The Not-So-Naughty Not!

If you move the "NOT" function in biconditional statements or exclusive disjunctions, it doesn't change their truth values in any cells in the column of a truth table! Observe below:

P Q ~P ~Q P = ~Q ~P = Q ~P ≠ Q P ≠ ~Q
TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE
TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE
FALSE TRUE TRUE FALSE TRUE TRUE FALSE FALSE
FALSE FALSE TRUE TRUE FALSE FALSE TRUE TRUE

I colored the compound statements in this truth table. The ones highlighted in the same color have the exact same truth values in each row. In other words, they're logically equivalent.

(P = ~Q) = (~P = Q)
TRUE
TRUE
TRUE
TRUE
(P ≠ ~Q) = (~P ≠ Q)
TRUE
TRUE
TRUE
TRUE

These 2 compound statements are tautologies, which are compound statements that are ALWAYS true! These special statements cannot be used to tell a lie! Logically equivalent statements can be used to create tautologies! (It especially works with biconditionals.)

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.

This math trick still works if you put more than 2 simple statements in a biconditional or exclusive disjunction!

R P Q ~R ~P ~Q P = Q = ~R P = ~Q = R ~P = Q = R
TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE TRUE
TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE
TRUE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE
FALSE TRUE TRUE TRUE FALSE FALSE TRUE TRUE TRUE
FALSE TRUE FALSE TRUE FALSE TRUE FALSE FALSE FALSE
FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE

Again, I colored the logically equivalent compound statements.

S R P Q ~S ~R ~P ~Q P ≠ Q ≠ R ≠ ~S P ≠ Q ≠ ~R ≠ S P ≠ ~Q ≠ R ≠ S ~P ≠ Q ≠ R ≠ S
TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE
TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
TRUE TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE
TRUE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
TRUE FALSE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE
TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE
TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
FALSE TRUE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE TRUE TRUE
FALSE TRUE FALSE TRUE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE
FALSE TRUE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE
FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE TRUE TRUE TRUE TRUE
FALSE FALSE TRUE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE
FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Another fact: biconditionals & exclusive disjunctions are negations of each other!

P ~P P = ~P P ≠ ~P
TRUE FALSE FALSE TRUE
FALSE TRUE FALSE TRUE

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© Derek Cumberbatch