๐ฃ Sheffer Stroke and Pierce Arrow
| Symbol | Operation | Formal Name | Expression |
|---|---|---|---|
| $$ | |||
| A \uparrow B | |||
| $$ | |||
| NAND | Sheffer stroke | ||
| $$ | |||
| (A \land B)' | |||
| $$ | |||
| $$ | |||
| A \downarrow B | |||
| $$ | |||
| NOR | Peirce arrow | ||
| $$ | |||
| (A \lor B)' | |||
| $$ | |||
- Henry M. Sheffer introduced $$ \uparrow $$ in 1913 as a single primitive capable of expressing all Boolean logic.
- Charles S. Peirce explored $$ \downarrow $$ earlier as a dual primitive.
- Both symbols represent functionally complete operations โ each alone can construct all logical gates.
๐น Reasons for Arrow Direction
The symbols chosen for visual and logical contrast, not physical necessity
๐ผ NAND โ Sheffer Stroke โ Strict, Upward, Energetic
Based on AND, which requires all inputs to be true โ a strict condition.
Motivation for Pointing Up The upward stroke $\uparrow$ symbolizes:
- Logical tension โ all conditions must be fulfilled
- Energetic lift โ negating strictness requires effort
- Constraint-breaking โ NAND lifts out of grounded certainty
NAND points up because it negates a strict, high-effort condition โ it follows from the rigidity of AND, in the complement way
๐ฝ NOR โ Peirce Arrow โ Loose, Downward, Grounded
Based on OR, which accepts any input being true โ a loose condition.
Motivation for Pointing Down The downward arrow $\downarrow$ symbolizes:
- Logical ease โ minimal fulfillment suffices
- Grounded collapse โ negating looseness returns to constraint
- Down-to-earth minimalism โ NOR flattens possibility
NOR points down because it negates a loose, low-effort condition โ it follows from the openness of OR, in the complement way
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