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.
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.
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.
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.
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.
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.
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.
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.
