Negating something twice brings you back to where you started. Saying 'not not P' is just another way of saying 'P'. It's like undoing a negative to get back to the original statement.
Any statement or proposition
Negation operator, meaning 'not'
The negation of the negation of P
If it's not not raining, then it is raining.
Simplifying logical expressions, making proofs clearer
Implication As Disjunction
Formula:
P→Q=¬P∨Q
Saying 'if P then Q' is the same as saying 'either not P or Q'. It's a way to express implications using 'or'.
Statements or propositions
Logical 'implies' operator
Negation operator
Logical 'or' operator
If it's raining, then the ground is wet equals either it's not raining or the ground is wet.
Rewriting logical implications, simplifying expressions in logic circuits
Contrapositive
Formula:
P→Q=¬Q→¬P
An implication is equivalent to its contrapositive. So 'if P then Q' means the same as 'if not Q then not P'.
Statements or propositions
Logical 'implies' operator
Negation operator
If the alarm rings, then I wake up equals if I don't wake up, then the alarm didn't ring.
Proof techniques, logical reasoning in arguments
Biconditional Definition
Formula:
P↔Q=(P→Q)∧(Q→P)
Saying 'P if and only if Q' means both 'if P then Q' and 'if Q then P' happen together.
Statements or propositions
Logical 'if and only if' operator
Logical 'implies' operator
Logical 'and' operator
I will go if and only if you go equals if I go then you go, and if you go then I go.