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 $q \dashv p$.

If the truth of one statement has an influence on that of another, but does not necessarily imply it logically, then we may say that the second statement depends upon the first.

Formal Definition

Let $A$ and $B$ be statement forms.

Then $A$ logically implies $B$ iff $\left({A \implies B}\right)$ is a tautology.

This is justified by Equivalence of Logical Implication and Conditional.

Also see

• Logical Equivalence

Sources

• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$: Definition $1.7$
• D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.7$