Disjunction and Implication

Theorems
This is sometimes referred to as the disjunctive syllogism or Modus Tollendo Ponens:
 * $$p \or q \dashv \vdash \neg p \implies q$$

This is sometimes referred to as the Rule of Material Implication:
 * $$\neg p \or q \dashv \vdash p \implies q$$

Both of the above come in negative forms:


 * $$\neg \left({p \implies q}\right) \dashv \vdash \neg \left({\neg p \or q}\right)$$
 * $$\neg \left({\neg p \implies q}\right) \dashv \vdash \neg \left({p \or q}\right)$$

Proof by Natural Deduction
By the tableau method:

Proofs which require the LEM
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.

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that: for all interpretations $$v$$.
 * $$v \left({p \or q}\right) = v \left({\neg p \implies q}\right)$$
 * $$v \left({\neg \left({\neg p \implies q}\right)}\right) = v \left({\neg \left({p \or q}\right)}\right)$$

Hence the result by the definition of interderivable.

We see that: for all interpretations $$v$$.
 * $$v \left({\neg p \or q}\right) = v \left({p \implies q}\right)$$
 * $$v \left({\neg \left({p \implies q}\right)}\right) = v \left({\neg \left({\neg p \or q}\right)}\right)$$

Hence the result by the definition of interderivable.