Some theorems, etc.

`2.`

**Addendum:** the `-`

symbol is a symbol like any another.

`9.`

**Addendum:** grouped symbols can be grouped adjacently or by being enclosed in another symbol (similar to the mereological notions of `proper parthood`

, `overlap`

, `underlap`

).

`11.`

A blank or empty symbol is a symbol.

- Axiom
**11.**entails that axioms**9.**and**9. Addendum**produce the same. - Symbols are bound in the manner of
*lambda calculus*. - The transition symbol
`|`

is a symbol like any other and can be omitted. - Axiom
**7.**can introduce a previously unconceived item into thought (via empty symbols). - The empty language (a blank page) and Spencer-Brown's
`The Void`

are subsets/subfragments. - Spencer-Brown's
*Laws of Form*can be assembled within (a subsets/subfragments can be produced that is isomporphic to the*Laws of Form*- a representation theorem). - That's another way to get to
*Classical Propositional/Sentential Logic*. *Line*,*arrow*, and*dot*(`.`

) ontologies and languages are subsets/subfragments.- The
`-`

symbol is a (kind of?) grouping under axiom**8.**(I've explored an alternative way of handling relationships elsewhere.)

- If we consider
*Thinking Notation*as a "fundamental" or "top-level" language/theory, and we associate two systems`A`

and`B`

(say, the*von Neumann*and*Zermelo*ordinals, respectively), then their being in 1-to-1 correspondance can be established via the axioms of*Thinking Notation*and basic substitution. Is that justificatory? (Can that be taken to undergird 1-to-1 correspondance and such mappings?) - Are
*Natural Languages*subfragments of*Thinking Notation*? - Are there mereological implementations/variants? Can a sequence/transition be modeled as a
`proper part`

? What are the advantages, disadvantages, are these isomorphic, etc.? - Consider:

`i.`

`A`

and `B`

are synonomous if the **Extension: No Limit** rule is allowed. Is that problematic?

`ii.`

Symbol binding occurs at the level of single symbol within a sequence, the entirety of a sequence, and all subsets/subfragments in between (due to the way that symbols are loosely defined). Is *that* problematic? Does a finer distinction need to be made?

`iii.`

I use **Optional Binding** in exploring *Connection Theory*. What happens when we deny *Lambda Calculus* uniform binding constraints in *Thinking Notation* itself?

`iv.`

Per my previous comments elsewhere, we observe that most subsystems of interest demonstrate additional properties like *closure*, *transitivity*, *commutativity*, etc. Do those additional restrictions mitigate any such worries that might arise from the properties of the more general system above? What exactly is that relationship (between more restricted subsystems) and top-level general observations we see above? (Are we just picking certain subsystems like we do with *ZFC Set Theory* - which selects a valid set-theoretic universe from an infinite milieu of alternative set formulations - or is there some interesting meta-theoretic relationship that can be induced, observed, described, etc. like some specific sequence of axioms? Do those two approaches arise from the same?)

- Thinking Notation
- Thinking Notation #2
- Thinking Notation #3
- Thinking Notation #4
- Thinking Notation #5
- Thinking Notation #6

##### post: 12/08/2022

##### update: 12/09/2022