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Usually in logic '=' or equal is an identity symbol so A=B really is used as 'A is B' (they are one and the same).

In mathematics the common usage of '=' and of A=B is that A is B which is a relation between the mathematical object A and itself (B) so equals is a relation that a mathematical object has with itself. In both these cases, equality is essentially a numerical identity relation that an object can only have with itself.

Why in common language do we encounter 'all people are equal' or 'we are equals' which suggest to numerically different objects that are equivalent.

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    In the second paragraph 'equal' does not mean 'identical'. It means that people have similar rights, status, etc., as in egalatarian society. Jan 31, 2023 at 20:43
  • ... and in the social context, "we are equals" can mean "we are as good as each other" (our abilities). Jan 31, 2023 at 20:52
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    Equality in mathematics is commonly used for any equivalence relation (up to symmetry, up to isomorphism, etc.), often without even citing "abuse of terminology". In predicate logic with equality "=" is explicitly axiomatized as anything reflexive, symmetric and transitive that supports intersubstitution of terms. It is the same with "all people are equal (before the law)" (i.e. up to legal treatment).
    – Conifold
    Feb 1, 2023 at 1:21
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  • "all people are equal" means "all men have equal rights" i.e. there is no "right" that belongs to Jim and does not belong to John. Feb 1, 2023 at 9:59

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Mathematical equality is not merely a "relation an object can have with itself". Equality is often, in fact, relative to an implicitly understood notion of isomorphism, and any distinction between objects that are "equal" and those who are "isomorphic" is often elided. Sometimes it is relevant to distinguish between objects that are isomorphic in one way but not another, e.g. all vector spaces of the same dimension are isomorphic (and people sometimes express this fact as "there is only one vector space of a given dimension", imbuing this isomorphism with the meaning of "equality"), but they can be non-isomorphic as representations of some group.

So already mathematical equality is not an absolute notion, but relative to the context in which the words are being used - you might view this as a manifestation of Frege's and Wittgenstein's context principle asserting that words/symbols only have meaning in the context of specific propositions. It is much the same with colloquial usages of "equal" (and, indeed, most other words).

When someone says "All people are equal", the context is usually something like a discussion of rights, and they're asserting all people are equal in the sense that the rights under discussion should apply to all people. If you insist on phrasing this in mathematical terms, the objects under consideration here are "a human together with a list of rights that human possesses" and we consider two of these equal if their list of rights are the same - the isomorphisms are bijections between the rights of two different humans.

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The question uses the technical term "relation", which has a formal meaning in mathematics. These days, we can use the language of relations to say that numeric equality is an example of an "equivalence relation", meaning that it fulfils certain conditions such as "transitivity". Relations can be modeled in set theory, if we want to. There are books about this.

And yet I have many cousins and other relations who are not mathematical objects.

There is a group of kids over by the bus stop, but they are not endowed with an associative binary operation having two-sided identity and inverses.

If I tell you my kitchen door is open, I'm not making a statement about its status in a topological space.

I can also say things like "those candidates are equally disreputable", without meaning to imply that there is a common numerical scale of reputation according to which the candidates have equal placement. What I am trying to say is that they are both bad and I don't care to distinguish between them in the matter of reputation.

When we speak of people being equal in a political sense, we are not making a mathematical assertion. We're saying that the people are on the same basis with respect to their participation in civic society. While that could be dressed up in mathematical language, people can perfectly well talk about political equality without having any mathematical knowledge at all.

Mathematical words are frequently borrowed from normal speech, so that the words chosen have some connection to the matter under investigation. In the case of "equal", this is so far in the past that we can't pinpoint an origin, but there are many more recent developments that are well-documented. We have come up with generalizations and distinctions and axioms, because in a mathematical context it pays to be precise. We've realized that "equality", once thought to be so basic that it doesn't need explanation, actually does demand more formal attention. That doesn't stop a term from being used in a non-mathematical context, where it doesn't carry the same rigorous definition.

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