# I need some help determining the validity of the following argument

“I got the highest grade on the last test and I have perfect attendance. If I get a cold, then I miss at least one class. I came down with a cold. Therefore, if I missed at least one class, then I did not get the highest grade on the last test".

• The argument is badly written... We have to assume at least an additional "axiom" : if I miss at least one class, then I have not perfect attendance. Feb 19 '19 at 9:39
• It conflates 'attendance' with 'grades'. Its certain that missing a day because of illness will mean loosing 100% attendance. But not certain grades will suffer. Feb 19 '19 at 13:39
• yes the premises are contradictory, what²s the issue with that ? Feb 20 '19 at 17:08

Hint

P1) Highest Grade and Perfect Attendance.

P2) If Get Cold, then Miss Class.

P3) Get Cold.

C) Therefore, if I Miss Class, then not Highest Grade.

From P2) and P3) we have : I Missed a Class.

Having said that, we can check with a valuation v such that v(Miss Class)= true and v(Highest Grade)= true.

In this way, the conclusion is false.

In addition we have v(Get Cold)= true.

In this way, all premises are satisfied and the conclusion is not: thus, the argument is not valid.

• Correct me If I'm wrong. I created a truth table and there are no critical rows, therefore, the argument should be vacuously true. For instance, if we check v(Miss Class)= true, v(highest grade)= true, and v (Get Cold)= true, then the premise v( Highest Grade and Perfect Attendance)= False and consequently, the argument can't be validated (or not) by making these assumptions. Feb 19 '19 at 14:40
• @Micaela - not clear... The argument is valid iff in every row where all premises are TRUE also the conclusion is TRUE. Feb 19 '19 at 14:47
• Right. In this case, there are no critical rows because the premises are contradictory. I found on the internet that "An argument with contradictory premises is guaranteed to be valid, no matter what its conclusion". However, I'm not sure if it is true that there are NO critical rows. I built a truth table based on the following premises: let P be "I got the highest grade on the last test", Q be "I have perfect attendance", and R be "I get a cold". The premises would be as follow: P /\ Q, R -> ~Q, R. Feb 19 '19 at 14:58
• @Micaela - correct: contradictory premises = valid. But the key-point is to read "miss at least one class" as "not perfect attendance", as said before. Obviously, this reading is not "available" to propositional logic alone. Feb 19 '19 at 15:12
1. You don't need perfect attendance to get the highest grade on the last test. This is where your argument goes down outright with no chance of recovery.

2. Depending on the school's rules, missing one class doesn't mean you don't have perfect attendance. The class may have been cancelled, or the school may say that perfect attendance means "present for the whole time of every class, except for at most three sick days with a notice from the pupil's doctor". This is more a bit of nitpicking, but still needs to be addressed.