After completing this section, you should be able to:
Computer languages use if-then or if-then-else statements as decision statements:
For example, the following representation of computer code creates an if-then-else decision statement:
Check value of variable i i .
In this imaginary program, the if-then statement evaluates and acts on the value of the variable i i . For instance, if i = 0 i = 0 , the program would consider the statement i < 1 i < 1 as true and “Hello, World!” would appear on the computer screen. If instead, i = 3 i = 3 , the program would consider the statement i < 1 i < 1 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.
The Countess of Lovelace, Ada Lovelace, is credited with writing the first computer program. She wrote an algorithm to work with Charles Babbage’s Analytical Engine that could compute the Bernoulli numbers in 1843. In doing so, she became the first person to write a program for a machine that would produce more than just a simple calculation. The computer programming language ADA is named after her.
Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 34 Ada Lovelace: English (1815-1852)” pp. 272-278, Math Makers: The Lives and Works of 50 Famous Mathematicians, Prometheus Books, 2019.
A conditional is a logical statement of the form if p p , then q q . The conditional statement in logic is a promise or contract. The only time the conditional, p → q , p → q , 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.
| p p | q q | p → q p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Notice that the only time the conditional statement, p → q , p → q , is false is when the hypothesis, p p , is true and the conclusion, q q , is false.
