Good Arguments

Section 2.4 Good Arguments

Subsection 2.4.1 Introduction

The political scene of ancient Greece was pretty wild. City-states separated by Greece’s rough terrain of mountains and the Mediterranean competed for power, and the wealthy and influential in each city wrestled for political influence for control of their community. Democracy allowed the people to decide on leadership and policy, and so rhetoric became the tool of competition among those seeking power.

This may be hard to believe, but sometimes politicians try to convince their audience to take a position with improper tactics. We would say that in these instances they make a flawed argument. Circa 300 BC, as the Greek politicians scrambled for influence, the great philosopher Aristotle began developing a system to analyze arguments. This system became what we know as "classical logic", and because of his work, Aristotle (382-322 BC)is known as the father of logic.

To know how to analyze arguments, we should define what an argument is. An argument is simply a sequence of logical statements. The conclusion of an arugment is the ending statement of the argument, and a premise is a logical statement used to support the conclusion.

For example: All cheaters sit in the back row. Monty sits in the back row. Therefore, Monty is a cheater.

Question 2.4.2.

Without any knowledge of how to validate an argument, do you think the previous argument is a valid one?

So an argument is a sequence of statements? Obviously we can make nonsensical sequences of statements, so how do we know if an argument is a good one or not? We say an argument is valid if the conclusion is logically supported by its premises, so the argument from Question 2.4.2 is not valid since non-cheaters could sit in the back row.

Subsection 2.4.2 Common Arguments

Subsubsection 2.4.2.1 Introduction

So how do we determine an argument to be valid? With logic.

Good understanding of logic can help us determine whether a given argument is valid or not. We already looked at some basic logic principles in Section 2.3, namely the conditional \(p \rightarrow q\) and the contrapositive \(\sim q \rightarrow \sim p\text{.}\) These two statements provide the framework of the two most basic, valid arguments.

Subsubsection 2.4.2.2 Modus Ponens

The modus ponens argument (translated from "affirming mode") consists of three parts

\(\displaystyle p \rightarrow q\)
\(\displaystyle p\)
Therefore, \(q\text{.}\)

The modus ponens argument is simply saying, "if \(p\) happens, then so does \(q\) and \(p\) happens (or is true). Therefore, \(q\) happens as well (or is also true)". This is definitely a valid argument.

Example 2.4.3. Modus Ponens.

For example: All men are mortal. Socrates is a man. Therefore Socrates is mortal.

Question 2.4.4. Modus Ponens.

Craft a modus ponens argument of your own.

Subsubsection 2.4.2.3 Modus Tollens

Like modus ponens, the modus tollens argument (translated from "method of denying") consists of three parts

\(\displaystyle p \rightarrow q\)
\(\displaystyle \sim q\)
Therefore, \(\sim p\text{.}\)

The modus tollens argument makes use of the contrapositive. It’s saying, "if \(p\) happens, then so does \(q\) and \(q\) does not happen (or is false). Therefore, \(p\) does not happen (or is also false)". This is also a valid argument.

Example 2.4.5. Modus Tollens.

For example: If it’s a bird, then it lays eggs. It does not lay eggs. Therefore, it’s not a bird.

Question 2.4.6. Modus Tollens.

Craft a modus tollens argument of your own.

Subsection 2.4.3 Common Errors

Subsubsection 2.4.3.1 Introduction

What about this example? If it’s a bird, then it lays eggs. It lays eggs. Therefore, it’s a bird. Is this a valid argument?

Subsubsection 2.4.3.2 Converse Errors

Nah. This is not a valid argument.

We just made converse error, and invalide argument of the form

\(\displaystyle p \rightarrow q\)
q
Therefore, \(p\text{.}\)

But we can think of several instances in which this is not a valid argument.

Question 2.4.7. Eggs and Birds?

We incorrectly concluded that since it lays eggs, it’s a bird. Give an example where this is false.

Sadly, converse errors are used all the time! Feel empowered to recognize them and point them out!

Subsubsection 2.4.3.3 Inverse Errors

What about this example? If the boy rides his skateboard wrecklessly, then he will break his arm. He did not ride his skateboard wrecklessly. Therefore, he did not break his arm.

There’s more than one way to break an arm, right?

We just made an inverse error with this invalid argument form

\(\displaystyle p \rightarrow q \)
\(\displaystyle \sim p\)
\(\displaystyle \sim q.\)

Remember, that an argument is valid if its conclusion is supported by its premises. The conclusion \(\sim q\) is not supported logically by the other premises.

Subsection 2.4.4 More Arguments

Determine if the following are valid arguments or not.

Question 2.4.8.

All cheaters sit in the back row. Monty sits in the back row. Therefore, Monty is a cheater.

Question 2.4.9.

All freshman must take writing. Caroline is a freshman. Therefore, Caroline must take writing.

Question 2.4.10.

All honest people pay their taxes. Darth is not honest. Therefore, Darth does not pay his taxes.

Question 2.4.11.

If an infinite series converges, then its terms go to 0. The terms of the infinite series \(\sum_{n=1}^{\infty} \dfrac{n}{n+1}\) do not go to 0. The infinite series \(\sum_{n=1}^{\infty} \dfrac{n}{n+1}\) does not converge.

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