Conditionnal and implication

Question

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?

Answer 1

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.

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

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.

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

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.

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

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.

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

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.

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

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

Answer 2

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.