Argument Strength

Any collection of statements with a conclusion is an argument. While the premises of an argument are always intended to support the conclusion, they don't always succeed. In this text, we study two ways that an argument can be successful:

1. The conclusion of the argument follows from the premises.
2. The premises of the argument provide evidence that the conclusion is likely to be true.

When an argument has a conclusion that follows from the premises, it can direct a reasoner to a certain conclusion. If you are certain that your keys are either in your office or locked in your car, and you learn that your keys are not in your office, then you can be certain that they are locked in your car. But often our reasoning does not lead us to certain conclusions. Suppose that 7 weeks into the semester, you note that your classmate, Ann, always brought her laptop to the lecture. It is natural to infer that she will bring her laptop to the next class. But you can't be certain of this conclusion (even though you are certain that she broght her laptop to all of the past lectures). After all, she might forget her laptop, or she may be sick and not come to class. Nonetheless, observing that Ann brought her laptop to past classes does support the conclusion that she will bring her laptop to the next class. In this text, we will study both ways that an argument can be successful. Deductive logic is the branch of logic that studies what follows with certainty. Inductive logic deals with uncertainty and how much one statement supports another.

Valid Arguments

Consider the argument where statements 1 and 2 are the premises and 3 is the conclusion:

1. Amsterdam is in The Netherlands.
2. Paris is in France.
3. College Park is in Maryland.

Even though all three statements are true, it is not a good argument. Indeed, adding a conclusion indicator to statement 3 sounds strange: "Amsterdam is in The Netherlands. Paris is in France. So, College Park is in Maryland." The reason it sounds strange is best understood by considering the form of the argument. The argument consists of three distinct statements, so its form is

$$X, Y \Rightarrow Z.$$

It is not hard to find an instance of this abstract argument in which the premises are both true and the conclusion is false (e.g., let $X$ be "Amsterdam is in The Netherlands", let $Y$ be "Paris is in France" and let $Z$ be "The moon is made of cheese"). Contrast the above argument with:

1. Either my keys are in my office or locked in my car
2. My keys are not in my office
3. So, my keys are locked in my car.

In this argument, the conclusion (statement 3) follows from the premises. While it is possible that the last statement is false (i.e., that my keys may not be locked in the car), the crucial point is that statement 3 cannot be false while both premises are true. While both arguments involve three statements (so are both instances of the above abstract argument), the key difference between them is that the second argument is an instance of an argument involving two basic statements. The following is an equivalent way of expressing the second argument with the two basic statements that make up the argument highlighted.

1. My keys are in my office or my keys are locked in my car.
2. It is not the case that my keys are in my office .
3. So, my keys are locked in my car.

Highlighting the two basic statements makes it clear that the second argument is an instance of the abstract argument discussed in the previous section:

$$X \text{ or } Y, \text{not }X \Rightarrow Y.$$

The key observation is that there are no instances of this abstract argument in which both premises are true and the conclusion is false. While it is of course possible that an isntance of $Y$ may be a false statement (i.e., that my keys may not be locked in the car), the crucial point is that the conclusion cannot be false while both premises are true (i.e., it cannot be true that my keys are either locked in my car or in my office, my keys are not in my office, and my keys are not locked in my car).

Valid Argument (Informally)

An argument is valid if it has a form that would make it impossible for the premises to be true and the conclusion false.

There are two important things to remember about this definition of validity. The first concerns the meaning of impossible. There are different ways that we use the word "impossible":

1. "It is impossible for an object to travel faster than the speed of light." or "It is impossible for me to hold my breath for longer than 15 minutes." These are physical impossibilities that depend on physical and/or biological facts.
2. "It is impossible that $2+2=5$." This is an arithmetic impossibility that depends on number theoretic facts, such as the definition of addition.
3. "It is impossible that Bob is both married and a bachelor." The impossibility here depends on the meanings of the properties involved (e.g., "being married" and "being a bachelor").

