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The word "if" is used in two meanings in mathematics:

Definition. A topological space is compact if every open cover has a finite subcover.

Theorem. A topological space is compact if it is finite.

Remark. There are infinite compact topological spaces.

In a definition "if" usually means "if and only if" (or more properly something like "this is what we mean by this word"). In a theorem "if" means implication; it is the usual logical if. Some people prefer to use "if and only if" in definitions, but this is not a universal practice so I cannot completely avoid the phenomenon.

Questions:

  1. Can students tell the difference between the two ifs? Research is preferred but experience is accepted.
  2. In your experience, has this double meaning of the word "if" actually lead to significant confusion? That is, is there a reason to worry that this might confuse someone?
  3. Has it been studied whether this issue slows down (or prohibits) learning what definitions are and how to use them?
  4. Would it be enough to remind students about this issue (once in every course, say) or should I do more? This will probably become unnecessary after a couple of years at the university, but it may be necessary at first.

Context: Consider bachelor level courses in universities. There are no separate "introduction to proofs" courses here, just courses about areas of mathematics. (The closest equivalent is "introduction to mathematics" which is the very first course and covers some basic proof techniques, but I cannot rely on students remembering all details from that short course.)

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    $\begingroup$ Great question. Anecdotally, I had professors who made an attempt to write iff for the definitional if each time, though force of habit sometimes made such quests not fully successful. There's also the alternative when lecturing to use the $\Rightarrow$ and $\Leftrightarrow$ arrows in lieu of the 'if ... then' and 'iff' linguistic constructions. $\endgroup$ – Michael Joyce Nov 24 '15 at 16:31
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    $\begingroup$ I have never distinguished between the two "ifs" and I suspect there is not chance of confusing them. You could always rewrite your definitions to avoid "if" eg: A topological space with the property that every open cover has a finite subcover is called compact $\endgroup$ – David Steinberg Nov 24 '15 at 18:18
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    $\begingroup$ I feel like a more fundamental question is "can students tell the difference between a definition and a theorem"? I would conjecture that when the answer to that is "yes", so is the proposed question. $\endgroup$ – Daniel R. Collins Nov 24 '15 at 23:59
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    $\begingroup$ @DanielR.Collins, I've encountered enough cases of confusing theorems with definitions that I occasionally make a point of saying "OK, but we can define anything whatsowever. Let's see why this makes sense" or something along those lines, and also explaining the importance/suprise of some theorem. $\endgroup$ – vonbrand Nov 25 '15 at 11:48
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    $\begingroup$ Our teacher has replaced the definition word if by the word when i.e. We say that a topological space is compact when every open cover has a finite subcover. $\endgroup$ – Thinkeye Nov 25 '15 at 20:45
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Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such:

It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive equality, for which mathematics has surprisingly mediocre accommodations, in contrast to computer science. (I'm not a fan of colon-equality or variants, but prefer clearer, less fragile, context-setting.)

In the case of definitional-iff, nowadays I prefer (as @DavidSteinberg suggests) entirely different language, often "Say that X is a Y when Z-property holds."

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    $\begingroup$ Your anecdote can be explained by mathematicians understanding a definition to be simply an assertion that is an instantiation of an axiom or theorem of an existential form. When you have real numbers $b,c$ and say "Let $a = b+c$." you are defining $a$ by instantiating the closure of the reals under its addition operation, naming the instance $a$, and thereafter it is a true assertion that $a = b+c$. $\endgroup$ – user21820 Nov 25 '15 at 3:12
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    $\begingroup$ Similarly for "iff" in definitions. When you say "We call an X a Y iff it satisfies Z.", we are simply creating the subtype of X that satisfies Z, and hence "An X is a Y iff it satisfies Z." is now a true assertion. In other words the execution of the definition is simultaneous with the desired true assertion, which assertion is what the mathematician actually wants anyway. $\endgroup$ – user21820 Nov 25 '15 at 3:16
  • $\begingroup$ (As for why I said "subtype" instead of saying "subset given by the axiom of comprehension", consider that mathematicians have no qualms saying "We call a set finite iff it is in bijection with an initial segment of $ω$." despite the type "set" not corresponding to a set under pain of contradiction.) $\endgroup$ – user21820 Nov 25 '15 at 3:18
  • $\begingroup$ Thanks! Comparing definitional/assertive iff to assignment/assertive equality reminds me of the notation $:\!\!\iff$ some people use in definitions in analogue to the assertive equality $:=$. I have also seen some professional mathematicians use $:=$ to mean both $:=$ and $=:$, and I find this more confusing than using $=$ for all kinds of equality. $\endgroup$ – Joonas Ilmavirta Nov 25 '15 at 12:05
  • $\begingroup$ @JoonasIlmavirta: Some people I know have used "$:\Leftrightarrow$" but not consistently. Anyway most important (at least pedagogically) is unambiguity; indeed it is worse if "$:=$" is used but with the defined object being on arbitrary side, than it is if "$=$" is used but with the defined object always being on the left. In the latter case the "Let" already unambiguously tells the reader what is being defined and so using the same equality symbol is a small issue. In contrast, using "if" in definitions in place of "iff" is a different and much larger issue. $\endgroup$ – user21820 Nov 25 '15 at 14:05
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iff and if

