The purpose of this handout is to define and exemplify a number of technical terms which will be used throughout the course. Read it carefully: this terminology will be assumed for the remainder of the course. By way of encouragement, it is a fact that most students find this material--the subject matter of logic and statistics--to be extremely challenging. For some, this will be the most difficult material encountered during the semester. What is wanted, and what you should focus upon, is a rough grasp of how the various terms are used by philosophers. As always, if you feel entirely helpless after lecture and after reading this handout several times over, contact me for additional discussion.
Philosophers use the word `argument' in a special way. Whereas most people would say that an argument is a dispute between two people--as can be seen everyday on talk-shows--philosophers say that an argument is composed of sentences and always consists of two parts. The first part is a group of statements collectively called the premises of the argument. The second part is a statement called the conclusion of the argument. We say that, in an argument, the conclusion follows from the premises. Or, in other words, the premises provide justification for believing that the conclusion is true. Anyone who presents an argument is basically saying that the conclusion must be accepted on the basis of the truth of the premises; the premises are assumed to b e true or are taken for granted. The whole point of the argument is to argue for, or establish, the truth of the conclusion. The following are two examples of arguments. See if you can distinguish between the premise(es) and the conclusion, or point, of each argument.
1) The light that we see from distant galaxies left them millions of years ago, and in the case of the most distant object that we have seen, the light left some eight thousand million years ago. Thus, when we look at the universe, we are seeing it as it was in the past.
--Stephen W. Hawking, A Brief History of Time: From the Big Bang to Black Holes (Toronto: Bantam Books, 1988), p. 28
2) [M]ales born into poverty are more likely to commit crimes as teenagers and adults than more privileged males. So a boom in births among poor mothers can be expected to exert upward pressure on the crime rate 15 to 20 years later.
--David E. Bloom and Neil G. Bennett, "Future Shock," The New Republic, June 19, 1989, p. 18
In (1), Hawking signals the argument's conclusion by using the word `thus'. In (2), the conclusion is marked by the word `so'. Such terms are called conclusion markers. Other conclusion markers are `therefore', `hence', `consequently', and `It follows t hat': there are many conclusion markers. Similarly, premises are often signaled by the use of premise markers. Examples include `because', `since', `assuming that', `supposing that', `given that', etc.
Now arguments, we have said, can be either good or bad. A good argument is one in which the truth of the conclusion is supported by or follows from the truth of the premises. A bad argument is one in which the truth of the conclusion does not follow fro m the truth of the premises.
Also, we have distinguished between two kinds of argument: inductive arguments and deductive arguments. The difference between the kinds of argument has to do with the connection between premises and conclusion. In a good deductive argument, the conclusion must be true provided that all the premises are true. In a good inductive argument, the conclusion may be true provided that all the premises are true. It is still possible in a good inductive argument for all the premises to be true and the conclusion to be false. It is not possible in a good deductive argument for all the premises to be true and the conclusion to be false. Loosely speaking, the reason for the distinction between inductive and deductive arguments is that the conclusion of an inductive argument contains more information than that contained in the premises, whereas the conclusion of a deductive argument contains less information than that contained in the premises. Thus the conclusion of a good inductive argument is more comprehensive than the premises, and so may be false. Conversely, the conclusion of a good deductive argument is less comprehensive than the premises, and so cannot be false provided that the premises are all true. Consider the distinction between inductive and deductive arguments in light of the following examples:
All whales are mammals
Willy is a whale
Therefore, Willy is a mammal
Crow A is black
Crow B is black
Crow C is black
Crow D is black
Therefore, all crows are black
In the deductive argument, the conclusion is only about a single animal of a certain kind, namely Willy the Whale. So there is not much information in the conclusion when compared to the premises, one of which is about all the animals of a certain kind. In the inductive argument, the conclusion contains much more information than the premises. The premises are each about particular crows, whereas the conclusion is about all crows. The conclusion might be false even though the premises are all true: crow E might be white.
