Because complex Boolean statements can get tricky to think about, we can create a truth table to break the complex statement into simple statements, and determine whether they are true or false. A table will help keep track of all the truth values of the simple statements that make up a complex statement, leading to an analysis of the full statement.

Some symbols that are commonly used for and, or, and not make using a truth table easier.

You can remember the first two symbols by relating them to the shapes for the union and intersection. [latex]A\wedge{B}[/latex] would be the elements that exist in both sets, in [latex]A\cap{B}[/latex]. Likewise, [latex]A\vee{B}[/latex] would be the elements that exist in either set, in [latex]A\cup{B}[/latex].
In the previous example, the truth table was really just summarizing what we already know about how the or statement work. The truth tables for the basic and, or, and not statements are shown below.  It is a good idea to try to memorize these basic truth tables.  It will help you when you are working with truth tables and more complex statements.

Truth tables really become useful when analyzing more complex Boolean statements.  But, what is the correct order needed when completing a truth table?

The Dominance of Connectives

The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem 1+3×2, and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.

Scientific Calculator

The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation—addition or multiplication—should be done first. Adding parentheses can change and/or clarify the order. The parentheses in the expression 1+(3×2) indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: 1+(3×2)=7.

As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the dominance of connectives. When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:

1. Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.
2. Next, we evaluate negations.
3. Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.
4. After evaluating all conjunctions and disjunctions, we evaluate conditionals.
5. Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.

The following image is a visual breakdown of the dominance of connectives.

When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an implication.

Implications are similar to the conditional statements we looked at earlier; [latex]p\rightarrow{q}[/latex] is typically written as “if p then q,” or “p therefore q.” The difference between implications and conditionals is that conditionals we discussed earlier suggest an action—if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.

Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.

In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.

For any implication, there are three related statements, the converse, the inverse, and the contrapositive.

Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.