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.
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 (a.k.a. Modus Ponens): $ \begin{array}{l} p \rightarrow q\\ p \\ \hline \therefore q\\ \end{array} $
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 (a.k.a. Modus Tollens): $ \begin{array}{l} p \rightarrow q\\ \sim q \\ \hline \therefore \sim p\\ \end{array} $
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.