3.1 Truth Values and Truth Functions

Truth Values and Functions

Truth values and truth functions give you a precise way to evaluate whether complex logical statements are true or false. By breaking compound statements into their simplest parts, you can determine the truth of the whole thing mechanically, without relying on intuition. This section covers truth values, truth functions, logical connectives, and how they all fit together.

Fundamental Concepts

A truth value is simply the truth or falsity of a statement. In formal logic, there are exactly two: T (true) and F (false). Every statement you work with gets assigned one or the other.

A truth function is an operation that takes one or more truth values as input and produces a single truth value as output. Think of it like a machine: you feed in T's and F's, and it spits out one T or F according to a fixed rule.

A truth-functional compound is a compound statement whose truth value is entirely determined by the truth values of its parts. This is a key idea. It means you don't need to know what the statements are about. You only need to know whether each component is true or false.

A truth table displays every possible combination of truth values for a set of component statements, along with the resulting truth value of the compound. It's your main tool for systematically analyzing truth-functional compounds.

Applying Truth Values and Functions

Here's the general process for evaluating a compound statement:

1. Identify the simple (atomic) statements and assign each a truth value. For example, let $p$ = "The sky is blue" (T).
2. Identify the logical connectives combining those statements into a compound, such as $p \land q$.
3. Apply the truth function for each connective to determine the compound's truth value.
4. To see all possible outcomes, build a truth table that lists every combination of T and F for the component statements.

For a compound with $n$ simple statements, the truth table will have $2^n$ rows. So two statements give you 4 rows, three statements give you 8, and so on.

Logical Connectives

Logical connectives are the operators that combine statements into compounds. Each connective defines a specific truth function.

Binary Connectives

A binary connective combines two statements to form a compound.

Conjunction ($\land$) means "and." A conjunction is true only when both components are true. If either part is false, the whole conjunction is false.

• Example: "The sun is shining and the birds are singing" ($p \land q$). This is true only if both $p$ and $q$ are true.

Disjunction ($\lor$) means "or" in the inclusive sense. A disjunction is true when at least one component is true. It's only false when both parts are false.

• Example: "I will go to the beach or I will go to the mountains" ($p \lor q$). This is true if you do one, the other, or both.

Unary Connective

A unary connective operates on just one statement.

Negation ($\lnot$) flips the truth value. If $p$ is true, $\lnot p$ is false, and vice versa.

• Example: If $p$ = "It is raining" (T), then $\lnot p$ = "It is not raining" (F).

Double negation ($\lnot \lnot p$) flips the value twice, bringing you back to the original. So $\lnot \lnot p$ is logically equivalent to $p$.

Conditional Statements

Material Conditional

The material conditional ($\rightarrow$) forms an "if...then" statement. The component before the arrow is the antecedent, and the component after is the consequent.

• Example: "If it rains, then the ground will be wet" ($p \rightarrow q$).

The truth conditions here trip up a lot of students. A material conditional is false in exactly one case: when the antecedent is true and the consequent is false. In every other case, it's true.

The rows where $p$ is false might seem strange. Why is "If it rains, then the ground is wet" true when it doesn't rain? The material conditional doesn't claim a causal connection. It only says: "It won't happen that $p$ is true while $q$ is false." When $p$ is false, that promise is never violated, so the conditional counts as true. This is sometimes called being vacuously true.

Biconditional

The biconditional ($\leftrightarrow$) means "if and only if." It's true when both components share the same truth value, and false when they differ.

• Example: "A triangle is equilateral if and only if all its sides are of equal length" ($p \leftrightarrow q$).

A useful way to think about it: a biconditional is equivalent to the conjunction of two conditionals going in opposite directions. That is, $p \leftrightarrow q$ means the same thing as $(p \rightarrow q) \land (q \rightarrow p)$. Both directions of the "if...then" must hold.