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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.

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3  
A different but related question is this "Alternative ways to say 'if and only if' " math.stackexchange.com/questions/39022/… –  Américo Tavares May 29 '11 at 23:49

3 Answers 3

up vote 17 down vote accepted

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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"q" is a necessary condition for "p", for the sake of symmetry. –  Yuval Filmus May 29 '11 at 23:47
    
thanks @Yuval...I added that, for the sake of symmetry! :) –  amWhy May 29 '11 at 23:59
    
+1 nice logical statements. –  B. S. Feb 17 '13 at 9:55
    
Note: I believe that 9, 10, and 11 don't hold in intuitionistic logic. –  Istvan Chung Nov 16 '13 at 22:45
    
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} –  Marnix Klooster Feb 20 at 19:10

"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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+1 for "q whenever p" –  Arjang May 29 '11 at 23:48

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. –  ncmathsadist May 30 '11 at 0:32

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