Deductivley valid argument (NEED HELP)

Question

. Is this argument deductively valid or invalid? Explain your answer.

P1. Every person that I know is truthful. 

P2. Some people who are truthful have told a lie. 

C1. Some person I know has told a lie. (From P1 and P2).

Answer 1

The argument is deductively INVALID.

To show why this argument is invalid, it is first helpful to revisit what makes for a valid argument. A valid argument is valid IFF (if and only if) it is impossible for all of the premises to be true and the conclusion false.

There are several methods for evaluating arguments for validity. Truth tables, Venn diagrams and proofs are all methods used to test an argument's validity. I am happy to teach these methods to students who wish to work on logic with me.

But we can also test for validity more informally. If you can come up with at least one scenario or "story" in which the truth values of the premises are all true and the truth value of the conclusion is false, the argument is INVALID. Invalid arguments may only have one possible "story" in which the conclusion would be false while the premises, or they might have more than one. But if we know there is AT LEAST one such scenario, we know that the argument is invalid.

In this case, it IS possible for all of the premises to be true and the conclusion false. This is easier to show with a Venn diagram, but I will try to explain this in text.

Consider a set of five "truthful" people: A, B, C, D, and E.

1. I know A, B, C and D.
2. It just so happens that A, B, C and D have never told a lie. This is not inconsistent with P2, "Some people who are truthful have told a lie." Why? Because it only applies to SOME truthful people, not to ALL. [If P2 stated instead that "Some of the truthful people I know have told a lie," the argument would actually be valid. But it does not.]
3. I am also a bit antisocial, and I only know four people, A, B, C and D.
4. Conclusion: P1 is TRUE under these conditions.

E, however, though truthful, has in fact told at least one lie. For our purposes, I only need one truthful person who has told at least one lie to exist anywhere in the world make P2 TRUE. Since E exists and is truthful and has told (at least one) a lie, P2 is TRUE.

Looking at this somewhat facetious case now, we can see that while I do not know E, E is nevertheless a truthful person who has told a lie. But, I don't know E! And so, not a single one of the (very few) four people I know has told a lie. In this scenario, we can see how P1 is true, P2 is also true, and without changing any truth values or adding anything else in, C1 (our conclusion) is false. Since we can describe at least one case (for this argument there are many possible counterexamples) in which the premises are true and the conclusion false, the argument is INVALID.