> _The Tardy Bus Problem_
>
> Given the following three statements as premises:
>
> (1) If Bill takes the bus, then Bill misses his appointment, if the bus is late.
>
> (2) Bill shouldn't go home, if (a) Bill misses his appointment and (b) Bill feels downcast.
>
> (3) If Bill doesn't get the job, then (a) Bill feels downcast and (b) Bill should go home.
>
> ...determine whether the following are valid to conclude:
>
> Q1: If Bill takes the bus, then Bill does get the job, if the bus is late.
>
> Q2: Bill does get the job, if (a) Bill misses his appointment and (b) Bill should go home.
>
> Q3: If the bus is late, then (a) Bill doesn't take the bus, or Bill doesn't miss his appointment, if (b) Bill doesn't get the job?
>
> Q4: Bill doesn't take the bus, if (a) the bus is late, and (b)Bill doesn't get the job.
>
> Q5: If Bill doesn't miss his appointment, then (a) Bill shouldn't go home, and (b) Bill doesn't get the job.
>
> Q6: Bill feels downcast, if (a) the bus is late, or (b) Bill misses his appointment?
>
> Q7: If Bill does get the job, then (a) Bill doesn't feel downcast, or (b) Bill shouldn't go home.
>
> Q8: If (a) Bill should go home, and Bill takes the bus, then (b) Bill doesn't feel downcast, if the bus is late.
>
> --
>
> I tried a Google search, and half of the few sites that turned up had a different premise: (3)(b) was changed to "Bill should not go home." The book I found it in is a logic book and has it the way found in the excerpt above, but it could still possibly be wrong. I made that error when copying it for the first time, and, also, one might note that, in all the other sentences, the contraction "shouldn't" is used instead of "should not." So maybe the book really is right and the internet sites are wrong.
>
> gabby

Answers:
Q1 : true
Q2 : true
Q3 : true
Q4 : true
Q5 : false
Q6 : false
Q7 : false
Q8 : true
---------------
And here's the detailed explanation. It assumes that a priori there is complete logical independence of the six fundamental statements used (i.e. without the three premises, there is no logical inconsistency in assuming any combination of truthhood or falsehood for each separately).

Let the following letters represent statements:

A = Bill takes the bus.
B = The bus is late.
C = Bill misses his appointment.
D = Bill feels downcast.
E = Bill should go home
F = Bill gets the job.

Then the 3 premises are:

(1) (A and B) implies C.
(2) (C and D) implies (not E).
(3) (not F) implies (D and E).
-----------------------
Q1: (A and B) implies F?

True:
1. The contrapositive of premise (2) is: E implies ((not C) or (not D))
2. Thus (D and E) implies (D and ((not C) or (not D))), which clearly implies (D and (not C)), which implies (not C)
3. Combining this with premise (3) gives: (not F) implies (not C).
4. Thus C implies F, and combined with premise (1) gives: (A and B) implies F.
-----------------------
Q2: (C and E) implies F?

True:
We already know C implies F, so therefore we know the weaker proposition (C and E) implies F.
-----------------------
Q3: This could be interpreted as either:

Q3A: (B and (not F)) implies ((not A) or (not C))

OR

Q3B: (B implies (not A)) or ((not F) implies (not C))

Q3A: True:
1. (B and (not F)) implies (B and D and E) by premise (3).
2. By step two of the reasoning for Q1, (D and E) implies (not C), so also (B and D and E) implies (not C).
3. Thus (B and (not F)) implies (B and D and E), which implies (not C), which implies ((not A) or (not C)).

Q3B: True:
1. (C implies F) was shown in step 4 of Q1, so we have the contrapositive (not F) implies (not C), which is the second
term in the "or" of Q3B. Thus it doesn't matter whether the first term is true (it is in fact false in some cases).

I *think* that the interpretation they were asking was Q3B, by the way it was phrased.
-----------------------
Q4: (B and (not F)) implies (not A)?

True:
1. The contrapositive of the true proposition Q1 is (not F) implies ((not A) or (not B)).
2. Thus (B and (not F)) implies (B and ((not A) or (not B))) which implies (B and (not A)) which implies (not A).
-----------------------
Q5: (not C) implies ((not E) and (not F))?

False: consider the combination:
A, B, C, F false;
D, E true.
These satisfy the premises but not Q5.
-----------------------
Q6: (B or C) implies D?

False: consider the combination:
A, C, D, E false;
B, F true.
These satisfy the premises but not Q6.
-----------------------
Q7: F implies ((not D) or (not E))

False: consider the combination:
A, B, C false;
D, E, F true.
These satisfy the premises but not Q7.
-----------------------
Q8: (E and A and B) implies (not D)?

True:
1. (E and A and B) implies (E and C) by premise (1).
2. The contrapositive of premise (2) is (E implies ((not C) or (not D))).
3. Thus (E and A and B) implies (E and C) which implies (C and ((not C) or (not D))) which implies (C and (not D)) which
implies (not D).

--grem"Even if Q3B was the correct interpretation, Q3A is still the BETS GAEM"linn