**Truth Tables**Worksheet

In order to analyze arguments, we need to understand the behavior of the connectives we looked at last time.

That is, we seek to understand the how the truth of a statement, for example $$p \rightarrow \sim q$$ varies for all the possible truth values of its individual statements (for the above example, it's $p$ and $q$).

A truth table does exactly the above.

**A Basic Example:**Negation

Fill in the following truth table for negation:

$$ \begin{array}{|c|c|} \hline p & \sim p \\ \hline T & \\ F & \\ \hline \end{array} $$

**Basic Examples**

Before we start analyzing the truth of compound statements, let keep a prototype example firmly in mind.

We'll let $p$ be the statement

**Mr. Holt is hungry,**and $q$ be the statement

**Mr. Holt is grumpy.**

**A Basic Example:**Conjunction

Fill in the following truth table for conjunction (and): $$ \begin{array}{|c|c|c|} \hline p & q & p \wedge q\\ \hline T & T &\\ T & F & \\ F & T & \\ F & F & \\ \hline \end{array} $$

**Example:**

**Mr. Holt is hungry, and Mr. Holt is grumpy.**

**A Basic Example:**Disjunction

Fill in the following truth table for disjunction (or): $$ \begin{array}{|c|c|c|} \hline p & q & p \vee q\\ \hline T & T &\\ T & F & \\ F & T & \\ F & F & \\ \hline \end{array} $$

**Example:**

**Mr. Holt is hungry, or Mr. Holt is grumpy.**

**A Basic Example:**The Conditional

Fill in the following truth table for the conditional: $$ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow q\\ \hline T & T &\\ T & F & \\ F & T & \\ F & F & \\ \hline \end{array} $$

**Example:**

**If Mr. Holt is hungry, then Mr. Holt is grumpy.**

**Example:**The Biconditional

$$ \begin{array}{|c|c|c|} \hline p & q & p \leftrightarrow q\\ \hline T & T &\\ T & F & \\ F & T & \\ F & F & \\ \hline \end{array} $$

**Example:**

**Mr. Holt is hungry if and only if Mr. Holt is grumpy.**

**Fact:**The biconditional $p \leftrightarrow q$ is defined as the conjunctions two conditionals: $p \rightarrow q$ and $q \rightarrow p.$

In terms of our example, this means that

**Mr. Holt is hungry if and only if Mr. Holt is grumpy.**

is really made up of two conditionals:

**If Mr. Holt is hungry, then Mr. Holt is grumpy.**and

**If Mr. Holt is grumpy, then Mr. Holt is hungry.**

Thus, in order to fill in the above truth table, we will need to expand it a bit as seen below...

**Example:**The Biconditional

Fill in the following truth table for the biconditional: $$ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & p \rightarrow q & q \rightarrow p & (p \rightarrow q)\wedge (q \rightarrow p) & q \leftrightarrow p \\ \hline T & T & & & &\\ T & F & & & & \\ F & T & & & &\\ F & F & & & &\\ \hline \end{array} $$

**Example:**

**Mr. Holt is hungry if and only if Mr. Holt is grumpy.**

**Example:**The Biconditional

The truth table for the biconditional in all its glory! $$ \begin{array}{|c|c|c|} \hline p & q & p \leftrightarrow q \\ \hline T & T & T\\ T & F & F\\ F & T & F\\ F & F & T\\ \hline \end{array} $$

**Fact of Life**

In order to continue, we need to simply know the truth tables for all the basic connectives: Conjunction, Disjunction, Conditional, and Biconditional.

With these, we are now ready to analyze any statement which contains logical connectives!

That is, we are one step closer to being able to logically analyze arguments!

**Example:**Construct a truth table for the statement $$p \wedge \sim q.$$

**Example:**Construct a truth table for the statement $$\sim (p \rightarrow \sim q).$$

**Compound Statements Consisting of Three Statements**$$ \begin{array}{|c|c|c|c|} \hline p & q & r & (\sim p \vee q) \leftrightarrow \sim r \\ \hline T & T & T & \\ T & T & F & \\ T & F & T & \\ T & F & F & \\ F & T & T & \\ F & T & F & \\ F & F & T & \\ F & F & F & \\ \hline \end{array} $$