I have to prove the validity or invalidity of the following argument :
Abbots and bishops are churchmen. No churchmen are either dowdy or elegant. Some bishops are elegant and fastidious. Some abbots are not fastidious. Therefore some abbots are dowdy.
It's clear that that no churchmen are either dowdy or elegant,so no bishop can be elegant, and hence the 3rd premis is a contradiction to the 2nd premis.So the premises of the argument are inconsistent and hence the argument form is valid , as it is impossible to find a substitution instance in which the conclusion is false and the premises are true at the same time.So the argument is valid.
Am I thinking in the right direction ? Also is the answer correct ? Any insight is appreciated. Thank you.
Yes, that's correct! Some people will say that 'either ... or ..' expresses an exclusive disjunction, rather than an inclusive disjunction, but 'not either A or B' clearly means 'not A and not B'.
And yes, as soon as an argument has an inconsistent set of premises, it is automatically valid, because it is impossible for it to have all true premises and a false conclusion, as you correctly point out.
Good job!