A Brief Logic Primer

 

I. Arguments

 

Logic concerns the relationship between premises and conclusions of arguments. To understand this, we need to understand what arguments are. “Argument” here does not mean fights, squabbles, or even a disagreement. It has a technical meaning in logic, namely:

 

An argument is a set of sentences where one or more (the premises) are intended to provide evidence for, or support for, another (the conclusion).

 

Here, for example, is an argument in this sense:

 

Jason must be at the movies. He was going to either see the movie “Spiderman” this evening with Sarah or stay home and study for his philosophy exam.  But I checked his apartment, and he wasn’t there.

 

The person who is offering this argument (Jason’s roommate Carl, say) is intending to provide a reason to believe that the first sentence is correct, namely, that Jason is at the movies. That claim—the claim for which evidence is offered—is the conclusion. He does this by citing two pieces of information as evidence: 1) that Jason is either home or at the movies; and 2) Jason is not at home. These are the premises. Carl is suggesting that if you accept these two pieces of information then you ought to accept the conclusion, because they give you reason to believe that the conclusion is true. (Notice that he hasn’t presented an argument for the premises themselves, but is just expecting that his audience will accept that they are true. If questioned, he might go on to give further arguments for these premises; but those would be different arguments.)

 

Here is the argument set out in what is called standard form:

 

            1. Jason is either at home studying or at the movies.

            2. Jason is not at home.

            3. Therefore, Jason is at the movies.

 

An argument in standard form is a numbered list of the assertions in an argument, with the premises set out first and the conclusion last. Some things to note:

-The first sentence in the passage above was the conclusion, not the last sentence. So the first sentence is listed last when presented in standard form. In general, the conclusion can appear anywhere in the argument. It might be the first sentence. It might appear in the middle. It might be the last sentence. It might even not be explicitly stated; the arguer might be expecting her audience to draw the obvious conclusion for her. In that case, when we present the argument in standard form, we would present the conclusion explicitly as the last sentence.

-An argument can have any number of premises (1 or more).

-An argument has only one conclusion. If there is an argument presented to give reason to believe that the premise of an argument is true, that is counted as another argument whose conclusion happens to be the same sentence as the premise in the original.

 

II. Evaluating Arguments: Two Considerations

 

When we are trying to determine whether to believe the conclusion of an argument, there are two, very different, questions that we have to ask. The first question concerns the premises themselves. In the argument above, for example, we have to accept that premises 1 and 2 are true. If Jason was in fact considering a third option (going to the bar, say) then premise 1 would be false. If he was studying on the balcony, but Carl didn’t check the balcony and so assumed that Jason wasn’t home, then premise 2 would be false. If either premise is false then we should obviously not rely on the argument as giving us a reason to believe the conclusion. So the first requirement in order for an argument to be successful is that the question:

            (a) Are all the premises true?

must be answered “yes”. (Notice the “all”; even if an argument has a thousand premises, if even one of those is false, and that premise is required in order to give a reason to believe the conclusion, then the argument is not successful.)

But that’s not the only question we need to ask. The premises might be true and still not provide a reason to believe the conclusion. For example, suppose that I substitute the conclusion in the argument above with the conclusion below:

 

            1. Jason is either at home studying or at the movies with Sarah.

            2. Jason is not at home.

            3. Therefore, pigs can fly.

 

Obviously these two premises give no reason to believe that the conclusion is true, even if those premises are true. We need in addition that there be the right kind of relationship between the premises and the conclusion. That relation is that the premises, if true, must give reason to believe that the conclusion is true. (This will be made more precise in a moment.) This relationship is obviously missing here; how things are with Jason lends no reason to believe anything about the aeronautical abilities of swine.

So the second requirement in order for an argument to be successful is that the question:

            (b) Do the premises, if true, give reason to believe that the conclusion is true?

must also be answered “yes.”

