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For those who commented on my previous questions, sorry for the lack of information and explanation. Clearly I did not do a good job of explaining myself so I deleted the question and hope this one goes better.

EDIT: here is what I have so far, can anyone tell me if it is right/wrong? http://www.box.net/shared/static/hrv9rh7ukt.jpg

I know how to perform standard tree method satisfiable tests but this question I am having trouble with because it has to do with elections.

Question:

Use the tree method to determine whether or not the following set of sentences is statisfialbe. If so, specify a satisfying case for its atomic sentences (ie aRjb, dRic, bRja, and cPid).

{bRja $\rightarrow$ dRic, dRic $\rightarrow$ $\neg$cPid, dRic $\rightarrow$ aRjb, $\neg$dRic $\rightarrow$ cPid, cPid $\rightarrow$ bRja, aRjb $\rightarrow$ cPid}

I think the format of the sentences is standard nomenclature but if it is not, ill will do my best to explain it. P refers to Social Preternce while R refers to Rule.

UPDATE:

I set the following

A = aRjb

B = bRja

C = cPid

D = dRic

So then the tree looks like:

  • $B \lor D$
  • $D \lor \neg C$
  • $D \lor A$
  • $D \lor C$
  • $C \lor B$
  • $A \lor C$

Maybe its because I haven't done even standard tree methods in a while but can anyone help me on how to check if it is satisfiable?

EDIT UPDATE:

bPia $\rightarrow$ cPia

aPic $\rightarrow$ (aPib $\lor$ bPic)

Conc, ($\neg$aPib $\land$ $\neg$bPia) $\land$ ($\neg$aPic $\land$ $\neg$cPia) $\rightarrow$ ($\neg$aPic $\land$ $\neg$cPia)

Can I solve it like the rest by substituting each "clause" with a letter?

So I would set bPia to B1, bPic to B2, cPia to B1, aPib to A1, aPic to A2, and solve like a normal tree?

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    Ok, that makes perfect sense to me. If you want you can just post something as an answer to the question so I can mark it as accepted for you. Thanks.2011-04-06

1 Answers 1

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It is apparent from looking at the formulas listed in the question:

  • $B \lor D$
  • $D \lor \neg C$
  • $D \lor A$
  • $D \lor C$
  • $C \lor B$
  • $A \lor C$

that making $A, B, C, D$ all true is a satisfying assignment.