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SamA on February 23, 2020

Hello @Dalaal,

I think that the first one is kind of a trick question. X comes with two necessary conditions that are contradictory: Y and not Y. This means that X is impossible, it cannot exist. Let's try taking the contrapositive of the premises.

P1: Y - -> not X
P2: not Y - -> not X

If Y exists or Y doesn't exist, either way we have the same conclusion: X does not exist.


P1: A exists
P2: not B - -> not C
P3: A - -> C

Taking the contrapositive of P2, we have C - -> B.
A - -> C - -> B
A exists, therefore B exists.

For the purpose of these drills, we want to use all of the premises. Because of this, I can tell that [B exists] is the answer they are looking for. It is the main conclusion.

However, you are correct that [C exists] is another possible conclusion. I would call it a sub-conclusion! Good thinking. Sometimes a logical reasoning question will ask you to identify a subsidiary conclusion, so it is important to understand what this means.

Anna20 on May 16, 2020

Thanks so much for the above explanation. Further to the above, could I clarify whether an invalid conclusion is the same as saying X does not exist?

To just check my understanding -
P1: A --> B
P2: B --> C
P3: A does not exist
C: B and C do not exist - is that the same conclusion as saying "no valid conclusion"?

In addition, I would be very grateful if you could please walk through the following:
P1: A --> not B
P2: C --> B
P3: not D --> A
C: ?

How do you connect the above using contrapositives - for the conclusion I got, not B --> not A --> D --> not C. Where did I go wrong here?

Many thanks again!

Anna20 on May 29, 2020

Please can I follow up on this. Thank you.

Anna20 on June 2, 2020

Please could I follow up on this. Thank you!

Brett-Lindsay on July 2, 2020

I'm trying to wrap my head around valid conclusions, too.

For the first argument, it almost seems impossible to conclude "B and C do not exist" because that would seem like we're using a necessary condition to calculate a sufficient condition.

I thought these S-->N conditions were a one-way street. Just because we don't have A doesn't mean that there isn't another sufficient condition to give us B. If there were, then we'd potentially have B and C.

It looks like an invalid argument.

Brett-Lindsay on July 2, 2020

@Anna2020 For the second one, I think it's possible to just write out the premises with their contrapositives and look for links (where one necessary condition is the same as a sufficient condition) and connect them. There's a bit of trial and error involved, but it seems to get faster with practice.

P1: A --> not B
B --> not A
P2: C --> B
not B --> not C
P3: not D --> A
not A --> D

You probably notice that there's a "not D" on the left side but never on the right side.
Let's put that on the far left:
not D --> A
Then you look for an A on the left side (P1) and add that to our chain:
not D --> A --> not B
Then you look for a "not B" at the left side (P2 contra) and add its necessary condition to the chain:
not D --> A --> not B --> not C
Then you notice that there are no "not C" statements on the left side. Looks like the chain is complete.
If necessary, we can take the contrapositive:
C --> B --> not A --> D
or use the original:
not D --> A --> not B --> not C

Hope that helps