(Material) Biconditional

A compound logical statement which is true exactly when both sides have the same truth value.

Example of a biconditional statement:

A number is even exactly when it is divisible by two without remainder.

• Logical biconditional

A biconditional is true if both sub-statements have identical truth values.

This is a special case ofthe conditional: For every true biconditional statement A ≡ B, it holds that both A → B and B → A must be true.

It follows that:

A → B   –   if A, then B.
∧ B → A   –   and, if B, then A.
∴ A ≡ B   –   Therefore: B if and only if A.

In the context of this website, the identity sign (≡, pronounced: “if and only if” or also: “is identical to”) is used for biconditionals. In other publications, symbols like ↔, ⇔, or occasionally the tilde symbol (~) are used. Especially in computer sciences and related fields, the neologism iff, (short for: “if and only if”) is common.

The biconditional is a very expressive logical operator, which also means that outside of formal systems (such as mathematics and logic) there are actually very few practical uses where such statements would hold.

In particular, this includes the following situations:

Obviously, biconditional statements are always valid if antecedent and consequent (A and B) are identical (e.g. “if it rains, it rains”).

This also applies to situations where the tautology only exists indirectly, for example through the definitions of the terms. An example from mathematics was already stated above: “if a number is divisible by 2, it is even”, which also entails: “if a number is not even, it is not divisible by 2”, since “even” and “divisible by 2” are synonymous by definition.

Such tautologies can be constructed via several intermediate steps, as the following example shows:

If today is Monday, then the day after tomorrow is Wednesday.

This is tautological, because “Wednesday” can be defined as “the day after Tuesday” and Tuesday in turn as “the day after Monday”. It follows that Wednesday is by definition “two days after Monday”. Thus the statement is tautological; Consequently the converse is also true:

If the day after tomorrow is Wednesday, then today is Monday.

Even independently of a true tautology, A and B can describe identical sets: In set theory this can be described as follows: 𝔸 ∖ 𝔹 = ∅ (the difference set of A with B is empty).

For example, the following sets:

𝔸 = All astronauts who flew to the moon.
𝔹 = Everybody who has taken rock samples from the moon.

Even though 𝔸 and 𝔹 do not necessarily describe the same set (it is possible to fly to the moon without bringing rock samples), the difference set between these groups is empty*. Therefore, the biconditional A ≡ B is valid.

The formal logical fallacy “Denying the antecedent” is based on a confusion of conditional and biconditional.

• Conditional
• Denying the antecedent

• Logical biconditional on Wikipedia