Bernard has a younger brother in logic Given:


*

*P is the set of all people,

*Man(x): x is a man,

*Woman(x): x is a woman,

*Child(x, y): x is a child of y,

*Younger(x, y): x is younger than y.


Express in the form of a formula: 'Bernard has a younger brother'.
I begin with stating some facts (assuming a traditional family composition):


*

*There is some $p \in P$ such that $p$ and Bernard share a common father and mother, 

*So there is some $m \in P$ and a $w \in P$ such that Man(m) and Woman(w) and these two, have given birth to both Bernard and $p$,

*Hence: Child(Bernard, m), Child(Bernard, w) and Child(p, m) and Child(p, w),

*Furthermore we know that this $p$ is younger than Bernard, thus: Younger(p, Bernard),


Putting it all together $$\exists m, p, w [(m, p, w) \in P:   Man(p) \land Younger(p, Bernard) \land (Man(m) \land Child(Bernard, m) \land (Woman(w) \land Child(Bernard, w) \rightarrow Child(p, m)\land Child(p, w))]$$
Does this correctly express this idea? Perhaps I need a bi-implication?
 A: No. The statement you have currently (assuming we interpret $(m,p,w)\in P:$ as $(m\in P)\wedge (p\in P)\wedge (w\in P)\wedge$ doesn't mean that Bernard has a younger brother. Rather it means that there are three people such that if one of them is a man who is Bernards father and another is a woman who is Bernards mother and the third is a man younger then Bernard, then the third is Bernards brother.
This will always be true no matter whether Bernard has a younger brother or not since we can just choose p be to be Bernard which makes the predicate Younger(p,Bernard) false and thus the whole implication true.
What you need is that all the things happen at once so instead of that implication you really do want another and. Then you will be saying that there exist three people ($\exists p\exists m\exists w (p\in P)\wedge (w\in P)\wedge (m\in P)$) and one of them is Bernard's father ($Man(m)\wedge Child(Bernard,m)$) and the other is Bernard's mother ($Woman(w)\wedge Child(Bernard,w)$) and the third of is Bernard's brother ($Man(p)\wedge Child(p,w)\wedge Child(p,m)$) and is younger then Bernard ($Younger(p,Bernard)$).
A bi-implication (I'm assuming equivalence) would make the statement mean that there exists a person who is a child of $m$ and $w$ if and only if he's Bernard's younger brother but that still says nothing about the existence of anyone since it only says that there exist 3 people who satisfy the right side (m and w are parents of p) if and only if they also satisfy the left side (p is a man younger than Bernard and m and w are a man and a woman who are Bernard's parents). That can easily be satisfied by having everyone be Bernard (m=p=w=Bernard) since then both sides of the equivalence are false and thus the equivalence is true.
Another edit to further clarify:
Existential quantifiers with implications are tricky in the sense that you want to make sure that there isn't something that can fulfill the implication vacuously.
As an example suppose $P(x),Q(x)$ are quantifier free. Then claiming $\exists x(P(x)\implies Q(x))$ will often not tell you much since whenever $\exists x \;\neg P(x)$ is true then automatically $\exists x (P(x)\implies Q(x))$ is true. 
You don't actually need $P(x),Q(x)$ to be quantifier free that I'm just trying to point out that an extra universal quantifier after the existential one changes things. A good example of that is $\forall x\exists y\forall z(P(x,y,z)\implies Q(x,y,z))$ in this case you need $\forall x\exists y\forall z\neg P(x,y,z)$ which constrains $y$ more. This form should remind you of the definition of limit since that's for many people the first prototypical situation when they encounter a (hopefully) $\Pi_3$ sentence.
A: Let the constant $b$ represent Bernard, for brevity.
$$\tag{1} \exists y \in P: \Big(\operatorname{Younger}(y,b) \wedge \operatorname{Man}(y) \wedge \exists x\in P:\big(\operatorname{Child}(b,x)\wedge \operatorname{Child}(y,x)\big)\Big)$$
"There is some person who is younger than Bernard, and is a man, and both Bernard and that someone are both a child of some same person."
Alternatively if half brothers are not allowed.
$$\tag{2} \exists y \in P: \Big(\operatorname{Younger}(y,b) \wedge \operatorname{Man}(y) \wedge \forall x\in P:\big(\operatorname{Child}(b,x)\to \operatorname{Child}(y,x)\big)\Big)$$
"There is some person who is younger than Bernard, and is a man, and is a Child of everyone of whom Bernard is a Child of."

Remark: Of course, here we are assuming that Bernard didn't spontaneously generate, and that being an orphan doesn't mean the negation of the existence of parents.
Edit: We're also assuming $\rm Man$ is not age dependent, and that being a boy qualifies.
Edit 2: We're also assuming for (2) that $\rm Child$ is defined biologically and not legally (by adoption or remarriage).  Though the later works for (1) which does not require a "traditional" family unit.
