce’s logical graphs, augmented by Spencer Brown and generalized to cactus graphs, made it possible for the first time to take on the extra complexities of differential propositional calculus.  This makes Peircean themes a constant throughout the development of differential logic.

Hope you and yours are safe and sound,

Jon

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Propositional Forms on Two Variables

To broaden our experience with simple examples, let’s examine the sixteen functions of concrete type and abstract type   Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of and on this family of forms.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each boolean function in several systems of notation.

Table A1.  Propositional Forms on Two Variables

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Differential Expansions of Propositions

Panoptic View • Enlargement Maps

The enlargement or shift operator exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example,

A suitably generic definition of the extended universe of discourse is afforded by the following set‑up.

For a proposition of the form the (first order) enlargement of is the proposition defined by the following equation.

The differential variables are boolean variables of the same type as the ordinary variables   Although it is conventional to distinguish the (first order) differential variables with the operational prefix this way of notating differential variables is entirely optional.  It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.

In the example of logical conjunction, the enlargement is formulated as follows.

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.

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for in the same way we did for   To that end, the value of at each may be computed in graphical fashion as shown below.

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the enlarged proposition

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element is drawn as a loop at the point

We may understand the enlarged proposition as telling us all the ways of reaching a model of the proposition from the points of the universe

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Differential Expansions of Propositions

Panoptic View • Difference Maps

In the previous section we computed what is variously described as the difference map, the difference proposition, or the local proposition of the proposition at the point where and

In the universe of discourse the four propositions indicating the “cells”, or the smallest distinguished regions of the universe, are called singular propositions.  These serve as an alternative notation for naming the points respectively.

Thus we can write so long as we know the frame of reference in force.

In the example the value of the difference proposition at each of the four points may be computed in graphical fashion as shown below.

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents.

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

The Figure shows the points of the extended universe indicated by the difference map namely, the following six points or singular propositions.

The information borne by should be clear enough from a survey of these six points — they tell you what you have to do from each point of in order to change the value borne by that is, the move you have to make in order to reach a point where the value of the proposition is different from what it is where you started.

We have been studying the action of the difference operator on propositions of the form as illustrated by the example which is known in logic as the conjunction of and   The resulting difference map is a (first order) differential proposition, that is, a proposition of the form

The augmented venn diagram shows how the models or satisfying interpretations of distribute over the extended universe of discourse   Abstracting from that picture, the difference map can be represented in the form of a digraph or directed graph, one whose points are labeled with the elements of and whose arrows are labeled with the elements of as shown in the following Figure.

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal previously unsuspected aspects of the proposition’s meaning.  We will encounter more and more of these alternative readings as we go.

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Differential Expansions of Propositions

Worm’s Eye View

Let’s run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way.  We begin with a proposition or a boolean function whose venn diagram and cactus graph are shown below.

A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like or   The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.

Let be the set of values
Let be the set of values

Then interpret the usual propositions about as functions of the concrete type

We are going to consider various operators on these functions.  An operator is a function which takes one function into another function

The first couple of operators we need to consider are logical analogues of two which play a founding role in the classical finite difference calculus, namely:

The difference operator written here as
The enlargement operator, written here as

These days, 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 its (first order) differential extension is constructed according to the following specifications.

where:

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say means “change ” and means “change ”.

Drawing a venn diagram for the differential extension requires four logical dimensions, but it is possible to project a suggestion of what the differential features and are about on the 2-dimensional base space by drawing arrows crossing the boundaries of the basic circles in the venn diagram for reading an arrow as if it crosses the boundary between and in either direction and reading an arrow as if it crosses the boundary between and in either direction, as indicated in the following figure.

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways propositions are formed on ordinary logical variables alone.  For example, the proposition says the same thing as in other words, there is no change in without a change in

Given the proposition over the space the (first order) enlargement of is the proposition over the differential extension defined by the following formula.

In the example the enlargement is computed as follows.

Given the proposition over the (first order) difference of is the proposition over defined by the formula or, written out in full:

In the example the difference is computed as follows.

At the end of the previous section we evaluated this first order difference of conjunction at a single location in the universe of discourse, namely, at the point picked out by the singular proposition in terms of coordinates, at the place where and   This evaluation is written in the form or and we arrived at the locally applicable law which may be stated and illustrated as follows.

The venn diagram shows the analysis of the inclusive disjunction into the following exclusive disjunction.

The differential proposition may be read as saying “change or change 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.

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Differential Expansions of Propositions

Bird’s Eye View

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form graphed as two letters attached to a root node, as shown below.

Written as a string, this is just the concatenation .

The proposition may be taken as a boolean function having the abstract type where is read in such a way that means and means

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition is true, as shown in the following Figure.

Now ask yourself:  What is the value of the proposition at a distance of and from the cell where you are standing?

Don’t think about it — just compute:

The cactus formula and its corresponding graph arise by replacing with and with in the boolean product or logical conjunction and writing the result in the two dialects of cactus syntax.  This follows because the boolean sum is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form.

Next question:  What is the difference between the value of the proposition over there, at a distance of and from where you are standing, and the value of the proposition where you are, all expressed in the form of a general formula, of course?  The answer takes the following form.

There is one thing I ought to mention at this point:  Computed over 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 is true?  Well, replacing with and with in the cactus graph amounts to erasing the labels and as shown below.

And this is equivalent to the following graph:

We have just met with the fact that the differential of the and is the or of the differentials.

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 adequate to handle the complexity of expressions evolving in the process.

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Cactus Language for Propositional Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

Table 1.  Syntax and Semantics of a Calculus for Propositional Logic

The simplest expression for logical truth is the empty word, typically denoted by or in formal languages, where it is the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression or, especially if operating in an algebraic context, by a simple   Also when working in an algebraic mode, the plus sign may be used for exclusive disjunction.  Thus we have the following translations of algebraic expressions into cactus expressions.

It is important to note the last expressions are not equivalent to the 3-place form

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Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions.  One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable -ary scope.  The syntactic formulas of this calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written and meaning exactly one of the propositions is false, in short, their minimal negation is true.  An expression of this form maps into a cactus structure called a lobe, in this case, “painted” with the colors as shown below.

The second kind of connective is a concatenated sequence of propositional expressions, written and meaning all the propositions are true, in short, their logical conjunction is true.  An expression of this form maps into a cactus structure called a node, in this case, “painted” with the colors as shown below.

All other propositional connectives can be obtained through combinations of these two forms.  As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface may be used for the logical operators.

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