Help needed with first-order logic representation

Question

I'm very new to first-order logic. I've been working on some tasks below, and would appreciate if somone could check if I have understood and solved the questions correctly

Task:

Assume that $B$, $F$ and $K$ are relational symbols so that

• $Bx$ interpreted as "$x$ is a biologist
• $Fx$ interpreted as "$x$ is a philosopher"
• $Kxy$ interpreted as "$x$ knows $y$"

Assume that $a, b$ and $c$ are constant symbols which represents Aristoteles, Bolzano and Copernicus. Find first-order logic formulas for following sentences:

1. Aristotle is both a biologist and a philosopher
2. All biologists are philosophers
3. No philosophers are biologists
4. Aristoteles knows a philosopher
5. Bolzano knows all philosophers
6. Copernicus knows only biologists

Here are my answers:

1. There exists one $x$ so that Aristoteles is a $Gx$ and a $Fx$, written as $\exists$$x(Bx(a)$ $\land$ $Fx(a)$

2. For all $x$ if $x$ are $Bx$ so $x$ are $Fx$, written as $\forall$$x$$(Bx(x$) $\rightarrow$ $Fx(x)$

3. It's not that there is an $x$ such that $x$ are $Fx$ and $x$ are $Bx$, written as $\lnot$$\exists$$x(Fx(x) \land Bx(x)$

4. There exists one $x$ such that Aristoteles knows a $Fx$, written as $\exists$$x(Kxy(a, Fx)$

5. Not sure

6. Not sure

I would appreciate any help and please feel free to correct me if I've done something wrong.

Thank you.

Answer 1

For $(1)$, we do not need a quantifier:

$(1) \quad B(a) \land F(a)$

For $(4)$, we need $\exists x(F(x) \land K(ax))$.

$(5) \quad \forall x(F(x) \rightarrow K(bx))$

$(6) \quad \forall x(\lnot B(x)\rightarrow \lnot K(cx))\equiv \forall x(K(cx) \rightarrow B(x))$. "For all x, if x is not a biologist, then Copernicus does not know x" $\equiv$ "For all x, if Copernicus knows x, then x is a biologist."

Also, be careful with parentheses on $(2), (3)$: you're missing closing parentheses. (Also on $(4)$, but I've included it above.)