Translation runs both ways. After Biconditionals and Longer Sentences, you should be able to read a formula aloud in English that keeps its exact meaning. This is how you check your own translations, and how you explain a proof to someone who does not read symbols.
Key idea
Why it works
| Formula | A faithful reading |
|---|---|
| $\neg (p \land q)$ | It is not the case that both $p$ and $q$ (not both $p$ and $q$). |
| $\neg p \land q$ | Not $p$, and $q$. |
| $(p \lor q) \to r$ | If either $p$ or $q$, then $r$. |
| $p \lor (q \land r)$ | Either $p$, or both $q$ and $r$. |
| $p \to (q \to r)$ | If $p$, then if $q$, then $r$. |
Worked examples
Worked example: Not both
- The main connective is the $\neg$ in front of the parentheses.
- It denies the conjunction as a whole.
Answer: It is not the case that both the oven is hot and the dough has risen.
Worked example: A conditional with a compound antecedent
- Main connective: $\to$.
- Antecedent: both $p$ and $q$.
Answer: If both the oven is hot and the dough has risen, then the bread bakes.
Worked example: Check a reading
- That English groups "either … or" first, so it says $(p \lor q) \land r$.
- The formula groups $q \land r$ instead.
Answer: No. A faithful reading is "Either the oven is hot, or both the dough has risen and the bread bakes."
A good habit: after you translate a sentence into symbols, read your formula back into English with these phrases and compare it with the original. If the two sentences would be true in different situations, one of the translations has a mistake, and the reading usually shows where the grouping went wrong.
Common mistake