"Special case" means "proper special case" or "weak special case"?

Question

Without references, Wikipedia says:

Concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa

Many mathematicians agree with Wikipedia that a concept has to be strictly more general than its special cases.

However, some mathematicians also believe that a special case can be an equivalent case. For example, perfect numbers are a special case of perfect numbers.

Although in many cases it is easy to tell whether the relation is strict or not, in some complicated math structures the strictness is not always obvious.

What is the precise definition of "special case"? Any reference or consensus?

(For example, in topology, the related terms such as "generalization" and "specialization" are precisely defined.)

Answer 1

Definitions are for mathematical objects. "Concepts" are informal; to the best of my understanding, there is no such thing as "concept theory" (or if there is, the term "concept" presumably has some highly technical meaning that is unfamiliar to me, and in any case it would not be relevant to this discussion). As a result, the words we use to describe concepts are always going to be a little bit squishy, and I think it is unlikely that anyone will be able to provide serious citations for a definition like this. Definitions might appear in introductory textbooks, but each textbook author will have their own conventions, and you will not find perfect consistency here.

Most people would probably understand perfect numbers to not be a "special case" of perfect numbers. When we say that X is a special case of Y, in addition to literally stating that X is a subset of Y, we're (usually) also trying to communicate the fact that X has some feature which makes it particularly easy, straightforward, or interesting to reason about in comparison to the rest of Y. This tends to imply that "the rest of Y" is nonempty, and so X = Y would usually not make much sense.

I can think of one example where we might write something like this, and that's when it is unknown whether X and Y are the same. For example, even perfect numbers might be called a "special case" of perfect numbers, because they have many interesting and well-understood properties that might not generalize to odd perfect numbers (e.g. their relationship with the Mersenne primes). It is conjectured that all perfect numbers are even. If that conjecture were proven, then the properties of even perfect numbers would be vacuously true of odd perfect numbers, and so even perfect numbers could no longer be plausibly characterized as a special case. But while the problem remains open, I would have no objection to describing them that way.

Answer 2

I agree with Kevin’s answer & comment, and would further argue that the phrase ‘special case’ simply means an example of particular noteworthiness.

As such,

• A ‘special case’ might even refer to an exception: for example, “on the reals, save for the special case $x=3,$ $f(x)$ is positive.”
• If a special case exists, then it makes sense that a non-special case potentially also exists (else we’d have the contradiction of a special case not being, well, special). Thus, the even perfect numbers are a special case of the perfect numbers, as the even case is remarkable (even if the odd case turns out to not exist).
• Calling $T(x)$ a special case of itself doesn’t make a lot of sense, as nothing is being highlighted here.

The phrase ‘special case’ isn't part of a theory, so doesn't have a mathematical definition. The “proper subset” connotation of special case is a special—and the most common—case of its general meaning “noteworthy/remarkable case”.

Answer 3

It is nice that Wikipedia has such articles that offers the possibility of clearer delineation of notions. However, it would be better if a more formal term, such as 'definition' or 'statement', were used instead of 'concept'. The word 'concept' has strong associations with mental contents (whether linguistic or non-linguistic) and the practice of language by a speakers' community, thus may uncautiously broaden the issue, and in formal settings, paradoxical cases may ensue (just recall Russell's paradox).

So, if a definition or a statement is put down in a sufficiently formal and precise fashion, there should not occur a confusion about its usage (speaking for the formal disciplines like mathematics). Otherwise, the confusion signifies that there is a deficiency in it.

Prudently, the Wikipedia article refers to instances, because instances are objective enough to settle the blurred lines. When there is no deficiency so that we can discern the instances that fall under the definition or the statement, we can carry on with set-theoretic comparisons.

The following excerpt from Manna and Pnueli's The Temporal Logic of Reactive and Concurrent Systems: Specification is a careful usage of terms, and I suppose, quite suggestive for the present topic:

As a side note, some authors of logic use "specialisation rules" for "instantiation rules"; but, as above discussion implies, not a good practice of terminology.