In each of these cases, the notion of impossibility depends on the content (i.e., on which objects are being talked about, or which properties are involved). For validity, the notion of impossibility depends only on the logical form of the statements involved. For example, any claim of the form "$X$ and not $X$" is (logically) impossible. It does not matter which statement is substituted for $X$ (or which objects $X$ is about). If $X$ is true, then "not $X$" is false and if "not $X$" is true, then $X$ is false. There is no situation in which both $X$ and "not $X$" are both true. Thus suggests the following equivalent definition of validity:

• An argument is valid if there are no counter-examples, where a counter-example to an argument is a situation in which the premises are all true and the conclusion is false.

The second thing to remember about the definition of validity is that a valid argument does not require that all of the premises are true or even that the conclusion is true. For example, the following argument is valid:

UMD has a campus in College Park or it has a campus on the Moon. UMD does not have a campus in College Park. So, UMD has a campus on the Moon.

Obviously, the conclusion is false. Yet this is a valid argument since it is an instance of abstract argument identified above:

$$X \text{ or } Y, \text{not }X \Rightarrow Y.$$

The crucial point is that since the conclusion is false, one of the premises must also be false (e.g., "UMD does not have a campus in College Park" is false). So, once an argument is classified as valid, there is still a question about whether the premises are true.

Sound Argument

An argument is sound when it is valid and all the premises are true.

To summarize: An invalid argument may have true or false premises with a true or false conclusion. A valid argument may have false premises with either a true or a false conclusion. The only combination that is ruled out is a valid argument with true premises and a false conclusion. Sound arguments always have true conclusions.

Putting everything together, there are two ways that you can use a valid argument:

1. Infer that the conclusion is true: If all of the premises are true, then the conclusion is also true.
2. Refute one of the premises: If the conclusion is false, then some premise is also false.

Inference

Consider the following simple version of a Sudoku puzzle.

1
	3

The Sudoko rule is that each column and each row must contain all the digits 1, 2 and 3. Based on this simple rule, the following inference determines the digit that can be added to the blue cell:

1. The blue cell contains either 1, 2 or 3.
2. The blue cell cannot contain 1.
   (Since there is a 1 in the row containing the blue cell).
3. The blue cell cannot contain 3.
   (Since there is a 3 in the column containing the blue cell).
4. So, the blue cell must contain 2.

This inference can be described schematically as follows:

$1$ or $2$ or $3$, not-$1$, not-$3$ $\Rightarrow 2$

In the above expression, "$1$" means "the cell contains the number $1$" (similarly for "$2$" and "$3$"). So, for instance, "$1$ or $2$ or $3$" is the statement "the cell contains $1$ or the cell contains $2$ or the cell contains $3$". The remaining cells can be filled in using the above inference and/or one of the following inferences:

$1$ or $2$ or $3$, not $1$, not $2$ $\Rightarrow 3$
$1$ or $2$ or $3$, not $2$, not $3$ $\Rightarrow 1$

The completed puzzle is:

1	2	3
3	1	2
2	3	1

Challenge

Try solving a more complicated Sudoku puzzle. Are there any other inference patterns that you used when solving the puzzle?

Challenge

Think of other examples of puzzles/games where similar pattern of inferences are applicable (e.g., tic-tac-toe or chess).

Information Updates

Suppose that Ann, Bob and Carla are at a restaurant having dinner. A waiter takes their drink order: Ann orders red wine, Bob orders white wine, and Carla orders a beer. A different waiter returns with a red wine, a white wine and a beer. This waiter is uncertain who ordered which drink, so he engages in the following dialogue. The waiter first asks "who ordered the red wine?" with Ann responding that she did. The waiter then asks a second question "who ordered the white wine?" with Bob responding that he did. Now the waiter does not have to ask a third question since he can infer that Carla must have ordered the beer. This inference fits the same pattern of reasoning identified above:

1. Carla ordered red wine, white wine or beer.
2. Carla did not order red wine.
3. Carla did not order white wine.
4. So, Carla ordered beer.

