Just what do we mean by logical consequence? Or rather, since this phrase is sometimes used in quite different contexts, what does a logician mean by logical consequence?
A few examples will help. First, let’s say that an argument is any series arguments, premises, and conclusions of statements in which one (called the conclusion) is meant to follow from, or
be supported by, the others (called the premises). Don’t think of two people arguing back and forth, but of one person trying to convince another of some conclusion on the basis of mutually accepted premises. Arguments in our sense may appear as part of the more disagreeable sort of “arguments”—
the kind parents have with their children—but our arguments also appear
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in newspaper editorials, in scholarly books, and in all forms of scientific and rational discourse. Name calling doesn’t count.
There are many devices in ordinary language for indicating premises and conclusions of arguments. Words like hence, thus, so, and consequently are identifying premises
and conclusions used to indicate that what follows is the conclusion of an argument. The words because, since, after all, and the like are generally used to indicate premises.
Here are a couple of examples of arguments:
All men are mortal. Socrates is a man. So, Socrates is mortal.
Lucretius is a man. After all, Lucretius is mortal and all men are mortal.
One difference between these two arguments is the placement of the con-clusion. In the first argument, the conclusion comes at the end, while in the second, it comes at the start. This is indicated by the words so and after all, respectively. A more important difference is that the first argument is good, while the second is bad. We will say that the first argument is logically valid, or that its conclusion is a logical consequence of its premises. The reason we logical consequence
say this is that it is impossible for this conclusion to be false if the premises are true. In contrast, our second conclusion might be false (suppose Lucretius is my pet goldfish), even though the premises are true (goldfish are notoriously mortal). The second conclusion is not a logical consequence of its premises.
Roughly speaking, an argument is logically valid if and only if the conclu-logically valid
arguments sion must be true on the assumption that the premises are true. Notice that this does not mean that an argument’s premises have to be true in order for it to be valid. When we give arguments, we naturally intend the premises to be true, but sometimes we’re wrong about that. We’ll say more about this possi-bility in a minute. In the meantime, note that our first example above would be a valid argument even if it turned out that we were mistaken about one of the premises, say if Socrates turned out to be a robot rather than a man.
It would still be impossible for the premises to be true and the conclusion false. In that eventuality, we would still say that the argument was logically valid, but since it had a false premise, we would not be guaranteed that the conclusion was true. It would be a valid argument with a false premise.
Here is another example of a valid argument, this time one expressed in the blocks language. Suppose we are told that Cube(c) and that c = b. Then it certainly follows that Cube(b). Why? Because there is no possible way for the premises to be true—for c to be a cube and for c to be the very same object as b—without the conclusion being true as well. Note that we can recognize that the last statement is a consequence of the first two without knowing that
Valid and sound arguments/ 43
the premises are actually, as a matter of fact, true. For the crucial observation is that if the premises are true, then the conclusion must also be true.
A valid argument is one that guarantees the truth of its conclusion on the assumption that the premises are true. Now, as we said before, when we actually present arguments, we want them to be more than just valid: we also want the premises to be true. If an argument is valid and the premises are also
true, then the argument is said to be sound. Thus a sound argument insures sound arguments the truth of its conclusion. The argument about Socrates given above was not
only valid, it was sound, since its premises were true. (He was not, contrary to rumors, a robot.) But here is an example of a valid argument that is not sound:
All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor.
The reason this argument is unsound is that its first premise is false.
Because of this, although the argument is indeed valid, we are not assured that the conclusion is true. It may be, but then again it may not. We in fact think that Brad Pitt is a good actor, but the present argument does not show this.
Logic focuses, for the most part, on the validity of arguments, rather than their soundness. There is a simple reason for this. The truth of an argument’s premises is generally an issue that is none of the logician’s business: the truth of “Socrates is a man” is something historians had to ascertain; the falsity of
“All rich actors are good actors” is something a movie critic might weigh in about. What logicians can tell you is how to reason correctly, given what you know or believe to be true. Making sure that the premises of your arguments are true is something that, by and large, we leave up to you.
In this book, we often use a special format to display arguments, which we
call “Fitch format” after the logician Frederic Fitch. The format makes clear Fitch format which sentences are premises and which is the conclusion. In Fitch format, we
would display the above, unsound argument like this:
All rich actors are good actors.
Brad Pitt is a rich actor.
Brad Pitt is a good actor.
