Conjunction and Implication

Context
Natural deduction

Theorems

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


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


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

This pair is sometimes called Modus Ponendo Tollens (MPT).


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

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

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

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

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

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

Proofs using the LEM
The remaining proofs depend on the Law of the Excluded Middle.

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

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

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