# Is there any reason for the heavy focus on binary relations in formal logic?

As a fan of C. S. Peirce, I'm surprised that, at least triadic relations, aren't investigated as much as binary relations are. What I mean is that with binary relations, they have already been classified by reflexivity, symmetry, transitivity, and the list continues e.g. in the semantics of modal logic. But as far as I can find, outside of Peirce, there isn't an equivalent investigation into triadic relations (Peirce seems to say that all relations beyond ternary ones "reduce" in some sense to unary, binary, or ternary relations).

Additionally, at one time I tried, but wasn't a mathematician enough, to classify all possible "ideas" that could at least be formulated in first order logic. For instance, a reflective relation was one where it is true that (x)(Rxx), and so I realized that even binary relations could be classified where multiple relations are involved. For instance, just as a symmetric relation satisfies (x)(y)(Rxy & Ryx) you can have relations that are only symmetrical along two predicates e.g. (x)(y)(Rxy & Syx). Does this make sense?

Basically, I'm asking if there is any reason why formal logic focuses on binary relations involving one predicate among the vast sea of other kinds of logical relations that also express our concepts? Am I just not finding the relevant literature or is this just a cultural and historical artifact?

• Interesting question. Do you have an example of a non-binary relation that cannot be expressed in terms of unary and binary ones? Between(x,y,z), for example, which is true just in case its second argument is in some sense between its first and third arguments, could be expressed as RightOf(y,x) & LeftOf(y,z). Commented Aug 18, 2014 at 19:14
• @HunanRostomyan: in this case, the core concept of between-ness is being brushed aside. For you also require Colinear(x,y,z); and Left-ness and Right-ness assumes frames of reference which would be defined via yet further objects. Commented Aug 18, 2014 at 19:29
• @NieldeBeaudrap Thanks. I agree with that. I'm interested nevertheless in the question of the theoretical dispensability of Between and n-ary (for n>2) relations in general in favor of relations of lower arity. An affirmative answer wouldn't mean that triadic relations shouldn't be classified in terms of high-level concepts, but it would explain to some small extent, at least, the lack of neglect. Assuming, of course, that these relations have been neglected by logicians. Commented Aug 18, 2014 at 19:41
• I think the textbook example is when person x gives gift y to person z. I don't see any way of reducing this to binary relations. Commented Aug 18, 2014 at 21:04
• Just a few corrections: a relation R on some set A such that [for all x in A, xRx] is called reflexive, not "reflective"; R is symmetric (not "symmetrical") if and only if [(for all x,y in A) (xRy => yRx)], where "=>" is supposed to be implication (no MathJax on philosophy.stackexchange!?!). Commented Jun 3, 2016 at 3:02

This is an important question and you make a number of points worth thinking about more deeply. I offer the following not as an answer, but as a formal prelude to more worthy answers. We start with relations:

§1. n-ary Relations as Sets of n-tuples

According to the standard set-theoretic account of relations, an n-ary relation R is nothing more or less than a set SR consisting of n-tuples of elements from n underlying domains. Some examples:

• The binary relation Taller, for example, is the set STaller = {(x, y) : x is taller than y}.
• The unary relation (or predicate) Prime, is the set SPrime = {(x) : x is prime}, where the weird 1-tuple (x) can be identified either directly with the individual x or the ordered pair (x, x).
• The ternary relation Closer, is the set SCloser = {(x, y, z) : x is closer to y than to z}.

As you rightly noted, the properties of binary relations have been paid a lot of attention to. We are exposed to reflexive, serial, euclidean, transitive, symmetric, equivalence, ... relations with increasing emphasis in philosophical logic courses, and so on. We rarely come across treatments of more than binary relations. While I do not know what the reason for that is, I suspect it has something to do with the theoretical dispensability of relations of higher arities. Here's why they're theoretically dispensable.