We have seen that requirement (a) can be satisfied without satisfying requirement (b): the premises can be true even though they don’t give reason to believe the conclusion. The reverse is true as well: (b) can be satisfied without satisfying (a). That is, it can be the case that the premises, if true, would give reason to believe that the conclusion is true, even though the premises are not, as a matter of fact, true. Here is an example:

 

            1. If pigs fly, then I’m a monkey’s uncle.

            2. Pigs fly.

            3. Therefore, I’m a monkey’s uncle.

 

The premises are false (and the conclusion is false as well). But suppose the premises were actually true—suppose that the ability of pigs to fly somehow means that I had a monkey for a niece or nephew, and suppose that pigs do fly—then we would have to accept the conclusion. When we ask question (b), we are engaging in a kind of imagining game: imagine that the world is such that the premises are true; would that then give you reason to believe that the conclusion is true as well? We can do this even if we have no idea whether the premises are true or false. For example:

 

1. The only extra-terrestrial life forms in the universe live on planets revolving around the star Alpha Centauri.

2. Mars does not revolve around Alpha Centauri.

3. Therefore, there are no extra-terrestrial life forms on Mars.

 

I’m sure you know that 2. is true, since Mars revolves around our sun and our sun is not Alpha Centauri. But you do not know (I bet!) whether 1. is true. But you can imagine that they are true. If they were true, then the conclusion would have to be true as well; the premises provide the right support for the conclusion. So we can determine whether requirement (b) is satisfied without paying any attention to the question whether requirement (a) is satisfied. We can also determine whether requirement (a) is satisfied without paying any attention to the question whether requirement (b) is satisfied; that is, we can investigate the question whether the premises are true or false without at all considering the question whether, supposing they were true, they would give reason to believe that the conclusion is true as well. Requirement (a) and (b) are completely independent issues: whether the premises (or the conclusion) are true or not, and whether the required relationship holds between the premises and the conclusion, are entirely separate matters.

That is what makes logic possible. Logic does not concern itself with the question whether the premises are true; that is, it does not concern itself with requirement (a). It’s zoologists, presumably, who would concern themselves with the aeronautical abilities of pigs, astronomers who are concerned with the question whether Mars revolves around Alpha Centauri, etc. Logic is, instead, only concerned with question (b): how, in general, can we determine whether the premises of arguments support their conclusions? That is, are there any general procedures that we can apply to particular arguments to determine whether the premises, if true, would provide reason to believe that the conclusion is true? Why do the premises support the conclusions of arguments and not others? These questions—questions concerning the evidential relationship between premises and conclusions of arguments—are the province of logic.

So logic does not determine whether you should believe the conclusion of an argument. That depends on the answer to both questions (a) and (b) being “yes”; and logic can only help with the second question. But it does take care of the second question, leaving you with the task of determining the answer to question (a). Question (b) is the question that concerns reasoning, because reasoning is the rational process by which we draw inferences from information—that is, conclusions from premises. So logic concerns the processes of reasoning. And reasoning is different from fact-gathering (requirement (a)). Reasoning is what we do with the facts we have gathered; it determines what we can glean from those facts. Sherlock Holmes does two things in his investigations: he gathers facts (by means of remarkable powers of observation); and he draws conclusions (who is the murderer, say) by reasoning from those facts. When Holmes does the second, he is using logic (“deduction”, as he would say, although that’s not quite right; see below).

 

III. First Relationship: Deductive Arguments

 

            What does it mean to say that the premises, if true, would give reason to believe that the conclusion is true? It turns out that there are two answers to this question, and so two fundamentally different kinds of argument. The first answer is: to say that the premises, if true, would give reason to believe that the conclusion is true is to say that that conclusion must be true if the premises are true; that is, it is impossible for  the premises to be true and the conclusion false. This is the strongest possible relation between the premises and the conclusion; it means that the premises guarantee that the conclusion is true, in the sense that it would be a logical contradiction to assert the premises and deny the conclusion.

