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RATIONAL EPISODES
Logic for the Intermittently Reasonable
By KEITH M. PARSONS
Prometheus Books
Copyright © 2010 Keith M. Parsons
All right reserved.
ISBN: 978-1-59102-730-0
Chapter One
WHAT IS LOGIC?
Why do we believe the things we do? We hold many of our beliefs because we were taught them at an early age and never thought to question them. We believe many things, even important things, for the slightest of reasons. A TV commercial we heard twenty years ago, which has long since faded from conscious memory, may still prompt us to prefer brand "X" over brand "Y." We believe many things because they flatter our vanity, soothe our fears, or pander to our biases. Politicians know this and that is why they are so good at pushing our buttons. Occasionally, though, we live up to our reputations as rational creatures and try to find out whether something is really so. That is, we try to think clearly and objectively about things and strive to base our beliefs on the best evidence available and not infer more than the evidence authorizes us to believe. In other words, we try to be like the wise man, who, as David Hume says, "proportions his belief to the evidence."
Also, we frequently try to influence other people's beliefs. Since humans are social creatures, and we have to act collectively to get important things done, and since actions depend on beliefs, we have to be interested in other people's beliefs. Another reason to be interested in other people's beliefs is that beliefs have consequences, often serious ones. If someone believes that people like you are evil, and that anyone who dies in the act of killing people like you will receive rich rewards in paradise, then you had better be on guard. Because beliefs matter, many people exert considerable effort to influence others' opinions.
Politicians, pundits, and advertisers often try to influence people's beliefs by manipulating them in various ways—by playing on their fears or prejudices, for instance. Sometimes, though, we try to influence other people's beliefs by reasoning with them rather than merely manipulating them. When we attempt to persuade people rationally we offer them arguments. When we draw conclusions from arguments or evidence we make an inference. Logic is about argument and inference. We study logic to learn how to distinguish good arguments from bad ones and to learn how to construct good arguments. Concomitantly, logic teaches us which inferences we should or should not draw from given arguments or evidence.
When we offer an argument to someone, we are trying to get that person to accept some conclusion. The reasons we offer in support of those conclusions are called premises by logicians. Since we communicate with language, and since language is organized into sentences, our arguments will be groups of sentences. In particular, our arguments will consist only of declarative sentences, that is, sentences that assert that something is either so or not so. Other types of sentences, those that ask questions or express commands, cannot constitute arguments (though they might sometimes suggest or imply arguments, e.g., when somebody asks "Do you really plan to wear that?" he or she signals willingness to argue that you shouldn't wear that). So an argument consists of a set of declarative sentences, one of which is the conclusion, and with one or more premises. The following is an argument; the premises are marked by the letter "P" and a number indicating which premise it is, and the conclusion is marked by the letter "C":
P-1: Few good people enter politics, and those who do soon cease to be good.
P-2: Joan Smith has decided to enter the mayor's race.
C: Either Joan Smith is not a good person, or she will soon cease to be one.
You may agree with the conclusion of this argument, or you may find it overly cynical. The point is that anyone offering you such an argument intends that the premises give you reasons for accepting the conclusion. You also often hear people saying that premises support a conclusion, or give grounds for the conclusion, or justify the conclusion. These are just different ways of saying that the premises are supposed to give us good reason to think that the conclusion is true.
Sometimes the premises of an argument really do support the conclusion, and sometimes they do not. A good argument is one where the premises support the conclusion; a bad argument is one where the premises fail to support the conclusion. Another way of putting it is that a good argument is one where the conclusion follows from the premises, but a bad argument is one where the conclusion does not follow from the premises. Logic gives us norms for distinguishing good arguments from bad.
Here is a good argument: P-1: All firemen are employees of the Department of Public Safety.
P-2: Jarrod is a fireman.
C: Jarrod is an employee of the Department of Public Safety.
Here is a bad argument:
P-1: All tenured faculty members of the History Department are PhDs in history.
P-2: Erin is a PhD in history.
C: Erin is a tenured faculty member in the History Department.
