A biconditional joins two conditionals: $p \leftrightarrow q$ says $p \to q$ and $q \to p$ together. With it you have all five connectives, so you can now translate long sentences. The method builds on Translating Conditionals.
Key idea
"$p$ if and only if $q$" and "$p$ just in case $q$" are $p \leftrightarrow q$. For a long sentence, find the main connective of the English first, then translate each part the same way, from the outside in.
Why it works
"You may enter if you have a ticket" gives $t \to e$; "you may enter only if you have a ticket" gives $e \to t$. Saying both at once, "you may enter if and only if you have a ticket", claims both conditionals, which is $e \leftrightarrow t$. Long sentences work like long formulas: the commas, "either" and "both" show which pieces belong together, just as parentheses do.
- Mark the simple statements and give each a letter.
- Find the connective that joins the two biggest pieces; commas and words like "either", "both" and "then" show the grouping.
- Write that main connective with parentheses around each side.
- Translate each side the same way.
Worked examples
Worked example: If and only if
Let $p$: the store is open; $q$: the light is on. Translate "The light is on if and only if the store is open."
- "If and only if" is $\leftrightarrow$.
- Left side: the light is on, $q$. Right side: $p$.
Answer: $q \leftrightarrow p$
Worked example: Three parts
Let $p$: it rains; $q$: the game is canceled; $r$: we visit the museum. Translate "If it rains and the game is canceled, then we visit the museum."
- The main connective is "if … then".
- The antecedent is "it rains and the game is canceled": $p \land q$.
- The consequent is $r$.
Answer: $(p \land q) \to r$
Worked example: Grouping by "either"
Translate "Either it rains or the game is canceled, and we do not visit the museum."
- The comma before "and" makes $\land$ the main connective.
- "Either … or" groups the first part: $p \lor q$.
- The second part is $\neg r$.
Answer: $(p \lor q) \land \neg r$
Worked example: Only if, with a compound
Translate "We visit the museum only if it rains and the game is canceled."
- "Only if" introduces the consequent.
- The consequent is the conjunction $p \land q$; the antecedent is $r$.
Answer: $r \to (p \land q)$
Common mistake
Translating word by word from left to right. "If it rains, then either the game is canceled or we visit the museum" is $p \to (q \lor r)$, but reading left to right you might write $(p \to q) \lor r$, which is a different claim (it is true whenever $r$ is true, even if it rains and the game is not canceled). Find the main connective of the English first.