Both the first argument concerning Jason’s whereabouts and the argument concerning extra-terrestrial life are examples of this sort of argument. In both cases, if the premises are true, the conclusion must be true as well. Here is another example:

 

1. All cats are mammals.

2. Some cats are black.

3. Therefore, some mammals are black.

 

It is impossible for these premises to be true and the conclusion false. So if the premises are true, then the conclusion must be true; the truth of the premises guarantees the truth of the conclusion. Someone who asserted the premises but denied the conclusion would be contradicting themselves. In this case, the premises are true; and so, since the premises guarantee the truth of the conclusion, we know that the conclusion is true as well. But the premises can guarantee the truth of the conclusion—in the sense that, if the premises are true the conclusion must be true—even if the premises are false. For example:

 

            1. All cats are dogs.

            2. Some cats are nuclear physicists.

            3. Therefore, some dogs are nuclear physicists.

 

The premises (and the conclusion) of this argument are false. But still, if the premises of the argument were true—if all cats are in fact dogs, and if some of those cat-dogs are nuclear physicists—then the conclusion would have to be true as well. The question we are asking is not whether the premises are true (or whether the conclusion is true); it is whether the premises, if true, would mean that the conclusion would have to be true.

Sometimes someone presents an argument where they intend that the premises guarantee the truth of the conclusion, but in fact they’re wrong; even if the premises were true, that would not ensure that the conclusion is true as well. Suppose, for example, someone were to argue as follows:

 

            1. All cats are mammals.

            2. Some mammals are black.

            3. Therefore, some cats are black.

 

This looks very similar to the argument before the last one; but in fact it is very different. The premises are true; and the conclusion is true as well. Nevertheless, it is possible for the premises to be true and the conclusion to be false. Suppose, for example, that no cats are in fact black, but that some dogs are. Then premise 1 would be true; it is still true that all cats are mammals. Premise 2 would be true; since dogs are mammals, and some dogs are black, it would be true that some mammals are black. But the conclusion would be false; no cats would be black. Now that is not how things are; but it is possible. And that means that the premises, if true, do not ensure that the conclusion is true; it is not the case that the conclusion must be true if the premises are true. We know that cats are black. But we could not know this because of this argument. If someone believed that some cats are black because of this argument, they would be wrong to do so, because the premises do not support the conclusion in the required way.

Someone who presents this argument has made a mistake. They think that the premises guarantee the truth of the conclusion. But they’re wrong. They intend to present an argument where the premises guarantee the truth of the conclusion; that is their aim, the standard to which this argument is held. But the argument fails to satisfy that aim, or meet that standard. So there are two features of this argument to notice:

 

      A1. The premises are supposed to, or intended to, guarantee the truth of the conclusion.

      B1. The premises do not, in fact, guarantee the truth of the conclusion.

 

With the previous arguments, however, the following is the case:

 

      A1. The premises are supposed to, or intended to, guarantee the truth of the conclusion.

      B2. The premises do, in fact, guarantee the truth of the conclusion.

 

Notice that these points are different then the further question whether:

 

      C. The premises are true.

 

Logic is not concerned with C at all, although in order to finally assess the argument, we would have to know whether C is true as well.

 

Arguments for which A1 is true are called deductive arguments. Here’s the definition:

 

A deductive argument is one where it is intended that the conclusion must be true if the premises are true.

 

To say that an argument is deductive is to specify what the relationship between the premises and the conclusion is supposed to be; it does not say whether the relationship between the premises is that way. So a deductive argument might be one where the premises do guarantee the truth of the conclusion (B2) or it might be one where the premises do not guarantee the truth of the conclusion (B1). What makes an argument deductive is that it is held to that standard; it is a further question whether it meets that standard.

Of course, someone who presents a deductive argument hopes that their argument will meet that standard. If it does, then it is valid:

 

A deductive is valid if the conclusion must be true if the premises are true; equivalently, if it is impossible for the premises to be true and the conclusion false.

