De Morgan's Laws (Logic)

Context
Natural deduction

Theorems

 * $$\lnot p \lor \lnot q \vdash \lnot \left({p \land q}\right)$$
 * $$\lnot \left({p \land q}\right) \vdash \lnot p \lor \lnot q$$


 * $$\lnot p \land \lnot q \vdash \lnot \left({p \lor q}\right)$$
 * $$\lnot \left({p \lor q}\right) \vdash \lnot p \land \lnot q$$


 * $$p \land q \vdash \lnot \left({\lnot p \lor \lnot q}\right)$$
 * $$\lnot \left({\lnot p \lor \lnot q}\right) \vdash p \land q$$


 * $$p \lor q \vdash \lnot \left({\lnot p \land \lnot q}\right)$$
 * $$\lnot \left({\lnot p \land \lnot q}\right) \vdash p \lor q$$

Their abbreviation in a tableau proof are collectively $$\textrm{DM}$$.

Proofs
These are proved by the tableau method:


 * $$\lnot p \lor \lnot q \vdash \lnot \left({p \land q}\right)$$:


 * $$\lnot p \land \lnot q \vdash \lnot \left({p \lor q}\right)$$:


 * $$\lnot \left({p \lor q}\right) \vdash \lnot p \land \lnot q$$:


 * $$p \land q \vdash \lnot \left({\lnot p \lor \lnot q}\right)$$:


 * $$p \lor q \vdash \lnot \left({\lnot p \land \lnot q}\right)$$:

Proofs that use the Law of the Excluded Middle
The following results require the Law of the Excluded Middle to prove, and hence are not accepted by the school of intuitionist logic.


 * $$\lnot \left({p \land q}\right) \vdash \lnot p \lor \lnot q$$


 * $$\lnot \left({\lnot p \lor \lnot q}\right) \vdash p \land q$$


 * $$\lnot \left({\lnot p \land \lnot q}\right) \vdash p \lor q$$

Comment
Note that both of these:


 * $$\lnot p \land \lnot q \vdash \lnot \left({p \lor q}\right)$$
 * $$\lnot \left({p \lor q}\right) \vdash \lnot p \land \lnot q$$

can be proved without resorting to the LEM.

All the others:


 * $$\lnot p \lor \lnot q \vdash \lnot \left({p \land q}\right)$$
 * $$p \land q \vdash \lnot \left({\lnot p \lor \lnot q}\right)$$
 * $$p \lor q \vdash \lnot \left({\lnot p \land \lnot q}\right)$$

are not reversible in intuitionist logic.