FORMAL / LOGICAL SYSTEMSPARA-LEMMA LOGIC 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: 1 1* 2 2* 2.5 2.5* 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 : 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 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 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. '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 |