In the previous lecture, I mentioned deductive and inductive arguments. Here we will be focusing on deductive arguments. Recall that deductive arguments are arguments whose conclusions are certain. That is, the premises do not provide evidence for the conclusion, but they show that the conclusion must be true.

But before we get directly into deduction, I should say a bit more about recognizing arguments in general. From last lecture, you should have a good idea what an argument is, and a decent idea of what an argument is not.

Recognizing Arguments

Most of us realize that a photograph in an ad is not itself an argument for anything. But other times it's more difficult to recognize an argument. Furthermore, arguments are hard to recognize because they are surrounded by irrelevant information like value judgments or digressions.

The first task is to find the conclusion. Remember conclusion indicators? If you see a conclusion indicator, it's most likely the conclusion (but remember that not every argument has a conclusion indicator). Next you find the premises: the reasons that support the conclusion. Then you figure out the way the premises are related to the conclusion. In the text, the authors give a method for diagramming arguments. I am not going to ask you to do this specifically, but feel free to try it if you think it will help.

The goal is to get to the point where you are able to look at a few sentences or a paragraph and determine whether it's an argument. One of the best ways to do this is to see if there is a controversial claim; if there is a controversial claim, it's most likely an argument. For example "the attacks against the US on 9/11 were orchestrated by the US government" is a controversial claim. If someone makes this claim, they better have premises to support it. But if someone says "the buildings fell because they were 50 years old and dilapidated" this is just an explanation. There is no controversial claim; the speaker is explaining why the buildings fell.

Now, on to deduction.

Deduction (The Prisoner and the Hats)

The following is an exercise for understanding deductive arguments. Try to follow the best you can. Imagine this scenario: Three prisoners are in a cell: call them prisoners X, Y, Z. They are friends. There is also a prison guard they talk to from time to time. X can see, Y is blind in one eye, and Z is completely blind. The prison guard brings in a basket along with 5 hats: 2 are red, 3 are white. The guard blindfolds the prisoners, and puts 3 of the 5 hats on their heads, and the other two in the basket. He tells the prisoners that he’ll let one go free if they can figure out what color hat is on their head with certainty once the blindfold is taken off. This means that it cannot be a guess, even a good guess, or he’ll execute them. Even when the prisoners are un-blindfolded, they cannot see the color of their own hats.

• Prisoner X is asked first if he knows with certainty the color of his hat. He says "no."
• Prisoner Y is asked the same. He says "no."
• Prisoner Z is asked the same and he, with certainty, says “White.”

How does he know? Before reading on, take a moment to reflect on how prisoner Z, the completely blind one, can know this.

Here's why Z knows. X must not see two red hats on the heads of Y and Z. If he did, he’d know he had a white hat since there are only two red hats, and he would go free. Y knows that X did not see 2 red hats. Thus, he knows that at least 1 of the 2 hats on his and Z’s head is white. If both are not red, at least 1 is white. Thus, if Y sees a red hat on Z’s head, he knows his must be white. But he could not have seen a red hat since he answered no. Since Y answered no, Z knows that his hat must be white. If it was red, Y would have answered yes. Despite the fact that he is blind, Z knows with certainty that his hat is white.

This is an example of deductive knowledge. Deductive knowledge means that, based on certain information, we know for certain that something is true. For example, we know that if there are 5 hats total and 3 are on the heads of the prisoners, then there must be 2 left over in the basket. Just as the blind man knows his hat is white.

Still, we make certain assumptions about this situation. We assume that the prison guard is not lying and saying there are two red hats when there are three. We also assume that X’s and Y’s answers are intelligent answers based on the information. But, assuming that these assumptions are true, then certain other things follow: this is deductive knowledge.

Computers run on logical rules and premises. But if the humans put in false information, then the computer could produce imperfect results. Let’s say that someone is using a computer to model the atmosphere. If he enters the incorrect wind velocity at some altitude, the computer, although running perfectly according to the rules of logic, is going to produce a false result. Computers, in fact, are logic machines.

Like the blind man the computer “assumes” certain things to be true. Don't worry if you don't entirely understand the prisoners and the hats example yet. For now, just understand that deductive arguments attempt to provide conclusions that follow from given information, conclusions that are certain.

Sometimes games help. Try out this free online version of the game Mastermind. It's a game of deduction. With each move you gain more information that leads you to a certain solution. For the rules, see the Mastermind wikipedia page.

Going Further With Deductive Arguments

Again, to drive home the point, deductive arguments aim to provide certainty to their conclusions. However, even if a deductive argument aims to do this, it sometimes fails. A deductive argument that successfully does this it is said to be valid. If a deductive argument aims to do this, but fails, it is invalid (notice that an invalid argument is inductive) . Below is a technical definition of valid.

Valid: a deductive argument where it is impossible for the premises to be true while the conclusion is false. In the box below are some examples of valid and invalid arguments.

In a deductive argument, we assume that the premises are true, even if they are obviously not true. This is to test whether or not it is valid. Note that in a valid argument, the premises do not actually have to be true.

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Valid:

1) All bachelors are unmarried.
2) Jose is a bachelor.
3) Thus, Jose is unmarried.

