Different ways to express If-Then

Different ways to write, or express, the conditional statement $p \rightarrow q$ besides "if $p$ then $q$."

1. "$p$ is a sufficient condition for $q$"; or
2. "$p$ only if $q$";
3. "$p$ implies $q$";
4. "$q$ whenever $p$"
5. "$q$ is a necessary condition for $p$" (i.e., "if not $q$, then not $p$", or $\lnot q \rightarrow \lnot p$);
6. "$q$ is a consequence of $p$";
7. "$q$ follows from $p$";
8. "$q$ if $p$".
9. "if not $q$, then not $p$."
10. "not $p$, or $q$"
11. "not ($p$ and not $q$)

Logically, we can write $(10)$ as $$(p \rightarrow q) \equiv (\lnot p \lor q)$$ and $(11)$ as $$(p \rightarrow q) \equiv \lnot(p \land \lnot q)$$

Those are just a few of the ways one can express "if $p$, then $q$." But some expressions may be more intuitive than others.

One final note: The term "unless" also relates to "if and only if" in the following sense: as in "$p$ unless $q$" is equivalent to "unless $q$, then $p$" which is equivalent to "if not $q$, then $p$".

"p only if q"

"q whenever p"

"q if p"

"q is a necessary condition for p"

"q unless not p"

The proposition $P\Rightarrow Q$ is logically equivalent to

$$\sim P \vee Q.$$