This argument is an instance of the above abstract argument ($R$ means "Clara ordered red wine", $W$ means "Clara ordered white wine", and $B$ means "Clara ordered beer"):

$R$ or $W$ or $B$, not $R$, not $W\ \Rightarrow\ B$

There is another perspective on this reasoning that focuses on the waiter's initial uncertainty and how that uncertainty is updated in response to new information. The first step is to represent the waiter's uncertainty. Initially, the waiter entertains different possible assignments of drinks to the three customers. Each possibility can be represented as a sequence of "$R$" (red wine), "$B$" (beer), and "$W$" (white wine). The first element of the sequence represents Ann's order, the second represents Bob's order and the last element represents Carla's order. For instance, the sequence $B\ R\ W$ means "Ann ordered the beer, Bob ordered the red wine and Carla ordered the white wine". Initially, the waiter entertains the following 6 possibilities (with the actual situation highlighted in gray):

The waiter asks two questions. Each answer reduces the waiter's uncertainty. After the first question, the waiter learns that Ann ordered red wine. This means that all situations except $R\ W\ B$ and $R\ B\ W$ are no longer possible. Then, after the second question, the waiter learns that Bob ordered white wine, so $R\ B\ W$ is no longer a possibility. This explains why the waiter does not need to ask a third question: After learning the answer to the two questions, all of the waiter's uncertainty is removed, and so, the only remaining possibility is that Carla ordered the beer.

Lecture

Slides

General Comments

The examples discussed above involve both inferring and noting when a statement follows from other statements. While these two concepts are related, there is an important difference. Inferring is an activity that a person or computer performs, whereas follows from is a relationship between propositions. It is one thing to note that a conclusion follows from some premises, but it is another thing to actually infer the conclusion from the premises. To illustrate, consider the following argument:

Every student at Maryland thinks PHIL 171 is the best Philosophy course. I am a student at Maryland. So, I think PHIL 171 is the best Philosophy course.

The last statement "I think PHIL 171 is the best Philosophy course" does indeed follow from the previous two statements. Of course, you can acknowledge this without actually inferring that you think PHIL 171 is the best Philosophy course. Indeed, you or someone you know may be evidence that the first statement is false.

Practice Questions

1. True or False: If an argument has a false conclusion, then it is invalid.
   True

   False

   ​

2. True or False: All sound arguments are valid.
   True

   False

   ​

3. True or False: If an argument has true premises and a true conclusion, then it is sound.
   True

   False

   ​

4. True or False: If the conclusion of a valid argument is true, then all of the premises must be true as well.
   True

   False

   ​

5. True or False: If the conclusion of a valid argument is false, then at least one of its premises is false.
   True

   False

   ​

6. Consider the following two arguments:

   i. If Lauren lived in Amsterdam, then Lauren speaks Dutch. Lauren did not live in Amsterdam. So, Lauren does not speak Dutch. ii. If Lauren lived in Amsterdam, then Lauren speaks Dutch. Lauren does not speak Dutch. So, Lauren did not live in Amsterdam.

   Which argument is valid?

   Only argument i. is valid.

   Only argument ii. is valid.

   Both arguments are valid.

   Neither argument is valid.

   ​

   Argument i. is not valid and argument ii. is valid. Argument i. can be criticized without knowing anything about where Lauren lived or what languages are spoken in Amsterdam. The key observation is that there are situations in which both premises are true while the conclusion is false. That is, one can imagine a situation in which it is true that if Lauren lived in Amsterdam, then she speaks Dutch, Lauren did not live in Amsterdam, and that Lauren does speak Dutch. Perhaps Lauren learned Dutch in school or lived in another Dutch city, such as Groningen. Argument ii. does not suffer from the same problem as argument i.: There is no way that both premises can be true while the conclusion is false. That is, it is not possible that Lauren lived in Amsterdam, does not speak Dutch and the statement "if she lived in Amsterdam, then she speaks Dutch" is true.

7. Can you find a valid argument with a false premise and a true conclusion?

8. Can you find a valid argument with a false premise and a false conclusion?

9. Can you find a valid argument in which all the premises are true and the conclusion is false?

10. Can you find a invalid argument with a true premises and a true conclusion?

11. Can you find an invalid argument with a true premises and a false conclusion?
12. Challenge

    Consider the following arguments:

    Argument 1
    John saw a man on the hill with a telescope.
    So, the man on the hill has a telescope.

    Argument 2
    Ann bumped into a man with an umbrella.
    So, Ann was holding an umbrella.

    Are these "good" arguments? That is, do the premises support the conclusion?