**Determining the Validity of Arguments**Worksheet

**Definition**: An

__argument__is a conditional statement whose "left side" consists of a conjunction of "premises" and a "right side" which is its conclusion.

The conclusion "follows" from the premises.

**The Structure of a Typical Argument**

$$ \begin{array}{l} \mbox{Premise 1}\\ \mbox{Premise 2}\\ \hline \therefore \mbox{Conclusion}\\ \end{array} $$

**Valid Arguments**

**Definition:**An argument is

__valid__if the conclusion necessarily follows from its premises. If the conclusion doesn't follow from the premises, the argument is said to be

__invalid.__

That is, if the statement $$\mbox{Premise 1} \wedge \mbox{Premise 2} \Rightarrow \mbox{Conclusion}$$ is a tautology (the last column of its truth table is all $T$s), then the argument is valid.

**Note on Notation**

The symbol $\Rightarrow$ is used for the conditional instead of $\rightarrow$ in order to distinguish the argument itself from premises which might also be conditionals.

**Example**

Decide if the following argument is valid. $$ \begin{array}{l} \mbox{If Mr. Holt is hungry, then he is grumpy.}\\ \mbox{Mr. Holt isn't grumpy.}\\ \hline \therefore \mbox{Mr. Holt isn't hungry.}\\ \end{array} $$

**Determining the Validity of An Argument**

**Step #1:**Translate argument into symbols:

$ \begin{array}{l} \mbox{If Mr. Holt is hungry, then Mr. Holt is grumpy.}\\ \mbox{Mr. Holt isn't grumpy.}\\ \hline \therefore \mbox{Mr. Holt isn't hungry.}\\ \end{array} $ becomes $ \begin{array}{l} p \rightarrow q\\ \sim q \\ \hline \therefore \sim p\\ \end{array} $

**Determining the Validity of An Argument**

**Step #2:**Write the argument as a conditional statement of the form $$\mbox{Premise 1} \wedge \mbox{Premise 2} \Rightarrow \mbox{Conclusion}.$$ In this case $$(p \rightarrow q) \wedge \sim q \Rightarrow \sim p$$

**Determining the Validity of An Argument**

**Step #3:**Construct a truth table for your argument. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline p & q & p \rightarrow q & \sim q & (p \rightarrow q) \wedge \sim q & \sim p & (p \rightarrow q) \wedge \sim q \Rightarrow \sim p \\ \hline T & T & T & F & F & F & T\\ T & F & F & T & F & F & T\\ F & T & T & F & F & T & T\\ F & F & T & T & T & T & T\\ \hline \end{array} $$ Since $(p \rightarrow q) \wedge \sim q \Rightarrow \sim p$ is a tautology, the argument is valid.

**Example**

Determine the validity of the following argument.

**The judge told Billy Bob that if he was caught making moonshine again that he would have to spend time in jail.**

Two months later, Billy Bob went to jail.

Therefore, he must have been making moonshine.

Two months later, Billy Bob went to jail.

Therefore, he must have been making moonshine.

**Step #1:**Translate argument into symbols: $$ \begin{array}{l} \mbox{If Billy Bob is caught making moonshine again, then he will go to jail.}\\ \mbox{Billy Bob went to jail.}\\ \hline \therefore \mbox{Billy Bob was caught making moonshine again.}\\ \end{array} $$ becomes $$ \begin{array}{l} p \rightarrow q\\ q \\ \hline \therefore p \\ \end{array} $$

**Step #2:**Write the argument as a conditional statement of the form $$\mbox{Premise 1} \wedge \mbox{Premise 2} \Rightarrow \mbox{Conclusion}.$$ $$(p \rightarrow q) \wedge q \Rightarrow p$$

**Step #3:**Construct a truth table for your argument. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline p & q & p \rightarrow q & (p \rightarrow q) \wedge q & (p \rightarrow q) \wedge q \Rightarrow p\\ \hline T & T & T & T & T \\ T & F & F & F & T \\ F & T & T & T & F \\ F & F & T & F & T \\ \hline \end{array} $$ Since $(p \rightarrow q) \wedge q \Rightarrow p$ is NOT a tautology, the argument is invalid.

