1.3.2 Semantics of Equality

Semantics of Equality

Beyond the convention of writing in infix notation there isn't much more to say about the syntax of the equality symbol, other than the fact that it's just a two place relation symbol. But what about its semantics?

This is where matters become more interesting. Although, syntactically, is just a 2-place relation symbol, it is a very particular one in this respect. In fact, (unlike , or , or any other two place relation symbol) we are not free to interpret it as we please. Thus, given any model , any assignment in , and any terms and , we shall insist that

Normal Models.

That is, the atomic formula is satisfied if and only if and have exactly the same interpretation. Therefore, really means equality in this model . A model that fullfill this requirement is called normal model .

In fact, is often regarded as a logical symbol on a par with or , for in normal models it has, just like those symbols, a fixed interpretation, and a semantically fundamental one at that.