Part I.
Mathematical and Exceptional Lemmas

Initially, para-lemma logic exists in two senses:

1. The sense of mathematics / primary Lemma Logic

* Necessary by theory (T = Theory)
** Math for philosophers (BAD)
*** Advanced math (GOOD)
**** Einstein-only (?)
***** Too crazy (MADEN)

2. The sense of qualifying the lemma

For example, in a set of lemma statements:


Or the like, lemmas can be used to act retroactively upon the
logical relations of statements in a list.
This is only arbitrary if
the list is arbitrary
, which may be seen as the first rule of
para-lemma logic.

We can see relations such as:

1 : 2 :: 1* : 2*

And similarly,

1 : 2* :: 3* : 4**

The clear distinction is that
stars always relate with stars, that is,
it is impossible to reach a comparison like this:

1 : 2* :: 3 : 4, which would be illegal.

So, that may serve as the second rule.

A star always produces a star within the direct comparison, or
otherwise across from the comparison. So, we get four major
types of comparisons assuming four numbers and up to four

1 : 2 :: 3 : 4

1 : 1* :: 2 : 2*

1* : 2* ::  3* : 4* (or, also: 1** : 2** ::  3** : 4**)

1* : 2** :: 3** : 4***

The advanced level (in Para-Lemma Logic) is to use the logical
relationships created by the original variables to construct
meanings for the lemmas themselves.

For example, the most basic level might be:

A : B :: C : D = No Lemma.

But equally A : D :: C : B = No Lemma.

This leads to
Categorical Deduction, but it also suggests a
mathematical problem of a double-horned dilemma about

As soon as lemmas are involved again, we get statements like:

A : A* :: B : B* which simply means that A : B :: A* : B*.

And, ultimately it ends up again at statements like:

A : B* :: C : D* which fit neatly into categorical deduction.          

However, the neat 1 : 1 relationship is not always present in these
more advanced comparisons.

Part II. The Third Sense: Infinite Extension

A theory that goes beyond these mathematical models of lemmas
may be had with functional theories, yielding infinitely extended
functions. Such a function is typically complex.

One word that could be used is 'interpretation', as in:
"Interpretation, interpretation*, interpretation**,
interpretation***, interpretation****...

interpretation*****, interpretation******..."

If the first interpretation is treated as itself a lemma (as in
0-d-equivalence-to-unity category theory), then this already
extends to seven lemmas!

They can be interpreted as follows, in a complex view of formal
category theory:

interpretation*: The formal qualification of a system. 'Strategy'.

interpretation**: The exceptions, empirical or otherwise, upon
the system. 'Techniques'.

interpretation***: The secondary formal existence of the system,
i.e. its systematic translation. 'Thoughts'.

interpretation****: The emergent applications of the system, e.g.
to empirical reality. 'Tools'.

interpretation*****: The entities or real objects of the system.

interpretation******: The meaning or higher significance of the
objects of the system, such as laws, principles, or cultured facts.

interpretation*******: The meaningful cultured environment of
objects interacting meaningfully. 'Beauty'.