What makes the first argument good and the second one bad? In the first argument, the premises support the conclusion in the strongest possible way, namely, if the premises of the first argument are true, then the conclusion has got to be true. So if you accept the two premises as true, you have to accept the conclusion also. Why? Well, it is just impossible for Jarrod to be a fireman, and for all firemen to be employees of the Department of Public Safety, and for Jarrod not to be an employee of the Department of Public Safety. But what do we mean by "impossible" here? Lots of things are impossible. It is impossible for a cow to jump over the moon. It is impossible for me to believe that a congressman can receive lavish gifts and favors from a lobbyist and not be influenced. When we say that it is impossible for the premises of an argument to be true and the conclusion false, do we mean "impossible" in any of these senses?
When we speak of impossibility in logic, that is, when we talk of logical impossibility, we mean impossibility in the strongest sense. We don't mean psychologically impossible, like believing that congressmen can be impartial when they have lived like royalty on a lobbyist's tab. We don't even mean physically impossible, that is, forbidden by the laws of nature, like a cow jumping with sufficient force to achieve escape velocity from the earth's gravity. We mean absolutely and strictly impossible in the sense that you cannot even coherently suppose that something logically impossible is true. Try to suppose that all firemen—including Jarrod, of course—are employees of the Department of Public Safety and that some fireman—Jarrod, in this case—is not an employee of the Department of Public Safety. Clearly, you cannot even coherently ask somebody to suppose that Jarrod both is and is not an employee of the Department of Public Safety.
Put more explicitly, the kind of impossibility we are talking about here, logical impossibility, is the impossibility involved when you assert a contradiction. You assert a contradiction when you say that both a statement and its denial are true. Yet perhaps the most basic law of logic is the Law of Noncontradiction, the law that a statement and its denial cannot both be true. Thus, by the law of noncontradiction, it cannot be true both that Jarrod is an employee of the Department of Public Safety and that Jarrod is not an employee of the Department of Public Safety. Yet, if you accept the two premises of the argument about Jarrod as true but reject the conclusion as false, you seem to be forced into saying precisely this. This means that it is logically impossible for the premises of the argument about Jarrod to be true and the conclusion false. When an argument is such that it is logically impossible for the premises to be true and the conclusion false, logicians say that the premises entail the conclusion. The entailment relationship is one of the most important ones in logic (more will be said about entailment in chapter 4).
Getting back to our sample argument from a few paragraphs ago, the one about Erin the history PhD, this argument is very different from the argument about Jarrod the fireman. With the argument about Erin it is entirely possible for the premises to be true and the conclusion false. It is possible—that is, there is no contradiction in supposing—that all the tenured faculty in the History Department have PhDs in history, and that Erin has a PhD in history, but is not a tenured member of the History Department. Maybe she was just hired as a tenure-track assistant professor but has not gotten tenure yet. Or maybe she is employed at another university, or maybe she works as a museum curator and does not hold a faculty appointment at all. The point is, quite simply, that the premises, even if they are both true, don't guarantee the conclusion.
Logicians call arguments of the first type valid arguments. Valid arguments are those where the premises entail their conclusions, that is, it is logically impossible for the premises all to be true and the conclusion false. In other words, validity is the property, which some arguments have, that IF their premises are all true, THEN their conclusion must be true. Notice carefully what the previous sentence just said: "Validity is the property, which some arguments have, that IF their premises are all true, THEN their conclusion must be true." This definition of "validity" does not say that valid arguments actually have true premises, and it doesn't say that valid arguments must have true conclusions. Actually, validity has nothing to do with whether the premises of a valid argument are in fact true. Validity says that IF all the premises are true, THEN the conclusion has got to be true. Otherwise, all bets are off. If one or more of the premises turns out not to be true, the conclusion need not be true, but that doesn't mean that the argument is not valid. Validity is about a specific kind of conditional (if ... then) relationship between an argument's premises and its conclusion. The relationship is that it is impossible for all the premises of a valid argument to be true and the conclusion false—and that's all it is. In logic, what a definition does not say is often just as important as what it does say.
