Recipe

Build a truth table for a compound proposition

Use when: you have a compound proposition with two or more atoms and need its truth value under every possible assignment

Goal: a complete table listing the formula's truth value on every row

Taught in Truth tables

1. 1

   Count the distinct atomic propositions in the formula; a table needs 2^n rows for n atoms.

   Example

   For (p \wedge q) \vee \neg r, the atoms are p, q, r, so n=3 and the table needs 2^3=8 rows.

   Why: this fixes exactly how many assignments must be checked, since a table with fewer rows would leave some combination untested.

2. 2

   List all 2^n combinations without repeats or omissions, in a consistent order.

   Example

   For p, q, r, the eight rows in order are TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.

   • two or three atoms — list the combinations by hand in a steady pattern, such as counting down from all-true.
   • four or more atoms — work systematically, cycling the last atom fastest, to avoid missing or repeating a row.

   Why: skipping or repeating a combination leaves the finished table an incomplete record of the formula's behavior.

3. 3

   On each row, apply the connective closest to the atoms first, then combine outward until one value remains.

   Example

   At p=T, q=F, r=T for (p \wedge q) \vee \neg r: first p \wedge q = T \wedge F = F and \neg r = \neg T = F, then F \vee F = F.

   • parentheses are present — evaluate inside them before combining with anything outside.
   • no parentheses — use the default order, \neg first, then \wedge, then \vee.

   Why: order of operations matters exactly as it does in algebra, since evaluating out of order can flip the final value.