Disjunction and Implication

Context
Natural deduction

Theorems
This is sometimes referred to as the disjunctive syllogism or Modus Tollendo Ponens:


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

This is sometimes referred to as the Rule of Material Implication:


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

Both of the above come in negative forms:


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

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

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

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

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

Proofs which require the LEM
$$\lnot p \Longrightarrow q \vdash p \lor q$$:

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

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

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

Note: it ought to be possible to prove this without resorting to Reductio Ad Absurdum (which depends on the Law of the Excluded Middle). This needs to be sorted out.