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Math Help - Sheffer truth table, operators?

  1. #1
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    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 Sheer 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.
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  2. #2
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    Quote Originally Posted by captainjapan View Post
    The late 19th century philosopher Charles Peirce...
    How does your textbook define the Peirce Operator?
    Definitions do vary.
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  3. #3
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    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
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    \begin{array}{rcc}<br />
   {\neg P} &  \equiv  & {P \uparrow P}  \\<br />
   {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)}  \\<br />
   {P \Rightarrow Q} &  \equiv  & {P \uparrow \left( {Q \uparrow Q} \right)}  \\ \end{array}
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