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Background

I am reading this book. In the chapter "The Ontology of Metaphysical Realism" the author says:

Many relations are such that pairs of objects enter into them only when taken in a certain order. Thus, being the father of is an asymmetrical relation: if one thing, a , is the father of another thing, b , then b is not the father of a . As logicians put it, it is the ordered pair, <a,b> ( a and b taken in just that order), that exhibits the relation. The three relations we have considered are all two-place or dyadic relations; but obviously there can be three-place, four-place, and, generally, n -place relations.

I understand the idea of an ordered pair when we talk about different objects a and b. However, we do not require that a and b be different. That is, we can write (a, a) and order a and a. What does this mean? How can an object be ordered with itself? On the Internet, I found only a formal definition of an ordered pair, and this did not help me because these definitions are simply expressions for which the characteristic property holds.

Question

Can you explain what (a, a) is with clear examples?

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    You can think of an ordered pair as a function from the set {0,1} to D, the domain of interest. Then if x is a member of D, it makes perfect sense to have a function that maps both 0 and 1 to x. That is, f={0->x,1->x} represents the ordered pair (x,x). Sep 20, 2022 at 15:39
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    A person is as tall as himself Sep 20, 2022 at 16:02
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    The word "order" is poorly chosen for an ordered pair. It isn't that the elements are ordered, it is that the positions are ordered. Sep 20, 2022 at 19:49
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    The expression (a, a) is a shorthand notation of the formal definition of ordered pair, of which there are many. If we take Kuratkowski's popular definition (a, b) = {{a}, {a, b}}, then we have (a, a) = {{a}}, so there is no formal ordering of "a" with itself. Expressions such as (a,a) represent an important property of binary relations - namely reflexivity. Without it, how would mathematicians represent relations such as "=" and "<=" as a set of ordered pairs.
    – nwr
    Sep 20, 2022 at 19:49
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    Consider the point (2,2) in the Cartesian plane. The 2s are identical, but they each represent a coordinate on a different axis. Like someone who lives at the corner of 2nd Street and 2nd Avenue. For that matter, consider a steel ball in physics class with a mass of 2 kilograms and a diameter of 2 meters. Its coordinates in the kg/m plane are (2,2).
    – user4894
    Sep 20, 2022 at 20:21

7 Answers 7

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We wish to understand what an ordered pair is. First, we let go of any preconceptions based on the words "ordered" and "pair". The phrase is just a name.

Then, we consider what is important about an ordered pair:

  1. There is a first component, which is some object from the universe of discourse.
  2. There is a second component, which is also some object from the universe of discourse.
  3. Whenever two ordered pairs have the same first component and the same second component, they are the same ordered pair.

And that's it.

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You are asking what the ordered pair (a,a) is.

Lets talk about what an ordered pair is first. As this thread exhibits, there are many different implementations of ordered pairs- via Kuratowski, Hausdorff, etc. All such implementations are such that (a,b) = (c,d) iff a = c, b = d. So we typically take an ordered pair to be any thing that satisfies the above property.

We can give an order to objects, ie a relation "<" that is reflexive, transitive, and anti-symmetric. Just to be clear, this notion is not the notion of order that we mean when we talk about ordered pairs.

There are many examples of such ordered pairs. In particular we have that most notions of "sameness" give us such an ordered pair. For example, take the notion of same size. My phone P is the same size as itself, ie (P,P) holds. My phone P is not the same size as my laptop L, so (P,L) does not hold.

Here's a nontrivial example, consider John J, Dill D. we can say that John loves himself, ie (J, J) holds. It might also be the case that (J,D) and (D,J) ie that John and Dill love each other. But it may not hold that (D,D), ie Dill might not love herself.

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  • How can an ordered pair satisfy the property (a,b) = (c,d), which requires two pairs? Sep 21, 2022 at 14:54
  • @HelloGoodbye the two "pairs" you see are syntactic, the ordered pair notion is semantic.
    – Papuseme
    Sep 21, 2022 at 16:03
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    Could you explain that in plain English, please? 😅 What do you mean when you say "two 'pairs' you see are syntactic" and "the ordered pair notion is semantic"? Sep 21, 2022 at 16:31
  • @HelloGoodbye say we are given two types A, B. We can construct the type pair (A,B) where pair a b are literally the marks on the page. We now define a notion of equality for pair, since syntactically no two pairs are ever equal. This notion of equality is semantic
    – Papuseme
    Sep 21, 2022 at 18:06
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Indistinguishability plays an important role in quantum and statistical phsyics, failing to account for it might produce some (physical) paradoxes, e.g., Gibbs paradox.

Distinguishability is then understood as follows: even if the objects are identical (i.e., they have the same mass, shape, etc.), we can follow their motion (e.g., in space) and thus distinguish them by their initial position. Thus, although the objects are identical, they are distinguishable by their positions, trajectories, history - i.e., by their relation to the surrounding world.

In quantum physics the notion of trajectory loses its sense, because the positions and momenta/velocities of an object are not simultaneously measurable (similar arguments can be made for other properties that one could use to distinguish objects.) This means that we cannot in principle know the origin of each object, and the only allowed physical states are those that are invariant to the exchange of objects. This has spectacular consequences, resulting in all the particles being divided in bosons and fermions, the emergence of the exchange interaction holding the objects together, etc.

Remark
@JoWehler in their answer say:

If you have a pair of two identical objects 'a' then there is only one possible order, namely the ordered pair (a,a).

Note that the number of possible orderings of two non-identical objects is 2 (for N objects it is N!), whereas for identical objects it is 1. This is the essence of the above mentioned Gibbs paradox, where a factor 1/N! had to be introduced ad-hoc, to correct for this discrepancy when the object are identical. This was later justified from the quantum viewpoint.

