Differential Propositional Calculus • 7
•
https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calcul…
Note. Please see the blog post linked above for the proper formats of the
notations used below, as they depend on numerous typographical distinctions
lost in the following transcript.
Formal Development —
The preceding discussion outlined the ideas leading to the differential extension
of propositional logic. The next task is to lay out the concepts and terminology
needed to describe various orders of differential propositional calculi.
Elementary Notions —
Logical description of a universe of discourse begins with a collection
of logical signs. For simplicity in a first approach we assume the signs
are collected in the form of a finite alphabet, ‡A‡ = {“a₁”, …, “aₙ”}.
The signs are interpreted as denoting logical features, for example,
properties of objects in the universe of discourse or simple propositions
about those objects. Corresponding to the alphabet ‡A‡ there is then
a set of logical features, †A† = {a₁, …, aₙ}.
A set of logical features †A† = {a₁, …, aₙ} affords a basis for generating
an n‑dimensional universe of discourse, written A° = [†A†] = [a₁, …, aₙ].
It is useful to consider a universe of discourse as a categorical object
incorporating both the set of points A = <a₁, …, aₙ> and the set of
propositions A↑ = {f : A → B} implicit with the ordinary picture of
a venn diagram on n features.
Accordingly, the universe of discourse A° may be regarded as an ordered
pair (A, A↑) bearing the type (Bⁿ, (Bⁿ → B)), which type designation may
be abbreviated as Bⁿ +→ B or even more succinctly as [Bⁿ]. For convenience,
the data type of a finite set on n elements may be indicated by either one
of the equivalent notations [n] or *n*.
Table 7 summarizes the basic notations needed to describe
ordinary propositional calculi in a systematic fashion.
Table 7. Propositional Calculus • Basic Notation
•
https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calcul…
Resources —
Logic Syllabus
•
https://inquiryintoinquiry.com/logic-syllabus/
Survey of Differential Logic
•
https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7
Regards,
Jon
cc:
https://www.academia.edu/community/LxkZB2
cc:
https://www.researchgate.net/post/Differential_Propositional_Calculus