Notice that if Jose is a bachelor he must be unmarried. The truth of the conclusion is guaranteed by the truth of the premises. The conclusion follows logically from the premises.

Valid:

1) All men are fathers.
2) Clark is a man.
3) Thus, Clark is a father.

Notice that in this one the premises are obviously false. But the point of this example is to show that, with validity, it's not about the actual truth or falsity of the premises. We assume that the premises are true to see whether or not the argument is valid, to see whether the conclusion follows logically from the premises. Again, even if the premises are not true, we assume they are true. If it's true that all men are fathers, then if Clark is a man, he must be a father.

Invalid:

1) In order for my car to start, it must have at least some gasoline.
2) My car has at least some gasoline.
3) Thus, my car will start.

Clearly, there are many other things that could happen to my car to prevent it from starting besides lack of gasoline. A lack of gasoline is one thing that can prevent my car from starting, but not the only thing. It is possible for the conclusion to be false based on the premises, thus the argument is invalid and not valid.

Invalid:

1) If it’s raining, the streets are wet.
2) The streets are wet.
3) Thus, it’s raining.

Again, it’s possible that the streets got wet another way. Maybe someone sprayed them with fire hoses. Or maybe someone set up sprinklers everywhere. There are other ways the streets can get wet besides as a result of rain, thus the truth of the premises does not guarantee the truth of the conclusion.

Another way to say all this is the following. Whether or not an argument is valid is about form. It’s not about content. When assessing an argument for validity, we assume its premises to be true, whether or not they actually are true (I repeat this because it seems counterintuitive, but try to really think about what it means)..

Validity shows us why it's the case that very rational, educated people can disagree. Look at politics, at academia, at religious discussions. Sometimes we think that if only a person were educated enough he or she would see the truth of some position. But this isn’t necessarily the case. Very educated people disagree all the time since truth is independent of validity. Take the following examples of the controversial case of abortion--one argument supports abortion and the other is against abortion. (Can you see the way these two arguments are being used in the video above with Stewart and Huckabee?)

In Support of Abortion:

1) If the Constitution implies that each person has a right to self-determination concerning matters of a person’s physical body, then abortion should be legal.
2) The Constitution does imply this.
3) Thus, abortion should be legal.

Against Abortion:

1) If the Constitution grants every person the right to self-determination, then the involuntary termination of life is wrong.
2) If the involuntary termination of life is wrong, then abortion should not be legal.
3) The Constitution does grant every person the right to self-determination.
4) Thus, abortion should not be legal. 

The previous arguments are both valid. Assuming that the premises are true, then the conclusion follows. The conclusions must be true if the premises are true. So where does that leave us if both arguments are valid, yet they contradict each other? Well, it brings us to soundness, or actual truth. But, as you'll find below, actual truth isn't necessarily a black and white thing.

Quick note: an implicit idea here is that reasoning well does not suggest that one is a moral person. There are very evil, very smart people. Take the paradigm case of Hitler. He was morally reprehensible, but we probably wouldn't accuse him of being dumb or incapable of using critical thinking to carry out his horrific agenda.

Soundness and Truth

We’ve been talking about an argument being valid and how this is independent of truth. You might have been wondering whether there is a method of assessing the actual truth of the premises of a deductive argument. There is! It's called "soundness."

Validity is for many easier to assess since it has to do only with the relation between the premises and not the actual truth of the premises. Assessing truth is a more difficult matter. Before getting directly into soundness, let's consider truth itself for a bit.

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How do we know if something is true? What do we really know? How many of you have left this town, this state, this country? How do you know the other countries exist? Because people tell you? Is a map correct? Is a model of the galaxy and solar system correct? How do you even know that the world is round?

These are questions we don't often ask ourselves, but they're relevant here. For philosophers and even scientists, truth is a difficult matter to assess.

The Flat Earth Society is a group that believes the earth is flat and not round, and that astronauts never reached the moon (if we can take their claims seriously). If you think this group is wrong, how do you know that it's wrong? Because historians say so, because scientists say so? How do you know what anything anyone is telling you is true? These are questions that can drive you crazy, and they are a motivating force behind movies like The Matrix.

Fortunately, there are ways of showing that some claims are more reasonable to believe than others. It is more reasonable to believe that the earth is round than that it is flat, and we'll discuss this in more detail when we get into inductive knowledge in the next lecture.

We're not going to go into detail about how we know certain things are true, but I think it's worth noting that truth isn't as easy a concept to pin down as it might seem. For the purposes of this class, we are going to understand truth in the commonsense, everyday way (this still does not imply that truth is always black and white). As the abortion arguments above illustrate, sometimes the assessment of truth is difficult.

But back to soundness. A sound argument is one which is valid and the premises are all true. In other words, a sound argument has a conclusion that must be true if the premises are true, but also the premises are actually true. Because the truth of claims can be debated, the soundness of arguments can be debated too--just as the truth of various claims are being debated by Stewart and Huckabee.

In the valid arguments in the box above, the first argument about bachelors is sound (the premises are true). In case you're wondering about the second premise, I have a friend named Jose and he is a bachelor, so that's what makes that premise true. The argument about fathers is unsound because the first premise is actually false: all men are not fathers.