First, second and monadic second order logic I have some confusion in understanding notions First, second and monadic second order logic. It will be great if some one can explain these concepts with examples (for beginners). Thank you.  
 A: This is a question what sort of objects we can quantify over.


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*In first-order logic, if $x$ is a free variable, it will always be interpreted as an object in the universe of the structure. So $\forall n(n+1>0)$, in the natural numbers with the usual interpretation of arithmetics will say that every natural number - an object in our universe - is not $0$, once we add $1$ to it. In other words, no natural number equals to $-1$.

*In second-order logic we are allowed to quantify over relations. So we can write something like this: $$\forall F(\forall x\exists y (F(x,y)\land\forall z(F(x,z)\rightarrow y=z))\land\forall x\forall w\forall y(F(x,y)\rightarrow\lnot F(w,y))\rightarrow\forall y\exists x F(x,y))$$ For every binary relation $F$, if $F$ is a function defined on the universe, and it happens to be injective, then $F$ is surjective. In other words, the universe is a finite set. Note that $F$ is not a function symbol in the language, it is a variable.

*In monadic second-order logic we are allowed to quantify over subsets of the universe, but not arbitrary relations. So we cannot quantify over functions, but we can still write somethings like "Every bounded set has a least upper bound" (which I will leave it for you as an exercise to write formally), or even "Every set which includes $0$ and closed under successors equals to the entire universe", which is the axiom of induction.
