Laws of inference are the laws that govern reasoning, i.e.: the correct arguments that follow the rules of logic.
Any reasoning is valid when it follows these laws and it is fallacious when it doesn't; if a valid reasoning is based on true premises, its conclusions are also necessarily true.
P or not P
Every statement is either true or false. / For every statement, either that statement is true, or its denial is true.
Ex.:
"Either it's raining or it's not."
not (P and not P)
No statement can be both true and false. / A statement and its denial cannot be both true.
Ex.:
"It can't be both raining and not raining."
not (not P) <=> P
If the denial of a statement is false, then that statement is true. / Denying the denial of a statement equals affirming that statement.
If a statement is true, then its denial is false. / Affirming a statement equals denying its denial.
Ex.1:
"The light is not off."
therefore: "The light is on."
Ex.2:
"The light is on."
therefore: "The light is not off."
(P implies Q), P => Q
If a statement implies another and the first one is true, then the second one is also true.
Ex.:
"When it rains, the road gets wet."
"It's raining."
therefore: "The road is wet."
(P implies Q), not Q => not P
If a statement implies another and the second one is false, then the first one is also false.
Ex.:
"When it rains, the road gets wet."
"The road isn't wet."
therefore: "It's not raining."
(either P or Q), P => not Q
If two statements are mutually exclusive and one of them is true, then the other one is false.
Ex.:
"This box can only contain either 1 apple or 1 pear"
"There's an apple in the box."
therefore: "There's no pear in the box."
(P or Q), not P => Q
If at least one of two statements is true and one of them is false, then the other one is true.
Ex.:
"This box contains 1 fruit that's either an apple or a pear"
"There's no apple in the box."
therefore: "There's a pear in the box."
not (P and Q) <=> (not P) or (not Q)
If two statements aren't both true, then at least one of them is false.
If at least one of two statements is false, then those statements aren't both true.
Ex.:
"This fruit is not a red apple."
therefore: "This fruit is not red, or it's not an apple."
Ex.:
"This fruit is not red, or it's not an apple."
therefore: "This fruit is not a red apple."
not (P or Q) <=> (not P) and (not Q)
If none of two statements is true, then both of them are false.
If two statements are both false, then none of them is true.
Ex.:
"I don't have an apple nor a pear."
therefore: "I don't have an apple and I don't have a pear."
Ex.:
"I don't have an apple and I don't have a pear."
therefore: "I don't have an apple nor a pear."
(work in progress...)