The most convenient way to study Boolean functions is writing them in the form of tables called the truth tables. They were invented by Charles Peirce in 1893. Truth tables have a great practical use in propositional logic since they allow to learn important examples of connectives as special Boolean functions.
Let $p_1,\ldots,p_n$ be $n\ge 1$ propositions and let $\circ$" be a "$k$-nary connective such that $\phi=p_1\circ \ldots \circ p_n$ is also a proposition. We have seen in the corresponding lemma that "$\circ$" induces a Boolean function $f_\phi:\mathbb B^n\to\mathbb B$ defined on the set of truth values $\mathbb B=\{1,0\}=\{true, false\}$:
A truth table of the Boolean function $f_\phi$ is a table listing all the possible valuations $[[]]_I$ of the $n$ propositions $p_1,\ldots,p_n$ and the resulting value of the boolean function $f_\phi$.
A schematic structure is the following. The $b_i$ in the right column are one of $1$ or $0$ each.
$[[p_1]]_I$ | $\ldots$ | $[[p_n]]_I$ | $[[p_1\circ\ldots\circ p_n]]_I$ |
---|---|---|---|
$0$ | $\ldots$ | $0$ | $b_1$ |
$1$ | $\ldots$ | $0$ | $b_2$ |
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |
$1$ | $\ldots$ | $1$ | $b_{2^n}$^{1} |
Definitions: 1 2 3 4 5 6
Examples: 7
Proofs: 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Sections: 22
In general, it is also possible that a truth table contains less than $2^n$ rows. This is especially the case if some of the propositions $p_i$ are Boolean constants. ↩