Possible examples of real life membership relations that are non-transitive ( not necessarily intransitive)?

Question

My question is not the same as this one : Are there real-life relations which are symmetric and reflexive but not transitive?. The answer " x has slept with y " was a good answer to this question ( more that 120 upvotes) . It wouldn't do for mine.

I'm not asking for arbitrary relations having such and such properties.

I'm looking for real life relations having a conceptual analogy with " membership". So here , the conceptual content of the examples am asking for really matters.

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• Let's admit, if you please, for the sake of the question, that the membership relation can hold between a concrete object and a set of concrete objects ( in spite of the fact that, properly, it only holds between sets, due to the fact that, in set theory, every object dealt with is a set).

I'm looking for real life analogous cases of the fact that the membership relation is not transitve.

• Maybe these ones :

(1) I am a member a football club. My football club belongs to a football league. But I am not a member of the football league.

(2) I am a british citizen, meaning that I belong to the british poeple. We are, say, in 1943, so currently the british people belongs to the Allies. But I am not one of the Allies.

(3) Letter ' $a$ ' belongs to the word ' cat'. The word ' cat' belongs to the sentence ' the cat is on the map'. But letter ' a ' is not a member of the sentence ' the cat is on the map'. ( Arguably, a sentence is not a set of letters, nor of sounds, but a set of words).

• I'm not totally satisfied with these examples. For they instantiate intransitivity, rather that non-transitivity strictly speaking.

Can you think of cases illustrating non-transitivity, I mean, the fact that in case $aRb$ and $bRc$ is true, $aRc$ can be true, but does not always nor automatically follow?

Answer 1

1. My friend Jane and I are both members of the Mathematics faculty at NoName University. NoName is an institutional member of the Gondwanaland Mathematical Society (the GMS). Jane's a member of the GMS, but I'm not.

2. Israel and Palestine are members of UNESCO. UNESCO is one of the constituent bodies of the United Nations. Israel is a member of the UN, but Palestine isn't.

Answer 2

I'll provide two programming examples twice. They are not about the "real"/physical world, as programming languages are abstractions invented by humans, but they illustrate the non-transitive property you want.

In the Smalltalk-80 programming language, every Class is an Object, but not every Object is a Class. This language has an intricate metaclass hierarchy (which you are invited to check in order to follow this explanation), which determines the following facts:

1. 'Hello, world!' is an Object.
2. Object is a Class.
3. However, 'Hello, world!' is not a Class. (It is just an ordinary non-class object.)

And also the following facts:

1. String is an Object.
2. Object is a Class.
3. String is a Class.

Explanation:

• String is a direct instance of String class, which extends Object class, which extends Class, which extends ClassDescription, which extends Behavior, which extends Object. Therefore, String is an (indirect) instance of Object.

• Object is a direct instance of Object class, which extends Class. Therefore, Object is an (indirect) instance of Class.

• (Reiterating part of the initial bullet point:) String is a direct instance of String class, which extends Object class, which extends Class. Therefore, String is an (indirect) instance of Class.

Regarding the 'Hello, world!' string:

• 'Hello, world!' is a direct instance of String, which extends Object. Therefore, 'Hello, world!' is an (indirect) instance of Object.

• However, 'Hello, world!' is not an (indirect) instance of Class, as Object does not extend Class.

In the Java programming language, the facts are similar. Only the explanation differs as Java has a much simpler metaclass hierarchy:

• String is a direct instance of Class, which extends Object. Therefore, String is an (indirect) instance of Object.

• Object is a direct instance of Class.

• (Reiterating part of the initial bullet point:) String is a direct instance of Class.

Regarding the "Hello, world!" string in Java, the explanation is identical to the one provided for the 'Hello, world!' string in Smalltalk-80.

All these facts can be made visible by running the following Java source code:

//https://replit.com/languages/java10
public class Main {
  public static void main(String args[]) {
    System.out.println("Hello, world!");
    System.out.println("\"Hello, world!\" is a direct instance of: " + "Hello, world!".getClass());
    //System.out.println("\"Hello, world!\" extends: " + "Hello, world!".getSuperclass()); //uncommenting it fails: "Hello, world!" is not a class
    System.out.println("String extends: " + String.class.getSuperclass());
    System.out.println("Is \"Hello, world!\" an (indirect) instance of Object? " + ("Hello, world!" instanceof Object));
    System.out.println("String is a direct instance of: " + String.class.getClass());
    System.out.println("Class is a direct instance of: " + Class.class.getClass());
    System.out.println("Class extends: " + Class.class.getSuperclass());
    System.out.println("Is String an (indirect) instance of Object? " + (String.class instanceof Object));
    System.out.println("Is Class an (indirect) instance of Object? " + (Class.class instanceof Object));
    System.out.println("Object is a direct instance of: " + Object.class.getClass());
    System.out.println("Is Object an (indirect) instance of Object? " + (Object.class instanceof Object));
    System.out.println("Object extends: " + Object.class.getSuperclass());
  }
}

Output:

"Hello, world!" is a direct instance of: class java.lang.String
String extends: class java.lang.Object
Is "Hello, world!" an (indirect) instance of Object? true
String is a direct instance of: class java.lang.Class
Class is a direct instance of: class java.lang.Class
Class extends: class java.lang.Object
Is String an (indirect) instance of Object? true
Is Class an (indirect) instance of Object? true
Object is a direct instance of: class java.lang.Class
Is Object an (indirect) instance of Object? true
Object extends: null