Properties of NOR

Theorem

Let $\downarrow$ signify the NOR operation.

The following results hold:

NOR with Equal Arguments

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

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

NOR is Commutative

$p \downarrow q \dashv \vdash q \downarrow p$

NOR is not Associative

$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$