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# Propositional Logic Problem

## Question

You are the detective in a crime.  From your research you know that Jill and John are lying however the others may be telling the truth or lying.  What can you deduce from the statements below?

Jill said:  IF John is not involved and Joe told the truth THEN Mike is not involved
John said:  IF Abby is involved or Sue is involved THEN Joe is involved
Abby said:  IF Cindy is not involved or Joe is not involved THEN Mike is involved or Jill is not involved
Cindy said:  IF Mike is involved and Joe told the truth THEN John is not involved or Jill is involved
Sue said:  IF Joe is involved or Sue is not involved THEN Cindy is involved or Abby is not involved
Mike said:  IF Cindy is not involved and Abby is involved THEN John is not involved or Jill is not involved
Joe said:  IF Jill is telling the truth or Cindy is lying THEN Sue is telling the truth or Mike is telling the truth

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Can I ask for a bit of clarification? There are a few things that could be interpreted in a couple of different ways

• When you say "or", do you mean AND/OR or Exclusive OR?
• When you say "IF" do you mean simply "IF", or "IF and only IF"?
• If the IF condition in a character's statement is not met (e.g. A says "If B is lying, then C is involved", but B is actually telling the truth), is that character considered to be telling the truth, lying, or neither? (obviously this becomes clear if the IF statements are interpreted as 'if and only if', but I'm asking in case they are not)
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I'm using the generally accepted usage of logical terms:

1.  "or" means and/or

2.  "if" means "if" ---- I do not mean "if and only if"

3.  Given: if A then B.  If A is false then nothing can be determined about B.  However the contrapositive condition should also be considered: if not B then not A.

4 hours ago, Evilhubert said:
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