Logical biconditional

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In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis (or antecedent) and p is a conclusion (or consequent).[1] This is often abbreviated p iff q. The operator is denoted using a doubleheaded arrow (↔), a prefixed E (Epq), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to (p → q) ∧ (q → p), or the XNOR (exclusive nor) boolean operator. It is equivalent to "(not p or q) and (not q or p)". It is also logically equivalent to "(p and q) or (not p and not q)", meaning "both or neither".

The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false.

In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the sets a and b coincide: they are identical. This does not mean that the concepts have the same meaning. Examples: "triangle" and "trilateral", "equiangular triangle" and "equilateral triangle". The antecedent is the subject and the consequent is the predicate of a universal affirmative proposition.

In the propositional interpretation, ab means that a implies b and b implies a; in other words, that the propositions are equivalent, that is to say, either true or false at the same time. This does not mean that they have the same meaning. Example: "The triangle ABC has two equal sides", and "The triangle ABC has two equal angles". The antecedent is the premise or the cause and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis.

A common way of demonstrating a biconditional is to use its equivalence to the conjunction of two converse conditionals, demonstrating these separately.

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.[citation needed] 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 the thesis the necessary condition of the hypothesis; that is to say, it is sufficient that the hypothesis be true for the thesis to be true; while it is necessary that the thesis be true for the hypothesis to be true also. When a theorem and its reciprocal are true we say that its hypothesis is the necessary and sufficient condition of the thesis; that is to say, that it is at the same time both cause and consequence.

Definition[edit]

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.

Truth table[edit]

The truth table for ~A \leftrightarrow B (also written as A ≡ B, A = B, or A EQ B) is as follows:

INPUT OUTPUT
A B A ~ \leftrightarrow ~ B
0 0 1
0 1 0
1 0 0
1 1 1


More than two statements combined by ~\leftrightarrow~ are ambiguous:

~x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow ... \leftrightarrow x_n may be meant as ~(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow ...) \leftrightarrow x_n,

or may be used to say that all ~x_i~ are together true or together false: (~x_1 \and ... \and x_n~)~\or~(\neg x_1 \and ... \and \neg x_n)

Only for zero or two arguments this is the same.

The following truth tables show the same bit pattern only in the line with no argument and in the lines with two arguments:

~x_1 \leftrightarrow ... \leftrightarrow x_n
meant as equivalent to
\neg~(\neg x_1 \oplus ... \oplus \neg x_n)

The central Venn diagram below,
and line (ABC  ) in this matrix
represent the same operation.
~x_1 \leftrightarrow ... \leftrightarrow x_n
meant as shorthand for
(~x_1 \and ... \and x_n~)
\or~(\neg x_1 \and ... \and \neg x_n)

The Venn diagram directly below,
and line (ABC  ) in this matrix
represent the same operation.


The left Venn diagram below, and the lines (AB    ) in these matrices represent the same operation.

Venn diagrams[edit]

Red areas stand for true (as in Venn0001.svg for and).

Venn1001.svg
The biconditional of two statements
is the negation of the exclusive or:
~A \leftrightarrow B~~\Leftrightarrow~~\neg(A \oplus B)

Venn1001.svg \Leftrightarrow \neg Venn0110.svg

Venn 0110 1001.svg
The biconditional and the
exclusive or of three statements
give the same result:

~A \leftrightarrow B \leftrightarrow C~~\Leftrightarrow
~A \oplus B \oplus C

Venn 1001 1001.svg \leftrightarrow Venn 0000 1111.svg ~~\Leftrightarrow~~

Venn 0110 0110.svg \oplus Venn 0000 1111.svg ~~\Leftrightarrow~~ Venn 0110 1001.svg

Venn 1000 0001.svg
But ~A \leftrightarrow B \leftrightarrow C
may also be used as an abbreviation
for (A \leftrightarrow B) \and (B \leftrightarrow C)

Venn 1001 1001.svg \and Venn 1100 0011.svg ~~\Leftrightarrow~~ Venn 1000 0001.svg

Properties[edit]

commutativity: yes

A \leftrightarrow B     \Leftrightarrow     B \leftrightarrow A
Venn1001.svg     \Leftrightarrow     Venn1001.svg

associativity: yes

~A ~~~\leftrightarrow~~~ (B \leftrightarrow C)     \Leftrightarrow     (A \leftrightarrow B) ~~~\leftrightarrow~~~ ~C
Venn 0101 0101.svg ~~~\leftrightarrow~~~ Venn 1100 0011.svg     \Leftrightarrow     Venn 0110 1001.svg     \Leftrightarrow     Venn 1001 1001.svg ~~~\leftrightarrow~~~ Venn 0000 1111.svg

distributivity: Biconditional doesn't distribute over any binary function (not even itself),
but logical disjunction (see there) distributes over biconditional.

idempotency: no

~A~ ~\leftrightarrow~ ~A~     \Leftrightarrow     ~1~     \nLeftrightarrow     ~A~
Venn01.svg ~\leftrightarrow~ Venn01.svg     \Leftrightarrow     Venn11.svg     \nLeftrightarrow     Venn01.svg

monotonicity: no

A \rightarrow B     \nRightarrow     (A \leftrightarrow C) \rightarrow (B \leftrightarrow C)
Venn 1011 1011.svg     \nRightarrow     Venn 1101 1011.svg     \Leftrightarrow     Venn 1010 0101.svg \rightarrow Venn 1100 0011.svg

truth-preserving: yes
When all inputs are true, the output is true.

A \and B     \Rightarrow     A \leftrightarrow B
Venn0001.svg     \Rightarrow     Venn1001.svg

falsehood-preserving: no
When all inputs are false, the output is not false.

A \leftrightarrow B     \nRightarrow     A \or B
Venn1001.svg     \nRightarrow     Venn0111.svg

Walsh spectrum: (2,0,0,2)

Nonlinearity: 0 (the function is linear)

Rules of inference[edit]

Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

Biconditional introduction[edit]

Biconditional introduction allows you 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, equally inferrable, "I'm alive if and only if I'm breathing."

 B → A   
 A → B   
 ∴  A ↔ B
 B → A   
 A → B   
 ∴  B ↔ A

Biconditional elimination[edit]

Biconditional elimination allows one to infer a conditional from a biconditional: if ( A B ) is true, then one may infer one direction of the biconditional, ( A B ) and ( B A ).

For example, if it's true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, I'm alive; likewise, it's true that if I'm alive, I'm breathing.

Formally:

 ( A ↔ B )  
 ∴ ( A → B )

also

 ( A ↔ B )  
 ∴ ( B → A )

Colloquial usage[edit]

One unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Another is "a if and only if b". Slightly more formally, one could say "b implies a and a implies b". The plain English "if'" may sometimes be used as a biconditional. One must weigh context heavily.

For example, "I'll buy you a new wallet if you need one" may be meant 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 not meant as a biconditional, since it can be cloudy while not raining.

See also[edit]

Notes[edit]

  1. ^ Handbook of Logic, page 81

References[edit]

This article incorporates material from Biconditional on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.