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Section 1.7 Arguments

Typically we think about an argument as a fight between people. However, a mathematical argument is different. A mathematical argument is a collection of statements called premises and a conclusion. Below is an example of an argument:

My state made it illegal to get an abortion after 6 weeks gestation. I just found out that I am pregnant and I am 8 weeks pregnant. Therefore, I cannot legally get an abortion in my state.

The premises are the first two sentences. The conclusion is the last sentence. Typically, conclusions are identified by such words as ''therefore'', ''thus'', ''hence'', etc. The goal of this section is to explain how to turn a written argument into a symbolic argument that can be evaluated using truth tables. It should be noted that we may need to be flexible with this translation and there are often many ways to interpret sentences as statements.

We begin with the last example: "My state made it illegal to get an abortion after 6 weeks gestation. I just found out that I am pregnant and I am 8 weeks pregnant. Therefore, I cannot legally get an abortion in my state." We can reword this to the following:

If I am more than 6 weeks pregnant, it is illegal to get an abortion in my state. I am more than 6 weeks pregnant. Therefore, it is illegal for me to get an abortion in my state.

We are now ready to identify two main statements: let \(p\) represent "I am more than 6 weeks pregnant" and let \(q\) represent "It is illegal to get an abortion in my state".

Once we have converted the sentences into simple statements connected through our logical operations (if-then statements, and's, or's, negations), we need to address the punctuation in an argument. We do this by taking all of the premises and creating one run-on sentence. In other words, we remove every period and replace it with the word "and". In our example we reduce the argument to two sentences. The first sentence consists of the run-on sentence of premises. The second sentence is the conclusion of the argument. We have:

If I am more than 6 weeks pregnant then it is illegal to get an abortion in my state and I am more than 6 weeks pregnant. Therefore, it is illegal for me to get an abortion in my state.
The last step needed before we convert the argument into symbolic notation is to turn these two sentences into one "if-then" statement. We do this in the following way:
If (insert run-on sentence consisting of all premises), then (insert conclusion).
We may need to be a bit flexible while reading this new one-sentence argument. In this example, it is very strange to read:
If If I am more than 6 weeks pregnant then it is illegal to get an abortion in my state and I am more than 6 weeks pregnant, then it is illegal for me to get an abortion in my state.
Instead, it is easier to read and understand the following equivalent statement:
If it is the case that if I am more than 6 weeks pregnant then it is illegal to get an abortion in my state and I am more than 6 weeks pregnant, then it is illegal for me to get an abortion in my state.

We are now ready to turn this argument into a symbolic argument. Let $q$ be “I am more than 6 weeks pregnant” and let $p$ be “It is illegal to get an abortion in my state”. We have the following symbolic argument:

\begin{equation*} [(q\implies p)\wedge q]\implies p \end{equation*}

We generalize this process below: Suppose an argument consists of the premises \(p_1,~p_2,\dots,p_n\) and the conclusion \(c\text{.}\) The corresponding symbolic form of this argument is:

\begin{equation*} (p_1\wedge p_2\wedge\dots \wedge p_n)\implies c \end{equation*}

Once an argument has been turned into its symbolic form, we can determine if the argument is valid and/or sound. These concepts are addressed in the next two sections.

Subsection 1.7.1 Valid Arguments

A mathematical argument is valid if it is always true. In other words, the final column of an argument’s truth table will consist of only true values. For the argument presented in the previous section,

\begin{equation*} [(q\implies p)\wedge q]\implies p \end{equation*}

we see the truth table is:

making this a valid argument.

Show that the following argument is not valid. "If access to abortion is restricted in a state, then the number of abortions in that state will decrease. The number of abortions decreased in that state. Consequently, access to abortion is restricted in that state.""

Show that the following argument is valid. "If a state restricts access to abortion, then the number of abortions will decrease in that state. The number of abortions did not decrease in that state. Therefore, the state did not restrict access to abortion.""

These last two examples highlight that the validity of an argument does not depend on the actual statements being used. Instead validity focuses only on the structure of the argument. This means that an argument can be valid, but may not be sound- a concept discussed in the next example.

Subsection 1.7.2 Sound Arguments

A valid argument is called sound if its premises are true. If a valid argument is not sound, we say that it is unsound. In the last example, the two premises are

  1. If a state restricts access to abortion, then the number of abortions will decrease in that state.

  2. The number of abortions did not decrease in that state.

Both premises would need to be supported by factual data to determine if they are either true or false. However, historically it has been shown that restricting access to abortion does not reduce the number of abortions. Instead, restricting access to abortion reduces the number of safe abortions performed [1.12.3]. Using this historical data, we could conclude that Premise 1 above is false. Thus making this argument unsound.

When a valid argument is sound, this means that the conclusion of the argument can be inferred from true statements. When a valid argument is unsound, this means the conclusion of the argument is being inferred from false statements thus making the actual argument less credible.