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