Disjunction


LEMMA    \(P \vee P \vdash P\)

Proof available

LEMMA    \(P \vdash P \vee P\)

Proof available

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

Proof available

LEMMA    \(P \vee Q \vdash Q \vee P\)

Proof available

LEMMA    \(Q \vee P \vdash P \vee Q\)

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

Proof available

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

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

Proof available

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

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

THEOREM   

False is a left identity element of disjunction.

$$\bot \vee P \dashv\vdash P$$

Proof available

THEOREM   

False is a right identity element of disjunction.

$$P \vee \bot \dashv\vdash P$$

Proof available

THEOREM   

True is a left zero element of disjunction.

$$\top \vee P \dashv\vdash \top$$

Proof available

THEOREM   

True is a right zero element of disjunction.

$$P \vee \top \dashv\vdash \top$$

Proof available

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