When an inductive argument is a good argument, we say that it is strong. When an inductive argument is a bad argument, we say that it is weak. What determines whether an inductive argument is weak or strong depends upon what kind of inductive argument it is. In the case of a generalization, the more instances one considers in the premises the stronger the argument. Had we examined 10,000 individual crows and they were all black, then we could be much more confident in our conclusion that all crows are black, even though the conclusion might still be false. Deciding when an inductive argument is strong or weak and thus determining how confident we can be in the truth of the conclusion of an inductive argument is the job of mathematical statistics. Distinguishing between good and bad deductive arguments, however, is the logician's task.
When a deductive argument is a good argument, we say that it is valid. When a deductive argument is a bad argument, we say that it is invalid. A deductive argument is valid just in case it is not possible for all the premises to be true and the conclusion to be false. Put another way, a deductive argument is valid just in case IF the premises are all true THEN the conclusion MUST be true. If a deductive argument is valid and we know that all of its premises are true, then we say that the argument is sound. Notice that, by implication, the conclusion of a sound argument is always true.
Since only inductive arguments can be strong or weak, and since only deductive arguments can be sound, valid, or invalid, we drop the phrase "strong inductive argument" in favor of the simpler phrase "strong argument". Similarly, "weak inductive argument" becomes "weak argument", "sound deductive argument" becomes "sound argument", "valid deductive argument" becomes "valid argument", and finally "invalid deductive argument" becomes "invalid argument." Thus if we say some particular argument is valid, we k now two things: first, that the argument is a deductive argument; and second, that it is not possible for the argument to have all true premises and a false conclusion simultaneously.
We have a great deal of terminology to keep straight with respect to deductive arguments. It is easy to confuse the notions of truth, validity, and soundness. To help you, first recall that one may say that a sentence is true (or false) but never an argument. Truth (or falsity) is always and only a property of sentences. On the other hand, one may say that an argument is valid but not a sentence. Validity is a property of arguments, and, as we discussed above, it is a property in particular of deductive arguments. Second, remember what validity concerns. Just as truth is understood in terms of a relation between a sentence and the world (or a fact of the world), validity is understood in terms of a logical relation between a group of sentences and a sentence. And third, only a deductive argument that legitimately shows its conclusion to be true is a deductive argument that is both valid and has premises that are true (i.e., a sound argument). Saying that an argument is sound means that you know two things: it is valid and has true premises. If such is the case, then we know that the conclusion itself must be true.
To test yourselves on these issues, try to determine whether the following statements about deductive arguments are true or false:
1) If the conclusion of an argument is true, then it must be sound.
2) If all the premises of an argument are true, then it must be valid.
3) An argument is valid when and only when its premises are true and its conclusion is true.
4) A sound argument is both valid and has all true premises.
5) A valid argument can have a false conclusion.
6) A sound argument can have a false premise.
7) A sound argument can have a false conclusion.
8) A valid argument with a false premise can have a true conclusion.
9) A valid argument can have all true premises and a false conclusion.
10) A valid argument can have all false premises and a false conclusion.
In responding to these statements, let us consider variations on the following argument:
If Cicero writes the Philipics, then Antony kills him.
Cicero writes the Philipics.
Therefore, Antony kills him.
As stated, the argument is sound. That is to say, it is valid and has all true premises. The Roman orator Cicero wrote the Philipics, an essay intended to expose Antony's excesses. Antony threatened to and did, in fact, kill Cicero for having written the Philipics.
The first of the ten statements must be false. In order to be sound, an argument must be valid and have all true premises. When this case obtains, we will have a true conclusion. But it would be incorrect to say that, given a true conclusion, the argument must be sound. Consider the following argument:
If Cicero writes the Philipics, then Antony kills him.
Antony kills him.
Therefore, Cicero writes the Philipics.
Notice that the argument has a true conclusion. But it is invalid, and hence not sound. It is invalid precisely because it is possible for the premises to be true and the conclusion false. Perhaps Antony accidentally runs over Cicero with a chariot and, whatever their previous squabbles, Cicero is killed before he can write the Philipics.