 

Such arguments are called valid deductive arguments (the above is the definition of “valid deductive argument”).

If the argument does not meet that standard (but is intended to, and so is deductive) it is called an invalid deductive argument. Here’s the definition:

 

A deductive argument is invalid if it is possible for the premises to be true and the conclusion false.

 

Notice that the only issue here is whether it is possible for the premises to be true and the conclusion false. As we saw, an argument may have premises and a conclusion that are, as a matter of fact, all true, but where it is possible for the premises to be true and the conclusion false (the argument above whose conclusion was “All cats are black”). That argument was therefore invalid. The question is not whether the premises, or the conclusion is true. The question is instead whether it is possible for the premises to be true and the conclusion false. Those are not the same questions.

So: there are deductive arguments: the premises are intended to guarantee the truth of the conclusion. Some of those are valid: the premises do guarantee the truth of the conclusion. Some are invalid: the premises do not guarantee the truth of the conclusion. We still haven’t asked whether the premises are true. Here are the possibilities:

 

I. Valid deductive argument, premises are all true.

II. Valid deductive argument, at least one premise is false.

III. Invalid deductive argument, premises are all true.

IV. Invalid deductive argument, at least one premise is false.

 

Arguments that fall into category II are not to be relied on, because an argument should not convince you if any of the premises are false (whether only one is false or all of them are). Arguments that fall into category III should not be relied on, because an argument should not convince you if the premises do not support the conclusion—which, for deductive arguments, means that the argument is invalid—even if the premises are true. Arguments that fall into category IV. should not be relied on for both of these reasons. But arguments in category I have true premises and it is impossible for their premises to be true and their conclusions false (because they are valid). Their conclusions must therefore be true as well. These are, of course, of course, the sort of arguments we try to present. These arguments—arguments in category I—are called sound arguments; the arguments in II, III, and IV are all called unsound. To be sound an argument must therefore satisfy the following two conditions:

 

            1. The premises are all true.

            2. The argument is a valid deductive argument.

 

Arguments in II violate condition 1; arguments in III violate condition 2; arguments in III violate both conditions.

 

IV: Second Relationship: Inductive Arguments

 

Not all arguments are deductive. That is, not all arguments are held to the standard that the premises must guarantee that the conclusion is true. Meteorologists gather all kinds of evidence concerning air pressure, the movement of the gulf stream, etc. and draw conclusions concerning what the weather will be tomorrow. Their arguments are quite good; notwithstanding various jokes at the meteorologists’ expense, meteorology is a very sophisticated science, and meteorologists’ predictions are often right. But not always; they do not guarantee accuracy, nor can they. That fact does not, however, mean that we should ignore all the weather reports as just bad reasoning.

The sun has risen every day for millions of years; that gives us very good reason to believe that it will also rise tomorrow. But it is possible that it doesn’t; the sun might explode, the earth might be blown up by aliens, etc. That history nevertheless gives you a very good reason to believe that the sun will be rise tomorrow.

If you hold a brick in your hand four feet from the ground and then let go, it will almost certainly fall to the ground. That is always what has happened before, after all, and that is what the laws of gravity say will happen. But it is possible that it does not. An unexpected gust of wind might push it up; a meteorite might hit it before it has a chance to fall; the laws of gravity might even be wrong, or change, or for some reason not apply to this brick. This is all extremely unlikely, however; the rational thing for you to expect is that the brick will fall.

Such arguments—and many, many others—are not intended to guarantee the truth of their conclusions. That is not the standard to which they are to be held. If it were—if we thought of such arguments as deductive—they would all be counted as invalid, and so as irrational. But that’s not right; these are perfectly good arguments. So not all arguments are deductive; not all arguments are supposed to be valid. There must be another standard.

