Conjunction and Implication

Theorems

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

This rule is sometimes called Modus Ponendo Tollens (MPT):


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

Proof by Natural Deduction
By the tableau method:

Proofs using the LEM
The remaining proofs depend (directly or indirectly) on the Law of the Excluded Middle.

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 $$v$$.
 * $$v \left({p \and q}\right) = v \left({\neg \left({p \implies \neg q}\right)}\right)$$
 * $$v \left({p \implies \neg q}\right) = v \left({\neg \left({p \and q}\right)}\right)$$

Hence the result by the definition of interderivable.

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

Hence the result by the definition of interderivable.