some study on logic

work_files/verum-nimply_logic.md

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+# Verum-Nimply Logic
+
+Verum-Nimply Logic is a logic system built upon a two gates which are functionally complete.
+
+## Atoms
+### Verum (⊤)
+Verum is a logic-gate modelling of the "Signal Source".
+| Output |
+|----|
+| 1 |
+
+### Falsum (⊥)
+Verum is a logic-gate modelling of the "Lack of Signal" or "No Connection".
+| Output |
+|----|
+| 0 |
+
+### Nimply (↛)
+Nimply is a logic-gate modelling of the "Signal Blocker".
+
+The source signal is connected to the Collector, and the blocking signal is connected to the Gate. The Nimply gate is not commutative.
+
+This document will use the convention where the Collector is placed on the lefthand side.
+
+| Collector | Gate | Emitter |
+| ---- | ---- | ---- |
+| 0 | 0 | 0 |
+| 0 | 1 | 0 |
+| 1 | 0 | 1 |
+| 1 | 1 | 0 |
+
+## Derivations
+
+### Diode
+= (p ↛ ⊥)
+
+### NOT p
+= (⊤ ↛ p)
+
+### p AND q
+= (p ↛ (NOT q))
+= (p ↛ (⊤ ↛ q))
+
+### p NAND q
+= NOT (p AND q)
+= NOT (p ↛ (⊤ ↛ q))
+= (⊤ ↛ (p ↛ (⊤ ↛ q)))
+
+### p OR q
+= NOT ( NOT(p) AND NOT(q) )
+= NOT ( (⊤ ↛ p) AND (⊤ ↛ q) )
+= NOT ( ( (⊤ ↛ p) ↛ (⊤ ↛ (⊤ ↛ p)) ) )
+= (⊤ ↛ ( ( (⊤ ↛ p) ↛ (⊤ ↛ (⊤ ↛ p)) ) ))
+Note: For the most cases, an OR gate can be substituted using merging wires and diodes.