# Is there a conjunction bias?

This is slightly related to question The unprecedented success of the “intersection” operator .

Apart from a set of maths books of null measure, most have the following property:

Objects definitions are presented as a conjunction of properties.

Most axiomatic are also clearly conjunctive in their presentation.
It is uncommon to have say "By definition a Zorglub is a red zorg or a white zorg".

Q1 : Do you agree with the bias (if not, give enough examples)?.

Q2: Is this bias mainly a discourse convention or does it lie deeper (where?) ?

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We generally want to talk about very specific things; conjunctions increase specificity, and hence they're useful. Maybe I'm missing the point---are you asking for more than this sort of answer? –  Scott Morrison Feb 16 '11 at 2:35
Conjunctivitis can be extremely contagious. –  Tom Goodwillie Feb 16 '11 at 4:35
@Scott: Specificity is relative (level) . A 'human being' is a man of a woman ( Yet of course you could argue that the right name is 'social human being' and that is certainly less natural as a concept as it's longer name shows).Yes I ask for more but your idea is good as one of the reason for the bias, yet it look to me as not being the only one. –  Jérôme JEAN-CHARLES Feb 16 '11 at 5:19
I don't think I agree with this bias- every time we say "by abuse of notation" we mean a disjunction. A common disjunction in mathematics would be define a zorg to be either a specific zorg or an equivalence class of zorgs, and similarly for maps between zorgs. For instance, a knot is a PL embedding of S^1 in S^3 or in R^3; or an ambient isotopy class thereof. –  Daniel Moskovich Feb 16 '11 at 14:31
When I first saw this question, my impression was that it was too broad to get useful answers. But I wasn't sure, so I let it lie for a while to see what kind of answers would be given. Now, 14 hours later, the response I like best is Tom Goodwillie's conjunctivitis joke...so I have voted to close. –  Pete L. Clark Feb 16 '11 at 16:26

## 1 Answer

A slightly tongue-in-cheek answer: definitions are the hypotheses of theorems. If the hypothesis of a theorem is a disjunction, you can always split the theorem up into two theorems with separate hypotheses, and doing so will often clarify the statement and proof anyway. So it's natural that single definitions are not usually disjunctive.

(Dually, if the conclusion of a theorem is a conjunction, then you can split it up into two theorems with separate conclusions. But I think we don't as often give names to the conclusions of theorems, unless they happen to coincide with a definition we gave for some other reason elsewhere.)

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@Mike : OK very nice point! It covers definitions that are tailor maid for a theorem (to clarify exposition) and this does not preclude them to be meaningful, There also the case of specialization that fit the conjunction bias yet does not seem to enter your case ? Do you agree? –  Jérôme JEAN-CHARLES Feb 18 '11 at 0:45
"Specialization" meaning what? –  Mike Shulman Feb 18 '11 at 5:35
I was not precise : A green zorg is a specialization of a zorg, and this fit the bill, this was my idea. A vector space on the reals can also clearly be called a specialization (of vector space), it fit the bill of conjunction (may be not so well): One says V IS a vector space on a field F AND F = R. I use specialization as adding/picking 1 or more specifics. –  Jérôme JEAN-CHARLES Feb 19 '11 at 1:06
But why do we specialize a definition in some way? Often, because there is some theorem we can prove in the specialized case but don't (yet) know how to prove in the general case. I didn't just mean definitions tailored for particular theorems; more generally we tend to make definitions so that we can then prove numerous theorems using those definitions as hypotheses. –  Mike Shulman Feb 19 '11 at 21:21
Ok agreed. You sometimes may also specialize a true general result to illustrate (obtain mental pictures). I think I use specialization in the sense of particularization. –  Jérôme JEAN-CHARLES Feb 20 '11 at 0:09