§2. n-tuples as nested ordered pairs

Since from the set-theoretic point of view, an n-ary relation is just a set of n-tuples, it is possible to reduce an arbitrary n-ary relation to a binary one. We start either by defining the ordered pair (a,b) in terms of sets (by Kuratowski's or Wiener's or some such method) or taking it as a primitive so long as the equivalence [ (a,b) = (c, d) iff a = c and b = d ] holds. Having defined the ordered pair, the 2-tuple, we can then define triples (a, b, c) as the pairs (a, (b, c)), quadruples as the pairs (a, (b, (c, d))), and in general n-tuples (a1, a2, ..., an) as the pairs (a1, (a2, (..., an)). Let's apply this to your example:

• The ternary relation GivesGift is the set G = {(x, y, z) : person x gives gift y to person z}, where we have two types DPeople of persons and DGifts of gifts and x, z ∈ DPeople and y ∈ DGifts. For the sake of concreteness let's assume DPeople = {Ruddy, Saul}, and DGifts = {Meaning and Necessity}. We express "Ruddy gives Meaning and Necessity to Saul" as GivesGift(Ruddy, Meaning and Necessity, Saul), which is true just in case (Ruddy, Meaning and Necessity, Saul) ∈ G. By the previous technique, we can represent G as G' = {(x, (y, z)) : person x gives gift y to person z}, and instead give the truth-conditions of that proposition as: true just in case (Ruddy, (Meaning and Necessity, Saul)) ∈ G'. The difference is that G is ternary and G' is binary, but they are functionally identical.

By the same technique, any other n-ary relation can be reduced to a binary one. Needless to say, this doesn't mean that non-binary relations aren't worth classifying and investigating.

• I have a really hard time with your claim that ternary relations are theoretically dispensable, but it is hard for me to articulate my trouble. I basically think that something has gotten lost in translation. I've been using logical relations as a formal representation of ideas. When I doubt that the giving(x,y,z) relation can be reduced to lower relations, I mean that something is lost if we say that giving(x,y,z) = giver(x) & gift(y) & recipient(z), because the connection between x, y, and z are lost. Similarly, something is lost if you say that giving(x,y,z) = gives-to(x,z) & gift(y). Commented Aug 20, 2014 at 16:16
• There is something "basic" about the ternary giving relation, as many other relations can be defined in terms of it. For instance, sending someone an email would be send-email(x,y,z) = sender(x) & email(y) & giving(x,y,z), and formally there is nothing basic about "giving" per se, but just that it is a ternary relation, at least as it seems to me. Maybe I get lost in the set theory somewhere, but I don't see how you have addressed the difficulty. How do you define a ternary relation into binary relations without losing anything? Or are you doing some math slight of hand that I didn't detect? Commented Aug 20, 2014 at 16:23
• @KevinHolmes I'm not convinced that what's lost is theoretically significant, since the truth conditions of ternary relations and their binary reductions seem to me plainly to agree. But I do understand your worry. Commented Aug 20, 2014 at 19:21
• @HunanRostomyan Of course if you have a pairing operation then all you need are binary relations. But that's cheating! If the domain of a model consists of people and gifts, then it doesn't contain any pairs of people and gifts: the pair (Meaning and Necessity, Saul) is neither a person nor a gift. Commented Jun 3, 2016 at 3:24
• @BrianO My contention is this: (a, b, c) = (a, (b, c)) = ((a, b), c). Commented Jun 3, 2016 at 3:25

I work in the field of fine art shipping, and clients often ask me how much it will cost to ship a certain artwork and then give me two dimensions. I try to suggest as gently as possible that, logically, shipping an artwork with two dimensions cannot be done, in the hopes of getting them to tell me the depth. Does the three dimensionality of space count as a ternary relation that can't be reduced? It may seem like cheating to introduce a real-world problem like this, but Kant considered space to be an a priori concept, so I'm thinking it's fair game.

I'd suggest its due to history, and ease of use. The first mathematical operations we are exposed to (beyond counting) is addition & multiplcation.

Here a sum such as 5+6+12+14 can be decomposed to a series of binary operations: ie 5+6, then 11+12 etc.

So a binary operation is the minimal 'arity' of these two operations.

In logic the logical operators can be written in terms of unary operators 'Nand' or 'Nor'; but their rules are complicated - so best to sacrifice efficiency in arity for ease of use.

I have seen formal rewriting systems of logic and even programmed small ones. The way I see it is a problem of complexity.

Logic is harder with more object states and interconnectedness. Changing one or correcting errors becomes hard when things can ripple and break easily.

Throw in multiple levels of details and problems like mind vs matter appear frequently when going towards a instance or concrete. The mathematical equivalent is breaking the Liskov subustition principal. Type means type of object (from his mathmatical perspective, a vector is an category of objects).

Another issue is view points which if formally represented result in a wicked problem. One object or role can end up having different, potentially contradictory representations. This is context and non intrinsic properties that have different semantics by role. An example is a customer. In marketing, they are demographic info. In accounting, they have a table with money values written for each transaction.