What are some different ways to write the conditional statement $p\implies q\,$, but in English?

There's the obvious "If p, then q", but are there any other ways to write it? I'm looking for another 3 or 4 ways to express this.


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 $p$".

  • 2
    $\begingroup$ "q" is a necessary condition for "p", for the sake of symmetry. $\endgroup$ – Yuval Filmus May 29 '11 at 23:47
  • $\begingroup$ thanks @Yuval...I added that, for the sake of symmetry! :) $\endgroup$ – Namaste May 29 '11 at 23:59
  • $\begingroup$ Note: I believe that 9, 10, and 11 don't hold in intuitionistic logic. $\endgroup$ – Lily Chung Nov 16 '13 at 22:45
  • $\begingroup$ The following equivalences may inspire other ways of expressing "if $p$ then $q$": \begin{align} (p \to q) & \;\equiv\; (p \equiv p \land q) \\ (p \to q) & \;\equiv\; (p \lor q \equiv q) \\ \end{align} $\endgroup$ – MarnixKlooster ReinstateMonica Feb 20 '14 at 19:10
  • $\begingroup$ I don't think (2) is correct. $\endgroup$ – Goldname Jan 5 '18 at 0:33

"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

$$\sim P \vee Q.$$

  • $\begingroup$ Convention: ~ has precedence over or. If that doesn't work for you, insert parens. $\endgroup$ – ncmathsadist May 30 '11 at 0:32

There appears to be some confusion in several answers above, I do not have sufficient reputation points to add a comment to the question and it seem rude to edit the answer p⟹q does not imply q⟹p

let p be "john drives to another city" let q be "john gets in a car"

If "John drives to another city" then "John gets in a car" but it does not follow that If "John gets in a car" then "John drives to another city"

For a numeric equivalent let p be x = 4 let q be x^2 = 16

If x=4 then x^2=16 but it does not necessarily follow that If x^2=16 then x=4

Hence only the following are true:

  • q whenever p
  • q if p
  • p is a sufficient condition for q
  • p implies q
  • q follows from p
  • q is a consequence of p
  • not p, or q
  • not (p and not q)

"p only if q" should be added as well. The sentence "p only if q" should not be confused with "p if q".


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