Recipe

Translate an English sentence into symbols

Use when: you're given an English sentence with one or more joining words (not, and, but, or, if, only if, unless, if and only if) and asked to write it in symbols

Goal: a formula, with a symbolization key, that makes the same claim the sentence makes, bracketed so no two different connectives stand side by side

Taught in If and only if, and longer sentences

1. 1

   Look for the joining word whose two sides are the biggest pieces of the sentence. Use commas, "either," and clause openers like "if" and "unless" as clues.

   Example

   "If it rains and the field is flooded, the game is cancelled unless the league moves it indoors." The comma after "flooded" splits the sentence at the "if … then": everything before it is the condition.

   • "It is not the case that" opens the sentence — the top-level connective is ¬ over everything after it.
   • the grouping truly can't be decided — note the ambiguity and give both readings.

   Why: the top-level connective decides the overall shape of the formula; every smaller choice happens inside it.

2. 2

   Build a key: one positive, complete, pronoun-free statement per distinct claim.

   Example

   R: It rains. F: The field is flooded. C: The game is cancelled. I: The league moves the game indoors.

   Why: every later step names claims by letter, so the letters have to mean one thing each.

3. 3

   Translate each side of the top-level connective as a sentence of its own. Use the rules from earlier sections: "unless" is "if not," "only if" puts its clause after the arrow, "neither" denies each part, and "but" is "and."

   Example

   Condition: "it rains and the field is flooded" is R \wedge F. Result: "the game is cancelled unless the league moves it indoors" is \neg I \to C.

   Why: a piece is itself a sentence, so the same rules apply at every level.

4. 4

   Put each compound piece in brackets and join the pieces with the top-level connective.

   Example

   (R \wedge F) \to (\neg I \to C)

   Why: brackets record which piece sits inside which; without them two different connectives would stand side by side.

5. 5

   Read the formula aloud, one bracket at a time, using the key. Check it against the original. For each conditional, name the one situation that breaks it.

   Example

   "If it rains and the field is flooded, then: if the league doesn't move it indoors, the game is cancelled." Broken only when it rains, the field floods, the game stays outdoors, and it isn't cancelled. That matches.

   Why: the readback is the only check that catches a piece placed on the wrong side of an arrow.