Truth tables and symbolic translation connect the language of ordinary sentences with the exact language of mathematical logic. In this page, we learn how to list truth values systematically, how to evaluate compound formulas, and how to translate English phrases such as "not", "and", "or", "if", and "if and only if." The focus keyword truth tables and symbolic translation is important because students often understand the individual connectives but lose accuracy when several connectives appear together. We will build the skill slowly through formulas, examples, and verified problems.

A truth table is a table that lists all possible truth-value assignments for the primitive statements in a formula and then computes the truth value of the compound statement in each case. In this course, we use $1$ for true and $0$ for false. Example with one statement: for $p$, the two possible values are $1$ and $0$, so $\neg p$ has the opposite values. Example with two statements: for $p$ and $q$, the four possible rows are $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$. Example with three statements: for $p$, $q$, and $r$, there are $2^3=8$ possible rows.