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Erica, Stanley and Robert were all witnesses in a car crash. Their statements are contradictory to one another, and all of them claimed that someone else lied. Erica claimed that Stanley lied, Stanley claimed that Robert lied, while Robert claimed that both of them lied. After thinking for a little bit, without any further questions the judged figured out who was telling the truth. Who was telling the truth?

I need to provide a step by step proof based on logic equivalences and deductions made based on the rules of logic.

Here's my take on the exercise:

p:Erica is telling the truth.

q:Stanley is telling the truth.

s:Robert is telling the truth.

Based on this representation we have:

p -> ¬q ( if Erica is telling the truth, then Stanley is not telling the truth )

q -> ¬s ( if Stanley is telling the truth, then Robert is not telling the truth )

s -> ¬p ^ ¬q ( if Robert is telling the truth, then both Erica and Stanley are not telling the truth)

Now we will examine the possibilities. If Erica is telling the truth we have:

1. p

2. p -> ¬q

3. ¬(¬s) - Since Stanley is lying, the opposite of what Stanley said is correct

4. s -> ¬p ^ ¬q

These 4 are conditions.

5. s - double negation on 3.

6. ¬p ^ ¬q - Modus Ponens of 4 and 5

7. ¬p - simplification of 6

8. ¬p ^ p - addition of 1 and 7.

Because of this contradiction it is clear that Erica is not telling the truth.

Similarly we will examine if Stanley is telling the truth.

1.q

2.p -> ¬q

3.q -> ¬s

4. ¬(¬p ^ ¬q) - Since Robert is lying, the opposite of what Robert said is true.

These 3 are conditions.

5. ¬p Modus Ponens of 1 and 2

6. ¬s Modus Ponens of 1 and 3

No contradiction, meaning Stanley was in fact telling the truth.

Lastly, let's examine the case if Robert was telling the truth.

1.s

2.¬(p->¬q) - since Erica is lying, the opposite is true

3.¬(q->¬s) - since Stanley is lying, the opposite is true

4. s -> ¬p ^ ¬q

These 4 are conditions.

5. ¬p ^ ¬q - Modus Ponens of 1 and 4.

6. ¬(¬p v ¬q) substitution of the implication in 2

7. p ^ q De Morgan's law on 6.

8. p simplification of 7

9. ¬p simplification of 5

10. p ^ ¬p addition of 8 and 9

Because of the contradiction it is clear that Robert also couldn't have been telling the truth.

Can my logic be justified? And is it possible to divide the exercise like so? If not, can you please show me the path to righteousness?(that might've been a bit too cheesy xD ) Thanks!