Arguments

The word "argument" can be used in different ways:

1. Ann and Bob are having an argument.
2. Ann is advancing the argument that Bob is innocent of the crime.
3. There is one argument in the function $f(x)=x^2$.

This class is focused on the second type of argument. One advances an argument by giving reasons designed to persuade the reader/hearer that a certain claim is correct. The main claim of an argument is called the conclusion. The evidence, or reasons, for accepting a conclusion are called premises. Putting everything together: The main claim, or proposal, is called the conclusion, the evidence for accepting the conclusion are called the premises, and the whole unit, the premises and conclusions taken together, is the argument.

Propositions

Arguments are made out of propositions. A proposition, or statement, is something that can be true or false. Note that some logic/philosophy texts use "claim" rather than "proposition" or "statement". Propositions are expressed using declarative sentences. A sentence is declarative if it makes a statement, that is, if it asserts that something is true/false. The following are examples of declarative sentences:

1. Amsterdam is in The Netherlands.
2. Helsinki is in Norway.
3. Textbooks are free in all of my courses.
4. The Terps beat the Buckeyes in football.
5. Attendance is mandatory in this class.
6. I have taken introduction to logic.

Note that the truth or falsity of sentences 3 - 6 depends on who is uttering the sentence or when the sentence is uttered. For example, sentence 6 is true when uttered by me, but may be false when uttered by someone else. Often the reason that the truth or falsity of a sentence depends on the context is because the sentence contains indexicals. An indexical is a linguistic expression whose reference can shift from context to context. Paradigmatic examples of indexicals are ‘I’, ‘here’, ‘today’, ‘yesterday’, ‘he’, ‘she’, 'they', 'my', and ‘that’.

Not all sentences are declarative. For instance, some sentences issue a command (such sentences are called imperatives):

• Show up to the lectures!

In addition, some sentences asked questions (such sentences are called interrogatives):

• Are you coming to class today?

The above two sentences do not express a proposition: It does not make sense to respond to the above command or the above question with "That's true".

I conclude with two final points that stress the importance of distinguishing propositions and declarative sentences. The first point is that many declarative sentences can express the same proposition:

• I have taken logic before.
• I took logic.
• This is not the first time I have taken logic.

All of these sentences mean the same thing. That is, they express the same proposition.

The second point is that some sentences are ambiguous: They may express more than one proposition. To illustrate, consider the following sentences:

1. I saw her duck.
2. Students hate annoying professors.
3. Lily, your iPad is on the table.

The first sentence could mean "I saw her pet duck" or "I saw her duck to avoid getting hit by the ball". The second sentence could mean "students hate to annoy their professors" or "students hate professors who are annoying". Finally, the third sentence may either be a declarative sentence expressing the location of Lily's iPad or a command telling her to take the iPad off the table.

Lecture

Slides

Challenge

Do the following sentences express a proposition?

1. The present king of France is bald.
2. This sentence is false.
3. This sentence is true.

Rational Belief

The objective of an argument is to compel listeners to believe the conclusion on the basis of the reasons given in support. Lawyers advance arguments to convince jurors that their clients are innocent. Politicians advance arguments to convince a group of people to vote for them or to convince their constituents that a policy should be implemented. There are many reasons why an argument may be persuasive. Setting aside issues in rhetoric, this course is focused on the reasoning that leads someone to believe the conclusion based on the supporting reasons.

Beliefs can represent the world more or less accurately...the more accurate the better. A standard view among philosophers is that the aim of belief is truth. That is, one should believe as many true things as possible. Of course, an easy way to maximize the number of true things that you believe is to believe everything. The problem with this approach is that it maximizes both the set of true beliefs and the set of false beliefs. The impetus to believe more and more true propositions needs to be balanced with an attempt to minimize false beliefs. This suggests a second dimension to evaluate beliefs: In addition to accuracy, beliefs can be evaluated as more or less rational. Consider the following examples:

• Ann has trouble distinguishing the twins Jonah and Lucas. She mistakenly believes that the person waving to her is Jonah rather than Lucas.
• Ann makes a mistake when calculating the tip for her bill. She mistakenly believes that she owes $25 instead of $22.50.

In each of the above examples, Ann has a belief that is not true. However, there doesn't seem to be anything irrational about these beliefs. Ann simply made a mistake. The philosopher David Christensen explains the different dimensions used to evaluate beliefs in Putting Logic in its Place: Formal Constraints on Rational Belief as follows:

Accuracy and rationality are linked, but they are not the same: a fool may hold a belief irrationally --- as a result of a lucky guess or wishful thinking --- yet it might happen to be correct. Conversely, a detective might hold a belief on the basis of a careful and exhaustive examination of all the evidence and yet the evidence may be misleading, and the belief may turn out to be wrong.

Rational beliefs are those that arise from good reasoning, whether or not that reasoning was successful in latching on to the truth. But, what is good reasoning?

