How do I determine if this argument is valid? I'm studying propositional logic the section on "valid arguments".

A self-assessment question reads

"Show whether or not the following argument is valid"

$\frac{P}{C}$

I don't know what function to apply to P to calculate C.

The solution in the back of the book shows the following truth table:

Variables   Premise Conclusion
P       C      P        C
F       F      F        F
F       T      F        T
T       F      T        F  &lt;--
T       T      T        T


So the conclusion is that the argument is invalid because the premise is true and the conclusion false.

I understand that of course, but how do I calculate Conclusion? Do I just guess what it should be?

My teacher had the following to say:


  If P is True, C is True. If P is False, C is False.
  
  Swap it for 'Premise - I am thirsty' &amp; 'Conclusion - I drink some water'
  
  The conclusion is dependant on the premises. If there is only one, they > will never differ and a Conclusion will never be False, with a True 
  Premise.


This answer flies in the face of the answer in the book.

I pointed this out to my teacher and got the following:


  My understanding is that P is the only variable; C is the conclusion that represents what P might be i.e True or False, once evaluated.


So... in conclusion I find myself still confused
The argument $\frac{P}{C}$ means "If we assume that $P$ is true, we can prove that $C$ is true." For example, $\frac{A}{A\lor B}$ because, if you can prove that $A$ is true, then $A\lor B$ is also true: there's no way that $A$ can be true but $A\lor B$ can be false at the same time.

In the argument you've been give, $P$ and&nbsp;$C$ are completely unrelated. There is a way that $P$ can be true and $C$ can be false at the same time: specifically, $P=\mathrm{true}$, $C=\mathrm{false}$. Therefore, the argument is not valid.

You don't calculate the conclusion, per se. rather, you look for a situation in which the premise is true but the conclusion is false. If you can find such a pair of premise and conclusion, the argument is invalid. Conversely, if you can prove that the premise being true always makes the conclusion true, the argument is valid.

I don't understand what your teacher is trying to say. It's not enough to show that the argument is valid for one particular instantiation of&nbsp;$P$ and one particular instantiation of&nbsp;$C$. "Valid" means it works for all instantiations. Also, the examples of $P$ being "I am thirsty" and $Q$ being "I drink some water" doesn't work.  There are situations where you are thirsty but you don't drink water (maybe you drink nothing; maybe you drink juice), and situations where you're not thirsty but you do drink water (maybe you're Mark Zuckerberg at a Senate hearing), as well as situations where you're thirsty and drink, or not thirsty and don't drink.