FORMAL / LOGICAL SYSTEMSPARA-LEMMA LOGICPart I.Mathematical and Exceptional LemmasInitially, 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 lemmaFor example, in a set of lemma statements:11*22*2.52.5*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 lemmas:1 : 2 :: 3 : 41 : 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 qualifiers.  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 ExtensionA 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. 'Truth'.interpretation******: The meaning or higher significance of the objects of the system, such as laws, principles, or cultured facts. 'Strength'.interpretation*******: The meaningful cultured environment of objects interacting meaningfully. 'Beauty'.          BACK TO SYSTEMS