Would it also be useful to include an ordered pair function in first order logic?

Typically, first-order logic is assumed to include an equality relation $=$, even though this is "non-logical," together with some postulates about equality.

Would it also be useful to include an ordered pair function $(*,*)?$ One could assume that $(x,y)=(x',y')$ precisely when $x=x$ and $y=y'$. Or perhaps it would be best to add denumerably many such functions, $(*)$, $(*,*)$, $(*,*,*)$, etc.

The upshot of doing (either) of these is that relations can now all be unary. This would prettify a lot of notation. For instance, the axiom schema of replacement would look much neater.

Thoughts, anyone?

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