6.7 Truth Assignment Test The material in this section is somewhat confusing to follow, and the Logicola exercises don’t fully explain what you are supposed to do. But if you play around a bit, you can figure out what is going on. Let me try to help with this. First off, this is just another way to test the validity of an argument. In the previous section (6.6), we constructed truth tables to test for validity. The truth table method (for testing validity) is completely clear and always works. But, since it examines every possible assignment of truth values to the atomic components of all the propositions in an argument, it is rather cumbersome and repetitive. The “Truth assignment test” developed in 6.7 takes a little more thinking, but it is considerably shorter. Let me begin by looking at an actual exercise from Logicola:

The idea is to use the workspace on the bottom to determine whether or not the argument is valid. The crucial point to keep in mind here is that in a valid argument, it is impossible for the premises all to be true while the conclusion is false. Alternately put, if it is possible for the premises all to be true with the conclusion false, then we know that the argument is invalid. So what we are trying to do in the bottom workspace is to create an assignment of truth values to the atomic components of the arguments that makes the premises true and the conclusion false. If we can do this, then the argument is invalid. If we can’t, the

argument is valid. It is crucial to keep this mind at all times, or you can forget what you are trying to do in the workspace. Look at the argument: (~P ⊃ ~C) P ∴C So, what we are trying to determine is whether or not it is possible for the premises to be true and the conclusion false. So, let’s assume this is so. That is, we assume that (~P ⊃ ~C), and P are true and that C is false, i.e., that P=1 and C=0. (Our goal will be to see if the first premise, (~P ⊃ ~C), comes out true on this assignment.) Let’s plug those values in on the bottom. Hit the tab key (or click inside the box) to enter the workspace. What I do at this point is simply replace all the instances of the atomic components with the truth values that we get by assuming the premises are true and the conclusion false. That is, in this case, I replace all instances of “P” with “1” and all instances of “C” with “0.” I begin by highlighting (“selecting”) an occurrence, say, of “P.” Thus:

And then I type in “1” in its place. Thus:

Then I do the same for “C.” (Remember, since I am assuming that the conclusion, C, is false, I will enter “0” for each occurrence of “C.” Thus:

Now, I can continue the process in the attempt to see if the first premise comes out true. (Remember, if it does, then I have found an assignment of truth values that makes the premises true and the conclusion false. So, if the first premise is true on this assignment, the argument is invalid.) Again, I highlight “~1” and replace it with “0,” and highlight “~0” and replace it with “1.” This gives me:

So, we know that on this assignment of truth values to the atomic components of the argument, the first premise is a conditional with a false antecedent and a true consequent. But we know that a conditional with a false antecedent and a true consequent is true, so this assignment of truth values to the atomic components of this argument makes all the premises true, and the conclusion false. So, this argument is invalid. I then select “Invalid,” on top, and get confirmation that my answer is correct:

I think this is a very helpful exercise simply because it is so easy to get confused along the way and forget what I am trying to show. (I made many mistakes the first time around in doing these exercises. You probably will too.) But the whole point is that if I can find an assignment of truth values to the atomic components that makes the premises true and the conclusion false, then I know the argument is invalid. (It is valid otherwise.) So, we are simply assuming the argument is invalid, and seeing if there is a possible truth value assignment that does this. Try some. They require thinking!