# 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  <--
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' & '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 $$C$$ 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.
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 $$P$$ and one particular instantiation of $$C$$. "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.