Adam I. Gerard

Thinking Notation #5

Some theorems, etc.

Symbols, Blank Symbols

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.

Observations, Thereoms, and Facts

  1. Axiom 11. entails that axioms 9. and 9. Addendum produce the same.
  2. Symbols are bound in the manner of lambda calculus.
  3. The transition symbol | is a symbol like any other and can be omitted.
  4. Axiom 7. can introduce a previously unconceived item into thought (via empty symbols).
  5. The empty language (a blank page) and Spencer-Brown's The Void are subsets/subfragments.
  6. 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).
  7. That's another way to get to Classical Propositional/Sentential Logic.
  8. Line, arrow, and dot (.) ontologies and languages are subsets/subfragments.
  9. The - symbol is a (kind of?) grouping under axiom 8. (I've explored an alternative way of handling relationships elsewhere.)

Some Questions

  1. 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?)
  2. Are Natural Languages subfragments of Thinking Notation?
  3. Are there mereological implementations/variants? Can a sequence/transition be modeled as a proper part? What are the advantages, disadvantages, are these isomorphic, etc.?
  4. Consider:

Compare two sequences

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?)