Advanced Biconditional Logical Indicators

Credit to aobonyo on Quizlet

“If and only if”

• Together, never apart.

• Both conditions are sufficient and necessary for one another

• Both conditions are either "in" or "out" together

Translate to simple English:
Alan attends the meeting if and only if Ben does.

1) If Ben does (attend the meeting), then Alan does.
2) Alan attends the meeting only if Ben does.

Translate to Lawgic:
Alan attends the meeting if and only if Ben does.

Group 1 (if): B → A
(Contrapositive: /A → /B)

Group 2 (only if): A → B
(Contrapositive: /B → /A)

Thus, Biconditional:
3) A ←—→ B , /A ←—→ /B

Biconditional Implication:
Alan attends the meeting if and only if Ben does.

A ←—→ B
/A ←—→ /B (Contrapositive)

Translate to simple English:
Ben goes to the park if Alan goes to the park, but not otherwise.

1) If Alan goes to the park, then Ben goes to the park.
2) If Alan does not go to the park, then Ben does not go.

Translate to Lawgic:
Ben goes to the park if Alan goes to the park, but not otherwise.

Group 1 (if): A → B
(contrapositive: /B → /A)

Group 1 (if, but not otherwise): /A → /B
(contrapositive: B → A)

Thus, Biconditional:
A ←—→ B, /A ←—→ /B

Biconditional Implication:
Ben goes to the park if Alan goes to the park, but not otherwise.

A ←—→ B
/A ←—→ /B

“Except”

• Always apart, never together.

• One condition is "out" while the other is "in" and vice versa.

Translate to simple English:
Alan goes to the park except when Ben goes to the Park.

1) Alan goes to the park unless Ben goes to the park.
(Group 3: Negate Sufficient)

With Group 3 rule applied, we can more simply say:
1) If Alan does not go to the park, then Ben goes.
2) If Ben does not go to the park, then Alan goes.

Translate to Lawgic:
Alan goes to the park except when Ben goes to the Park.

/A --> B
/B --> A (Contrapositive)

Biconditional Implication:
Alan goes to the park except when Ben goes to the Park.

/A <---> B or /B <---> A

“Either/ or…, but not both”

• Always apart, never together.

• One condition is "out" while the other is "in" and vice versa.

• “…,but not both” precludes possibility of “either/or” meaning “both”

Translate to simple English:
Either Alan or Ben goes to the park, but not both.

1) Alan or Ben goes to the park.
2) Alan and Ben cannot both go to the park

Translate to Lawgic:
Either Alan or Ben goes to the park.

1) Group 3: /A --> B
(contrapositive: /B --> A)

2) Group 4: A --> /B
(contrapositive: B --> /A)

BIconditional Implication:
Either Alan or Ben goes to the park, but not both.

/A <---> B

/B <---> A