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
Best Answer
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!