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Ali, Bob, Celia and Danny were making a decision conditional. Bob said that he will attend a given event if Ali does. Celia said that she will go if Bob does. Danny said that he will go if Celia does. Eventually exactly two of them attended the event, which two? 

Let, $$A=Ali, B=Bob, C= Celia, D=Danny $$ Then, $$\left( A\Rightarrow B \right)\wedge\left( B\Rightarrow C \right) \wedge\left( C\Rightarrow D\right) $$ $$A\Rightarrow B\Rightarrow C\Rightarrow D$$ $$\therefore A \Rightarrow D$$ Did only Ali and Danny attend the event in the end?

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    Why not use $A$,$B$,$C$,and $D$?! – 2017-01-26
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    I asked myself that question too haha! I thought it was a convention to use $P, Q,$ and so on... I will edit my post. – 2017-01-26
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    Yes, we often use $P$'s and $Q$, but it's ok to use any letters! And it's always a good idea to pick letters that go with what they represent. :) – 2017-01-26

2 Answers 2

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OK, your

$$\left( P\Rightarrow Q \right)\Rightarrow \left( Q\Rightarrow R \right) \Rightarrow \left( R\Rightarrow S\right) $$

is wrong, since that should be:

$$\left( P\Rightarrow Q \right)\land \left( Q\Rightarrow R \right) \land \left( R\Rightarrow S\right) $$

And please let's just use $A$, $B$, $C$, and $D$:

$$\left( A\Rightarrow B \right)\land \left( B\Rightarrow C \right) \land \left( C\Rightarrow D\right) $$

Now, if $A$ goes, then obviously $B$ goes, and thus $C$, and thus $D$ ... so all four are going: not good! So $A$ does not go.

Similarly, if $B$ goes, then $C$ and $D$ go, which is still too many, so $B$ does not go either.

So, $C$ and $D$ go: Celia and Danny!

You may be wondering how you could do this more formally:

OK, we know that exactly two people are going, so we have:

$(A \land B \land \neg C \land \neg D) \lor$

$(A \land \neg B \land C \land \neg D) \lor$

$(A \land \neg B \land \neg C \land D) \lor$

$(\neg A \land B \land C \land \neg D) \lor$

$(\neg A \land B \land \neg C \land D) \lor$

$(\neg A \land \neg B \land C \land D)$

(sorry, there is really no easier way to represent this in standard propositional logic)

OK, we definitely have:

$A \rightarrow B$

$B \rightarrow C$

$C \rightarrow D$

And we have one of the six disjuncts from that really long statement above. OK, let's explore these six possibilities:

  1. $(A \land B \land \neg C \land \neg D)$

Then we have $B$ and $\neg C$ but with $B \rightarrow C$ we get $C$, and that contradicts $\neg C$. So, this is not an option.

  1. $(A \land \neg B \land C \land \neg D)$

Then we have $A$ and $\neg B$ but with $A \rightarrow B$ we get $B$, and that contradicts $\neg B$. So, this is not an option.

  1. $(A \land \neg B \land \neg C \land D)$

Again we have $A$ and $\neg B$ and with $A \rightarrow B$ we get $B$, and that contradicts $\neg B$. So, this is not an option.

  1. $(\neg A \land B \land C \land \neg D)$

Now we have $C$ and $\neg D$ but with $C \rightarrow D$ we get $D$, and that contradicts $\neg D$. So, this is not an option.

  1. $(\neg A \land B \land \neg C \land D)$

Now we have $B$ and $\neg C$ but with $B \rightarrow C$ we get $C$, and that contradicts $\neg C$. So, this is not an option.

  1. $(\neg A \land \neg B \land C \land D)$

No problem here!! So this is it: $C$ and $D$ go and $A$ and $B$ do not.

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    So if I understand correctly this type of question has nothing to do with the conjunction of the 3 conditionals being true or false? Since two conjuncts out of three are false, isn't the statement as a whole false? What would be a general formulation of such problem? – 2017-01-26
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    @Hermitian No, that's not it. The 3 conditionals are all given, so they are all true. It's just that exactly two out of $A$, $B$, $C$, and $D$ are true. Now, to represent that is actually not easy, but I'll add something to my answer. – 2017-01-26
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    You made it crystal clear! I was seeing this as a chain of events that was dependent on the fact that $A$ had to occur for $B$ to occur and so on, so I thought that neither $B$, $C$ nor $D$ could occur if $A$ had not occurred. Yet when these word problems are put it into propositional logic terms the answer becomes almost obvious, I guess this is the best way to avoid silly mistakes... – 2017-01-26
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Case 1 -

If Ali goes then Bob goes. And Bob goes then Celia goes. And if Celia then Danny also. So all 4 goes this is not the case.

Case 2 -

If Bob goes then Celia and Danny goes. It means 3 of them goes.

Case 3 -

If Celia goes and Danny goes. Exactly two. So its answer.