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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

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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.

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 $P$ 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.

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  • $\begingroup$ Ok so because P, and C are unrelated the argument can never be valid. That's what I am taking away from this. Thank you $\endgroup$ – Aethalides Apr 23 at 17:37
  • $\begingroup$ @Aethalides Yes. You can never conclude something from a premise that has no relation to it. $\endgroup$ – David Richerby Apr 23 at 19:16

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