To get the free dessert, one must order an entree and a salad. But anyone who orders either an entree or a salad can ...

Mehran on September 1, 2016

@shubbyaluk "must" introduces a necessary condition so the first sentence is diagrammed as follows:

P: FD ==> E & S
not E or not S ==> not FD

"Anyone" introduces a sufficient condition so the second sentence would be diagrammed as follows:

P: E or S ==> FSD
not FSD ==> not E & not S

Similarly, the conclusion would be diagrammed as follows:

C: not FSD ==> not FD
FD ==> FSD

This is a valid contrapositive transitive argument:

not FSD ==> not E & not S ==> not FD
FD ==> E & S ==> FSD

Notice that if you have order both an entree and salad (i.e. E & S), that you clearly have satisfied the sufficient condition of the second premise, which is ordering either an entree or a salad (i.e. E or S).

Now let's take a look at (B).

"To be elected class president, one must be well liked and well known."

P: ECP ==> WL & WK
not WL or not WK ==> not ECP

"Anyone who is well liked or well known has something better to do than run for president."

P: WL or WK ==> SB
not SB ==> not WL & not WK

"Therefore, no one who has something better to do will be elected class president."

C: SB ==> not ECP
ECP ==> not SB

This is a flawed argument. "Something better to do" is the necessary condition in our second premise.

You cannot conclude anything from the existence of the necessary condition, so (B) is eliminated.

Hope that helps! Please let us know if you have any other questions.