# Equality as logical

### Do we take the identity relation x = y as logically primitive? Must we?

## Equality as a logical primitive

In the standard semantics of first-order logic, as we defined it in previous posts, the equality relation = gets special treatment—it is not just another binary relation symbol, to be interpreted differently in various models, but rather our semantics specifies in the atomic case of the satisfaction relation that it is to be interpreted as actual identity. Namely, a model satisfies the statement that two individuals are equal if and only if they are in fact equal. Thus, we have treated = in the disquotational manner, much like how we treat the other logical symbols such as ∧, ¬, and ∃. In this sense, we treat = as a logical primitive.

I should like to explore the question of whether this special treatment of equality was required. Can we somehow express true equality in the equality-free fragment of first-order logic? Let us define that an assertion φ in a first-order language is *equality-free*, if the equality symbol = does not appear in it.

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## Models of clones

There is a standard construction in mathematics of the *real line with two origins*, which refers to the partially pre-ordered structure which is just like the real line order, except that there are two distinct number 0s—we might denote them 0 and 0*—which are related to all the other numbers just as 0 is, and which are ⩽ related to one another just as 0 is related to itself. This structure sometimes serves as an example or counterexample in connection with natural questions or conjectures.

More generally, let us consider for any given model all the various models of clones that we might construct over it. These are the models arising when we have duplicated some or perhaps all of the individuals of the original model with an assortment of exact copies, clones exhibiting all the same properties as the originals, as much as is possible except for them all being distinct from one another.

Specifically, I define that a structure *M** is a *model of clones* over a model *M*, if

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