Truth Tables: Statements

Gig Φ Philosophy
(at-a-glance overviews of philosophical concepts)

Published October, 2022

Truth Functions

Beyond Validity

After translating statements into propositional logic, a number of assessment methods can be used to verify the potential truth values of those statements as well as the validity of arguments. As discussed previously, validity concerns the truth-preserving structure of a deductive argument if the premises are true. 

Given the often complex nature of arguments and their constituent parts, it can be difficult to determine just from the statements themselves whether or not they are true, let alone truth-preserving throughout. For instance, if we take the last example from the previous section, it quickly becomes apparent that the truth of the statement depends on a variety of factors, as in the example explored above.

Challenge of Truth

Even the truth of more simplistic conditional statements can be difficult to determine when we do not yet know whether any of the conditions hold. It becomes more challenging when there are so many conditions that need to be met, and even more so when we combine a compound statement like the one above with many others in an argument. 

Luckily, the formal structure of propositional logic allows us to assess the potential truth values of even the most complex statements and arguments. 

The next section will explore the basic rules for determining the potential truth values of statements, as well as the use of those truth values to determine the validity of arguments.

Truth Values & Tables

By truth value, we mean the attribution of truth (T) or falsity (F) to a given statement. Simple statements are assigned each possible truth value. 

 This resource will use the classical notion of truth value, with only two possible values. It is worth noting that in other logical systems (e.g., quantum theory and computer science), there may be more than two possible truth values which allow for different models of entailment, identity, and infinity.

The truth value of compound statements will then be determined by the truth values of the simpler statements that make them up. 

Truth tables are then constructed to determine the truth values of each statement and argument validity. A truth table is a representation of the ways that a statement can express truth values. In this sense, truth tables are mechanical and follow rules for their construction. Which is what we shall explore next.

How to Construct Truth Tables for Statements

IN 2 EASY STEPS

This section will examine the rules for constructing truth tables for simple and compound statements, as well as instructions for how to classify and compare statements. 

STEP 1: SETUP TABLE

Columns = Variables

First, we need to identify the scope of the table being constructed. We do so by determining the number of simple statements present in the proposition. Remember that each simple statement will be represented by a variable. Each variable will get its own column in the table. For example:

Rows = Truth Values

Next, we will sketch out the rows where we will place the truth values, once they are determined. We do this to ensure we explore every possible truth value to be assigned. Remember that each simple statement will be assigned each possible truth value (T and F). Each truth value will get its own row. 

For compound statements with multiple variables, an easy trick for determining how many rows will be in the table is to multiply 2 (the number of possible truth values) to the power of the number of variables. For example:

STEP 2: ASSIGN VALUES

Simple Statements

Finally, we will assign the appropriate truth values. In order to make sure all possible arrangements are accounted for in the table, begin on the far left column and divide the rows in half (assigning ‘T’ to the first half of the rows, and assigning ‘F’ the second half). Then move to the next column to the right and divide in half again, rotating equally between ‘T’ and ‘F’ assignments. For example:

Compound Statements

If you are determining the truth value of a simple statement (one variable), this will be all you need. If you are determining the truth value of compound statements and / or the validity of arguments, you will need to continue.

In order to determine the truth values for compound statements, it is worth noting the rules for each of the following tables. Each will correspond to the logical operator(s) present in each compound statement.

RULES: TRUTH VALUES

NEGATION

Notice that the truth values for P have simply been negated. All “not-P” truth values are just the opposite of what they would have been for P.

CONJUNCTION

Notice that the conjunction of “P and Q” is only true if both components are true. 

It is false if either or both components are false.

DISJUNCTION

Notice that the disjunction of “either P or Q” is only false if both components are false. 

It is true if either or both components are true.

CONDITIONAL

Notice that the conditional statement of “if P, then Q” is only false when the antecedent is true and the consequent is false. 

It is true in all other instances, even when the antecedent condition does not hold.

MATERIAL EQUIVALENCE

Notice that the material equivalence of “P if and only if Q” is only true when both components have the same truth value. 

It is false if the truth values are different.