# Upper And Lower Postulates

by Steve Beckett of the Berlin Quintographical Society

The distinction between upper and lower postulates (or postulatory objects) is a concept that might come up in non-empirical quintography, first introduced by Johannes Klingemann in 2011.

# Definition

Let *S* be a system of *n* postulates, and **P** a particular postulate in *S*.

Let *T* be the Hinzelmann transformation of the system *S* \ **P**. For all **T** ∈ *T*, C(**T**, **P**) holds, where C is Charles' predicate, if and only if **P** is an upper postulate of the system *S*.

Similarly, for all **T** ∈ *T*, C(**T**, **P**) does not hold, if and only if **P** is a lower postulate of the system *S*.

# Implications

## Alternative definition of necessity of images

The linking of the Hinzelmann transformation and Charles' predicate in this manner has profound implications for the nature of hypothetical quintography. Considering the second Hinzelmann law, it can be proven that upper postulates are equivalent to facets of a system of postulates, and that lower postulates are never equivalent to images.

## Postulatory decision

By the third and fourth Hinzelmann laws, plus the Klappmann-Jones theorem, it can be derived that if system *S* is extended by an additional postulate **Q**, if the Kamotsky value of *S* is Kevin congruent to that of *S* ∪ {**Q**}, then **Q** is not consistent with **S**. This is incredibly important as it allows a strong form of postulatory decision to be made without using the statistical-numerological methods of generation theory.