Disjunction and Implication

Context
Natural deduction

Theorems

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

(The above are sometimes referred to as the disjunctive syllogism or Modus Tollendo Ponens.)


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


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


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

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

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

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

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

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