Here, the sentences above the short, horizontal line are the premises, and the sentence below the line is the conclusion. We call the horizontal line the
Fitch bar. Notice that we have omitted the words “So . . . must be . . .” in the Fitch bar conclusion, because they were in the original only to make clear which
sen-tence was supposed to be the conclusion of the argument. In our conventional
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format, the Fitch bar gives us this information, and so these words are no longer needed.
Remember
1. An argument is a series of statements in which one, called the conclu-sion, is meant to be a consequence of the others, called the premises.
2. An argument is valid if the conclusion must be true in any circum-stance in which the premises are true. We say that the conclusion of a logically valid argument is a logical consequence of its premises.
3. An argument is sound if it is valid and the premises are all true.
Exercises
2.1
➶|✎
(Classifying arguments) Open the file Socrates’ Sentences. This file contains eight arguments separated by dashed lines, with the premises and conclusion of each labeled.
1. In the first column of the following table, classify each of these arguments as valid or invalid. In making these assessments, you may presuppose any general features of the worlds that can be built in Tarski’s World (for example, that two blocks cannot occupy the same square on the grid).
Sound in Sound in
Argument Valid? Socrates’ World? Wittgenstein’s World?
1.
2.
3.
4.
5.
6.
7.
8.
2. Now open Socrates’ World and evaluate each sentence. Use the results of your evaluation to enter sound or unsound in each row of the second column in the table, depending on whether the argument is sound or unsound in this world. (Remember that only valid arguments can be sound; invalid arguments are automatically unsound.)
Valid and sound arguments/ 45
3. Open Wittgenstein’s World and fill in the third column of the table.
4. For each argument that you have marked invalid in the table, construct a world in which the argument’s premises are all true but the conclusion is false. Submit the world as World 2.1.x, where x is the number of the argument. (If you have trouble doing this, you may want to rethink your assessment of the argument’s validity.) Turn in your completed table to your instructor.
This problem makes a very important point, one that students of logic sometimes forget. The point is that the validity of an argument depends only on the argument, not on facts about the specific world the statements are about. The soundness of an argument, on the other hand, depends on both the argument and the world.
2.2
✎
(Classifying arguments) For each of the arguments below, identify the premises and conclusion by putting the argument into Fitch format. Then say whether the argument is valid. For the first five arguments, also give your opinion about whether they are sound. (Remember that only valid arguments can be sound.) If your assessment of an argument depends on particular interpretations of the predicates, explain these dependencies.
1. Anyone who wins an academy award is famous. Meryl Streep won an academy award.
Hence, Meryl Streep is famous.
2. Harrison Ford is not famous. After all, actors who win academy awards are famous, and he has never won one.
3. The right to bear arms is the most important freedom. Charlton Heston said so, and he’s never wrong.
4. Al Gore must be dishonest. After all, he’s a politician and hardly any politicians are honest.
5. Mark Twain lived in Hannibal, Missouri, since Sam Clemens was born there, and Mark Twain is Sam Clemens.
6. No one under 21 bought beer here last night, officer. Geez, we were closed, so no one bought anything last night.
7. Claire must live on the same street as Laura, since she lives on the same street as Max and he and Laura live on the same street.
2.3
✎
For each of the arguments below, identify the premises and conclusion by putting the argument into Fitch format, and state whether the argument is valid. If your assessment of an argument depends on particular interpretations of the predicates, explain these dependencies.
1. Many of the students in the film class attend film screenings. Consequently, there must be many students in the film class.
2. There are few students in the film class, but many of them attend the film screenings.
So there are many students in the film class.
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3. There are many students in the film class. After all, many students attend film screen-ings and only students in the film class attend screenscreen-ings.
4. There are thirty students in my logic class. Some of the students turned in their homework on time. Most of the students went to the all-night party. So some student who went to the party managed to turn in the homework on time.
5. There are thirty students in my logic class. Some student who went to the all-night party must have turned in the homework on time. Some of the students turned in their homework on time, and they all went to the party.
6. There are thirty students in my logic class. Most of the students turned in their home-work on time. Most of the students went to the all-night party. Thus, some student who went to the party turned in the homework on time.
2.4
✎
(Validity and truth) Can a valid argument have false premises and a false conclusion? False premises and a true conclusion? True premises and a false conclusion? True premises and a true conclusion? If you answer yes to any of these, give an example of such an argument. If your answer is no, explain why.
Section 2.2