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If $P$ and $Q$ are statements,

$P \iff Q$

and

The following are equivalent:

$(\text{i}) \ P$

$(\text{ii}) \ Q$

Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while others use the former. Is it simply out of convention or "etiquette" that one formulation is preferred? Or is there something deeper? Thanks!

Community
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Bachmaninoff
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3 Answers3

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As Brian M. Scott explains, they are logically equivalent.

However, in principle, the expression $$(*) \qquad A \Leftrightarrow B \Leftrightarrow C$$ is ambiguous. It could mean either of the following.

  1. $(A \Leftrightarrow B) \wedge (B \Leftrightarrow C)$

  2. $(A \Leftrightarrow B) \Leftrightarrow C$

These are not equivalent; in particular, (1) means that each of $A,B$ and $C$ have the same truthvalue, whereas (2) means that either precisely $1$ of them is true, or else all $3$ of them are true. Also, you can check for yourself that, perhaps surprisingly, the $\Leftrightarrow$ operation actually associative! That is, the following are equivalent:

  • $(A \Leftrightarrow B) \Leftrightarrow C$
  • $A \Leftrightarrow (B \Leftrightarrow C)$.

In practice, however, (1) is almost always the intended meaning.

goblin GONE
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    I would never write the second one without parentheses. Also because there's also a third possible interpretation: $A\iff(B\iff C)$. As a general rule, for a nested binary operator $@$, parentheses should only be omitted iff $(A @ B) @ C$ and $A @ (B @ C)$ are equivalent. – celtschk Jul 10 '13 at 07:04
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    @celtschk, biconditional is associative - see the last sentence of my answer. – goblin GONE Jul 10 '13 at 07:09
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    Ah, I missed that. That's indeed surprising. Although on second thought, it's perhaps not that surprising; after all, it makes sense that equivalence is an equivalence relation :-) – celtschk Jul 10 '13 at 07:15
  • @celtschk, okay I've just reorganized a little bit to make things clearer. – goblin GONE Jul 10 '13 at 07:17
  • Yes, with that rearrangement, the statement is no longer easy to miss. – celtschk Jul 10 '13 at 07:19
  • @celtschk, thank you, I think it's better now. Not quite sure what you're getting at with regards to equivalence being an equivalence relation. While that's certainly true, its not clear to me the connection with associativity. – goblin GONE Jul 10 '13 at 07:21
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    Well, just ignore that remark; I was temporarily confusing associativity and transitivity. I shouldn't write comments before the second coffee, I guess ;-) – celtschk Jul 10 '13 at 09:15
  • @celtschk, no worries. Yeah they're conceptually kind of similar. – goblin GONE Jul 10 '13 at 09:16
  • Perhaps "exactly equivalent" has some technical meaning that I'm not aware of, but to me it implies "equivalent in every way", which I would deny. They are logically equivalent, but conceptually different. – Ray Jul 10 '13 at 12:24
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    Actually I now noticed that in three-valued Łukasiewicz logic they are not equivalent: $((F\iff U)\iff U) = (U\iff U) = T$, but $(F\iff(U\iff U))=(F\iff T)=F$ – celtschk Jul 11 '13 at 09:14
  • @celtschk, yes that is a good point. See also the answer given here. – goblin GONE Jul 11 '13 at 09:52
  • This answer implies that associativity for an operation is not a sufficient condition to drop parentheses around an expression in infix notation. There exist a lot of people who seem to believe such. – Doug Spoonwood Jul 12 '13 at 16:57
  • Yes, I would be very surprised if anyone wrote $A\iff B \iff C$ and meant $(A\iff B)\iff C$; since TFAE is overwhelmingly a more common and useful notion in mathematics, I would expect $A\iff B \iff C$ to mean that. – Mike Shulman Apr 13 '15 at 17:36
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They are exactly equivalent. There may be a pragmatic difference in their use: when $P$ and $Q$ are relatively long or complex statements, the second formulation is probably easier to read.

Brian M. Scott
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    Also, TFAE is very nice for when there is more than one claim (especially when there is no obviously preferable way of ordering them in an IFF chain). – anon Jul 10 '13 at 06:05
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"TFAE" is appropriate when one is listing optional replacements for some theory. For example, you could list dozen replacements for the statements, such as replacements for the fifth postulate in euclidean geometry.

"IFF" is one of the implications of "TFAE", although it as $P \rightarrow Q \rightarrow R \rightarrow P $, which equates to an iff relation.

wendy.krieger
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