An initial universe of discourse, 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:
The initial alphabet, is extended by a first order differential alphabet, resulting in a first order extended alphabet, defined as follows:
The initial basis, is extended by a first order differential basis, resulting in a first order extended basis, defined as follows:
The initial space, 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:
Finally, the initial universe, is extended by a first order differential universe or tangent universe, at each point of resulting in a first order extended universe or tangent bundle universe, defined as follows:
This gives the type:
A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe and the first order differential proposition we have arrived, in concept at least, at the foothills of differential logic.
Table 11 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner.
Table 11. Differential Extension : Basic Notation