Propositional Logic:
Introduction to Propositional Logic

If you've done any programming so far, you might be familiar with working with if statements and while loops. These both work with propositional logic to decide whether to branch the program, or continue looping a set of instructions.

Learning about propositional logic can really help with writing better programming logic, and also help you debugging your logic.

Propositions

Some examples of propositional statements are:

These are statements that can result in true (Fran DOES have a cat!) or false (Actually, the cat is orange). We could use these statements in a program, with an if statement...

string catColor;
cout << "What color is the cat? ";
cin >> catColor;

if ( catColor == "black" )
{
  // proposition was true...
}
else
{
  // proposition was false...
}

We can essentially think of it as a yes/no question of "Is the cat black?", where we could have different sets of operations to perform if the statement is true (the cat is black) or if the statement is false (the cat is not black).

On the flip side, we could not write a program with questions like "What is your favorite color?" -- That is not a yes/no question, and the computer wouldn't understand what that type of question is. Instead, if we really wanted to define behavior based on favorite color, we would need a series of if / else if statements asking about "is favorite color red?", "is favorite color blue?", and so on...

string favColor;
cout << "What is your favorite color? ";
cin >> favColor;

if ( favColor == "red" )
{
  // Code for red as favorite color...
}
else if ( favColor == "orange" )
{
  // Code for orange as favorite color...
}
else if ( favColor == "yellow" )
{
  // Code for yellow as favorite color...
}
// ... and so on ...

Compound Propositions

We can also create a compound proposition using the logical operators AND, OR, and NOT.

AND ∧

In C++, Java, and C#, the && symbol is used for a logical "AND". In discrete math, we often use ∧.

If we have two propositional statements, e.g., p and q, then the compound proposition p ∧ q is only true when both sub-statements are also true. If even one of the sub-statements are false, then the entire compound statement is false.

OR ∨

In C++, Java, and C#, the || symbol is used for a logical "OR". In discrete math, we often use ∨.

If we have two propositional statements, e.g., p and q, then the compound proposition p ∨ q is true when one or both sub-statements are also true. The result is only false when all sub-statements are false.

NOT ¬

In C++, Java, and C#, the ! symbol is used for a logical "NOT". In discrete math, we often use ¬ (or sometimes ~).

NOT operates on one statement. Let's say we have the proposition p. If p is true, then the result of ¬p is false. If p is false, then the result of ¬p is true.

if ( a != b ) // ...

bool done = false;
while ( !done )   // While the program is NOT done...
{
  // ... then show main menu, process stuff, whatever ....
}

And now... A dramatic workplace play: The printer is offline - AND - The printer is out of paper.

Jim: Hey Janet, we can't print documents; the printer is offline AND the printer is out of paper.

Janet: Alright Jim. Thanks for the heads up.

( ... later ... )

Rahaf: Hey Janet, looks like the printer is offline, but it has paper.

Janet: Has paper?? Jim lied to me!!

( ... later ... )

Rose: Hey Janet, the printer is online but it's out of paper.

Janet: Well, which is it?!?!

( ... later ... )

Anuraj: I just printed out some docs... of course the printer is online and it has paper!

Janet: That's the complete opposite!!

Propositional variables

When working with propositional logic, we commonly shorten a propositional statement to a variable. For example, we can represent "she is in Discrete Math" with d, and "she is majoring in Computer Science" with c. We could then write out statements like:

• d ∧ c
  She is in Discrete Math AND she is majoring in Computer Science.
• d ∧ ¬c
  She is in Discrete Math AND she is NOT majoring in Computer Science.
• ¬d ∧ c
  She is NOT in Discrete Math AND she is majoring in Computer Science.
• ¬d ∧ ¬c
  She is NOT in Discrete Math AND she is NOT majoring in Computer Science.

Propositional variables should be one letter in length. (though when programming, you should use longer variable names :)

(( Work in progres... ))