Conditionals are the trickiest part of translation, because English has many ways to say "if … then" and some of them put the parts in reverse order. You met the basic idea in Hypothetical Propositions; here we make it exact. Throughout, $p$ stands for one simple statement and $q$ for another.
| English | Symbols |
|---|---|
| if $p$, then $q$; $q$ if $p$; $q$ provided that $p$; whenever $p$, $q$ | $p \to q$ |
| $p$ only if $q$ | $p \to q$ |
| $p$ is sufficient for $q$ | $p \to q$ |
| $q$ is necessary for $p$; $p$ requires $q$ | $p \to q$ |
| $p$ unless $q$ | $\neg q \to p$, which is equivalent to $p \lor q$ |
Key idea
Why it works
Worked examples
Worked example: Only if
- "Only if" introduces the consequent: having a ticket.
- So the antecedent is getting in.
Answer: $p \to q$
Worked example: Necessary
- A necessary condition is the consequent.
- Getting in needs a ticket: if you get in, you have one.
Answer: $p \to q$ (the same claim as the last example)
Worked example: Unless
- "$X$ unless $Y$" means "if not $Y$, then $X$".
- If it does not rain, the game goes ahead.
Answer: $\neg s \to r$, equivalent to $r \lor s$
Why does "unless" act like "or"? "The game goes ahead unless it rains" is broken only when it does not rain and the game still does not go ahead. $r \lor s$ is false in exactly that case too. So you may translate "unless" as $\lor$; both forms are correct.
Common mistake