Logic problems for Math, Identifying invalid arguments

[aburchett]

Identify which argument is invalid.

If the elephant roars, then it does not want company.
The elephant wants company.
Therefore, the elephant did not roar.

If I shower in the morning, then I do not smell bad.
I did not shower in the morning.
Therefore, I smell bad.

Either the water is filtered or it contains lead.
The water is not filtered.
Therefore, it contains lead.

If the panda yawns, then she is not alert.
If the panda is not alert, then she will not smile for the camera.
Therefore, if the panda yawns, then she will not smile for the camera.

If the panda yawn, then she is not alert.
The panda yawned.
Therefore she is not alert.

 

[DrSteve]

\(\displaystyle p\rightarrow q\) together with \(\displaystyle p\) yields \(\displaystyle q\). (modus ponens or law of detachment)

\(\displaystyle p\rightarrow q\) together with \(\displaystyle \neg q\) yields \(\displaystyle \neg p\). (modus tollens or law of contrapositive)

\(\displaystyle p\vee q\) together with \(\displaystyle \neg q\) yields \(\displaystyle p\). (law of disjunctive inference)

\(\displaystyle p\rightarrow q\) together with \(\displaystyle q\rightarrow r\) yields \(\displaystyle p\rightarrow r\). (chain rule)

 

[lookagain]

Identify which argument is invalid.

1) If the elephant roars, then it does not want company.
The elephant wants company.
Therefore, the elephant did not roar.

2) If I shower in the morning, then I do not smell bad.
I did not shower in the morning.
Therefore, I smell bad.

3) Either the water is filtered or it contains lead.
The water is not filtered.
Therefore, it contains lead.

4) If the panda yawns, then she is not alert.
If the panda is not alert, then she will not smile for the camera.
Therefore, if the panda yawns, then she will not smile for the camera.

5) If the panda yawn, then she is not alert.
The panda yawned.
Therefore she is not alert.

aburchett,

DrSteve gave you laws for these.

I will talk you through them (mine are redundant) in an informal way as an alternative:

1) This is an example of a contrapositve, which has the same truth
value as the original. "If A, then B" is the same as (has the same truth value as)
"if not B, then not A."

In my sense, I see this as:

If the elephant roars, then it is "chasing away" any company with its roar.
But you find out the elephant does want company, so it wouldn't make
sense to roar to any potential company at that time then, because it would
keep the company away. That roaring would be counteractive to it wanting
the company. Therefore, the elephant won't roar. ----> Valid


2) Just because you did not shower, that doesn't mean that you smell bad.
For instance, you may have bathed instead, or you may have put on some
fragrances to cover up any potential smells, etc.

Then this argument is \(\displaystyle invalid.\)


3) It has to be one or the other as there are only two choices. It cannot be both
at the same time. So, because you are told that it is not filtered, then it must
be the case that it contains lead, as that is the only other option. ----> Valid


4) It was stated as a "chain rule." I will optionally call it a "transitive rule."
If A, then B. If B, then C. So, if A, then C.
A leads to B leads to C ----> Valid


5) If it is the case that the panda yawns, you are told that she is not alert.
So, if you are told just the first part, you know that the second part is true.

It may be that if the panda yawns, she is also exhibiting some other behavior,
but you are not interested in those hypothetical behaviors for this argument. ----> Valid