**Valid Arguments Versus Fallacies**

The above argument is not valid. In fact, is an example of a well-known fallacy called

__The Fallacy of the Converse.__

On the other hand, our first example was a valid argument. In fact, this line of argumentation is so common that it has a name too:

__The Law of Contraposition.__

**Common Valid Arguments**

**The Law of Detachment**$ \begin{array}{l} p \rightarrow q\\ p \\ \hline \therefore q\\ \end{array} $

*(a.k.a. Modus Ponens):***Example:**$ \begin{array}{l} \mbox{If Mr. Holt is hungry, then Mr. Holt is grumpy.}\\ \mbox{Mr. Holt is hungry.}\\ \hline \therefore \mbox{Mr. Holt is grumpy.}\\ \end{array} $

**Common Valid Arguments**

**The Law of Contraposition**$ \begin{array}{l} p \rightarrow q\\ \sim q \\ \hline \therefore \sim p\\ \end{array} $

*(a.k.a. Modus Tollens):***Example:**$ \begin{array}{l} \mbox{If Mr. Holt is hungry, then Mr. Holt is grumpy.}\\ \mbox{Mr. Holt isn't grumpy.}\\ \hline \therefore \mbox{Mr. Holt isn't hungry.}\\ \end{array} $

**Common Valid Arguments**

**The Law of Transitivity (a.k.a. Hypothetical Syllogism):**$ \begin{array}{l} p \rightarrow q\\ q \rightarrow r\\ \hline \therefore p \rightarrow r\\ \end{array} $

**Example:**$ \begin{array}{l} \mbox{If Mr. Holt is hungry, then Mr. Holt is grumpy.}\\ \mbox{If Mr. Holt is grumpy, then Mr. Holt assign lots of homework.}\\ \hline \therefore \mbox{If Mr. Holt is hungry, then Mr. Holt will assign lots of homework.}\\ \end{array} $

**Common Valid Arguments**

**The Law of Disjunctive Syllogism:**$ \begin{array}{l} p \vee q\\ \sim p \\ \hline \therefore q \\ \end{array} $

**Example:**$ \begin{array}{l} \mbox{Billy Bob will tile his kitchen green or his bathroom pink.}\\ \mbox{Billy Bob didn't tile his kitchen green.}\\ \hline \therefore \mbox{Billy Bob tiled his bathroom pink.}\\ \end{array} $

We now move on to common fallacies, the first of which, we've seen already.

**Common Fallacies**

**Fallacy of the Converse:**$ \begin{array}{l} p \rightarrow q\\ q \\ \hline \therefore p \\ \end{array} $

**Example:**$ \begin{array}{l} \mbox{If Billy Bob is caught making moonshine again, then he will go to jail.}\\ \mbox{Billy Bob went to jail.}\\ \hline \therefore \mbox{Billy Bob was caught making moonshine again.}\\ \end{array} $

**Common Fallacies**

**Fallacy of the Inverse:**$ \begin{array}{l} p \rightarrow q\\ \sim p \\ \hline \therefore \sim q \\ \end{array} $

**Example:**$ \begin{array}{l} \mbox{If Billy Bob is caught making moonshine again, then he will go to jail.}\\ \mbox{Billy Bob wasn't caught making moonshine again.}\\ \hline \therefore \mbox{Billy Bob didn't go to jail.}\\ \end{array} $

**Common Fallacies**

**Fallacy of the Inclusive OR:**$ \begin{array}{l} p \vee q\\ p \\ \hline \therefore \sim q \\ \end{array} $

**Example:**$ \begin{array}{l} \mbox{Billy Bob will tile his kitchen green or bathroom pink.}\\ \mbox{Billy Bob tiled his kitchen green.}\\ \hline \therefore \mbox{Billy Bob didn't tile his bathroom pink.}\\ %\mbox{Billy Bob will go out with Linda Lou or Peggy Sue.}\\ %\mbox{Billy Bob is going out with Linda Lou.}\\ \hline %\therefore \mbox{Billy Bob is not going out with Peggy Sue.}\\ \end{array} $

**Using Common Forms to Decide the Validity of an Argument**

There are two ways we can decide the validity of an argument:

1) Use a truth table.

2) Try to recognize the argument as one of the tried-and-true forms we saw above.

**Using Common Forms to Decide the Validity of an Argument**

**Example:**Decide is the following argument is valid and say which form of argument or fallacy you used to make your determination. $$ \begin{array}{l} \mbox{At a cheapskate hotel, if you pay for access, you can surf the internet.}\\ \mbox{You didn't pay for access.}\\ \hline \therefore \mbox{You didn't surf the internet.}\\ \end{array} $$

We can represent the above argument as
$$
\begin{array}{l}
p\rightarrow q\\
\sim p\\ \hline
\therefore \sim q\\
\end{array}
$$
We see that this is the Fallacy of the Inverse.

This argument is invalid.

This argument is invalid.

**One Final Note:**Strange as it may seem, an argument can be valid even when one or more of the premises are false. $$ \begin{array}{l} \mbox{Salem is in Oregon or California}\\ \mbox{Salem is not in Oregon.}\\ \hline \therefore \mbox{Salem is in California.}\\ \end{array} $$ The above argument is valid since the conclusion follows logically from the premises (one of which is a bold-faced lie).

It goes to show that the believability of the premises is also an important consideration when analyzing an argument.