Using truth tables to evaluate an argument and provide counterexamples. So, I have been given an two compound conditional statements to use as premises with the aim of reaching a conditional conclusion. I need to determine the validity of this argument. Truth tables are somewhat confusing for me, but this is my best effort at figuring this problem out:
$$\begin{array}{c|c|c||c|c|c|c|c|c}
J & K & L & 1^{st} Prem: J\to(K\to L) & 2^{nd} Prem: K\to(J\to L) & Concl: (J \lor K)\to L\\\hline
F & F & F & T & T & T \\\hline
F & F & T & T & T & T \\\hline
F & T & F & T & T & F \\\hline
F & T & T & T & T & T \\\hline
T & F & F & T & T & F \\\hline
T & F & T & T & T & T \\\hline
T & T & F & F & F & F \\\hline
T & T & T & T & T & T \\\hline
\end{array}$$
From what I can tell, this argument is invalid because there are two instances where both premises are true but the conclusion is false. I need to come up with a counterexample and I am not sure where to begin
 A: For a counterexample you look at the very rows where the premises are all true but the conclusion is false. In particular, look back at the reference columns on the left. For example, row 3 is one of these rows, and in this row $K$ is true, but $J$ and $L$ are false.  Typically, that is enough to be the counterexample: you just say: "A counterexample to the validity of this argument would be when $K$ is true, and $J$ and $L$ are false, for then all ther premises are true, but the conclusion is false."
Sometimes, however, they want you to provide a more concrete scenario, i.e. where $J$, $K$, and $L$ have some meaning so that the invalidity of the argument is even more Obvious. So: pick something for $J$, $K$, and $L$ that makes the premises true (in our world), but the conclusion false. OK, how about:
$J$ : Bob is an adult male
$K$ : Bob is unmarried
$L$ : Bob is a bachelor
Clearly, the two premises are now true, since effectively they say that if you are an adult male and you are unmarried, then you are a bachelor. The conclusion, however, says that you are a bachelor as soon as either you are an adult male, or you are unmarried ... and that is not true. I could be an adult male, but not a bachlor, since I am married. Or: I could be unmarried, but still not a bachlor, since I am female.
Anyway, the point is: look at the reference columns for the rows where the premises are true and the conclusion is false to get your counterexample!
A: You can substitute anything in for J, K, and L, so long as the truth values of your statements match the assigned values in the row where the premises are both true and the conclusion is false. If you use line 3, pick any statements for J, K, and L, where those statements are J (false), K (true), and L (false). The statements you use can be anything. The statements you choose do not have to relate to each other or even be about the same topic since the validity of the argument is not related to anything other than the truth values of the premises/conclusion. It is best to use well-known truths or falsehoods. The guy above said to use Bob is an adult male, bob is a bachelor, etc. I don't know who Bob is so I have no way to determine the truth value of a statement like "Bob is a bachelor.' However, everyone who is reading this proof will know that 'All humans are animals' is true, or that 'All odd numbers are divisible by 2' is false.. Using well known truths or falsehoods will give a much stronger counter-example. Instead of saying Bob is unmarried(since we have no way to verify this), use something that is a well-known truth. Something like 
J = "all odd numbers are evenly divisible by 2"(which is well known to be false)
K = "all humans are animals" (which is well known to be true)
L = "all snakes are mammals" (which is well known to be false)...
These 3 statements aren't even related and it does not matter for a good counterexample. All that matters is that the atomic statements you choose for J, K and L are chosen so that their truth value matches the truth values in your line 3, which causes it to be the case that using them causes 2 true premises and a false conclusion. The fact that the argument itself allows for an opportunity for us to assign statements that give us true premises and false conclusion shows it is invalid- and that's exactly the point of a counter-argument. Hope this helps. 
I will also point out that the truth table in the original post is not complete. You have to put a T or F under each logical operator and each atomic statement in your table. Each box therefore requires 2 truth values in this example. One for the main logical operator and one for the atomic statement(in parentheses).
