With the connectives from The Main Connective and Scope you can now translate real sentences. This lesson handles the three easiest connectives, not, and and or, and the English phrases that hide them.
| English | Symbols |
|---|---|
| $p$ and $q$; both $p$ and $q$; $p$, but $q$; $p$, although $q$; $p$, yet $q$ | $p \land q$ |
| $p$ or $q$; either $p$ or $q$ | $p \lor q$ |
| not $p$; it is not the case that $p$ | $\neg p$ |
| neither $p$ nor $q$ | $\neg p \land \neg q$, or $\neg (p \lor q)$ |
| not both $p$ and $q$ | $\neg (p \land q)$, or $\neg p \lor \neg q$ |
Key idea
Why it works
Worked examples
Worked example: Neither … nor
- "Neither … nor" denies both parts.
- Not $p$, and not $q$.
Answer: $\neg p \land \neg q$ (equivalently $\neg (p \lor q)$)
Worked example: Not both
- The denial covers the conjunction "both $p$ and $q$".
- Write the conjunction in parentheses and deny it.
Answer: $\neg (p \land q)$
Worked example: Three parts
- The comma before "but" marks the main connective, $\land$.
- The first part is $p$.
- The second part is "neither $q$ nor $r$": $\neg q \land \neg r$.
Answer: $p \land (\neg q \land \neg r)$
Worked example: Both denied, joined by or
- Each part is a negation: $\neg p$ and $\neg q$.
- "Either … or" joins them.
Answer: $\neg p \lor \neg q$, which says the same thing as "not both".
Notice what translation leaves out. "Maya plays chess, but Leo does not" suggests a contrast, as if we expected Leo to play. Logic keeps only the claim that both parts are true. The hint of surprise is real in conversation, but it does not change when the sentence is true, so $\land$ is the right symbol.
Common mistake