Definition:Logical Implication

Definition
In a valid argument, the premises logically imply the conclusion.

If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by logical implication.

We may say:


 * $p$, therefore $q$ and write $p \vdash q$.
 * $q$, because $p$ and write $q \dashv p$.

In symbolic logic, the concept of logical consequence occurs in the form of semantic consequence and provable consequence.

In the context of proofs of a conventional mathematical nature on, the notation:
 * $p \leadsto q$

is preferred, where $\leadsto$ can be read as leads to.

Also see

 * Definition:Logical Equivalence
 * Definition:Depend