Equivalence of Logical Implication and Conditional

Theorem

$\paren {p \implies q} \dashv \vdash \paren {p \vdash q}$

That is, the conditional is logically equivalent to logical implication.

Proof

This directly follows from:

• The Modus Ponendo Ponens: $p \implies q, p \vdash q$
• The Rule of Implication: $\left({p \vdash q}\right) \vdash p \implies q$.

$\blacksquare$

Caution

This is not to say that the conditional and the logical implication are the same thing.

If $p \not \vdash q$ it does not mean that $\neg \left({p \implies q}\right)$.

The latter statement is true only when $p$ is true and $q$ is false.

The former statement just says that it is not always true that when $p$ is true then $q$ is true.