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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.

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

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  • $\begingroup$ Thank you for your answer. Then I assume $(2)$ and $(3)$ is correct? $\endgroup$ – Dabbish Oct 28 '13 at 1:24
  • $\begingroup$ Yes indeed. The English is awkward, but your symbolizations for $(2), (3)$ are spot on, save for the missing parentheses. $\endgroup$ – Namaste Oct 28 '13 at 1:26
  • $\begingroup$ 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 $\endgroup$ – Dabbish Oct 28 '13 at 1:29
  • $\begingroup$ No, it would be $\forall x(B(x) \rightarrow F(x))$ $\endgroup$ – Namaste Oct 28 '13 at 1:30
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    $\begingroup$ 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. $(3)\quad \lnot \exists x(F(x) \land B(x))$. $\endgroup$ – Namaste Oct 28 '13 at 1:31

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