Binary relations are obvious, and I see the need to have 3-ary relations such as "being in between things". Are there natural language examples of relations greater 3?

Edit: without combining binary or tertiary relations

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    Just take a conjunction (or any other binary operation) of two binary relations. A is a son of B, and C is a son D. According to Quine's reduction theorem, a predicate of any arity can be reduced to a combination of binary predicates.
    – Conifold
    Sep 13, 2019 at 9:53
  • Any examples without combining binary or tertiary relations? And I guess "being in between" can be reduced to binaries, but there is a sense of it being a single emergent property of its own.
    – csp2018
    Sep 13, 2019 at 9:56
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    That does not exist, but one can make up examples where it is less obvious. A hit B with a C attached to a D. "Attached to" can be iterated indefinitely.
    – Conifold
    Sep 13, 2019 at 9:59
  • Thanks! Also interesting how it's ambiguous what is attached to D with this syntax.
    – csp2018
    Sep 13, 2019 at 10:02

3 Answers 3


If you're asking about predicate valency then there are a few examples of tritransitive/quadrivalent verbs with four arguments. For example:

I1 bet you2 five dollars3 that it would rain4.

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    @jobermark So what? This is an academic site and it's unreasonable to expect discussions to be kept at a high school level. The argument/adjunct distinction has been common in linguistics for decades. I don't think we can have a fruitful discussion when you call aktionsart a 'contrivance'. Sep 15, 2019 at 0:26
  • @jobermark I really don't know why you're angry with me. I haven't said that history is dead or what you know is obsolete. In the comments on your answer I presented two diagnostic tests for why the temporal adverbs are adjuncts not arguments. If that's not relevant to your philosophy of language just say so. And, if possible, tell us what framework you are using. Sep 15, 2019 at 0:36
  • @jobermark You never said your framework doesn't distinguish arguments from adjuncts. I thought you were trying to say that the temporal adverbs were arguments the whole way through. Sep 15, 2019 at 0:41
  • @curiousdanii You can leave out the direct object too, and have it implied. And roles are not strictly positional in any language. I can make an appositive and put the subject at the end -- the same response I made last time you offerd this same complaint
    – user9166
    Sep 15, 2019 at 0:47
  • @jobermark I didn't at the beginning understand why you thought clausal adjuncts should count for 5-ary verb relations. If I understand correctly now you're considering the relation only in terms of functional roles and not at all in terms of grammar, syntax or what constituents a verb requires or licenses. Sep 15, 2019 at 1:11

I can't quickly think of ones very obviously corresponding to a word in a natural language (such as "between" for ternary). The best I can think of is "surrounded by", since presumably (living on a two-dimensional surface) one needs at least three things to be surrounded by -- one step up from between!

In principle one could make up a word for all kinds of quaternary relations such as:

  • The relation between the four people on a double date.
  • The relation between four athletes competing for one of the three spots on the podium.
  • The relation between four points a, b, c, and d, where a and b want to be connected to each other and c and d want to be connected to each other but the line segments between them intersect. (Such a relation may be useful to talk about a specific kind of problem in geopolitics.)

A list of more than three items could be viewed as a relationship where each item in the list is connected by "and". For example,

"Today I spent $20 on food, $40 on housing, $10 on entertainment and $30 dollars on transportation."

I could symbolize my spending (S) as one part for food (f), one part for housing (h), one part for entertainment (e) and one part for transportation (t) and then write the relation S with variables as S(f,h,e,t).

I could break this up into four unary relations all connected by "and", but from a natural language perspective if I were telling someone what I spent, I would likely combine them into a 4-ary relation.

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