The upshot is that you can have valid arguments with true premises and a true conclusion, or false premises and a false conclusion, or false premises and a true conclusion. The only thing you cannot have is a valid argument with true premises and a false conclusion. With invalid arguments, all combinations are possible. Invalid arguments can have true premises and a false conclusion. A valid argument therefore serves as what we might call a "truth preserver." That is, if we have a valid argument with true premises, we can be completely confident that the conclusion is true also. With the argument about Jarrod the fireman, if the premises are true, we can be assured that the conclusion is also. But with the argument about Erin, even if the premises are true, they do not tell us whether the conclusion is true or not. So if you care about truth, validity matters.
How can we tell if an argument is valid? Do we have to examine each one on a case-by-case basis and see if it seems valid to us? That would be an impossibly tedious task. Fortunately, logicians have identified a number of argument forms such that any argument having that form must be valid. However, not every valid argument has an easily identifiable form. Consider the following argument:
Pretzels are salty
Pretzels have flavor.
This example is obviously valid. The premise could not possibly be true and the conclusion false since saltiness is a kind of flavor, but logicians have a hard time identifying a general form for this argument. In this text we shall just ignore arguments that are obviously valid but have no easily identifiable form. We shall focus exclusively on arguments that are valid because they have the right argument forms.
What, then, is an argument form? The best way to answer at this point is with examples. Recall the argument about Jarrod the fireman:
P-1: All firemen are employees of the Department of Public Safety.
P-2: Jarrod is a fireman.
C: Jarrod is an employee of the Department of Public Safety.
Now consider another argument:
P-1: All sauropods were dinosaurs.
P-2 Apatosaurus was a sauropod.
C: Apatosaurus was a dinosaur.
Though these arguments are about entirely different things—one is about Jarrod the fireman and the other is about Apatosaurus the sauropod—they share a common form. This is the form they share (I'm allowing "was" and "were" in the argument about dinosaurs to be variants of "are" and "is"):
P-1: All Ss are Ps.
P-2: x is an S.
C: x is a P.
This is a valid argument form, that is, every argument that has this form is valid. If all Ss are Ps, and if x is an S, x is absolutely guaranteed to be a P also, no matter what S, P, and x refer to.
The form of an argument therefore can be thought of as the argument's "logical skeleton." Roughly speaking, it is what is left of an argument when we take away the terms that make reference to specific things or kinds of things and replace them with placeholders (like letters). We also leave in place the terms that do the logical work—terms like "all," "some," "and," "not," "or," and "if." We shall give a more precise characterization of logical form in the next chapter, but I think enough has been said to impart a rough, intuitive sense of what logical form is.
Contrast the argument about Jarrod the fireman with the argument about Erin the history PhD:
P-1: All tenured faculty members of the History Department are PhDs in history.
P-2: Erin is a PhD in History.
C: Erin is a tenured faculty member of the History Department.
Here is another argument.
P-1: All humans are mortal.
P-2: Socrates is mortal.
C: Socrates is human.
This argument is invalid. A friend of mine once had a dog she named Socrates. Socrates the dog was mortal, but he was not human. So even if all humans are mortal, and Socrates the dog is mortal, it doesn't follow that Socrates is human. Notice that this argument has the same form as the argument about Erin. The form of both arguments is this:
P-1: All Ss are Ps.
P-2: x is a P.
C: x is an S.
This is not a valid argument form. We cannot say that every valid argument has a valid argument form. The argument about pretzels we considered earlier does not have a valid form, yet it is intuitively valid. What we can say is that every argument that does have a valid argument form is valid, and that is what we shall be interested in.
Once the idea of logical form is grasped, we can talk about substitution instances. A substitution instance of an argument form is just any argument that has that form. For instance, the argument about Jarrod the fireman and Apatosaurus the sauropod dinosaur are both substitution instances for this argument form:
P-1: All Ss are Ps.
P-2: x is an S.
C: x is a P.
Any argument that is a substitution instance of a valid argument form will be a valid argument. In doing proofs, which we shall be doing soon, it is essential to be able to recognize substitution instances of valid argument forms.
(Continues...)
Excerpted from RATIONAL EPISODES by KEITH M. PARSONS Copyright © 2010 by Keith M. Parsons. Excerpted by permission of Prometheus Books. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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