Introduction
Differential logic is the component of logic whose object is the description of variation — for example, 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 treats the principles governing 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.
cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)
supplies the groundwork for any number of further extensions, beginning with the first order differential extension 
The construction of 
can be described in the following stages.
is extended by a first order differential alphabet 
resulting in a first order extended alphabet 
defined as follows.
is extended by a first order differential basis 
resulting in a first order extended basis 
defined as follows.
is extended by a first order differential space or tangent space 
at each point of 
resulting in a first order extended space or tangent bundle space 
defined as follows.
![A^\bullet = [ a_1, \ldots, a_n ]](https://s0.wp.com/latex.php?latex=A%5E%5Cbullet+%3D+%5B+a_1%2C+%5Cldots%2C+a_n+%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is extended by a first order differential universe or tangent universe ![\mathrm{d}A^\bullet = [ \mathrm{d}a_1, \ldots, \mathrm{d}a_n ]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7Bd%7DA%5E%5Cbullet+%3D+%5B+%5Cmathrm%7Bd%7Da_1%2C+%5Cldots%2C+%5Cmathrm%7Bd%7Da_n+%5D&bg=ffffff&fg=000000&s=0&c=20201002)
at each point of 
resulting in a first order extended universe or tangent bundle universe ![\mathrm{E}A^\bullet ~=~ [ \mathrm{E}\mathcal{A} ] ~=~ [ \mathcal{A} ~\cup~ \mathrm{d}\mathcal{A} ] ~=~ [ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n ].](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BE%7DA%5E%5Cbullet+%7E%3D%7E+%5B+%5Cmathrm%7BE%7D%5Cmathcal%7BA%7D+%5D+%7E%3D%7E+%5B+%5Cmathcal%7BA%7D+%7E%5Ccup%7E+%5Cmathrm%7Bd%7D%5Cmathcal%7BA%7D+%5D+%7E%3D%7E+%5B+a_1%2C+%5Cldots%2C+a_n%2C+%5Cmathrm%7Bd%7Da_1%2C+%5Cldots%2C+%5Cmathrm%7Bd%7Da_n+%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
![[ \mathbb{B}^n \times \mathbb{D}^n ] ~=~ (\mathbb{B}^n \times \mathbb{D}^n\ +\!\!\to \mathbb{B}) ~=~ (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).](https://s0.wp.com/latex.php?latex=%5B+%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En+%5D+%7E%3D%7E+%28%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En%5C+%2B%5C%21%5C%21%5Cto+%5Cmathbb%7BB%7D%29+%7E%3D%7E+%28%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En%2C+%5Cmathbb%7BB%7D%5En+%5Ctimes+%5Cmathbb%7BD%7D%5En+%5Cto+%5Cmathbb%7BB%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002)
we arrive at the foothills of differential logic.
Inquiry Into Inquiry 