# Is “iff” the same as equality if each member is a predicate?

"Iff" - if and only if ($\Leftrightarrow$ or $\leftrightarrow$, although the first usually carries a "meta" meaning, something that is not evaluated) - is used in $2x=3\Leftrightarrow x=\frac32$, because both $2x=3$ and $x=\frac32$ are boolean expressions. When talking about the members of the equations equivalences, and not the members of the equations themselves, this makes no sense, because as in $2x=3$, neither member (not $2x$ nor $3$) is a predicate. In other words, if you say «$2+2=4$», I'd say «OK», but saying the $2+2$ itself is meaningless, as the result is neither void (as in a command or question) nor boolean (as a usual affirmative or informative sentence).

In this sense, equivalence (iff) between predicates is the same as equality. But why have another symbol like $\leftrightarrow$, why can't equations' equivalences be written as $(2x=3)=\left(x=\frac32\right)$? Is it that for any two predicates $p$ and $q$ relying respectively on tuples (or ordered multisets or whatever you want to call them) of variables $P$ and $Q$, $(p(P)\leftrightarrow q(Q))\Leftrightarrow(p(P)= q(Q))$?

Also, why does $\LaTeX$ distinguish between \Leftrightarrow (resulting in $\Leftrightarrow$) and \iff (resulting in a longer arrow $\iff$)? Surely the people who made it thought there should be a reason in Mathematics why these symbols seeming to carry the same meaning would be separate glyphs.

What kind of conventions are there regarding "iff" and equality? Where can these be considered the same concept? What are, fundamentally, the differences between them?

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The $\LaTeX$ part of the question could probably be answered better on the TeX.SE site... –  anorton Sep 3 '13 at 22:40
@anorton, I asked it here because it is related. If all the other questions are answered well, those will likely be automatically answered. Whoever made this was thinking specifically about Mathematics. –  JMCF125 Sep 3 '13 at 22:42
True. I'm just saying that if you can't get an answer about that here, you can probably get one on TeX.SE. There are a lot of people on that site who seem to do nothing but study typographical differences in characters all day. :) –  anorton Sep 3 '13 at 22:45
In my opinion, there's nothing logically wrong with writing $(2x=3) = (x=3/2),$ indeed the sentence $\forall x[(2x=3) = (x=3/2)]$ can be viewed as a theorem of the real numbers. However, for reasons of readability, I'd prefer to use $\iff$. –  goblin Sep 4 '13 at 14:59
@JMCF125, I think so. But it depends. In first-order logic, there is a distinction between $=$ and $\iff$. But, I do not think it is an essential distinction, and I'm sure we can think of more unified logics where they coincide. –  goblin Sep 4 '13 at 15:06

For your first question, there is a lot to say about the issue (which is related to the issue of intensional vs. extensional equality) but one important difference is simply that logical equivalence and equality apply to different types of things; the former applies to logical formulas and the latter applies to terms. If we were to use the symbol "$=$" for both, then the rules of any formal system for manipulating expressions with "$=$" would be more complicated because they would have to split into cases depending on which type of thing the "$=$" applied to.

Another issue with your proposed expression $(2x=3)=(x=3/2)$ is the use of parentheses to serve as quotation marks for logical formulas, which could invite confusion with the more usual use of parentheses.

For the second question, I think the reason is that \Leftrightarrow is a generic symbol that can be used for lots of different things, and \iff is for logical equivalence, which is a binary operator with low precedence in the order of operations, so it is elongated in order to provide a large visual separation between the things on either side.

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"Iff" - if and only if... is used in $2x=3\Leftrightarrow x=\frac32$, because both $2x=3$ and $x=\frac32$ are boolean expressions.

Yes. These expressions are either true or false.

When talking about the members of the equations equivalences, and not the members of the equations themselves, this makes no sense, because as in $2x=3$, neither member is a predicate.

In mathematics, equality is a relation (a binary predicate) on some set of objects (the rational numbers in your example). The objects in a set are never themselves predicates or other Boolean expressions.

In this sense, equivalence (iff) between predicates is the same as equality.

No. Equality is a relation on a set. "Iff" is not a relation on any set. It is a logical operator connecting a pairs of Boolean expressions to form another, as are the "And" and "Or" operators.

"$x=2$" makes sense. "$x\Leftrightarrow 2$" makes no sense.

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«The objects in a set are never themselves predicates or other Boolean expressions.»: wait, what? Why not? Why can't they be functions to booleans (AKA predicates)? Does any set in any set-theory have this limitation? –  JMCF125 Sep 4 '13 at 14:59
@JMCF125 I don't know of any set theory that allows for sets of logical predicates. –  Dan Christensen Sep 4 '13 at 15:03
«"$x=2$" makes sense. "$x\Leftrightarrow2$" makes no sense.»: exactly, because $x$ and $2$ are not predicates. But what if they weren't? Would it be indifferent (or determined by convention) to use one or the other? –  JMCF125 Sep 4 '13 at 15:04
Do you know a set theory that explicitly says that is NOT allowed? Because that sounds like assuming "I don't know any set that allows numbers beyond a googolplex times the Graham's number". :) –  JMCF125 Sep 4 '13 at 15:07
@JMCF125 $P(Q(x))$ amounts to same kind of thing -- a predicate with a predicate as an argument. –  Dan Christensen Sep 4 '13 at 16:32