Paradoxes of Material Implication

Context
Natural deduction

Theorems

 * $$q \vdash p \Longrightarrow q$$

"If something is true, then anything implies it."


 * $$\lnot p \vdash p \Longrightarrow q$$

"If something is false, then it implies anything."

The following is known as the Self-Distributive Law:


 * $$p \Longrightarrow \left({q \Longrightarrow r}\right) \dashv \vdash \left({p \Longrightarrow q}\right) \Longrightarrow \left({p \Longrightarrow r}\right)$$

We also have, interestingly, this result:


 * $$\left({p \Longrightarrow q}\right) \Longrightarrow r \vdash \left({p \Longrightarrow r}\right) \Longrightarrow \left({q \Longrightarrow r}\right)$$

... but:


 * $$\left({p \Longrightarrow r}\right) \Longrightarrow \left({q \Longrightarrow r}\right) \not \vdash \left({p \Longrightarrow q}\right) \Longrightarrow r$$

"Given any two statements, one of them implies the other."

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

Proofs
$$q \vdash p \Longrightarrow q$$:

$$\lnot p \vdash p \Longrightarrow q$$:

$$p \Longrightarrow \left({q \Longrightarrow r}\right) \vdash \left({p \Longrightarrow q}\right) \Longrightarrow \left({p \Longrightarrow r}\right)$$:

$$\left({p \Longrightarrow q}\right) \Longrightarrow \left({p \Longrightarrow r}\right) \vdash p \Longrightarrow \left({q \Longrightarrow r}\right)$$:

$$\left({p \Longrightarrow q}\right) \Longrightarrow r \vdash \left({p \Longrightarrow r}\right) \Longrightarrow \left({q \Longrightarrow r}\right)$$:

$$\left({p \Longrightarrow r}\right) \Longrightarrow \left({q \Longrightarrow r}\right) \not \vdash \left({p \Longrightarrow q}\right) \Longrightarrow r$$:

$$\vdash \left({p \Longrightarrow q}\right) \lor \left({q \Longrightarrow p}\right)$$:

Comment
These counter-intuitive results have caused debate and confusion among philosophers for millennia.

In particular, the result $$\lnot p \vdash p \Longrightarrow q$$ is known as a vacuous truth. It is exemplified by the (rhetorical) argument:

"If England win the Ashes this year, then I'm a monkey's uncle."