Statement 2 must be false as well. An argument's validity is independent of the truth of the premises. Some invalid arguments can certainly have true premises. Consider the previous argument as an example. The premises are both true, and the conclusion is in fact true. But the argument itself is invalid. It does not follow that Cicero must have written the Philipics just because Antony threatened to and did, in fact, kill him. Cicero need not have written the Philipics to be killed by Antony.
The third statement is also false. Referring to the above example, you should notice that each premise is true and conclusion is true. The argument is, nonetheless, invalid. Thus it is possible for an invalid argument to have true premises and a true conclusion. Again, the validity of an argument is a separate issue from the actual truth or falsity of the premises and conclusion.
Statement 4 is, of course, true in virtue of the definition of a sound argument.
The fifth statement is true provided that the argument has at least one false premise. Reflect on this argument:
If Cicero writes the Philipics, then Antony praises him.
Cicero writes the Philipics.
Therefore, Antony praises him.
It is certainly valid but has a false conclusion. Notice, though, that the argument must also have a false premise. If it didn't, it wouldn't be valid. We know without doubt that an argument is invalid when the premises are all true and the conclusion is false. This is our test for invalidity.
Conversely, the tests for validity are complicated. All we know so far is that, in order for an argument to be valid, its conclusion must be true given that the premises are all true. But how, you may be wondering, can we talk about truth and falsity in such a freewheeling way and still maintain that validity is something independent of truth? Put another way, aren't we defining validity in terms of truth and falsity? The point to appreciate is that the truth we're talking about is not actual truth. M ore explicitly, we are saying that we will temporarily assume the truth of the premises to see if their being true makes the conclusion true. Whether the premises are true is a matter for the historian or the scientist. We are interested here in instilling correct intuitions about validity, truth, and soundness. What you need to gather from this handout more than anything else are these intuitions.
Proceeding, then, to statement 6, we recall from the definition of soundness that the argument must be both valid and have all true premises. Hence, this statement is false.
Similarly, statement 7 is false. Validity guarantees that, if the premises are true, the conclusion must also be true. Sitting at home, we check the validity of an argument. We then go out into the real world, discover that the premises are in fact true, and claim that the argument is sound. This is also just to say that we've checked the world correctly (i.e., that we know for certain that each of our premises is true) and that we've checked validity correctly.
Surprisingly, statement 8 is true. A valid argument with a false premise does not guarantee that the conclusion will be true, nor does it guarantee that the conclusion will be false. Consider the following example:
If Cicero doesn't write the Philipics, then Antony kills him.
Cicero doesn't write the Philipics.
Therefore, Antony kills him.
Statement 9 is utterly false. This is actually the one case in which we know that the argument is invalid.
Statement 10, on the other hand, is true. Reflect on the following argument:
If Antony praises Cicero, Cicero does not write the Philipics.
Antony praises Cicero.
Therefore, Cicero does not write the Philipics.
This is surely a valid argument. But each premise is false, as is the conclusion. In summary, then, a valid argument can have false premises and a false conclusion; false premises and a true conclusion; or true premises, in which case it must have a true conclusion and is said to be sound. The one case we are excluding is the one in which the premises are all true and the conclusion is false.
Finally, compare your understanding of all these special terms with the following Glossary. You are encouraged to use this as a reference throughout the semester.
A complex of statements--one of which, the conclusion, is intended to follow from the rest of the statements, called the premises.
The sentence in an argument whose truth is thought to follow from or be established by the truth of the premises.
An argument whose conclusion is intended to follow with absolute certainty from the premises.
An argument whose conclusion is intended to follow with some probability, and not with certainty, from the premises.
A sentence in an argument that is assumed to be true for the sake of the argument. It forms the basis of the argument.
A deductive argument is said to be sound when it is both valid and has, in fact, all true premises.
An inductive argument is said to be strong when its conclusion can be asserted with a fair degree of certainty.
A deductive argument is said to be valid when it cannot have all true premises and a false conclusion simultaneously. If the premises are all true, the conclusion must be true. If the conclusion is false, at least one of the premises must be false.
An inductive argument is said to be weak when its conclusion cannot be asserted with a fair degree of certainty.
Adapted from "Introduction to Logic: A Study Guide", Don Berkich and Keith Coleman, University of Kansas Press 1991.