Disjunction


THEOREM    Disjunction is idempotent. $$P \vee P \dashv\vdash P$$

Proof available

THEOREM    \(P \vdash P \vee P\)

Proof available

THEOREM    \(P \vee P \vdash P\)

Proof available

THEOREM   

Disjunction is commutative.

$$P \vee Q \dashv\vdash Q \vee P$$

Proof available

LEMMA    \(P \vee Q \vee R \vdash P \vee (Q \vee R)\)

Proof available

LEMMA    \(P \vee (Q \vee R) \vdash P \vee Q \vee R\)

Proof available

THEOREM   

Disjunction is associative.

$$P \vee Q \vee R \dashv\vdash P \vee (Q \vee R)$$

Proof available

LEMMA   

Disjunction is left self-distributive.

$$P \vee (Q \vee R) \dashv\vdash (P \vee Q) \vee (P \vee R)$$

Proof available

LEMMA   

Disjunction is right self-distributive.

$$(Q \vee R) \vee P \dashv\vdash (Q \vee P) \vee (R \vee P)$$

Proof available

THEOREM   

Disjunction is self-distributive.

Proof available