# Help needed with first-order logic representation

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

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

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.

• Hint for 5: We rephrase that to "For all philosophers, Bolzano ______" – apnorton Oct 28 '13 at 1:21
• Hint for 6: What does it imply about $x$ if Copernicus knows $x$? – apnorton Oct 28 '13 at 1:23

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.)

• Thank you for your answer. Then I assume $(2)$ and $(3)$ is correct? – Dabbish Oct 28 '13 at 1:24
• Yes indeed. The English is awkward, but your symbolizations for $(2), (3)$ are spot on, save for the missing parentheses. – Namaste Oct 28 '13 at 1:26
• haha, English is not my first language, so I apologize, regarding the $(2)$ question would $\forall$$x$$(Bx(x))$ $\rightarrow$ $Fx(x)$ be more correct? I added 1 more paranthese – Dabbish Oct 28 '13 at 1:29
• No, it would be $\forall x(B(x) \rightarrow F(x))$ – Namaste Oct 28 '13 at 1:30
• Oh, I'm sorry, I saw your double use of x in (2) and (3): B(x): x is a biologist, F(x): x is a philosopher. $(3)\quad \lnot \exists x(F(x) \land B(x))$. – Namaste Oct 28 '13 at 1:31