# De Morgan's Laws (Logic)

This proof is about De Morgan's Laws in the context of propositional logic. For other uses, see De Morgan's Laws.

## Theorem

### Disjunction of Negations

#### Formulation 1

$\neg p \lor \neg q \dashv \vdash \neg \left({p \land q}\right)$

#### Formulation 2

$\vdash \left({\neg p \lor \neg q}\right) \iff \left({\neg \left({p \land q}\right)}\right)$

### Conjunction of Negations

#### Formulation 1

$\neg p \land \neg q \dashv \vdash \neg \left({p \lor q}\right)$

#### Formulation 2

$\vdash \left({\neg p \land \neg q}\right) \iff \left({\neg \left({p \lor q}\right)}\right)$

### Conjunction

#### Formulation 1

$p \land q \dashv \vdash \neg \left({\neg p \lor \neg q}\right)$

#### Formulation 2

$\vdash \left({p \land q}\right) \iff \left({\neg \left({\neg p \lor \neg q}\right)}\right)$

### Disjunction

#### Formulation 1

$p \lor q \dashv \vdash \neg \left({\neg p \land \neg q}\right)$

#### Formulation 2

$\vdash \left({p \lor q}\right) \iff \left({\neg \left({\neg p \land \neg q}\right)}\right)$

## The Intuitionist Perspective

Note that this:

$\neg p \land \neg q \dashv \vdash \neg \left({p \lor q}\right)$

can be proved in both directions without resorting to the LEM.

All the others:

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

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

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

are not reversible in intuitionistic logic.