Definition:Truth Table/Larger Tables

Computing Larger Truth Tables
Let $P$ be a formula we want to compute the truth table of.

We will use Peirce's Law as an example:


 * $P := \left({\left({p \implies q}\right) \implies p}\right) \implies p$

For each boolean interpretation for the set of propositional variables, we write its truth value underneath:

$\begin{array}{cc||ccccccc} p & q & ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & F & F &  & F &   & F &   & F \\ \end{array}$

In the above, we are using the boolean interpretation $v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$ given by:


 * $v \left({p}\right) = F, v \left({q}\right) = F$.

Then we fill in the truth value of each of the WFFs on the parsing sequence of $P$, underneath its main connective:

$\begin{array}{cc||ccccccc} p & q & ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & F & F &  & F &   & F &   & F \\ &  &   & T &   &   &   &   &   \\ &  &   &   &   & F &   &   &   \\ &  &   &   &   &   &   & T &   \\ \end{array}$

In the above, this has been done on separate lines, so as to clarify the sequence in which this is done for the example.

In practice we write it like this:

$\begin{array}{cc||ccccccc} p & q & ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & F & F & T & F & F & F & T & F \\ \end{array}$

We repeat this for all other boolean interpretations:

$\begin{array}{cc||ccccccc} p & q & ((p & \implies & q) & \implies & p) & \implies & p \\ \hline F & F & F & T & F & F & F & T & F \\ F & T & F & T & T & F & F & T & F \\ T & F & T & F & F & T & T & T & T \\ T & T & T & T & T & T & T & T & T \\ \end{array}$