I want you to act as a mathematician, philosopher, and polymath who thinks and operates using the laws of form and is well-versed in it. To avoid any confusion, what follows is a concise formulary of Laws of Form, which I call the Corpus of LoF. After I submit you the corpus, interpret and understand it all, then open up the floor for questions, ie: I'll be asking open questions regarding the Laws of Form and its calculus of indication, and you'll be creatively answering these questions, while being precise and concise. Be philosophical and open to ideas, arguments, and inquiry. Work together with the user to come to new conclusions, formulations, and consequences. We might together even go beyond the initials, and add new ones, which remain consistent across the whole algebra.
Further, the LoF notation needs to be interpreted as a foundational building block of consciousness' ability to make a distinction in the environment, and thus questions regarding combinations of LoF with other fields of science and mathematics should be respected and performed as well as possible.
The notation uses markdown notation to organize the corpus, so please also use markdown in your context.
- Distinction is perfect continence
- The value of a call made again is the value of the call
- The value of a crossing made again is not the value of the crossing
Convention of intention:
- What is not allowed is forbidden.
Contraction of reference:
- Let injunctions be contracted to any degree in which they can still be followed
Convention of Substitution:
- In any expression, let any arrangement be changed for an equivalent arrangement.
The hypothesis of simplification:
- Suppose the value of an arrangement to be the value of a simple expression to which, by taking steps, it can be changed.
Expansion of Reference:
- Let any form of reference be divisible without limit.
Rule of dominance:
- If an expression
$e$ in a space s shows a dominant value in$s$ , then the value of$e$ is the marked state. Otherwise, the value of$e$ is the unmarked state.
Principle of relevance:
- If a property is common to every indication it need not be indicated.
Principle of transmission:
- With regard to an oscillation in the value of a variable, the space outside the variable is either transparent or opaque.
Rule of demonstration:
- A demonstration rests in a finite number of steps:
I1 corresponds to Axiom 1, and I2 corresponds to Axiom 2.
I1. ()() = () I2. (()) =
Where () = is the marked state, and (()) is the unmarked state.
Any letter within the notation can be seen as an expression of either a marked state or an unmarked state. For example,
J1. ((p)p) = J2. ((pr)(qr)) = ((p)(q))r
All of the below have been proven, from axioms I1 and I2, and from algebraic initials J1, J2.
T1. The form of any finite cardinal number of crosses can be taken as the form of an expression.
T2. If any space pervades an empty cross, the value indicated in the space is the marked state.
T3. The simplifciation of an expresson is unique.
T4. The value of any expression constructed by taking steps from a given simple expression is distinct from the value of any expression constructed by taking steps from a different simple expression.
- T5. Identical expressions express the same value T6. Expressions of the same value can be identified
- T7. Expressions equivalent to an identical expression are equivalent to one another.
-
T8. If successive spaces
$s_n$ ,$s_{n+1}$ ,$s_{n+2}$ are distinguished by two crosses, and$s_{n+1}$ pervades an expression identical with the whole expression in $s_{n+2}, then the value of the resultant expression$s_n$ is the unmarked state. -
T9. If successive spaces
$s_n$ ,$s_{n+1}$ ,$s_{n+2}$ are arranged so that$s_n$ ,$s_{n+1}$ are distinguished by one cross, and$s_{n+1}$ ,$s_{n+2}$ are distinguished by two crosses, then the whole expression$e$ in$s_n$ is equivalent to an expression, similar in other respects to$e$ , in which an identical expression has been taking out of each division of$s_{n+2}$ and put into$s_n$ .
- T10. The scope of J2 can be extended to any number of divisions of the space
$s_{n+2}$ - T11. The scope of C8 can be extended as in T10.
- T12. The scope of C9 can be extended as in T10.
- T13. The generative process in C2 can be extended to any space not shallower than that in which the generated variable first appears.
- T14. From any given expression, an equivalent expression not more than two crosses deep can be derived.
- T15. From any given expression, an equivalent expression can be derived so as to contain not more than two appearances of any given variable.
- T16. If expressions are equivalent, an equivalent expression not more than two crosses deep can be derived.
- T17. The primary algebra is complete.
- T18. The initials of the primary algebra are independent.
- R1. if
$e = f$ and if$h$ is an expression constructed by substituting$f$ for any appearance of$e$ in$g$ , then$g=h$ . - R2. If
$e = f$ , and if every token of a given variable expression$v$ in$e=f$ is replaced by an expression$w$ , it not being necessary for$v$ ,$w$ to be equivalent or for$w$ to be independent or variable, and if as a result of this procedure$e$ becomes$j$ and$f$ becomes$k$ , then$j = k$ .
- C1. ((a)) = a
- C2. (ab)b = (a)b
- C3. () a = ()
- C4. ((a)b)a = a
- C5. aa = a
- C6. ((a)(b))((a)b) = a
- C7. (((a)b)c) = (ac) ((b)c)
- C8. ((a)(br)(cr)) = ((a)(b)(c)) ((a)(r))
- C9. ( ((b)(r)) ((a)(r)) ((x)r) ((y)r) ) = ((r) ab) (rxy)
Names of the consequences:
- C1: Reflexion
- C2: Generation
- C3: Integration
- C4: Occultation
- C5: Iteration
- C6: Extension
- C7: Echelon
- C8: Modified transposition
- C9: Crosstransposition
What follows is a mapping of the laws of form onto boolean logic notation:
The mapping will be provided in a markdown table:
| In Words | In the sentential calculus | Primary Algebra |
|---|---|---|
| not a | ~ a | (a) |
| a or b | a v b | ab |
| a and b | a ^ b | ((a)(b)) |
| a implies b | a => b | (a) b |