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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.

  • 3
    A different but related question is this "Alternative ways to say 'if and only if' " http://math.stackexchange.com/questions/39022/alternative-ways-to-say-if-and-only-if2011-05-29

5 Answers 5

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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 $p$".

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    It is actually true: (2) states exactly what $p\to q$ states.2018-09-30
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"p only if q"

"q whenever p"

"q if p"

"q is a necessary condition for p"

"q unless not p"

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

$\sim P \vee Q.$

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    Convention: ~ has precedence over or. If that doesn't work for you, insert parens.2011-05-30
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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)
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"p only if q" should be added as well. The sentence "p only if q" should not be confused with "p if q".