Can someone please explain the difference between first/second/third order logic, using examples? I am currently struggling to get my head around how to recognise the difference between first/second/third order logical arguments.
I feel that the easiest way for me to understand the difference would be if someone could give me an example of each


*

*In plain English

*In terms of predicate logic


Any help would be much appreciated.
 A: First-order, second-order and third-order logic are all logical languages with universal and existential quantifiers. The difference lies in what quantifiers speak about. 
In first-order logic one can quantify over individuals. An example of a first-order formula is the commutativity axiom for a group $(G,*,e)$:
$$\forall x. \forall y. x * y = y * x$$
In second-order logic one can quantify over sets of individuals. An example of a second-order formula is the axiom defining the existence of least upper bounds for a complete partial order $(D,\sqsubseteq)$, which says that every subset $Y$ has a least upper bound:
$$\forall Y. Y \subseteq D \Rightarrow (\exists z. (\forall y. y \in Y \Rightarrow y \sqsubseteq z) \wedge (\forall x. (\forall y. y \in Y \Rightarrow y \sqsubseteq x) \Rightarrow z \sqsubseteq x))$$
In third-order logic one can quantify over sets of sets of individuals. An example of a third-order formula is the axiom for topological spaces which states that the union of a family of open sets is an open set.
