# Is identity included in the “key” in predicate logic?

So, my exam is in a few days. We've been told to practice setting up a key in predicate logic. . From what I've understood, a typical key looks something like this:

$Lxy$: $x$ likes $y$

$Rx$: $x$ is rich

$j$: John

But, something my lecturer said about identity made me wonder whether or not I should include it in the key (if I am going to use identity, that is). He said that, for example, $x=y$ is the predicate $=xy$ ($x$ is identical to $y$, or something similar), only that since having the "$=$" in the middle is what we're used to (before logic), that's what we do in logic as well.

Thus, I have the impression that it is the same as a predicate (like "$Lxy$"), and therefore wonder whether I should have it in the key or not? If not, why not? :)

Thanks!

• It is not clear to me what a "key" is ... In general, we can develop first-order logic without equality (i.e."$=$") or adding it to the language. But if you add it to f-o logic, you will treat it syntactically as a predicate (i.e.$x=y$ is treated as an abbreviation for the "official" $=xy$, and this is a well-formed formula of the language). But semantically you cannot treat eqaulity as the other predicates, suitable of different interpretation. If it is present in the language, its meaning must be fixed : it is the relation I =<x,x> of identity in the domain. – Mauro ALLEGRANZA Jan 5 '14 at 17:08
• key, Mauro, is being used to define the domain, the predicates that are being used: e.g. "Let P(x) denote 'x is a police officer', let Q(x) denote 'x is a queen' ..." etc. – Namaste Jan 5 '14 at 17:11
• Not usually, no; it's often clear from the context. If "identity" is loosened to include, say, equivalence of some form or another, then it's sometimes taken as $Ixy$, for example, but this is a minor issue. You're unlikely to run up against such things just yet. – Shaun Jan 5 '14 at 17:12

No need to include it in your key: it seems clear that your instructor accepts $x = y$ as denoting $=_{xy}\quad$
So you are free to simply use "$x = y$" to denote $x = y$.