There is. These arguments are inductive. What they have in common is that the arguer intends that the premises, if true, make it likely, or probable that the conclusion is true; that is, it is unlikely, or improbable, that the premises are true and the conclusion is false.

 

An inductive argument is one where it is intended that the conclusion is probably true if the premises are true.

 

The premises of an argument might make it probable that the conclusion is true even if the premises are true and the conclusion false. The meteorologist might have very good reason to believe that it will rain tomorrow (she might say that there is a “90% chance” of rain tomorrow), but it not in fact rain tomorrow. The meteorologist was still correct; given the information she had, it was very likely that it would rain, even though it didn’t. Inductive arguments are not supposed to guarantee the truth of their conclusions; they are not deductive arguments.

 

To say that an argument is inductive, like saying that an argument is deductive, is to specify the intended relationship between the premises and the conclusion. It is not to say that that is, in fact, the relationship between the premises and the conclusion. Just as some deductive arguments are valid and some are invalid, some inductive arguments are strong and some are weak. Here are the definitions:

 

An inductive argument is strong if the conclusion is probably true if the premises are true; equivalently, it is unlikely for the premises to be true and the conclusion false.

 

An inductive argument is weak if the premises’ being true does not make it likely that the conclusion is true.

 

Here is an example of a strong inductive argument:

 

1. In a randomly chosen sample of 1000 WMU students, 80% grew up in southwestern Michigan.

2. Therefore, approximately 80% of the WMU student population grew up in southwestern Michigan.

 

This argument is strong because the sample is representative of the population (because it is a random sample from that population), and it is a large sample. Here is an example of a weak inductive argument:

 

1. In a randomly chosen sample of 1000 WMU students, 80% are under 30 years old.

2. Therefore, approximately 80% of the population of Kalamazoo is under 30 years old.

 

This argument is weak because the sample is not representative of the population; it is biased. WMU students are far more likely to be under 30 than are people in Kalamazoo in general.

 

Notice that the strong argument is still one where the premises could be false. It is possible, though very unlikely, that the randomly chosen sample just happened to pick out all the students from southwestern Michigan, and that in fact the rest of the population is from elsewhere. Again, that is extremely unlikely; but it is possible. So the premise could be true and the conclusion false. So if we apply the valid/invalid distinction to it, we would have to say that it is invalid, and so a bad argument. Nevertheless, it is a good argument. So we should not apply the valid/invalid distinction to it; that distinction only applies to deductive arguments, and this is not a deductive argument; it is an inductive argument. Only deductive arguments are valid or invalid; only inductive arguments are strong or weak.

 

So: there are inductive arguments. some are strong inductive arguments; some are weak inductive arguments. All these categories concern only the relationship between the premises and the conclusion. We still haven’t said anything about the truth of the premises. As with deductive arguments, there are four possibilities:

I. Strong inductive argument, premises are all true.

II. Strong inductive argument, at least one premise is false.

III. Weak inductive argument, premises are all true.

IV. Weak inductive argument, at least one premise is false.

 

Arguments that fall into category II are not to be relied on, because an argument should not convince you if any of the premises are false (whether only one is false or all of them are). Arguments that fall into category III should not be relied on, because an argument should not convince you if the premises do not support the conclusion—which, for inductive arguments, means that the argument is weak—even if the premises are true. Arguments that fall into category IV. should not be relied on for both of these reasons. But arguments in category I have true premises and it is unlikely for their premises to be true and their conclusions false (because they are strong). Their conclusions is therefore probably true. These are, of course, of course, the sort of arguments we try to present. These arguments—arguments in category I—are called cogent arguments; the arguments in II, III, and IV are all called uncogent. To be cogent an argument must therefore satisfy the following two conditions:

 

            1. The premises are all true.

            2. The argument is a strong inductive argument.

 

Arguments in II violate condition 1; arguments in III violate condition 2; arguments in IV violate both conditions. Notice that, unlike a sound argument, a cogent argument might still have a false conclusion. What we do know is that the conclusions of all cogent arguments are probably true; but, as we saw, a conclusion can be probably true and still be false.

