equality – just a relation between *two* objects?

Question

I just read the wikipedia article about "equality". Why do they treat equality = as a relation linking just two objects? It seems like they are treating = just as a symbol (which can be written between two expressions), satisfying some axioms (axioms of equivalence relation + substitution scheme), like in a formal system. But in the natural language, one could also formulate things like “The three objects $\mathrm{e}^{i\pi}$, $\cos \pi + i\sin\pi$ and $-1$ are equal”. (Note that this is equivalent but not identical to asserting that $\mathrm{e}^{i\pi}=\cos \pi + i\sin\pi= -1$, where $a = b = c$ is a shortcut for “$a = b$ and $b = c$“.)

Thus in my opinion, this notion of equality doesn't reflect human thought.

Answer 1

Of course, equality is property of two object, so I believe it natural to be treaten as a binary predicate symbol. However this really is no limitation to its use whatsoever. The formal language behaves as the natural one. You would agree that $=$ is binary but you can use it more generaly, lets say for three objects. You can do the same in formal language.

Compare to set theory. You start only with one symbol, that is the elementhood relation $\in$, it is very useful to start as small as possible (you e.g. desire to have a simple, but expressible formal system). But you can define new notions like relations, set of natural numbers and so on. Similarly, if you like, you can define ternary equality. The formal setting of predicate logic is perfectly capable of doing so. simply add an axiom $\forall y,x,z( =^3(x,y,z)\Leftrightarrow x=y\wedge y=z)$. It is just like agreeing on how to use equality in natural language. Yet it seems quite useless in this case.

Answer 2

I dispute your claim that "this is equivalent but not identical to . . .". Why do you think these expressions are not identical? Personally, if I were asked to tell whether three objects were equal, I would first check whether the first two were equal, and then - if they were - I would check whether the first was equal to the third. So to my mind equality between three objects really does reduce to equality between two objects, and there is no semantic mismatch.

Certainly, given that equality between two objects is sufficient to discuss equality between three (or $n$) objects, the onus is on you to explain why the latter is a truly separate concept.

Answer 3

The reason why equality is only defined for two elements is because it is only necessary to define it for two elements; using transitivity of equality and induction, any statement that $n$ objects are equal can always be reduced to $\binom{n}{2}$ statements that two objects are equal.

A natural definition for a set $A$ of $n$ elements to be equal is that, for all $m < n$, every subset $B \subset A$ of $A$ which has $m$ elements is equal.

Certainly we would want any definition to satisfy this property, so it makes sense to choose this property for the definition.

This leads to an inductive definition of equality of sets with $n$ elements for each $n \in \mathbb{N}$.

For $n=1$, we have from the symmetry of equality, $a=a$, that every one element set is equal.

The definition of equality for a two element set $a=b$ is the standard one.

Now consider an arbitrary three element set $\{a,b,c\}$ such that $a=b=c$.

By transitivity of equality, we have that $a=b$, $b=c$ $\implies a=c$. Therefore, by transitivity, $$a=b=c \iff a=b, b=c, a=c \iff a=b, b=c$$ Similar arguments hold for any $n>3$. In other words, the question of equality for a set $A$ with $n$ elements reduces to the question of equality for all of its subsets with $n-1$ elements, which reduces to the question of equality for all of its subsets with $n-2$ elements, ..., which reduces to the question of equality for all of its subsets with 2 elements. Therefore, we gain essentially gain nothing by considering equality to be an $n$-ary relation instead of a $2$-ary relation.

For an infinite set, we define it to be equal if all of its subsets with finitely many elements are equal (again since any definition of equality for an infinite set should satisfy this property, so it makes sense to choose this property for the definition).

Then it follows that for each finite subset its equality reduces to the question of equality of its subsets with 2 elements.

Thus even for an infinite set the question of whether or not it is equal reduces to whether or not it is equal pairwise, i.e. whether or not all of its subsets with 2 elements are equal, because of the transitivity of addition. Similar arguments will not work for relations which are not transitive.