Conventions on definitional if(f)

Question

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance:

Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such that $ B(x,\epsilon) \subseteq A$.

Since this is a definition, there are obviously no cases when the reverse conditional fails so it would be true to write 'iff' instead. But it seems strange to me that it's not common to write the formally stronger statement. I suppose the reasoning is (a) the lack of ambiguity mentioned above (b) potentially writing 'iff' might look as though one were stating an equivalent condition that should not be taken as the definition, e.g.

Observation 2: A set $A$ is open in $(X,d)$ iff $X\setminus A$ is closed in $(X,d)$.

Am I right that this is the convention? Is it acceptable/understandable to write 'iff' for definitions? Apologies if this is not a well-enough-formed question for the local standards.

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It's also occured to me that there might be space in the notation to adapt the definitional '$:=$' to give '$:\!\mathrm{iff}$' to be used in such cases, eg.

Definition 3: A set $A$ is open in $(X,d)$ :iff $\forall x \in A$, $\exists \epsilon \gt 0$ such that $ B(x,\epsilon) \subseteq A$.

Or indeed:

Let $A := \{1,2,3\}$ and $B:=\{1,2\}$. Then each $b \in B$ is also in $A$. Now $$\forall b \in B, b \in A \quad \mathrm{iff\!:} \quad B \subseteq A$$ so $B \subseteq A$ by definition.

Has this been used? Would it be sensible usage? Can I claim it as a great notational victory and tell people about it at parties?

Answer 1

Sidestepping the philosophical stuff that's about to ensue, let me say this. Since "if" in a definition is correct already, it would be unattractive to replace it with a more restrictive condition "iff." In mathematics, a rule of thumb is to not overcomplicate something by using a stronger thing when a weaker thing already suffices. Basically, you have nothing to gain but dirty looks from those who believe "iff" is incorrect :)

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No, it is conventionally not really right use the biconditional when first defining terms. I finally managed to dig up this exchange at the math wikiproject which contains some insights on preferring "if". I am aware of another exchange on the topic in 2006 where an editor vehemently advocated "iff," but I don't think that author or his arguments matched (the expertise of) the ones given in this more recent discussion I am linking. (Even at the 2006 discussion, Ryan Reich showed up to weigh in on preferring "if".)

I think the links I provided have ample evidence to show that the most popular convention is to use "if" and not "iff." One very experienced mathematician at the math wikiproject went so far as to say that the use of iff in definition is "a hallmark of amateurish mathematical writing that almost never appears in quality publications."

(Incidentally Wikipedia also has a little bit addressing this, and I know that the mathematics project Manual of Style includes lines about not using iff in definitions.)

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It is fine to use the biconditional when showing that another condition is equivalent to the condition you used when first defining your term.

When you make a definition, you are relabeling a (potentially complex) set of conditions with a simpler name. I don't think it is really a logical "if", it is more of a definitional linguistic "if". Some logician may show up and blow me out of the water by saying that there really is no difference, but I'll still go out on a limb and try to describe why using "iff" sounds fishy to me.

It's tempting to conflate the logical biconditional with the linguistic relation of being "synonymous." However, you have to remember that when we are writing biconditionals we are within the framework of some logical calculus. The terms that are referred to in this calculus have to be defined before we can incorporate them in logical statements.

Another thing to realize is that you don't really need "if" to write definitions. You can say things like "we define a square to be a simple polygon which satsifies (conditions)." Or: "There are seven days in the week Monday, ... Friday. The two days Saturday and Sunday are defined to be weekend days.

There isn't really any "if A, B" or "A if B" going on here: the act of defining takes place just outside of the logical framework.

Answer 2

The distinction between if and iff is that if can be a subset relation, while iff is a set equality relation.

The role of a definition is to bring things into view of a theory, so it needs to deal with failure in the theory as well. Correspondingly, while a definition might be inspired by the theory, it makes no assumptions that are left to the theory.

Here we shall use quotes to mark out what is being defined. For example, '1 foot = 12 inches' is a statement. Writing '1 foot = 12 "inches" ' is the definition of an inch as 1/12 foot. When quotes are set around a relation, the items are elsewhere defined, and the relation is said to be true: '1 foot "=" 12 inches' supposes the foot and inch are separately defined (eg as fractions of a yard), and the equality is said to be true. It can be set around a number, when one wants to discuss that relationship, eg 'One metre = "39.37" inches'.

A statement 'A iff B' equates to '(A if B) = (B if A)'. This is a relationship which can not be used as a definition. One can use statements like

• ("A" if B) and (B if "A") requires both to be true.
• ("A1" if B) and (B if "A2") leaves 'A1 = A2' to be set by the theory
• ("A if B) , leaving B if A to theory.

The reason that the equality fails, is that it is not the role of definitions to make assumptions about either A or B. Instead, it must assume that the sets A and B are not identical, and 'A if B' suggests that A is a subset of B.

Defining both parts of the relation, means that some X can only become A if both B arises from it, and it arrises from B. But it is well in the scope of the theory to find B from A if A comes from B. So the first definition is actually redundant, and the second part might be discarded.

The definitions by A1 and A2, is useful to test the variations of A are identical. There is a test that inertial mass ($F = ma$) and gravitational mass ($a = GM/r^2$), these are known to be exact to 14 places. This experiment could be written as A1"="A2.

In practice, one either defines ("A" if B) or defines (B if "A"), and let the theory set the other value.