The Logical Biconditional = Material Equivalence = XNOR = "If and only if"

[Petros Agapetos]

The Logical Biconditional / Material Equivalence
P <=> Q <--equivalent to--> P if and only if Q
P <=> Q <--equivalent to--> P is materially equivalent to Q
P is both a necessary and a sufficient condition for Q.
P <=> Q <--equivalent to--> "P xnor Q", where xnor = exclusive nor, inputs true only when P and Q are either simultaneously true or simultaneously false. Outputs true only if inputs are the same truth value (both P and Q are true or both false), outputs false if the inputs P and Q have different truth values.

P -->Q: "If P then Q"

'If P then Q' can be expressed in the following equivalent ways:

P is sufficient for Q (P => Q)
Q is necessary for P (Q <= P)
~Q is sufficient for ~P (~Q => ~P)
~P is necessary for ~Q (~P <= ~Q)


{P} {Q} {P or Q} {P and Q} {P xor Q} {P xnor Q}
__________________________________________________ _____
T T T T F T
T F T F T F
F T T F T F
F F F F F T


[P V Q]: the disjunction "P or Q" outputs true (T) only if either P is true or Q is true that is [P or Q] includes the option where both P is true and Q is true, and outputs false only if both P and Q are false.

[P & Q] <--> the conjunction "P and Q" <--> outputs true (T) only if both P and Q are true, and outputs false (F) otherwise.

[P xor Q] <--> the exclusive disjunction "P xor Q" outputs true (T) only if exactly one of P or Q is true, and excludes the option where both P and Q are true. [P xor Q] outputs true if the truth values of P and Q are different, and outputs false when the truth values of P and Q are the same.

[P xnor Q] <--> the exclusive nor "P xnor Q" outputs true if P and Q are both either simultaneously true or simultaneously false; outputs true (T) only if the truth values of P and Q are the same, and outputs false otherwise. [P xnor Q] is the logical complement of [P xor Q].

The exclusive nor [P xnor Q] can be stated in the following equivalent ways:
P xnor Q
P iff Q
P if and only if Q
P is a necessary and sufficient condition for Q
Q is a necessary and sufficient condition for P
P <=> Q, where "<=>" is equivalence / logical equality (same truth value outputs in truth table), and it is called the logical biconditional or material equivalence and indicates simultaneous necessity and sufficiency.

[Aldaris]

Man, next time use TeX when making a table.

[Samnium]

I hate logic, first mathematical chapter that you see in higher education (It's not that I hate logic, it's one of the bases of mathematics, it's more the way that the chapter is taught)

[Kamal900]

My brain is fucked..