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Consider the following conversation:

"Gerda," said Hans, "we must know if Petra went to Berlin." "Well," said Gerda, "we know that if she didn't then she went to Cologne. And we know that she didn't go to both Cologne and Dusseldorf." "Yes, yes," said Hans, "that's all true. And we also know that she went to at least one of Dusseldorf or Essen." "Exactly," said Gerda, "and if she went to Essen she didn't go to Cologne. So your question is answered, Hans."

(a) Gerda clearly thinks that she and Hans have enough information to resolve the issue of Petra's whereabouts. Decide on the conclusion Gerda thinks is correct and symbolize the argument which leads to that conclusion. Remember to give a clear key.

So far I have..

B = Petra went to Berlin
C = Petra went to Cologne
D = Petra went to Dusseldorf
E = Petra went to Essen.
~ = negation

E -> ~C , (D v E) v (D & E) , ~(C & D) , ~ B -> C / B.

What I am confused on is that, is this the way to "symbolize the argument which leads to that conclusion." ?

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How to symbolize the argument ?

You are on the right track. With the sentence letters you have introduced to symbolize the different statements, you have to write down the premises of the arguments (you have quite done it) and you have to verify the possible conclusion.

The conclusion must be one of : B and ¬ B : "Petra went to Berlin or not (and if not, she went to Cologne, by first premise)".

Thus :

"if she didn't (Petra didn't went to Berlin) then she went to Cologne"

must be :

¬ B → C

and

"she didn't go to both Cologne and Dusseldorf"

is

¬ (C ∧ D)

and

"she went to at least one of Dusseldorf or Essen"

is

D ∨ E

and

"if she went to Essen she didn't go to Cologne"

is

E → ¬C.

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I can't tell you exactly what - your instructor, I'm assuming - wants here, but I do know that what you have so far is only a set of premises and a conclusion. Judging by context, I would imagine "symbolizing the argument" should involve manipulating those premises step-by-step until they look like the conclusion, like you might in a mathematical proof. And, at least back in my own logic classes, that meant writing down each step, and then beside it writing the premises (or prior step) and the logical rule by which it was deduced.

I don't want to do too much for you, since this sounds a bit like a homework problem that you might want to be able to work out yourself, but feel free to ask if you'd like clarification or an example or something.

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