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That is, we can write (a, a) and order a and a. What does this mean? How can an object be ordered with itself?

This is a formal definition. The pair in (a, a) is not any pair of actual objects possibly referred to by a. In (a, a), you have only one variable but you have two occurrences of it, the first one and the second one. This is what allows the expression to avoid ambiguity.

That being said, and although a dyadic relation obviously involves a pair of objects, interpreting relations in terms of pairs or n-uplets is just more mathematics, not formal logic. There is a long history of logicians discussing what is called "relational syllogisms" and this seems a more interesting perspective even if it doesn't seem to have gone very far.

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  • If I understand your answer correctly, you are saying that the pair (a, a) does not order objects and is just an expression with two occurrences of one variable. Am I right? Sep 20, 2022 at 17:13
  • Can you also explain what "avoid ambiguity" means? Sep 20, 2022 at 17:15
  • @AlexanderChaikov "Am I right?" Yes. Objects are what they are. You cannot order them without physically moving them. 2. "avoid ambiguity" (a, a) is not ambiguous because we can distinguish between the two occurrences of a, that is, we can tell which is the first and which is the second. Sep 20, 2022 at 17:30
  • I'm sorry for the additional questions, I just want to understand. When we say that (a, b) is a pair, are we talking about the objects that stand for a and b, or are we talking about the symbols a and b? You say that we can distinguish between the two occurrences of a, that is, you are talking about symbols. However, I thought that pairs order objects a and b. Can you clarify how you understand pairs? Sep 20, 2022 at 17:51
  • @AlexanderChaikov "I thought that pairs order objects a and b" Mathematical objects are concepts, not physical objects, so you cannot order them. The only ordering there is can only be on the page, so to speak, that is, in the string of characters "(a, b)". This is also why this way of talking is fallacious. Sep 20, 2022 at 18:06
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First, a quote:

... there is a related distinction that needs to be mentioned in connection with the type-token distinction. It is that between a thing, or type of thing, and (what is best called) an occurrence of it—where an occurrence is not necessarily a token. The reason the reader was asked above to count the words in Stein's line in front of the reader's eyes, was to ensure that tokens would be counted. Tokens are concrete particulars; whether objects or events they have a unique spatio-temporal location. Had the reader been asked to count the words in Stein's line itself, the reader might still have correctly answered either ‘three’ or ‘ten’. There are exactly three word types, but although there are ten word tokens in a token copy of the line, there aren't any tokens at all in the line itself. The line itself is an abstract type, as is the poem in which it first appeared. Nor are there ten word types in the line, because as we just said it contains only the three word types, ‘a,’ ‘is’ and ‘rose,’ each of which is unique. So what are there ten of? Occurrences of words, as logicians say: three occurrences of the word (type) ‘a,’ three of ‘is’ and four of ‘rose’. Or, to put it in a more ontologically neutral fashion: the word ‘a’ occurs three times in the line, ‘is’ three times and ‘rose’ four times. Similarly, the variable ‘x’ occurs three times in the formula ‘∃x (Ax & Bx)’.

Conifold pointed out in a comment to a post of mine about types that we have the option of considering inter alia multisets, here. Using mustaches, we get {a, a} as the multiset of a, of multiplicity 2 then, and this is akin to there being two occurrences of a in the (text) string "{a, a}."

Now once we switch from "{}" to "()," here, the definition will go back to set theory "proper," in the sense that we might say that the expression "(a, a)" is a degenerate case of an ordered pair. It is like the end-state of a singleton operation repeated twice "in time" without, of course, making an actual difference in order. Other phrases we might use: trivial order, order of identity, monordinal (never heard that one, but I wouldn't put it past 'em; anyway, for "monadic order"), etc.

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  • In no way can (a,a) be described as a degenerate case. Is the point (2,2) in the plane of somehow lesser status than (4,47)? An ordered pair may be taken as a function from the set {1,2} to some target set, say the reals. If f(1) = a and f(2) = a, in what way is this degenerate? It's just a non-injective function. I don't understand your remark.
    – user4894
    Sep 21, 2022 at 2:25
  • @user4894, I'm probably using the word "degenerate" wrong, or in a weird context, anyway. So consider the set for 1!. This just goes to 1, so there is only one way to order 1 by itself. 2! goes to 2, but 3! goes to 6, though, etc. By convention they usually hold that 0! = 1, though they do have a good reason, I must admit. Anyway, if my use of the word "degenerate" goes through, it is, again, only in some weird context (some deep labyrinth in order theory). And once you bring surreals in, things get even more complicated (for me, anyway; also some simplifications, luckily). Sep 21, 2022 at 2:50
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If you have a pair of two identical objects 'a' then there is only one possible order, namely the ordered pair (a,a).

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Thanks everyone, it seems after reading all the answers I seem to understand. My question has been answered by Arno and Papuseme, but I would like to add a few notes.

So what I'd like to say is that it's important to distinguish between an object and how we look or how we use that object. For example, imagine a school where there are not enough teachers and the physics teacher teaches mathematics. We can say "physics teacher" and "math teacher", but they are the same person.

Ordered pairs allow us to convey the objects themselves along with the meaning in which they are used. For example, we want to create an object that would convey to us complete information about the position of points on the plane. We can use the ordered pair. We agree that the first coordinate of the pair is X component of the point, and the second coordinate of the pair is the Y component of the point. Then (1, 1) definitely tells us where the point is located, but we did not have two numbers 1. We used one number 1 for different purposes.

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