In my experience, students who have a solid grasp of first-order logic have absolutely no problem with the inconsistent use of "if" in definitions. The problem is that most students don't, and as I've personally observed, often confuse between "if" and "iff" precisely because of such notational issues. Therefore the answer to your first two questions is "No" and "Yes" for many students.

The reason it is extremely inconsistent to use "if" when "iff" is meant is because we commonly have the following two types of definitions:

We say that an X is A iff it satisfies P.

If an X is A, then define f(X) to be Y. [Here "f(X)" may be meaningless for an X that is not A!]

If you want concrete examples:

We say that a real number is rational iff it is the ratio of two integers.

If a set of real numbers $S$ is finite, define $\max(S)$ to be the maximum element of $S$.

Clearly, in the second type of definition, "if" is certainly not "iff". In a technical sense it is not even the material implication, but to satisfactorily deal with that we would have to use some form of type theory, which is beyond the scope of this answer.

As for your third question, I don't know any formal study conducted on this issue, but I have encountered some very ambiguous definitions before, where usage of "if" made it unclear whether "iff" was meant. At the moment I can't recall them, but I think it was an "if" that was buried in some complicated definition. Also, when definitions are understood as "iff" statements, then many proofs become trivial at least at the lower levels since often simply unfolding definitions yields the proof after one or two other small logical steps. In my experience, many students do not understand how to unfold the definitions both forward and backward to meet in the middle, and hence find many logically trivial theorems very hard to prove!

For your last question, you should simply use "iff" whenever you mean "iff", and tell students that some people don't and so they have to be careful when they read other mathematical writings and always check that they are certain which is meant.

Let and take

On an unrelated note, there is a similar notational inconsistency between the two common uses of "let". One is an instantiation of an existential assertion and the other is the creation of a new context of universal quantification:

[Where $S$ is a non-empty set:] Let $a \in S$. [This instantiates the non-emptiness of $S$.]

Take any $r \in \mathbb{Q}$ such that $r^2 = 6$. [Many people use "Let $r \in \mathbb{Q}$ such that $r^2 = 6$.".]

I personally strongly disapprove of using the same notation in both cases, since in the second example it is totally false to assert that there is such an $r$, and empirically students often do not know the logical structure of proofs and have great difficulty distinguishing between the two types if "let" is used in both cases. Some authors are careful to distinguish them, and one way is to use "take" or "given" for universal quantification.

Side remarks

A number of logicians I've spoken to agree with me on the above two notation issues, but they also say that they don't know what can be done about it, and their main reason is that "most of the textbooks and papers use it". I don't think that's too important in considering whether or not to continue using it, and indeed some of them thereafter try to unambiguously distinguish these in their own teaching.

The main problem is that teachers already know what they mean when they write "if" or "let", but they don't realize that most students don't know. The few that eventually grasp it fully are usually the only ones that become teachers of the next generation... And unlike a compiler that tells the programmer in his face that his program is meaningless (syntax error), mathematics teachers try hard to guess the students' meaning, usually giving benefit of doubt, and hence many students get away with random guesswork, simply because the teachers know how to modify their proofs to make it work.

Incidentally, you will never find the same kind of error in the programming world, because instantiation is equivalent to an assignment of a variable with the result of a function call, whereas quantification corresponds to a for-each loop. Similarly, programmers were the first to realize the great importance and utility of scoping, indicated by either braces or indentation, usually both. In the same way it would be excellent for mathematics students to learn this, and one way would be through teaching natural deduction in Fitch-style. Sadly, this is almost never done.

Lest you disbelieve me, I've regularly overheard students discussing among themselves how to prove a theorem, and invariably it goes:

To find limits we always write down "Let $ε > 0$." first...

The question is about injection, so write down "$f(a) = f(b)$"...

And worse still:

Let's try induction. If $n$ is true, ..., therefore $n+1$ is true.

Proof by induction: If $n = n+1$, ...

Imagine if people who have learn programming for 3 months still wrote:

int n; input(n);
if( n==true ) { ... }
if( n==n+1 ) { ... }

They don't, but why? It's because their compilers force them to learn something that mathematics teachers don't ensure their students learn, which is actual logical reasoning including conscious type-checking.

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