NOR with Equal Arguments

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Theorem

Let $\downarrow$ signify the NOR operation.


Then for any proposition $p$:

$p \downarrow p \dashv \vdash \neg p$

That is, the NOR of a proposition with itself corresponds to the negation operator.


Proof 1

By the tableau method of natural deduction:

$p \downarrow p \vdash \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \downarrow p$ Premise (None)
2 1 $\neg \paren {p \lor p}$ Sequent Introduction 1 Definition of Logical NOR
3 1 $\neg p$ Sequent Introduction 2 Rule of Idempotence: Disjunction

$\Box$


By the tableau method of natural deduction:

$\neg p \vdash p \downarrow p$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Premise (None)
2 1 $\neg \paren {p \lor p}$ Sequent Introduction 1 Rule of Idempotence: Disjunction
3 1 $p \downarrow p$ Sequent Introduction 2 Definition of Logical NOR

$\blacksquare$


Proof by Truth Table

Apply the Method of Truth Tables:

$\begin {array} {|ccc||cc|} \hline p & \downarrow & p & \neg & p \\ \hline \F & \T & \F & \T & \F \\ \T & \F & \T & \F & \T \\ \hline \end{array}$


As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\blacksquare$


Also see


Sources