(Corrected 4.01.09)
Both Peirce's alpha graphs and Bricken's boundary logic may be seamlessly accommodated into my logic algebra. For this I'll adopt following conventions into my notation:
1) Letters g and f signify any propositional formula (or graph).
2) Negation of a formula is expressed by enclosing this formula between ‹ and ›.
3) Appearance of a formula in odd depth of negation is marked by enclosing this formula between ≺ and ≻; in even depth (incl 0) -- between ≼ and ≽; in any depth (incl 0) -- between ⋘ and ⋙.
4) Contents of the negated formula are joined by conjunction or disjunction depending on immediate context of the negated formula, i.e. [‹g f›] = ‹[g f]› and (‹g f›) = ‹(g f)›.
5) When formula is true independently of the context, context markers may be dropped.
6) Default context (Blank sheet) is conjunctive (i.e. Peircean).
Now we can demonstrate equivalences of three systems:
| Peirce α | Bricken BL | Tom |
|---|---|---|
| Basic formulas | ||
| g | g | g |
| ‹g› | ‹g› | ‹g› |
| gf | ‹‹g›‹f›› | (gf) ⇔ [‹‹g›‹f››] |
| ‹‹g›‹f›› | gf | (‹‹g›‹f››) ⇔ [gf] |
| ‹g‹f›› | ‹g›f | (‹g‹f››) ⇔ [‹g›f] |
| Transformations | ||
| ⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ T | ‹›‹› ⇔ ‹› ⇔ T | (‹‹›‹››) ⇔ (‹‹››) ⇔() ⇔ ○ ⇔ ⇔ [‹›] ⇔ [‹›‹›] |
| ‹›‹› ⇔ ‹› ⇔ F | ⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ F | [‹‹›‹››] ⇔ [‹‹››] ⇔ [] ⇔ □ ⇔ (‹›) ⇔ (‹›‹›) |
| ‹›g ⇔ ‹› | ‹‹›g› ⇔ | (‹›g) ⇔ □ ⇔ [‹‹›g›] |
| ‹‹›g› ⇔ | ‹›g ⇔ ‹› | (‹‹›g›) ⇔ ○ ⇔ [‹›g] |
| ‹‹g›› ⇔ g | ‹‹g›› ⇔ g | ‹‹g›› ⇔ g |
| ≼gf≽ ⇒ ≼g≽ | ≺gf≻ ⇒ ≺g≻ | (≼gf≽) ⇒ (≼g≽) [≺gf≻] ⇒ [≺g≻] |
| ≺g≻ ⇒ ≺gf≻ | ≼g≽ ⇒ ≼gf≽ | (≺g≻) ⇒ (≺gf≻) [≼g≽] ⇒ [≼gf≽] |
| g⋘f⋙ ⇔ g⋘gf⋙ | g⋘f⋙ ⇔ g⋘gf⋙ | g⋘f⋙ ⇔ g⋘gf⋙ |
|