Definition:Conditional

The Conditional is a binary connective written symbolically as $$p \Longrightarrow q$$ whose behaviour is as follows:

$$p \Longrightarrow q$$ is defined as "If $$p$$ is true, then $$q$$ is true."

This is known as a conditional statement, conditional proposition or just a conditional.

"$$p \Longrightarrow q$$" is voiced "if $$p$$ then $$q$$".

You may encounter different symbols which mean the same thing as $$p \Longrightarrow q$$, for example:
 * $$p \to q$$;
 * $$p \supset q$$, in which usage "$$\supset$$" is called the "hook" or "horseshoe" sign.

It is usual to use "$$\Longrightarrow$$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in a more "mathematical" context. There are other uses in mathematics for the other symbols.

Semantics of the Conditional
We have stated that $$p \Longrightarrow q$$ means "If $$p$$ is true, then $$q$$ is true."

Alternatively, it can be said as:


 * "$$q$$ is true if $$p$$ is true."
 * "(The truth of) $$p$$ implies (the truth of) $$q$$."
 * "$$p$$ is true only if $$q$$ is true."

The latter one may need some explanation. $$p$$ can be either true or false, as can $$q$$. But if $$q$$ is false, and $$p \Longrightarrow q$$, then $$p$$ can not be true. Therefore, $$p$$ can be true only if $$q$$ is also true, which leads us to our assertion.


 * "$$p$$ may be true unless $$q$$ is false."
 * "$$q$$ is true whenever $$p$$ is true."
 * "$$q$$ is true provided that $$p$$ is true."
 * "$$p$$ is true therefore $$q$$ is true."
 * "$$q$$ is true because $$p$$ is true."
 * "$$p$$ is stronger than $$q$$" (and, by the same coin, "$$q$$ is weaker than $$p$$").
 * "$$q$$ is subalternate to $$p$$."
 * "$$q$$ is subimplicant to $$p$$."

We are at liberty to write this the other way round. $$q \Longleftarrow p$$ means the same as $$p \Longrightarrow q$$.

The language of the conditional
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:


 * Antecedent: In a conditional $$p \Longrightarrow q$$, the statement $$p$$ is the antecedent. Some authors use the term "premise" (or "premiss"), but we already have a use for the term premise. Other authors use the term "hypothesis", but this word has other applications (see hypothesis), so we prefer not to use it in this context.


 * Consequent: In a conditional $$p \Longrightarrow q$$, the statement $$q$$ is the consequent. Some authors use the term "conclusion", but we already have a use for that term conclusion.


 * Sufficient Condition: If $$p \Longrightarrow q$$, then $$p$$ is a sufficient condition for $$q$$. That is, if $$p \Longrightarrow q$$, 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.


 * Necessary Condition: If $$p \Longrightarrow q$$, then $$q$$ is a necessary condition for $$p$$. That is, if $$p \Longrightarrow q$$, then it is necessary that $$q$$ be true for $$p$$ to be true. This is because unless $$q$$ is true, $$p$$ can not be true.

Fallacies concerning the conditional
If we know that $$q$$ is true, and that $$p \Longrightarrow q$$, this tells us nothing about the truth value of $$p$$. This also takes some thinking about. Here is a plausible example which may illustrate this.

Let $$P$$ be the statement "$$x$$ is a whole number divisible by 4."

Let $$Q$$ be the statement "$$x$$ is an even whole number."

It is straightforward to prove the implication $$P \Longrightarrow Q$$. (We see that if $$P$$ is true, that is, that $$x$$ is a whole number divisible by 4, then $$x$$ must be an even whole number, so $$Q$$ is true.) However, $$Q$$ can quite possibly be an even number that is not divisible by 4, for example, $$x = 6$$. In this case, $$Q$$ is true, but $$P$$ is false.

To suppose otherwise is to commit a fallacy. So common are the fallacies that may be committed with regard to the conditional that they have been given names of their own:


 * Affirming The Consequent: If a conditional holds, and its consequent is true, it is a fallacy to assert that the antecedent is true. That is: $$p \Longrightarrow q, q \not \vdash p$$.


 * Denying The Antecedent: If a conditional holds, and its antecedent is false, it is a fallacy to assert that the consequent is false. That is: $$p \Longrightarrow q, \lnot p \not \vdash \lnot q$$.

Converse
The converse of the conditional $$p \Longrightarrow q$$ is the statement $$q \Longrightarrow p$$.

The converse of a true conditional is not necessarily true, and the converse of a false conditional is not necessarily false.

Inverse
The inverse of the statement $$p \Longrightarrow q$$ is the statement $$\lnot p \Longrightarrow \lnot q$$.

The inverse of a true conditional is not necessarily true, and the inverse of a false conditional is not necessarily false.

Contrapositive
The contrapositive of the conditional $$p \Longrightarrow q$$ is the statement $$\lnot q \Longrightarrow \lnot p$$.

The Rule of Transposition gives that a statement and its contrapositive have the same truth value.

Relationship between Inverse, Converse and Contrapositive
Notice that:
 * The inverse of a conditional is the converse of its contrapositive;
 * The inverse of a conditional is the contrapositive of its converse;
 * The converse of a conditional is the inverse of its contrapositive;
 * The converse of a conditional is the contrapositive of its inverse.