I am required to answer the following question:
Use the product rule to show that there are $2^{2^{n}}$ different truth tables for propositions in n variables.
This is my current answer:
According to the product rule, if there are n distinct possibilities for one event, and m distinct possibilities for a second event, then there are $n\cdot m$ possible choices.
There are n rows, each of which can only contain 2 values, true or false. By the product rule, this means that the amount of options for rows will equal 2 times itself n times, which will equal $2^{n}$.
Furthermore, each of these rows can be one of two values, true or false. If $2^{n}$ rows have 2 options, then by the product rule, the amount of options will equal 2 times itself, $2^n$ times. This equals $2^{2^{n}}$.
However, I am not sure if this holds. Can every row come out True or False? Isn't that not true since the row of the truth table itself depends on the elements of the row and how we are evaluating it?
The $2^n$ rows of the truth table are fixed as the possible inputs of $n$ variables, from all-zeros to all-ones. However, after those rows are fixed, what values they map to are completely arbitrary in this case – the analysis is not limited to a single logical function that fixes the values corresponding to each possible input. There are logical functions sending all inputs to true, or all to false, but those are only two such functions out of $2^{2^n}$.
For example, with two variables: $$\begin{array}{c|c}00&a\\01&b\\10&c\\11&d\end{array}$$ The left column is fixed, but for each entry in that left column we are free to choose 0 or 1 as the corresponding entry in the right column, yielding 16 different logic functions.