Introduction to Logic
Truth Table Analysis of Arguments
We are almost done with truth tables. The next thing we can use them for in Logic is determining whether an argument in propositional logic is valid or invalid.
The key is knowing what it means to say that an argument is valid, and knowing how to map that onto a truth table.
What it means is that if the premises are all true, it is impossible for the conclusion to be false. It doesn’t mean that the premises are all true, but that if they are, they will necessitate a true conclusion.
This can also fruitfully be approached from the side of invalidity. “Invalid” means that even if the premises are all true, they do not necessitate that the conclusion is true.
So, if you found a line on a truth table for an argument, on which the conclusion was F, but all the premises were T, the argument would be invalid. And if you found a line on which the conclusion was T and all the premises were T also, but then on another line, the conclusion was F and the premises were T, the argument would be invalid also. That is, there would have to be no line on which the premises were all T and the conclusion F, in order for the table to show it is Valid.
The truth table approach means that if an argument is not invalid, it is valid. So you should check an argument for invalidity, and to do that, you have to be clear on what to look for.
What do you think you’d have to say about an argument that had inconsistent premises?
Let’s see some examples.
|(p||>||q)||/ ~p||// ~q|
First, note that I’ve introduced slashes. That indicates we have an argument here. A single slash is used to separate premises from each other. A double slash comes before the conclusion, which will always be presented as the right-most statement.
|(p||>||q)||/ ~p||// ~q|
This is correctly filled in. The values for the premises are marked by being bolded, and the values of the conclusion are in lower case. There are four possibilities, i.e., four lines to read through. On the first line, not both premises are true, so the F conclusion does not mean it is invalid yet. On the second line, both premises are F, so that line tells us nothing. On the third line both premises are T, and the conclusion is F -that means it’s invalid, because this is a case where a true conclusion is not generated despite the premises all being true. That’s all we need to know, we can stop reading it. But let’s read the fourth line anyway, because it’s instructive. On the fourth line, both premises are T and so is the conclusion. What do we make of that?
Nothing. It means nothing, since the third line shows that this argument does not guarantee T conclusions from T premises, but allows the possibility of F conclusions. That makes the form untrustworthy, not one hundred percent reliable, invalid.
I hope you recognize that form; it’s known by the phrase that describes what its premises are doing. (Denying the antecedent.)
This one has a name you should also know:
The invalidity here is also shown on the third line, where both premises are T and the conclusion is F. The first line, on which both premises and the conclusion are all T, does not indicate anything that would counteract what the third line shows, namely that this form allows the possibility of T premises and a F conclusion.
This is Affirming the Consequent.
Here’s another to consider:
|~||(G||∙||M )||/ M||v||~G||// ~G|
The premises say “Not both G and M. Either M or else G is false.” The conclusion is “Not G.”
This could be
It is false that both Garfield and Marmaduke are dogs. Either Marmaduke’s a dog or else Garfield’s not one. Therefore Garfield’s not one.
What does the table show? It shows that whenever the conclusion is F, at least one premise is F also. On the first and second lines, the conclusion is F. But on the first line, the first premise is F (look under the ~), and on the second line the second premise is F (look under the v). So that shows that it’s not invalid, therefore it’s valid.
The next two lines show all premises T and the conclusion T. But even if we didn’t look closely at them, we could say it’s valid because the only chance of it being invalid is when the conclusion is F, and we’ve already seen all the possibilities of that.
Here are some exercises you can practice on:
1. P ≡ ~N // N v P
2. K ≡ ~L / ~(L ∙ ~K) // K >L
3. Z // E > (Z > E)
4. C ≡ D / E v ~D // E > C
5. A > (B v C) / ~C v B // A >B
6. J > (K > L) / K > (J > L) // (J v K) > L
7. If Sartre’s an existentialist, then Wittgenstein wrote the Tractatus, therefore if Wittgenstein wrote the Tractatus, Sartre’s an existentialist.
8. Hurley is President, so either he’s President or Ackermann is Dean.
9. If “time flies” is a metaphor, it’s not literally true. If it’s not literally true then time does not fly, so if “time flies” is a metaphor, then time does not fly.
10. If the argument from design is weak, it’s a weak analogy. In order to be a weak analogy, it must make an unwarranted comparison, so the argument from design makes an unwarranted comparison.
11. Winters are cold and summers are hot, so either summers are hot or the moon is made of green cheese.
12. Russell was either a realist or an empiricist. If the former, then he was not an idealist, so he was not an empiricist.
13. If he loves her, he’ll marry her. Therefore if he doesn’t love her he won’t marry her.
14. If humans can settle on the moon, then they can settle on Mars. If they can settle on Mars, they can settle on Jupiter. So if the moon can be settled, so can Jupiter.
15. This argument is invalid if and only if it can have true premises and a false conclusion. Therefore it is invalid, since it has a false conclusion.
16. The fact that animals are less intelligent than us does not imply that we may disregard their welfare. If we disregard their welfare, then we are inhumane, and no better than animals ourselves. Thus, if we disregard their welfare, it is false that they are less intelligent than us.
17. If Sweden is in North Africa, then either Egyptians are blue-eyed or Swedes are dark and handsome. Sweden is not in North Africa, so Egyptians are not blue-eyed.
18. Unless you have both a sledge and a wedge, you are not going to be splitting any wood. You have a sledge but you don’t have a wedge, so either you’re not going to be splitting any wood or else you’re going to go buy one.
19. If she loves him, she’ll marry him, therefore if she doesn’t marry him, she doesn’t love him.