 

V. Summary

 

Here’s a map of the distinctions:

 

Deductive Arguments

            Valid Deductive Arguments

                        Valid Deductive Arguments with all true premises: sound

                        Valid Deductive Arguments with at least one false premise: unsound

            Invalid Deductive Arguments

                        Invalid Deductive Arguments with all true premises: unsound

                        Invalid Deductive Arguments with at least one false premise: unsound

Inductive Arguments

            Strong Inductive Arguments

                        Strong Inductive Arguments with all true premises: cogent

                        Strong Inductive Arguments with at least one false premise: uncogent

            Weak Inductive Arguments

                        Weak Inductive Arguments with all true premises: uncogent

                        Weak Inductive Arguments with at least one false premise: uncogent

 

Notice that this map does not allow inductive arguments to be either valid or invalid, or inductive arguments to be either strong or weak.

 

VI. Exercises

 

Part I: Identify whether the following arguments are deductive or inductive.

1) Because triangle A is congruent with triangle B, and triangle A is isosceles, it follows that triangle B is isosceles.

2) The plaque on the leaning tower of Pisa says that Galileo performed experiments there with falling objects. So Galileo must have performed those experiments there.

3) The rainfall in Seattle has been more than 15 inches every year for the past thirty years. Therefore, the rainfall next year will probably be more than 15 inches.

4) Since Agatha is the mother of Raquel and the sister of Tom, it follows that Tom is the uncle of Raquel.

5) Since Agatha is the mother of Raquel and the sister of Tom, it follows that Tom is the cousin of Raquel.

6) Paying off terrorists in exchange for money is not a wise policy, since such action will only lead them to take more hostages in the future.

7) Either classical culture originated in Greece, or it originated in Egypt. Classical culture did not originate in Egypt. So it originated in Greece.

8) Most famous movie stars are millionaires. Leonardo Di Caprio is a famous movie star. Therefore, he is probably a millionaire.

9) No ads for toothpaste are works of art. Some Shakespearean sonnets are works of art. Therefore some Shakespearean sonnets are not ads for toothpaste.

10) If you go to WMU you are a university student. You are a university student. Therefore, you go to WMU.

 

Part II: The following arguments are deductive. Determine whether they are valid or invalid. Then determine if they are sound or unsound.

1) Since Moby Dick was written by Shakespeare, and Moby Dick is a science fiction novel, it follows that Shakespeare wrote a science fiction novel.

2) If Bill Clinton was impeached, then there was a scandal during his presidency. There was a scandal during his presidency. Therefore, he was impeached.

3) Since the Spanish American War occurred before the American Civil War, and the American Civil War occurred after the Korean war, it follows that the Spanish American War occurred before the Korean War.

4) Some fruits are green. Some fruits are apples. Therefore, some fruits are green apples.

5) The United States Congress has more members than there are days in the year. Therefore, at least two members of Congress have the same birthday.

 

Part III: The following arguments are inductive. Determine whether they are strong or weak. Then determine if they are cogent or uncogent.

1) The ebb and flow of the tides has been occurring every day for millions of years. But nothing lasts forever. So the tides will probably die out in the next few years.

2) Every map of the United States shows that Canada is next to Hawaii. Therefore, Canada is next to Hawaii.

3) Paleontologists have unearthed the fossilized bones of huge reptiles, which we call dinosaurs. Tests suggest that these creatures roamed the earth more than 50 million years ago. Therefore dinosaurs probably did roam the earth then.

4) Constructing the great Pyramids at Giza required lifting massive stone blocks to great heights. The ancient Egyptians probably used some sort of antigravity device to do this.

5) Most professors at WMU live in Moscow, Russia. Marc lives in Moscow, Russia. Therefore Marc is a professor at WMU.