Based on Zalamea. See also Zeman, Sowa and Dau.
p, q, f -- propositional letters
P, Q, F -- predicate letters
Alpha
| Peirce's modified notation | My (modified) notation |
|---|---|
| pq -- conjunction | (pq) |
| ‹p› -- negation | ‹p› |
| ‹p‹q›› -- implication | [‹p›q] |
| ‹‹p›‹q›› -- disjunction | [pq] |
A1 Erasure: Any evenly enclosed graph may be erased.
A2 Insertion: Any graph may be inserted in any oddly enclosed region.
A3 Iteration: Any graph may be iterated (i.e. repeated) in a strict region of that graph.
A4 Deiteration: Any graph whose occurrence could result from iteration may be deiterated (i.e. erased).
A5 Double cut: A double cut may be inserted or erased around any graph in any region.
A1 (ER): pq ⇒ p (pq) ⇒ p
A2 (IN): ‹p› ⇒ ‹pq› ‹p› ⇒ ‹pq›
A3, A4 (IT and DI): p‹q› ⇔ p‹pq› (p‹q›) ⇔ (p‹pq›)
A5 (DC): p ⇔ ‹‹p›› p ⇔ ‹‹p››
Beta
LI -- Line of Identity
B1 Erasure: Any evenly enclosed portion of an LI may be erased.
B2 Insertion: Any portions of LI's may be joined in an oddly enclosed region.
B3 Continuous iteration: Any LI may be extended towards strict regions. Any LI may branch in its region.
B4 Continuous deiteration: Any LI may be retracted towards regions with lesser cuts.
B1: P---Q ⇒ P- -Q (PQ) ⇒ (P)(Q)
B2: ‹P- -Q› ⇒ ‹P---Q› ‹(P)(Q)› ⇒ ‹(PQ)›
B3, B4: P--- ‹q› ⇔ P-‹- q› (P)‹q› ⇔ (P‹q›)?
Some X is R to every Y ⇒ To every Y some X is R.
-‹-‹-R2-›-› ⇒ ‹‹-R2-›-› (X1[‹Y2›R12]) ⇒ [‹Y2›(X1R12)]?
Gamma
{f} -- possibly not f (‹f›)
‹{f}› -- not possibly not f = necessarily f ‹(‹f›)› = [f]
{‹f›} -- possibly not not f = possibly f (‹‹f››) = (f)
‹{f}› ⇒ ‹‹f›› ⇒ f ⇒ ‹‹f›› ⇒ {‹f›}
G1: In an even area, any Alpha cut may be half-erased to become a Gamma cut.
‹› ⇒ {} ‹› ⇒ (‹›)
G2: In an odd area, any Gamma cut may be half-completed to become an Alpha cut.
‹{}› ⇒ ‹‹›› ‹(‹›)› ⇒ ‹‹››
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