# Logic

## Truth Tables for the Conditional and Biconditional


### Learning Objectives

After completing this section, you should be able to:
1. Use and apply the conditional to construct a truth table.
2. Use and apply the biconditional to construct a truth table.
3. Use truth tables to determine the validity of conditional and biconditional statements.
Computer languages use if-then or if-then-else statements as decision statements:
1. If the hypothesis is true, then do something.
2. Or, if the hypothesis is true, then do something; else do something else.
For example, the following representation of computer code creates an if-then-else decision statement:
Check value of variable .
If , then print "Hello, World!" else print "Goodbye".
In this imaginary program, the if-then statement evaluates and acts on the value of the variable . For instance, if , the program would consider the statement  as true and “Hello, World!” would appear on the computer screen. If instead, , the program would consider the statement  as false (because 3 is greater than 1), and print “Goodbye” on the screen.
In this section, we will apply similar reasoning without the use of computer programs.

### Use and Apply the Conditional to Construct a Truth Table

A conditional is a logical statement of the form if , then . The conditional statement in logic is a promise or contract. The only time the conditional,  is false is when the contract or promise is broken.
For example, consider the following scenario. A child’s parent says, “If you do your homework, then you can play your video games.” The child really wants to play their video games, so they get started right away, finish within an hour, and then show their parent the completed homework. The parent thanks the child for doing a great job on their homework and allows them to play video games. Both the parent and child are happy. The contract was satisfied; true implies true is true.
Now, suppose the child does not start their homework right away, and then struggles to complete it. They eventually finish and show it to their parent. The parent again thanks the child for completing their homework, but then informs the child that it is too late in the evening to play video games, and that they must begin to get ready for bed. Now, the child is really upset. They held up their part of the contract, but they did not receive the promised reward. The contract was broken; true implies false is false.
So, what happens if the child does not do their homework? In this case, the hypothesis is false. No contract has been entered, therefore, no contract can be broken. If the conclusion is false, the child does not get to play video games and might not be happy, but this outcome is expected because the child did not complete their end of the bargain. They did not complete their homework. False implies false is true. The last option is not as intuitive. If the parent lets the child play video games, even if they did not do their homework, neither parent nor child are going to be upset. False implies true is true.
The truth table for the conditional statement below summarizes these results.

### Use and Apply the Biconditional to Construct a Truth Table

The biconditional, , is a two way contract; it is equivalent to the statement  A biconditional statement,  is true whenever the truth value of the hypothesis matches the truth value of the conclusion, otherwise it is false.
The truth table for the biconditional is summarized below.

### Check Your Understanding


### Key Concepts

1. The conditional statement, if  then , is like a contract. The only time it is false is when the contract has been broken. That is, when  is true, and  is false.
       
        
2. The biconditional statement,  if and only if , it true whenever  and  have matching true values, otherwise it is false.
       
        
3. Know how to construct truth tables involving conditional and biconditional statements.
4. Use truth tables to analyze conditional and biconditional statements and determine their validity.

### Video

1. Logic Part 8: The Conditional and Tautologies
2. Logic Part 11B Biconditional and Summary of Truth Value Rules in Logic
3. Logic Part 13: Truth Tables to Determine if Argument is Valid or Invalid