Material conditional


THEOREM    \(P \Rightarrow (Q \Rightarrow R) \vdash (P \Rightarrow Q) \Rightarrow (P \Rightarrow R)\)

Proof available

THEOREM    \((P \Rightarrow Q) \Rightarrow (P \Rightarrow R) \vdash P \Rightarrow (Q \Rightarrow R)\)

Proof available

THEOREM   

The material conditional operation is left self-distributive.

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

Proof available

THEOREM   

The material conditional relation is reflexive.

$$\vdash P \Rightarrow P$$

Proof available

THEOREM (Hypothetical syllogism)   

The material conditional relation is transitive.

$$P \Rightarrow Q,\ Q \Rightarrow R \vdash P \Rightarrow R$$

Proof available

THEOREM   

Material conditional is left-distributive over disjunction.

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

Proof unavailable

THEOREM   

Material conditional is right-distributive over conjunction.

$$P \Rightarrow Q \wedge R \dashv\vdash (P \Rightarrow Q) \wedge (P \Rightarrow R)$$

Proof unavailable

THEOREM    \(P \wedge Q \Rightarrow R \dashv\vdash (P \Rightarrow R) \vee (Q \Rightarrow R)\)

Proof available

THEOREM    \(P \vee Q \Rightarrow R \dashv\vdash (P \Rightarrow R) \wedge (Q \Rightarrow R)\)

Proof available

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