Can anything not be equal to itself?

Question

Consider the statement

1=1

This is the law of identity translated to arithmetic.

More generally we could say

x=x

Whatever x is; it must be equal to itself.

Is this always true? Can there be situations when it doesn't hold?

In Jowetts introduction to Platos Theatatus he writes that

of the three laws of thought, the first being A=A, is an identical proposition - that is a mere word or symbol claiming to be a proposition; ...[they] are untrue; because they exclude degrees, mixed modes and double aspects under which truth often presents itself to us.

For, x=x, referring only to itself as symbol, is merely formal and does not refer; once we ask it to refer the situation becomes more complex; when x is a river; what does it mean for a river ton be equal to itself? It changes from moment to moment.

Still, this is not what I'm asking about; I'm asking how far can one loosen the classical meaning of equality; the law of thought that Jowetts mentions and still be rendered meaningful in logic.

One possibility can be shown by examining the following situation:

A blue box on a table in front of which Jameel is sat on a chair. I ask him to close his eyes; and when he has I turn the box around; and when I ask him is it the same box; he being of a suspicious mind says well it looks like the same box; perhaps you've exchanged it for an identical box; or perhaps you've turned it upside down...

This is at least two determinable situations; by permutation and by rotational symmetry; which can be brought together into one idea of symmetry.

In a way one has expanded the notion of equality.

What other possibilities are there?

Answer 1

As @JamesKingsley alludes, there are many conventions in mathematics for defining equality. I would claim that one of those, the mapping via equivalence classes is the only reasonable definition of equality. Equality is always relative to some standard of accepted variation.

All definitions only describe equivalence classes, so equality is not an actual relationship, but an equivalence based upon the definition. Two real numbers are equal if the distance between them is zero, so 1 = 0.999... but clearly in some sense they are different (and asserting such a thing can cause a little war in grade-school classrooms.) And the circumference of a circle radii is equal to the fraction of the enclosing square it covers. They are not equal in any realistic sense, how are a proportion and a linear measure comparable? But we do measure them both, and we can see those measures converges to the same real number. In the classical proof, we can watch the difference go to zero.

We would often be wise to pull back to this perspective. It eliminates tons of classical questions at one blow. If I cut off a chicken's leg, is it still the same chicken? If I cut it in two which half is the original chicken and which half is the excised part, or is there no longer a chicken there? These stop being philosophical questions and become linguistic ones.

All of it relies on your definition of a chicken, even though you do not really know that definition in any real detail, yourself. In Wittgenstein's sense, they are rules of the game, and no one knows all of those rules, or the game itself would stop being useful.

If you accept that equality is about accepted variation, and therefore equivalence classes, everything has to be in the same equivalence class as itself. But that is a convention, not an essential attribute of existence.

Answer 2

I'm having trouble understanding what it would mean for something to not equal to itself - equals is a synonym for is, and obviously, a thing must be itself.

Moving in the other direction, there are lots of examples where a thing can be "equal" to something else:

1. As many programmers will know, in Java there are two different notions of equality. In one case, equality is about the object itself, but in another case equality is about the object's properties (the equals method of an object). This is analogous to your chair example.

2. In measure theory in math, there is the idea of two sets being equal almost everywhere. The idea is that the two sets are the same except for a set of points whose measure is 0.

3. In general in mathematics, there is a notion of an equivalence class, which is used in several different ways, but always following the same idea: for any purpose, the two objects are interchangeable, and thus "equal" for a particular purpose. For example, the rotation of a square by spinning it all the way around once, and spinning it all the way around twice, are "equal" in the sense that they result in a square with the same orientation.

Answer 3

With remarkable coincidence I have discovered Lewis Carroll's essay to Mind from 1895 "What the Tortoise Said to Achilles" immediately before stumbling on your question here.

In that essay the Tortoise asks Achilles to convince him that if (A) and (B) are true then (Z) is true, in reference to Euclid's first proposition on the construction of an equilateral triangle:

(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.

but unfortunately Achilles ends up in an infinite regress.

Maybe it is possible to adapt the Tortoise race-course (as he called it) to your question; something like:

(A) A thing is equal to itself.
(B) X is a thing.
(Z) X is equal to itself.

Answer 4

There is,as you say, the law of identity.

Something a definition of negation must conform to:

(def) What is so is not what is not so. (Aristotle)

Or using a free variable and the identity sign:

(def) not(x = not x)