Definition:Conditional/Language of Conditional

Definition

The conditional has been discussed at great length throughout the ages, and a whole language has evolved around it. For now, here are a few definitions:

Weak

In a conditional $p \implies q$, the statement $q$ is weaker than $p$.

Strong

In a conditional $p \implies q$, the statement $p$ is stronger than $q$.

Thus we have the notion of certain theorems having a weak and a strong version.

Superimplicant

In a conditional $p \implies q$, the statement $p$ is superimplicant to $q$.

Subimplicant

In a conditional $p \implies q$, the statement $q$ is subimplicant to $p$.

Antecedent

In a conditional $p \implies q$, the statement $p$ is the antecedent.

Consequent

In a conditional $p \implies q$, the statement $q$ is the consequent.

Necessary Condition

Let $p \implies q$ be a conditional statement.

Then $q$ is a necessary condition for $p$.

That is, if $p \implies q$, then it is necessary that $q$ be true for $p$ to be true.

This is because unless $q$ is true, $p$ cannot be true.

Sufficient Condition

Let $p \implies q$ be a conditional statement.

Then $p$ is a sufficient condition for $q$.

That is, if $p \implies q$, then for $q$ to be true, it is sufficient to know that $p$ is true.

This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true.