Validity is something we have left largely undiscussed until now. Before we move on to begin studying Categorical logic, our first encounter with Deduction in any depth, it will be helpful to get a more precise sense of what validity amounts to.
The first point is to keep in mind that in deduction, the entire question of validity is settled in terms of the FORM of the argument, whereas in induction, the CONTENT of the argument is decisive.
What does it mean to say that Validity is a matter of form? It means that the relations set up within the argument, between its parts (which may be categories in categorical syllogisms, or sentences in hypothetical and disjunctive syllogism, as well as between premises and conclusion) either guarantee or fail to guarantee that an inference goes through. For instance, consider this argument:
Some fruits are red and some fruits are apples, so some fruits are red apples.
A very attractive piece of reasoning, to which most people will happily give their approval, on the grounds that they recognize the obvious truth of the premises and the conclusion. The form of the argument is easy enough to see: Some A is B, Some A is C, so Some A is BC.
It’s just like this one: Some fruits are blue, and some fruits are apples, so some fruits are blue apples.
Now we have a problem, because we know that these premises are true, but we’ve wound up with a false conclusion, and we know that “valid” means that a conclusion follows from premises with necessity. But that’s not happening in this example –this conclusion does not follow at all.
The moral of this is that the argument about the red apples is just as bad as the one about the blue apples. Despite the fact that it feels “good” because its premises and conclusion happen to be true, the form is not making the conclusion have to be true as a result of the premises. It is an invalid argument!
Let’s look at another example:
If George Washington was assassinated, then he’s dead. But he wasn’t assassinated, so he’s not dead.
What this example shows us is that no argument of the form If p then q, Not p, so Not q can be trusted. This form allows it to happen that true premises lead to a false conclusion, so this form has to be rejected as Invalid, not trustworthy.
This particular form is one that probably most people would fall for; something about it seems “right” to an unsuspecting ear. But this form is known in Logic as the formal fallacy Denying the Antecedent (because it consists of an argument with a conditional statement as a premise, plus the denial of the antecedent of that conditional; from that combination of things, nothing ever follows as a matter of necessity).
So if you hear someone argue “If the lights are on, someone must be home, but the lights aren’t on, so no one is home,” you’d be able to say: “That’s just as bad (or good) as if you said If Washington was assassinated, he’s dead, but he wasn’t, so he isn’t, which anyone can see is completely ridiculous.”
In responding this way, we say you have refuted the other person’s argument with a counterexample. A counterexample is another argument of exactly the same form as the original one, with true premises and an obviously false conclusion. A counterexample demonstrates the invalidity of an argument form, because it demonstrates that it is possible for that form to allow a false conclusion to follow from true premises. To be valid means for an argument to never allow for anything but a true conclusion to follow from true premises. “Valid” does not mean that the premises are true, and it does not mean that the conclusion is true either; it means that IF the premises are accepted as true, the conclusion follows inescapably. Another way to say it: it is impossible for the form to have true premises and a false conclusion. A counterexample is an example that shows that that claim is not true of a given form.
Another very common formal fallacy is worth exhibiting here. It’s called Affirming the Consequent, and we can use George Washington to help us see it. “If Washington was assassinated, he’s dead. And he is dead, so he must have been assassinated.” The form in this case is simply: If p, then q. q, therefore p.
Here are a few invalid arguments. See if you can spell out their forms, and generate counterexamples to show they are invalid.
Here’s a brief audio clip reviewing the basic ideas:
1. All oranges are fruits and all oranges are citrus, so all citrus are fruits.
2. No actors are politicians, so no politicians are insensitive, since no actors are insensitive.
3. Some students are people who shower every day, because some students are people with jobs and some people with jobs shower every day.
4. All Boy Scouts are Christians so all Christians know how to tie square knots, since all Boy Scouts know how to tie square knots.
5. All people who are envious are either spiteful or greedy, so everyone is greedy, since some envious people are spiteful.
6. All people who assist others in suicide are guilty of murder. Accordingly some compassionate people are not guilty of murder inasmuch as some people who assist others in suicide are compassionate.
7. Some farm workers are not people who are paid decent wages, because no illegal aliens are people paid decent wages and some illegal aliens are not farm workers.
8. All community colleges with low tuition are either schools with large enrollments or institutions supported by taxes. Therefore all community colleges are institutions supported by taxes.
9. If energy taxes are increased, then either the deficit will be reduced or global warming will be taken seriously. If the deficit is reduced, inflation will be checked. Therefore if energy taxes are increased, inflation will be checked.
You may find you tune in to the task somewhat intuitively, but if you need some guidance, here’s what to do: Rewrite the statements by replacing “content words” (like “people” and “envious”) with letters; that way the form shows up. That is, leave words like “all” and “who are” because they are the structures that set up the linkages. “Then ” is not always stated, but only implied in some cases, so realize that you may need to supply it in some cases
Make sure you are clear on which statement is the conclusion before you go any further!
Now, just use your imagination to come up with an alternative example of new content you can “plug in” where those letters are, and of course, make sure that if you replace the letter “A” with the word “animals” one time, you also replace it with “animals” every time it shows up (in the same example). See if you can generate a counterexample.
You’ll have to test what you come up with by asking yourself: 1) are all the premises I’ve generated true? and 2) is the conclusion I’ve generated false? If the answer to either question is “no,” then you have not shown anything yet, and you’ll have to ditch your example and try another one. You only refute an argument this way when you successfully find an example with all true premises and a false conclusion. No other combination of truth values accomplishes anything.
I highly recommend you not get fancy here; just use simple ideas like “cat,” “dog,” “animal,” and “mammal” if you are dealing with a categorical syllogism, and very simple and obviously true or false statements if you are dealing with disjunctive or hypothetical syllogisms (like “Garfield is a cat,” “Garfield is a dog,” “Meatloaf is made of wood,” “Meatloaf is made of meat,” etc.). Don’t make up interesting and creative examples about your roommate or last night’s dinner, because the rest of us will not be able to tell if they are true or false! Make up very obvious and boring example statements like “No robins are mammals.” Also, be sure to avoid using a word that has more than one meaning –or, if it has more than one meaning, but sure to use it in the same sense each time.
This link will open a powerpoint addressing this material: 5 Counter examples