A byconditional (called also an equivalence) is a sentence of the form

A if and only if B

or in symbols:

A ↔ B

A byconditional is true just in case the two clauses are of the same truth value (i.e. either both true or both false).

A byconditional is equivalent to a conjunction of two conditionals of this form:

A ↔ B             is equivalent to          (A → B) and (B → A)

Where does the expression “if and only if” come from?
Remember that “A, only if B” is symbolized as A → B. “A, if B” is symbolized as B → A. Taken together, i.e. in a conjunction, they give us “A if and only if B”.

Since we know that A ↔ B is equivalent to (A → B) and (B → A), and we know that (A → B) and (B → A) is equivalent to (~B → ~A) and (~A → ~B) (we just replace each conditional by its contrapositive), we can infer that

A ↔ B             is equivalent to             ~A ↔ ~B

Prof. Bazett of the University of Cincinnati introduces you to biconditionals in the following video (bi-conditionals combine two conditionals):

### Biconditional Statements | “if and only if” Video Summary

• 0:13 – There are cases where both the original conditional and its converse are true. In these cases, we can have the biconditional.
• 0:24 – The biconditional uses a double-sided arrow. It means that both P → Q and Q → P are true.
• 0:45 – For example: If I study hard, then I will pass. We can diagram this as: P → Q.
• 0:58 – It can also be the other way around: If I pass, then I studied hard. We can diagram this as Q → P.
• 1:11 – Combining these statements make a single conjunctive statement. It becomes: If I study hard, then I will pass, and if I pass, then I studied hard.
• 1:50 – Combining them in a shorter way, the statement becomes: I will pass if and only if I study hard. Diagram this as: pass ↔ study.
• 2:09 – The if and only if () phrase serves a shorthand for referring to biconditionals which means that both directions of a conditional statement are true.

This video is purpose-built for the LSAT and expands into combining biconditionals and contrapositives.

### Biconditionals Video Summary

• 0:09 – Consider the statement: The party was a success if and only if everyone had a good time.
• 0:21 – This is a combination of two conditional statements. One is: The party was a success if everyone had a good time. The other is: The party was a success only if everyone had a good time.
• 1:02 – In this case, the sufficient condition is also a necessary condition, and the necessary condition is also a sufficient condition. This type of statement is called a biconditional.
• 1:26 – Biconditionals always use the phrase “if and only if.”
• 1:30 – Looking at “if,” that conditional statement has the necessary condition first and the sufficient condition second. Meanwhile, looking at “only if,the sufficient condition is first and the necessary condition is second.
• 1:45 – Diagram the first statement as: If the party was a success then everyone had a good time. (success → good time; contrapositive: ~good time → ~success)
• 2:21 – Diagram the second statement as: If everyone had a good time then the party was a success. (good time → success; contrapositive: ~success → ~good time)
• 2:51 – Infer that if either of the conditions is negative then the other one has to be negative as well.
• 3:29 – Use the two-headed arrow (↔) to symbolize when a conditional statement is a biconditional (the conditions are both sufficient and necessary). (success ↔ good time; contrapositive: ~success ↔ ~good time)
• 4:20 – Recap: A biconditional is two conditionals taken together where the sufficient condition is also a necessary condition and vice versa. The conditions in a biconditional always have to go together.

Next LSAT: Jun 10/Jun 11

We know that by claiming a byconditional:

A ↔ B

we claim that the two clauses A and B have the same truth value (i.e. they are either both true or both false).

How do we say that some two clauses A and B have the opposite truth value?

In one of these ways (it is not important which one, because they are all equivalent):

~A ↔ B
A ↔ ~B
B ↔ ~A
~B ↔ A

#### Biconditional Examples

The company gains profit if and only if it earns more than it spends. The company does not earn more than it spends.

If the above statements are true, which one of the following must also be true?
(A) If the company does not earn more than it spends, then it does not gain profit.
(B) The company does not gain profit.

(A) If the company does not earn more than it spends, then it does not gain profit.
&
(B) The company does not gain profit.

 Statement Symbols Valid/Invalid Description 1. If the company gains profit, then it earns more than it spends. company profit → earn more Given Premise 1 2. If the company earns more than it spends, then it gains profit. earn more → company profit Given Premise 2 3. If the company does not earn more than it spends, then it does not gain profit. ~earn more → ~company profit Valid Contrapositive of premise 1 4. If the company does not gain profit, then it does not earn more than it spends. ~company profit → ~earn more Valid Contrapositive of premise 2

Premise 1:

If the machine works (MW), then the parts are installed correctly (IC).

MW → IC

Premise 2:

If the parts are installed correctly (IC), then the machine will work (MW).

IC → MW

### Conclusion

Biconditional:

The machine will work if and only if the parts are installed correctly.

MW ↔ IC

Premise 1:

If I graduated with honors (GH), then I got a grade of 95% or higher (95).

GH → 95

Premise 2:

If I got a grade of 95% or higher (95), then I will graduate with honors (GH).

95 → GH

### Conclusion

Biconditional:

I will graduate with honors if and only if I get a grade of 95% or higher.

GH ↔ 95

Premise 1:

If she attended the birthday party (BP), then she finished all her tasks for that day (FT).

BP → FT

Premise 2:

If she finished all her tasks for that day (FT), then she will attend the birthday party (BP).

FT → BP

### Conclusion

Biconditional:

She will attend the birthday party if and only if she finishes all her tasks for that day.

BP ↔ FT

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