Field Picture

Let us summarize the outlook on differential logic we’ve 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 to considering a larger universe of discourse   An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe

The operators we’ve examined so far are the enlargement or shift operator and the difference operator   The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept.  Given a space a field of a specified type over is formed by associating with each point of an object of type   If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition becomes a scalar field, that is, a field of values in

For example, consider the logical conjunction shown in the following venn diagram.

Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over

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

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator acts on the set of sixteen functions are shown below.  Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

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

Transforms Expanded over Ordinary and Differential Variables

A first view of how the shift operator acts on the set of sixteen functions was provided by Table A3 in the previous post, expanding the expressions of over the set of ordinary variables.

A complementary view of the same material is provided by Table 4 below, this time expanding the expressions of over the set of differential variables.

Enlargement Map Expanded over Differential Variables

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

Transforms Expanded over Ordinary and Differential Variables

As promised last time, in the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators and act on that set.  There being some advantage to singling out the enlargement or shift operator in its own right, we’ll begin by computing for each of the functions

Enlargement Map Expanded over Ordinary Variables

We first encountered the shift operator when we imagined ourselves being in a state described by the truth of a certain proposition and contemplated the value of that proposition in various other states, as determined by a collection of differential propositions describing the steps we might take to change our state.

Restated in terms of our initial example, we imagined ourselves being in a state described by the truth of the proposition and contemplated the value of that proposition in various other states, as determined by the differential propositions and describing the steps we might take to change our state.

Those thoughts led us from the boolean function of two variables to the boolean function of four variables as shown in the entry for in the first three columns of Table A3.

Let’s catch a breath here and discuss the full Table next time.

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

Propositional Forms on Two Variables

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as expressed in several notations.  In future discussions the two Tables will be described as the Index Order and the Orbit Order of propositions, respectively, “orbits” being the usual term in mathematics for similarity classes under a group action.  For ease of comparison, here are fresh copies of both Tables on the same page.

Recalling the discussion up to this point, we took as our first example the boolean function corresponding to the logical conjunction and examined how the differential operators and act on   Each operator takes the boolean function of two variables and gives back a boolean function of four variables, and respectively.

In the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators and act on that set.  There being some advantage to singling out the enlargement or shift operator in its own right, we’ll begin by computing for each function in the above Tables.

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