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 discussion of logical difference calculus in the personal correspondence between Boole and De Morgan and Peirce, too, made use of differential operators in a logical context, but the exploration of those 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.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

Cactus Language for Propositional Logic (cont.)

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

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

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 that 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 is a parenthesized sequence of propositional expressions, written to mean exactly one of the propositions is false, in short, their minimal negation is true.  An expression of that form is associated with a cactus structure called a lobe and is “painted” with the colors as shown below.

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

All other propositional connectives can be obtained through combinations of the above 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.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

Introduction

Differential logic is the component of logic whose object is the description of variation — focusing on the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system, its differential component governs the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by differential propositional calculi.  A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

• Logic Syllabus
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization