# Thread: Sheffer truth table, operators?

1. ## Sheffer truth table, operators?

The late 19th century philosopher Charles Peirce...

a) Express ¬p using only the Sheffer stroke operation (up arrow).

b) Express p v q using only the She®er stroke operation (up arrow). Justify your answer (e.g. using a truth table).

c) Explain why the set of operators (up arrow) is functionally complete .

d) Express the Sheffer stroke operation p q using only the Peirce arrow CS173 Discrete Mathematical Structures operation. Explain why the set of operators {↑} is functionally complete.

2. Originally Posted by captainjapan
The late 19th century philosopher Charles Peirce...
How does your textbook define the Peirce Operator?
Definitions do vary.

3. The late 19th century philosopher Charles Peirce (rhymes with hearse,' not fierce') wrote about a set of logically dual operators and, in his writings, coined the term `Ampheck' to describe them. The two most common Ampheck operators, the Peirce arrow (written ) and the Sheffer stroke (written ), are defned by the following truth table.

p q p ↑ q p ↓ q
T T F F
T F T F
F T T F
F F T T

4. $\displaystyle \begin{array}{rcc} {\neg P} & \equiv & {P \uparrow P} \\ {P \wedge Q} & \equiv & {\left( {P \uparrow Q} \right) \uparrow \left( {P \uparrow Q} \right)} \\ {P \vee Q} & \equiv & {\left( {P \uparrow P} \right) \uparrow \left( {\ Q\uparrow Q} \right)} \\ {P \Rightarrow Q} & \equiv & {P \uparrow \left( {Q \uparrow Q} \right)} \\ \end{array}$