In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements
P
Q
P
Q
P
Q
P
Q
Nowadays, notations to represent equivalence include
\leftrightarrow,\Leftrightarrow,\equiv
P\leftrightarrowQ
(P → Q)\land(Q → P)
(P\landQ)\lor(\negP\land\negQ)
Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.
In the conceptual interpretation, means "All 's are 's and all 's are 's". In other words, the sets and coincide: they are identical. However, this does not mean that and need to have the same meaning (e.g., could be "equiangular trilateral" and could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).
In the propositional interpretation,
P\leftrightarrowQ
A common way of demonstrating a biconditional of the form
P\leftrightarrowQ
P → Q
Q → P
P → Q
\negP → \negQ
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
It is often said that the hypothesis is the sufficient condition of the thesis, and that the thesis is the necessary condition of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time.
Notations to represent equivalence used in history include:
=
=
x,y
x=y
=
\equiv
\sim
\rightleftarrows
\sim
\leftrightarrow
\rightleftarrows,\sim
\sim
\leftrightarrow
E
Q
\supset\subset
\Leftrightarrow
\subset\supset
and so on. Somebody else also use
\operatorname{EQ}
\operatorname{EQV}
Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The following is a truth table for
A\leftrightarrowB
When more than two statements are involved, combining them with
\leftrightarrow
x1\leftrightarrowx2\leftrightarrowx3\leftrightarrow … \leftrightarrowxn
may be interpreted as
(((x1\leftrightarrowx2)\leftrightarrowx3)\leftrightarrow … )\leftrightarrowxn
or may be interpreted as saying that all are jointly true or jointly false:
(x1\land … \landxn)\lor(\negx1\land … \land\negxn)
As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:
The left Venn diagram below, and the lines (AB ) in these matrices represent the same operation.
Red areas stand for true (as in for and).
| < | --- START LEFT TABLE IN TABLE ---> | < | --- START CENTRAL TABLE IN TABLE ---> | < | --- START RIGHT TABLE IN TABLE ---> |
|---|
Yes
A\leftrightarrowB | \Leftrightarrow | B\leftrightarrowA | |
| \Leftrightarrow |
Yes
~A | ~~~\leftrightarrow~~~ | (B\leftrightarrowC) | \Leftrightarrow | (A\leftrightarrowB) | ~~~\leftrightarrow~~~ | ~C | |||
~~~\leftrightarrow~~~ | \Leftrightarrow | \Leftrightarrow | ~~~\leftrightarrow~~~ |
Biconditional doesn't distribute over any binary function (not even itself), but logical disjunction distributes over biconditional.
No
~A~ | ~\leftrightarrow~ | ~A~ | \Leftrightarrow | ~1~ | \nLeftrightarrow | ~A~ | |
~\leftrightarrow~ | \Leftrightarrow | \nLeftrightarrow |
Monotonicity
No
A → B | \nRightarrow | (A\leftrightarrowC) | → | (B\leftrightarrowC) | |||
| \nRightarrow | \Leftrightarrow | → |
Truth-preserving: Yes
When all inputs are true, the output is true.
A\landB | ⇒ | A\leftrightarrowB | |
| ⇒ |
Falsehood-preserving: No
When all inputs are false, the output is not false.
A\leftrightarrowB | \nRightarrow | A\lorB | |
| \nRightarrow |
(2,0,0,2)
Nonlinearity: 0 (the function is linear)
See main article: Rules of inference. Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.
See main article: Biconditional introduction. Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A if and only if B.
For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:
B → A A → B ∴ A ↔ B
B → A A → B ∴ B ↔ A
Biconditional elimination allows one to infer a conditional from a biconditional: if A ↔ B is true, then one may infer either A → B, or B → A.
For example, if it is true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, then I'm alive; likewise, it's true that if I'm alive, then I'm breathing. Or more schematically:
A ↔ B ∴ A → B
A ↔ B ∴ B → A
One unambiguous way of stating a biconditional in plain English is to adopt the form "b if a and a if b"—if the standard form "a if and only if b" is not used. Slightly more formally, one could also say that "b implies a and a implies b", or "a is necessary and sufficient for b". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition[15]). In which case, one must take into consideration the surrounding context when interpreting these words.
For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.