The Staircase of Truth

Here's another math trick that's about logic instead of numbers! Can you guess how this math trick got its name?

P	Q	~P	~Q	P & Q	P & ~Q	~P & Q	~P & ~Q
TRUE	TRUE	FALSE	FALSE	TRUE	FALSE	FALSE	FALSE
TRUE	FALSE	FALSE	TRUE	FALSE	TRUE	FALSE	FALSE
FALSE	TRUE	TRUE	FALSE	FALSE	FALSE	TRUE	FALSE
FALSE	FALSE	TRUE	TRUE	FALSE	FALSE	FALSE	TRUE

Look at the truth values of the columns with an ampersand(&). This punctuation mark does the logical operation known as conjunction, when people use the word "AND".

Conjunctions are only true when every input inside them is true; if just 1 input is false, then it makes the whole conjunction false!

It also matters where the tildes are, if you examine the compound statements closely.

The tilde (~) represents the negation function in logic. When you use the word not in a sentence, you are using the negation function. It flips the truth value of a statement.

Let's see this math trick with 3 simple statements!

R & P & Q

R & P & ~Q

R & ~P & Q

R & ~P & ~Q

~R & P & Q

~R & P & ~Q

~R & ~P & Q

~R & ~P & ~Q

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

Now there are 8 columns with a compound statement! If there were 4 simple statements, then there would be 16 compound statements! The number of compound statements would increase in powers of 2; the number of simple statements would be the exponent! And, of course, 2 is the base!

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