It was raining → the roads were slippery

R → S
if R then S

R: it was raining; S: the roads were slippery

He was home ← the light was on

H ← L
L → H
if L then H

H: he was home; L: the light was on

Ann and Bill helped ← Carol was away

(Ann helped ∧ Bill helped) ← Carol was away

(A ∧ B) ← C
C → (A ∧ B)
if C then both A and B

A: Ann helped; B: Bill helped; C: Carol was away

Sam will help ∧ Tom will help if we ask him

Sam will help ∧ (Tom will help ← we will ask Tom to help)

S ∧ (T ← A)
S ∧ (A → T)
both S and if A then T

A: we will ask Tom to help; S: Sam will help; T: Tom will help

it was warm → they ate outside provided it didn’t rain

it was warm → (they ate outside ← it didn’t rain)

it was warm → (they ate outside ← ¬ it rained)

W → (O ← ¬ R)
W → (¬ R → O)
if W then if not R then O

O: they ate outside; R: it rained; W: it was warm

the new project was approved → Carol started work on the new probject and so did Dave if he was finished with the last one

the new project was approved → (Carol started work on the new probject ∧ Dave started work on the new probject if he was finished with the last one)

the new project was approved → (Carol started work on the new probject ∧ (Dave started work on the new probject ← Dave was finished with the last project))

A → (C ∧ (D ← F))
A → (C ∧ (F → D))
if A then both C and if F then D

A: the new project was approved; C: Carol started work on the new probject; D: Dave started work on the new probject; F: Dave was finished with the last project

If he found the instructions, Tom set up the new machine ∧ if Tom didn’t find the instructions, he packed up the old machine

(Tom found the instructions → Tom set up the new machine) ∧ (Tom didn’t find the instructions → Tom packed up the old machine)

(Tom found the instructions → Tom set up the new machine) ∧ (¬ Tom found the instructions → Tom packed up the old machine)

(F → S) ∧ (¬ F → P)
both if F then S and if not F then P

F: Tom found the instructions; P: Tom packed up the old machine; S: Tom set up the new machine

¬ I’ll see it → ¬ I’ll believe it

I won’t see it → I won’t believe it

If I don’t see it, I won’t believe it

It was sunny → ¬ (it rained ∨ it snowed)

It was sunny → ¬ it rained or snowed

It was sunny → it didn’t rain or snow

If it was sunny, it didn’t rain or snow

¬ the set works ← ¬ (the set is plugged in ∧ ¬ the set is broken)

¬ the set works ← ¬ (the set is plugged in ∧ the set isn’t broken)

¬ the set works ← ¬ (the set is plugged in and isn’t broken)

The set doesn’t work if it isn’t both plugged in and unbroken

¬ (Adams will back out ∨ Brown will back out) → (the deal will go through ← ¬ (Collins will have trouble with financing ∨ Davis will have trouble with financing))

¬ Adams or Brown will back out → (the deal will go through ← ¬ (Collins or Davis will have trouble with financing))

¬ Adams or Brown will back out → (the deal will go through ← neither Collins nor Davis will have trouble with financing)

¬ Adams or Brown will back out → the deal will go through provided neither Collins nor Davis has trouble with financing

If neither Adams nor Brown backs out, the deal will go through provided neither Collins nor Davis has trouble with financing