Notation: Representing Arguments

There are two key features of an argument. The fist is that an argument consists of a group of statements. The second is that exactly one of the statements in an argument must be identified as the conclusion. Typically, the conclusion of an argument is indicated by one of the following phrases:

therefore        hence       for this reason
thus       implies that       entails that       so
it must be that       we may infer       wherefore
it follows be that       we may conclude that
consequently       as a result       accordingly

While every argument must have a conclusion, not all arguments include one of the above phrases. Consider the following argument:

The University of Maryland deserves increased expenditures in the years ahead. The University provides excellent educational opportunities for students with diverse backgrounds, and is committed to being a preeminent national center for research and for graduate education. Furthermore, at current funding levels the University cannot fulfill its potential.

The first sentence expresses the conclusion of the argument.

For much of this course, it will be convenient to represent arguments in a standard form: List the premises separated by a comma (or a period when expressing the argument in English), then the symbol "$\Rightarrow$" followed by the conclusion. So, the standard form of the above argument is:

The University provides excellent educational opportunities for students with diverse backgrounds, and is committed to being a preeminent national center for research and for graduate education. Furthermore, at current funding levels the University cannot fulfill its potential. $\Rightarrow$ The University of Maryland deserves increased expenditures in the years ahead.

More abstractly, we will use capital letters (possibly with indices) to represent the premises and conclusion of an argument. So, the expression

$$P_1, P_2, P_3, P_4\Rightarrow C$$

is an argument consisting of 4 premises (represented by $P_1$, $P_2$, $P_3$, and $P_4$) and the conclusion represented by $C$.

Lecture

Slides

Arguments

1. Read "$\Rightarrow$" as "therefore" or "so". So, $P_1, P_2\Rightarrow C$ is read as "$P_1$ and $P_2$ therefore $C$", or "$P_1$ and $P_2$ so $C$".
2. The expression "$P\Rightarrow Q$" describes an argument. It does not mean that the argument is good (i.e., that the premise $P$ supports the concluion $Q$). We will introduce additional notation to evaluate arguments.
3. While every argument must have a conclusion, it will be convenient to allow arguments to have no premises. So, the expression "$\Rightarrow Q$" expresses an argument consisting of no premises and the conclusion $Q$.

Practice Questions

1. True or False: Any collection of sentences is an argument.
   True

   False

   ​

   There are two problems with this statement. First, arguments are defined to be collections of propositions not sentences. Second, only *declarative* sentences can be used to express arguments. The following is true: Any collective of declarative sentences expresses an argument.
2. True or False: Every argument must have a conclusion.
   True

   False

   ​

   A defining feature of an argument is that it has a conclusion. The conclusion is the statement that the argument is supporting, what is being "argued for".
3. True or False: An argument must be expressed using one of the above conclusion indicators.
   True

   False

   ​

   Conclusion indicators are not required in an argument. For example, consider the argument "Freedom of the press is the most important of our constitutionally guaranteed freedoms. Without it our other freedoms would be immediately threatened. Furthermore, it provides the opportunity for the advancement of new freedom." This is an argument, but it doesn't contain any of the above conclusion indicators. The conclusion is "Freedom of the press is the most important of our constitutionally guaranteed freedoms."
4. True or False: An argument must have at least one premise.
   True

   False

   ​

   It is useful when studying logic to allow arguments that do not have any premises (though every argument must have a single conclusion). Of course, typically, good arguments will include some supporting evidence (i.e., some premises).
5. Which of the following expresses an argument with one premise and one conclusion?
   Since Edison invented the phonograph, there have been many technological developments.

   Since Edison invented the phonograph, he deserves credit for a major technological development.

   ​

   The word "since" is sometimes used to indicate a premise, as in the second sentence. In the first sentence, "since" is used as a preposition to introduce the time in the past when Edison invented the phonograph.

6. Suppose that $S1, S2\Rightarrow S3$. What can you conclude?

   $S1, S2$, and $S3$ represent propositions.

   $S1, S2$, and $S3$ are all true.

   $S3$ is true, but $S1$ and $S2$ may be either true or false.

   ​

   The notation "$S1, S2\Rightarrow S3$" means that $S1$, $S2$, $S3$ is an argument where $S3$ is the conclusion, and $S1$ and $S2$ are the premises. You cannot conclude anything about the truth or falsity of any of the statements. The only thing you can conclude is that $S1$, $S2$, and $S3$ are propositions.

7. Suppose that $S1, S2\Rightarrow S3$. What can you conclude?

   $S3$ is the premise.

   $S3$ is a conclusion.

   ​

   The statement coming after "$\Rightarrow$" is the conclusion.

8. Suppose that $S1, S2\Rightarrow S3$. What can you conclude?

   $S1$ and $S2$ are evidence that $S3$ is true.

   You can infer from $S1$ and $S2$ that $S3$ is true.

   None of the above.

   ​

   The notation "$S1, S2\Rightarrow S3$" means that $S1$, $S2$, $S3$ is an argument where $S3$ is the conclusion, and $S1$ and $S2$ are the premises. You cannot conclude that the argument is good or bad, only that $S1$ and $S2$ potentially support the conclusion.