The fundamental logical concepts – biconditional

Definition: A biconditional is a compound statement formed by combining 2 conditionals under “and.”  Biconditionals are true when both statements have the exact same truth value.

A biconditional actually contains 2 conditional statements. AB means (AB)˄(BA). Either both are true or neither are true.

How can this we see this?
From 2 conditional statements we can infer a biconditional.
AB   (if AB is true)
BA   (if BA is true)
AB   (then we infer that AB is true)

Example
If I’m breathing, then I’m alive
If I’m alive, then I’m breathing
Therefore I’m breathing if and only if I’m alive. Or also: I’m alive if and only if I’m breathing.

And the other way around is also possible: from a biconditional we can infer 2 conditionals.
   AB   (iff AB is true)
AB   (then we infer that AB is true)

   AB
BA   (then we infer that BA is true)

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.

Different interpretations in language:
A biconditional is of the form (AB)˄(BA), expressed as A if B and B if A, or AB, A if and only if B.
Or we say, A implies B and B implies A.
Sometimes ‘if’ is used as a biconditional, depending on the context.

Examples
I’ll buy you a new wallet if you need one.
If the speaker means with ‘if’, whether or not the wallet is needed, then ‘if’ is meant as a biconditional.

It is cloudy if it is raining.
Since it can be cloudy while it’s not raining, ‘if’ here is not meant as a biconditional. (And also not as a conditional ‘conjunction’, but as a time ‘conjunction’. So correct would be ‘when’.)

biconditional  biconditional meaning

A

B

 If and only if it rains, the roofs get wet

T

T

T

If and only if it rains, the roofs get wet (if it rains, the roofs get wet; and if the roofs get wet, it rains)

T

F

F

It is not the case that if it rains, then the roofs don’t get wet

F

F

T

It is not the case that if it doesn’t rain, then the roofs get wet

F

T

F

If and only if it doesn’t rain, the roofs don‘t get wet (if it doesn’t rains, the roofs don‘t get wet; and if the roofs don‘t get wet, it doesn’t rain)
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