Biconditional
The biconditional is a logical connective that establishes an “if and only if” relationship. It is typically symbolized using a double-sided arrow.
a ↔ b
Recall from the lesson on the Implication, that the implication was made up of two parts: the sufficient condition and the necessary condition.
In the same way that the arrow of an implication pointed from the sufficient condition to the necessary condition, the double-arrow can be understood as symbolizing that each statement is both sufficient and necessary for the other statement.
Thus, if one statement is true, the other must also be true. Furthermore, if one statement is false, the other must also be false.
Let us return to our example for this: “The dog barks (B) if and only if the cat runs away (R)”.
B ↔ R
If the dog barks, what does that entail? Yup, the cat runs away.
If the cat runs away, what does that entail? You guessed it, the dog barks.
On the other side of the logical spectrum:
If the dog does NOT bark, what does that entail? According to our biconditional, it means the cat did NOT run away.
Similarly, if the cat does not run away, it means the dog did NOT bark.
Since the truth-property of one statement dictates the truth-property of the other statement, the biconditional is also sometimes referred to as “equivalence”.
Speaking of equivalence, since we have covered most the necessary logical connectives at this point, we can state that:
a ↔ b
Is equivalent to:
(a ∧ b) ∨ (¬a ∧ ¬b)
At this point, you should know how to (mostly) read that: a disjunction made up of two conjunctions, wherein the first conjunction asserts that both a and b are true; and the second conjunction asserts that both are… Well, you might be able to already guess it, but we still need to cover the negation symbol.
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app now:
The biconditional is a logical connective that establishes an “if and only if” relationship. It is typically symbolized using a double-sided arrow.
a ↔ b
Recall from the lesson on the Implication, that the implication was made up of two parts: the sufficient condition and the necessary condition.
In the same way that the arrow of an implication pointed from the sufficient condition to the necessary condition, the double-arrow can be understood as symbolizing that each statement is both sufficient and necessary for the other statement.
Thus, if one statement is true, the other must also be true. Furthermore, if one statement is false, the other must also be false.
Let us return to our example for this: “The dog barks (B) if and only if the cat runs away (R)”.
B ↔ R
If the dog barks, what does that entail? Yup, the cat runs away.
If the cat runs away, what does that entail? You guessed it, the dog barks.
On the other side of the logical spectrum:
If the dog does NOT bark, what does that entail? According to our biconditional, it means the cat did NOT run away.
Similarly, if the cat does not run away, it means the dog did NOT bark.
Since the truth-property of one statement dictates the truth-property of the other statement, the biconditional is also sometimes referred to as “equivalence”.
Speaking of equivalence, since we have covered most the necessary logical connectives at this point, we can state that:
a ↔ b
Is equivalent to:
(a ∧ b) ∨ (¬a ∧ ¬b)
At this point, you should know how to (mostly) read that: a disjunction made up of two conjunctions, wherein the first conjunction asserts that both a and b are true; and the second conjunction asserts that both are… Well, you might be able to already guess it, but we still need to cover the negation symbol.
Want an interactive way to learn logic?
Enhance your critical thinking skills with Symbols Logic. Unlock access to the first studycard and the first level of all activities for free. Get the app now: