Inquiry Into InquiryDifferential Logic • 14
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 • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi