I have a question I need to express in FOL, "Some pianist is more lazy than all other pianists." I've been racking my brain but I can't quite get it.
P(x)... x is a pianist L(x,y)... x is more lazy than y
My best guess is AxEy(~x=y<->(P(y)&P(x)&L(y,x)))
Thank you!
First off, this is clearly an existential sentence, so it should start with $\exists x$, not with $\forall x$
In fact, the sentence is saying about some pianist, so the basic form will be $$\exists x (P(x) \land \phi(x))$$
where $\phi(x)$ is what we want to express about this pianist $x$.
Now, what we want to say about this pianist $x$ is that $x$ is lazier than all other pianists. To do that, think about it this way: if you take any pianist $y$ other than $x$, it will be the case that $x$ is lazier than $y$. So, we can express $\phi(x)$ as:
$$\forall y ((P(y) \land y \neq x) \to L(x,y))$$
So, plug that into the basic form above, and the result is:
$$\exists x (P(x) \land \forall y ((P(y) \land y \neq x) \to L(x, y)))$$